20 May morning study (swaps, forwards, futures)
by Barbara Valbuzzi, CFA
Barbara Valbuzzi, CFA (Italy).
LEARNING OUTCOMES
The candidate should be able to:
demonstrate how interest rate swaps, forwards, and futures can be used to modify a portfolio’s risk and return
demonstrate how currency swaps, forwards, and futures can be used to modify a portfolio’s risk and return
demonstrate how equity swaps, forwards, and futures can be used to modify a portfolio’s risk and return
demonstrate the use of volatility derivatives and variance swaps
demonstrate the use of derivatives to achieve targeted equity and interest rate risk exposures
demonstrate the use of derivatives in asset allocation, rebalancing, and inferring market expectations
✅ MANAGING INTEREST RATE RISK WITH SWAPS
Learning Outcome
demonstrate how interest rate swaps, forwards, and futures can be used to modify a portfolio’s risk and return
There are many ways in which investment managers and investors can use swaps, forwards, futures, and volatility derivatives.
The typical applications of these derivatives involve modifying investment positions for hedging purposes or for taking directional bets, creating or replicating desired payoffs, implementing asset allocation and portfolio rebalancing decisions, and even inferring current market expectations.
The following table shows some common uses of these derivatives in portfolio management and the types of derivatives used by investors and portfolio managers.
Common Uses of Swaps, Forwards, and Futures
Typical Derivatives Used
Modifying Portfolio Returns and Risk Exposures (Hedging and Directional Bets)
Interest Rate, Currency, and Equity Swaps and Futures; Fixed-Income Futures; Variance Swaps
Creating Desired Payoffs
Forwards, Futures, Total Return Swaps
Performing Asset Allocation and Portfolio Rebalancing
Equity Index Futures, Government Bond Futures, Index Swaps
Inferring Market Expectations for Interest Rates, Inflation, and Volatility
Fed Funds Futures, Inflation Swaps, VIX Futures
It is important for an informed investment professional to understand how swaps, forwards, futures, and volatility derivatives can be used and their associated risk–return trade-offs.
Therefore, the purpose of this reading is to illustrate ways in which these derivatives might be used in typical investment situations.
Sections 2–4 of this reading show how swaps, forwards, and futures can be used to modify the risk exposure of an existing position.
Sections 5–6 provide a discussion on derivatives on volatility.
Sections 7–9 demonstrate a series of applications showing ways in which a portfolio manager might solve an investment problem with these derivatives. The reading concludes with a summary.
Changing Risk Exposures with Swaps, Futures, and Forwards
Financial managers can use swaps, forwards, and futures markets to quickly and efficiently alter the underlying risk exposure of their asset portfolios or anticipated investment transactions. This section covers a variety of common examples that use swaps, futures, and forwards.
Managing Interest Rate Risk
Interest Rate Swaps
An interest rate swap is an over-the-counter (OTC) contract between two parties that agree to exchange cash flows on specified payment dates—one based on a variable (floating) interest rate and the other based on a fixed rate (the “swap rate”)—determined at the time the swap is initiated.
The swap tenor is when the swap is agreed to expire.
Both interest rates are applied to the swap’s notional value to determine the size of each payment.
Normally, the resulting two payments (one fixed, one floating) are in the same currency but will not be equal, so they are typically netted, with the party owing the greater amount paying the difference to the other party.
In this manner, a party that currently has a fixed (floating) risk or other obligation can effectively convert it into a floating (fixed) one.
Interest rate swaps are among the most widely used instruments to manage interest rate risk.
In particular, they are designed to manage the risk on cash flows arising from investors’ assets and liabilities.
Interest rate swaps and futures can also be used to modify the risk and return profile of a portfolio.
This is associated with managing a portfolio of bonds that generally involves controlling the portfolio’s duration.
Although futures are commonly used to make duration changes, swaps can also be used, and we shall see how in this reading.
Finally, interest rate swaps are used by financial institutions to hedge the interest rate risk exposure deriving from the issuance of financial instruments sold to clients.
Example 1 shows how an interest rate swap is used to convert floating-rate securities into fixed-rate securities.
Here the firm initially expects continuing low interest rates, so it issues floating-rate bonds. But after concluding that rates are likely to increase, the firm seeks to convert its interest rate risk to a fixed obligation, even though doing so means making higher payments up front.
EXAMPLE 1
Using an Interest Rate Swap to Convert Floating-Rate Securities into Fixed-Rate Securities
An investment firm has sold £20 million of three-year floating-rate bonds that pay a semiannual coupon equal to the six-month market reference rate plus 50 bps. A few days later, the firm’s outlook changes substantially, and it now expects higher rates in the future. The firm enters into an interest rate swap with a tenor of approximately three years and semiannual payments, where the firm pays a fixed par swap rate of 1.25% and receives the six-month reference rate. The swap settlement dates are the same as the coupon payment dates on the floating-rate bonds. At the first swap settlement date, the six-month reference rate is 0.75%.
Analysis:
At the first coupon payment and swap settlement date, the six-month reference rate is 0.75% (annualized). This means that on the swap the investment firm will make a net payment of £50,000 as follows:
Receive based on the reference rate: 0.75% × £20 million × (180/360) = £75,000.
Pay based on the fixed rate: 1.25% × £20 million × (180/360) = £125,000.
Net payment made by the firm to swap dealer: £125,000 − £75,000 = £50,000.
At the same time, the first semiannual coupon payment on the securities will be (0.75% + 0.50%) × £20 million × (180/360) = £125,000.
The total payment made by the investment firm on the securities and the swap is £175,000 (= £125,000 + £50,000).
Now assume that as we move forward to the second coupon payment and swap settlement date, interest rates have increased and the six-month reference rate is 1.50%.
On the swap, the investment firm will receive a net payment of £25,000 as follows:
Receive based on the new reference rate: 1.50% × £20 million × (180/360) = £150,000.
Pay based on the fixed rate: 1.25% × £20 million × (180/360) = £125,000.
Net payment received by the firm: £150,000 − £125,000 = £25,000.
The coupon payment on the securities will be (1.50% + 0.50%) × £20 million × (180/360) = £200,000.
The total payment made by the investment firm on the securities and the swap is again £175,000 (= £200,000 − £25,000).
The investment firm has effectively fixed its all-in borrowing costs. Since this fixed cost is synthesized by a combination of the underlying debt position and the derivative contract, it can be described as a synthetic fixed security.
Why should the investment firm decide to pay a fixed rate of 1.25%, on a semiannual basis, for the remaining life of the securities when the reference rate is only 0.75% today?
The reason is that the firm’s outlook is now for higher rates in the future, as expressed by market participants in the upward-sloping yield curve.
An upward-sloping yield curve reflects that investors require higher risk premium compensation for holding longer-term securities.
The agreed-on fixed rate on the swap is based on the term structure of rates at the time the deal is initiated.
If the term structure changes, the new fixed rate agreed on by the counterparties on a swap with the same residual time to maturity as the original one will be different from the original rate.
This means that the market value of the swap will become positive or negative.
In particular, the investment firm in Example 1 has managed to fix the interest rate on future payments but has given away the opportunity to benefit from possible lower interest rates in the future.
If the term structure of interest rates has a parallel shift downward, meaning that all rates across tenors decrease, the value of the swap will become negative from the perspective of the fixed-rate payer, depending on the new swap market fixed rate.
The investment firm has managed to achieve the desired fixed profile of future cash flows, but it might incur a loss if the firm wants to unwind the interest rate swap before maturity.
Alternatively, if rates rise, as now expected, the swap can be unwound at a profit by the same reasoning in reverse: The value of the swap becomes positive from the fixed-rate payer’s view; fixed-rate payment paid is less than floating-rate payment received.
This explanation introduces the concepts of marking to market of the swap and how swaps can be used in fixed-income portfolio management with the objective to hedge the changes in value of a portfolio with fixed cash flows.
When a bond portfolio is fully hedged, its value is immunized with respect to changes in yields. This can be stated as ΔP = (Ns)(ΔS), where ΔP is the change in the value of the bond portfolio and ΔS is the change in the value of the swap for a given change in interest rates. The notional principal of the swap (Ns) will be determined as Ns = ΔP/ΔS. To reduce changes in value of a fixed-rate portfolio, the manager will want to lower the overall duration by exchanging part of this fixed-rate income stream for a floating-rate stream. This can be done by entering an interest rate swap where the portfolio manager will pay the fixed rate and receive the floating rate.
It is important to keep in mind that most of the time, the hedging instrument and the asset or portfolio to be hedged are imperfect substitutes.
The result is a market risk, called basis risk or spread risk—the difference between the market performance of the asset and the derivative instrument used to hedge it.
When using an interest rate swap to hedge, it is possible that the changes in the underlying rate of the derivative contract, and thus in the value of the swap, do not perfectly mirror changes in the value of the bond portfolio.
Furthermore, the composition of the bond portfolio could bear additional market risks other than interest rate risk.
For instance, suppose a portfolio of corporate bonds is hedged with an interest rate swap. In this case, even if interest rate risk is hedged, the investor is still exposed to credit spread risk.
The main underlying assumptions we will use are that the change in value of the bond portfolio can be approximated by using the concept of modified duration, the yield curve is flat, and it is affected only by parallel shifts.
Furthermore, we assume here that the portfolio and the derivative contract used to hedge are perfect substitutes.
A measure for the change in the value of the bond portfolio (ΔP) for a change in interest rates is given by the portfolio’s modified duration, MDURP. The same measure calculated for the interest rate swap, MDURS, is used to determine the change in the value of the swap, ΔS. The target modified duration for the combined portfolio is MDURT, and MVP is the market value of the bond portfolio.
By properly choosing the notional value and the tenor of the swap, the portfolio manager can achieve a combination of the existing portfolio and the interest rate swap that sets the overall portfolio duration to the target duration: (MVP)(MDURP) + (NS)(MDURS) = (MVP)(MDURT).
The equivalence ΔP = (NS)(ΔS) becomes (MVP)(MDURT − MDURP) = (NS)(MDURS). To find the swap notional principal, NS, we need to solve for the following formula:
𝑁𝑆=(𝑀𝐷𝑈𝑅𝑇−𝑀𝐷𝑈𝑅𝑃𝑀𝐷𝑈𝑅𝑆)(𝑀𝑉𝑃)1
The modified duration of a swap (MDURS) is the net of the modified durations of the equivalent positions in fixed- and floating-rate bonds. Thus, the position of the pay-fixed party in a pay-fixed, receive-floating swap has the modified duration of a floating-rate bond minus the modified duration of a fixed-rate bond, where the floating- and fixed-rate bonds have cash flows equivalent to the corresponding cash flows of the swap. A pay-fixed, receive-floating swap has a negative (positive) duration from the perspective of a fixed-rate payer (receiver), because the duration of a fixed-rate bond is positive and larger than the duration of a floating-rate bond, which is near zero. Moreover, the negative duration of this position to the fixed-rate payer/floating-rate receiver makes sense in that the position would be expected to benefit from rising interest rates.
EXAMPLE 2
Using an Interest Rate Swap to Achieve a Target Duration
Consider a portfolio manager with an investment portfolio of €50 million of fixed-rate German bonds with an average modified duration of 5.5. Because he fears that interest rates will rise, he wants to reduce the modified duration of the portfolio to 4.5, but he does not want to sell any of the securities. One way to do this would be to add a negative-duration position by entering into an interest rate swap where he pays the fixed rate and receives the floating rate. A two-year interest rate swap has an estimated modified duration of −2.00 from the perspective of the fixed-rate payer.
Demonstrate how the manager can use this interest rate swap to achieve the target modified duration.
Solution:
The portfolio manager’s goal is for the bonds and the swap to combine to create a portfolio with a market value of €50 million and a target modified duration of 4.5. This relationship can be expressed as follows:€50,000,000(5.50) + (NS)(MDURS) = €50,000,000(4.50),where
NS = Interest rate swap’s notional principal
MDURS = Interest rate swap’s modified duration, set equal to −2.00
So, the notional principal of this interest rate swap that the manager should use is determined using Equation 1, as follows:
NS = [(4.50 − 5.50)/(−2.00)] × €50,000,000 = €25,000,000.
MANAGING INTEREST RATE RISK WITH FORWARDS AND FUTURES
Learning Outcome
demonstrate how interest rate swaps, forwards, and futures can be used to modify a portfolio’s risk and return
The market in short-term interest rate derivatives is large and liquid, and the instruments involved are forward rate agreements (FRAs) and interest rate futures.
A forward rate agreement is an OTC derivative instrument that is used mainly to hedge a loan expected to be taken out in the near future or to hedge against changes in the level of interest rates in the future.
In fact, with advanced settled at maturity, an FRA will settle only the discounted difference between the interest rate agreed on in the contract and the actual rate prevailing at the time of settlement, applied on the notional amount of the contract.
In general, managing short-term interest rate risk with an interest rate forward contract can also be done with an interest rate futures contract.
Forwards, like swaps, are OTC instruments and are especially useful because they can be customized, but they do have counterparty risk.
In contrast, exchange-traded interest rate futures contracts are standardized and guaranteed by a clearinghouse, so counterparty risk is virtually zero.2
Forward rate agreements and interest rate futures are widely used to hedge the risk associated with interest rates changing from the time a loan or a deposit is anticipated until it is actually implemented. Example 3 demonstrates how interest rate futures are used to lock in an interest rate.
EXAMPLE 3
Using Interest Rate Futures to Lock in an Interest Rate
Amanda Wright, the chief investment officer (CIO) of a US-based philanthropic foundation is expecting a donation of $30 million in two months’ time from a member of the foundation’s founding family.
This significant donation will then be invested for three months and subsequently will be divided into smaller grants to be made to medical and educational institutions supported by the foundation.
The current (i.e., spot) three-month reference rate is 2.40% (annualized).
The CIO expects interest rates to fall, and she decides to hedge the rate on the deposit with Eurodollar futures.
To provide background information, Eurodollar futures are cash settled on the basis of the market reference rate for an offshore deposit having a principal value of $1 million and a three-month maturity.
These contracts are quoted in terms of the “IMM index” that is equal to 100 less the annualized yield on the security.
A 1 bp (0.01% or 0.0001) change in the value of the futures contract equates to a $25.00 movement in the contract value.
Thus, the basis point value (BPV) of a $1 million face value, 90-day money market instrument is given by
𝐵𝑃𝑉=Face value×(Days360)×0.01%=$1,000,000×(90360)×0.01%=$25
Analysis:
Wright buys 30 of the Eurodollar futures contracts at 97.60, locking in a forward rate of 2.40%. After two months, the donation is received and the CIO initiates the deposit at the then-lower spot rate of 2.10%. She unwinds the hedge at a futures price of 97.90, which is 30 bps higher than where the position was initiated.
The foundation will receive $180,000 from the deposit plus the hedge, as follows:
Interest obtained on the deposit: 2.10% × $30 million × (90/360) = $157,500.
Profit on the hedge is 30 bps (30 × $25 = $750), which for 30 contracts corresponds to $22,500 (= $750 × 30).
This corresponds to the return on an investment at the initial three-month reference rate of 2.40%, or 2.40% × $30 million × (90/360) = $180,000. This calculation demonstrates that by buying the Eurodollar futures, Wright did indeed lock in a forward rate of 2.40%.
Institutional investors and bond traders can decide to use interest rate futures or fixed-income futures (also referred to as “bond futures”) contracts, which are longer dated, to hedge interest rate risk exposure.
The choice will depend on the maturity of the bond or portfolio to be hedged. Since they are listed, interest rate futures have a limited number of maturities.
Furthermore, the nearest months’ contracts have higher liquidity than the longer tenors.
For these reasons, interest rate futures (e.g., Eurodollar futures) are commonly used to hedge short-term bonds, with up to two to three years remaining to maturity.
When using interest rate futures to hedge a short-term bond, an effective and widely adopted technique to construct the hedge is to use a strip of futures contracts.
Having measured the responsiveness of the bond to an interest rate change, it is now necessary to measure the sensitivity of each cash flow to changes in the relevant forward rate.
Then, one can calculate the number of futures contracts needed to hedge the interest rate exposure for each cash flow.
Fixed-income futures contracts remain, however, the preferred instrument to hedge bond positions, given that their liquidity is very high. This is especially true for US Treasury bond futures.
Fixed-Income Futures
Portfolio managers that want to hedge the duration risk of their bond portfolios usually use fixed-income futures.
They are standardized forward contracts listed on an exchange that have as underlying a basket of deliverable bonds with remaining maturities within a predefined range.
The most liquid contracts include T-note and T-bond futures listed on the Chicago Board of Trade or the Chicago Mercantile Exchange.
Contracts expire in March, June, September, and December, and the underlying assets include Treasury bills, notes, and bonds.
In Europe, the most liquid and most heavily traded fixed-income futures are traded on the Eurex, and these are the Euro-Bund (FGBL), Euro-Bobl (FGBM), and Euro-Schatz (FCBS).4
These futures contracts have German federal government–issued bonds with different maturities as underlying. The Schatz is also known as the short bund futures contract because the maturities of the underlying bonds range from 21 to 27 months.
In contrast, maturities of underlying bonds range from 4.5 years to 5.5 years for the Bobl futures contract and are even longer (between 10 years and 30 years) for the Bund futures contract.
Bond futures are used by hedgers to protect an existing bond portfolio against adverse interest rate movements and by arbitrageurs to gain from price differences in equivalent instruments.
A fixed-income futures contract has as its underlying reference assets a basket of deliverable bonds with a range of different coupon levels and maturity dates.
Most futures contracts are closed before delivery or rolled into the next contract month.
However, in the case of delivery, the futures contract seller has the obligation to deliver and the right to choose which security to deliver.
For this reason, the duration of a futures contract is usually consistent with the forward behavior of the cheapest underlying deliverable bond.
This is called the cheapest-to-deliver (CTD) bond, the eligible bond that the seller will most likely choose to deliver under the futures contract if he decides to deliver (rather than close out the futures position).
The price sensitivity of the bond futures will, therefore, reflect the duration of the CTD bond.
Within the underlying basket of bonds, the seller will deliver the CTD bond, the one that presents the greatest profit or smallest loss at delivery.
To provide a guide for choosing the CTD bond, the concept of the conversion factor (CF) has been introduced.
Given that the short side has the option of delivering any eligible security, a conversion factor invoicing system that allows for a less biased comparison in choosing among deliverable bonds has been established.
In fact, the amount the futures contract seller receives at delivery will depend on the conversion factor that, when multiplied by the futures settlement price, will generate a price at which the deliverable bond would trade if its coupon were the notional coupon of the futures contract specification (e.g., 6% coupon and 20 years to maturity).
The principal invoice amount at maturity is given in the following equation:5
Principal invoice amount = (Futures settlement price/100) × CF × Contract size.2
The cheapest-to-deliver bond is determined on the basis of duration, relative bond prices, and yield levels. In particular, a bond with a low (high) coupon rate, a long (short) maturity, and thus a long (short) duration will most likely be the CTD bond if the market yield is above (below) the notional yield of the fixed-income futures contract.
The notional yield is usually in line with the prevailing interest rate.
The pricing discrepancy between the price of the cash security and that of the fixed-income futures is the basis.
It is determined by the spot cash price less the futures price multiplied by the conversion factor.
The possibility of physical delivery of the underlying asset guarantees convergence of futures and spot prices on the delivery date.
In fact, the no-arbitrage condition requires the basis to be zero on the delivery date; otherwise, substantial arbitrage profits can be made.
However, basis traders look for arbitrage opportunities by capitalizing on relatively small pricing differences.
If the basis is negative, a trader would make a profit by “buying the basis”—that is, purchasing the bond and shorting the futures.
In contrast, the trader would make a profit by “selling the basis” when the basis is positive; in this case, she would sell the bond and buy the futures. Example 4 demonstrates how to determine the CTD bond for delivery under a Treasury bond futures contract.
EXAMPLE 4
Delivery on a Fixed-Income Futures Contract
A trader has sold 10-year US Treasury bond futures contracts expiring in June and now has the obligation to deliver and the right to choose which security to deliver (the CTD bond). The futures contract reference security is a US Treasury bond with 20 years to maturity and a coupon of 6%. The T-bond futures contract size is $100,000. The futures contract settlement price is $143.47. The trader now needs to determine which of the two bonds in the following table is cheapest to deliver.
Bond A
Bond B
Cash Bond
T 4½ 02/15/36
T 5 05/15/37
Cash Dirty Price
$120.75
$128.50
Bond Purchase Value
$120,750
$128,500
Futures Settlement Price
143.47
143.47
Conversion Factor
0.8388
0.8883
Contract Size
$100,000
$100,000
Principal Invoice Amount
$120,342.64
$127,444.40
Delivery Gain/Loss
−$407.36 = $120,342.64 − $120,750
−$1,055.60 = $127,444.40 − $128,500
Analysis:
The trader will try to maximize the difference between the amount received upon delivery, given by the futures contract settlement price (divided by 100) times the conversion factor times $100,000, and the cost of acquiring the bond for delivery, given by its market price plus any accrued interest (i.e., the dirty price). Note that this example assumes no accrued interest.
The conversion factors for both bonds are less than 1 since both bonds have a coupon lower than 6%, the coupon for the futures contract standard. Bond A can be purchased for $120,750 and Bond B for $128,500, both per $100,000 face value. These purchase prices are compared with the amounts received upon delivery. Principal invoice amounts are calculated using Equation 2, as follows:
Principal invoice amount = (Futures settlement price/100) × CF × $100,000.
Bond A: 143.47/100 × 0.8388 × $100,000 = $120,342.64.
Bond B: 143.47/100 × 0.8883 × $100,000 = $127,444.40.
The cheapest to deliver is Bond A, the 4½% T-bond with a maturity date of 02/15/36, since the loss on delivering Bond A ($407.36) is less than the loss on delivering Bond B ($1,055.60).
Continuing with the previous analysis where we hedged a portfolio of fixed-rate securities, we now determine the hedge ratio (HR) expressed as the number of fixed-income futures contracts to be sold or purchased. The relation ΔP = (HR)(ΔF) is still valid; note that we saw it previously in the context of swaps as ΔP = (Ns)(ΔS), where ΔP is the change in the value of the bond portfolio and ΔF is the change in the value of the fixed-income futures. The “ideal” hedge balances any change in value in the cash securities with an equal and opposite-sign change in the futures’ value.
With futures, however, we have to consider the cheapest-to-deliver bond price and the conversion factor. Because the basis of the CTD bond is generally closest to zero, any change in the futures price level (ΔF) will be a reflection of the change in the value of the CTD bond adjusted by its conversion factor. By considering the relative price movement of the bond futures contract to the cheapest-to-deliver bond, we have ΔF = ΔCTD/CF. By substituting into the equation ΔP = (HR)(ΔF), the hedge ratio becomes
𝐻𝑅=𝛥𝑃𝛥𝐶𝑇𝐷(𝐶𝐹)3
In the case where the bond to hedge is the CTD, then a hedge ratio based on the conversion factor is likely to be quite effective (given that the price of a fixed-income futures contract tends to track closely with that of the cheapest-to-deliver bond).
However, for other securities with different coupons and maturities, the number of bond futures that are used to hedge against price changes of a fixed-rate bond is calculated on the basis of a duration-based hedge ratio. Moreover, the relationship between the bond’s price and its yield can also be stated in terms of basis point value and the portfolio’s target modified duration, MDURT, such that the portfolio’s target basis point value (BPVT) is
BPVT = MDURT × 0.01% × MVP4
In the special case where the objective is to completely hedge the portfolio, BPVT = 0. The effect of the basis point value hedge ratio (BPVHR) is then conceptualized as BPVP + BPVHR × BPVF = 0. Thus, BPVHR = −BPVP/BPVF, which uses the basis point value of the portfolio to be hedged (BPVP) and that of the futures contract (BPVF), whereBPVP = MDURP × 0.01% ×MVP5and
BPVF = BPVCTD/CF6
In Equation 6, the numerator is BPVCTD, the basis point value of the cheapest-to-deliver bond under the futures contract, and the denominator is CF, its conversion factor. The basis point value of the cheapest-to-deliver bond is determined, in a manner analogous to Equation 4 and Equation 5, asBPVCTD = MDURCTD × 0.01% × MVCTD7where MVCTD = (CTD price/100) × Futures contract size.
Finally, for small changes in yield, by substituting into the equation BPVHR = −BPVP/BPVF, where BPVF becomes BPVCTD/CF, in the special case of complete hedging, BPVHR in terms of number of futures contracts is
𝐵𝑃𝑉𝐻𝑅=−𝐵𝑃𝑉𝑃𝐵𝑃𝑉𝐶𝑇𝐷×Conversion factor8
EXAMPLE 5
Hedging Bond Holdings with Fixed-Income Futures
A portfolio manager is holding €50 million (principal) in German bunds (DBRs) and wants to fully hedge the value of the bond investment against a rise in interest rates. The portfolio has a modified duration of 9.50 and a market value of €49,531,000. Moreover, the manager wishes to fully hedge the bond portfolio (so, BPVT = 0) with a short position in Euro-Bund futures with a price of 158.33. The cheapest-to-deliver bond is the DBR 0.25% 02/15/27 that has a price of 98.14, modified duration of 8.623, and conversion factor of 0.619489. The size of the futures contract is €100,000.
Determine the following:
The BPVP of the portfolio to be hedged
Solution to 1:
The basis point value of the portfolio (BPVP), stated in terms of the change in value for a 1 bp (0.01%) change in yield, is calculated using Equation 5, as follows:
BPVP = MDURP × 0.01% × MVP
Portfolio Principal€50,000,000Portfolio Market Value
€49,531,000
Modified Duration
9.50
BPVP = 9.50 × 0.0001 × €49,531,000 = €47,054.45.
Thus, the portfolio to be hedged has a BPVP of €47,054.45 per €50 million notional.
The BPVCTD of the futures contract hedging instrument
Solution to 2:
The basis point value of the CTD bond underlying the futures contract (BPVCTD) is calculated using Equation 7, as follows:
BPVCTD = MDURCTD × 0.01% × MVCTD
Futures Hedge
Euro-Bund Futures Price
158.33
Contract Size
€100,000
Cheapest-To-Deliver Bond
DBR 0¼ 02/15/27 Gov’t.
Modified Duration
8.623
Bond Price
98.14
Conversion Factor
0.619489
BPVCTD = 8.623 × 0.0001 × [(98.14/100) × €100,000] = €84.63.
So, the BPV of the CTD bond (BPVCTD) is €84.63.
The number of Euro-Bund futures contracts to sell to fully hedge the portfolio
Solution to 3:
Using Equation 8 and the Solutions to 1 and 2, we have:
𝐵𝑃𝑉𝐻𝑅=−𝐵𝑃𝑉𝑃𝐵𝑃𝑉𝐶𝑇𝐷×𝐶𝐹=−€47,054.45€84.63×0.619489=−344.437≈−344
Therefore, the number of Euro-Bund futures to sell to fully hedge the portfolio is 344 contracts.
In the real world, however, the hedging results are imperfect because (1) the hedge is done with the cheapest-to-deliver bond, and since the CTD bond can change over the holding period, the duration of the futures contract can also change; (2) the relationship between interest rates and bond prices is not linear, owing to convexity; and (3) the term structure of interest rates often changes via non-parallel moves.
Reconsidering Example 2 from before, in which the manager whose portfolio has a modified duration of 5.5 years wants to lower the duration to 4.5 years, the general principle is the same. What needs to be determined is the number of futures contracts that are required to reduce the portfolio’s modified duration to the target level. In this more general case, where MDURT (and BPVT) is non-zero, stated in terms of basis point value and BPVHR, we have BPVP + BPVHR × BPVF = BPVT.
Solving for BPVHR and substituting for BPVF, we have the more general version of Equation 8:
𝐵𝑃𝑉𝐻𝑅=(𝐵𝑃𝑉𝑇−𝐵𝑃𝑉𝑃𝐵𝑃𝑉𝐹)=(𝐵𝑃𝑉𝑇−𝐵𝑃𝑉𝑃𝐵𝑃𝑉𝐶𝑇𝐷/𝐶𝐹)=(𝐵𝑃𝑉𝑇−𝐵𝑃𝑉𝑃𝐵𝑃𝑉𝐶𝑇𝐷)×𝐶𝐹9
EXAMPLE 6
Decreasing Portfolio Duration with Futures
Consider the portfolio manager from Example 5 who now decides to decrease the portfolio’s modified duration from 9.50 to 8.50. The yield curve is flat. Additionally, we have already demonstrated that given the portfolio’s market value of €49,531,000, the BPVP is €47,054.50. Finally, assume the CTD bond underlying the Euro-Bund futures is the same as before, DBR 0.25% 02/15/27, with a BPVCTD of €84.63 and a conversion factor of 0.619489.
Determine the following:
The BPVT of the portfolio to be hedged
Solution to 1:
Using Equation 4 with a MDURT of 8.50, the portfolio’s target basis point value (BPVT) will be
BPVT = 8.50 × 0.0001 × €49,531,000 = €42,101.35.
The number of Euro-Bund futures contracts to sell to reduce the portfolio’s modified duration to 8.50
Solution to 2:
To achieve the target modified duration of 8.50, the portfolio manager must implement a short position in Euro-Bund futures. Using the same cheapest-to-deliver bond with a BPVCTD of €84.63 and a conversion factor of 0.619489, the number of Euro-Bund futures to sell to decrease the portfolio’s duration is calculated using Equation 9:
𝐵𝑃𝑉𝐻𝑅=(€42,101.35−€47,054.50€84.63)×0.619489=−36.26≈−36 futures contracts
Therefore, the number of Euro-Bund futures to sell to achieve the target portfolio duration of 8.50 is 36 contracts.
MANAGING CURRENCY EXPOSURE
Learning Outcome
demonstrate how currency swaps, forwards, and futures can be used to modify a portfolio’s risk and return
Currency swaps, forwards, and futures can be used to effectively alter currency risk exposures. Currency risk is the risk that the value of a current or future asset (liability) in a foreign currency will decrease (increase) when converted into the domestic currency.
Currency Swaps
A currency swap is similar to an interest rate swap, but it is different in two ways: (1) The interest rates are associated with different currencies, and (2) the notional principal amounts may or may not be exchanged at the beginning and end of the swap’s life.6
Currency swaps help the parties in the swap to hedge against the risk of exchange rate fluctuations and to achieve better rate outcomes. In particular, a cross-currency basis swap exchanges notional principals because the goal of the transaction is to issue at a more favorable funding rate and swap the amount back to the currency of choice. Firms that need foreign-denominated cash can obtain the funding in their local currency and then swap the local currency for the required foreign currency using a cross-currency basis swap. The swap periodically sets interest rate payments, mostly floating for floating, separately in two different currencies. The net effect is to use a loan in a local currency to take out a loan in a foreign currency while avoiding any foreign exchange risk. In fact, the exchange rate is fixed, as illustrated in Example 7.
EXAMPLE 7
Cross-Currency Basis Swap
Consider a Canadian private equity (PE) firm that is executing a leveraged buyout (LBO) of a small, struggling US-based electronics manufacturer. The goal is to turn around the company by implementing new robotics technology for making servers and infrastructure devices for “bitcoin mining.” Exit from the LBO via initial public offering is expected in three years. To execute the LBO and provide working capital for US operations, the PE firm needs USD40 million. The rate on a US dollar loan is the semiannual US dollar reference floating rate plus 100 bps. The PE firm discovers that it can borrow more cheaply in the local Canadian market and decides to fund the LBO in Canadian dollars (CAD) by borrowing CAD50 million for three years at the semiannual Canadian dollar reference floating rate plus 65 bps. Then it contacts a New York–based dealer and requests a quote for a three-year cross-currency basis swap with semiannual interest payments to exchange the CAD50 million into US dollars. The three-year CAD–USD cross-currency basis swap is quoted at −15 bps at a rate of USD/CAD 0.8000 (expressed as US dollars per 1 Canadian dollar). The swap agreement provides that both parties pay the semiannual reference floating rate, but the Canadian dollar rate also includes a “basis.” Here the basis is the difference between interest rates in the cross-currency basis swap and those used to determine the forward exchange rates. If covered interest rate parity holds, a forward exchange rate is determined by the spot exchange rate and the interest rate differential between foreign and domestic currencies over the term of the forward rate. However, usually covered interest rate parity does not hold and thus gives rise to the basis.
The basis is quoted on the non-USD leg of the swap. “Paying” the basis would mean borrowing the other currency versus lending US dollars, whereas “receiving” the basis implies lending the other currency versus borrowing US dollars. The three-year CAD–USD cross-currency swap in this case is quoted at −15 bps. This means that the Canadian PE firm, the “lender” of the Canadian dollars in the swap, will receive the Canadian dollar reference rate, assumed to be 1.95%, minus 15 bps every six months in exchange for paying the US dollar reference rate for the US dollars it has “borrowed.” Given that the PE firm pays the Canadian dollar floating rate plus 65 bps on its bank loan, the effective spread paid becomes 80 bps (= 65bps + 15 bps). This compares with a spread of 100 bps if instead it borrowed in US dollars.
Analysis:
We now examine the cash flows in the cross-currency basis swap, where N is the notional principal of the Canadian dollar leg of the swap and S0, agreed at the start, is the spot exchange rate for all payments (at inception, on interest payment dates, and at maturity). For the Canadian PE firm, this means that
N = CAD50 million and S0= USD/CAD 0.8000.
Flows at the inception of the swap.
At inception, the Canadian PE firm delivers Canadian dollars (N) in exchange for US dollars (at a rate of N × S0).
At each payment date, the PE firm makes a floating-rate payment in US dollars and receives a floating-rate payment in Canadian dollars that is passed on to the local Canadian lender. At maturity, the PE firm returns the USD40 million to the dealer and in return receives the CAD50 million, which it uses to pay off its lender.
Periodic payments.
At each swap payment date, the Canadian PE firm receives interest on Canadian dollars (N) in exchange for paying interest on US dollars (N × S0). Importantly, the “basis” component (of −15 bps) will be included along with the semiannual Canadian dollar reference floating rate.
Suppose that on the first settlement date the semiannual reference floating rate in Canadian dollars is 1.95% and the basis is −15 bps. Therefore, the Canadian dollar rate on the swap is 1.80% (= 1.95% − 15 bps), and we assume the US dollar rate is 2.50% (the semiannual reference floating rate). For the PE firm, the first of a sequence of periodic cash flows resulting from the swap amounts to:
The interest rate payment on the PE firm’s loan is CAD50 million × (1.95% + 0.65%) × (180/360) = CAD650,000. Considering the CAD450,000 received on the swap (A), the PE firm’s net payment is CAD200,000.
At USD/CAD 0.8000, this net payment of CAD200,000 corresponds to a payment of USD160,000, which when added to the USD500,000 paid on the swap (B) totals USD660,000. Importantly, note that had the Canadian PE firm taken out the US loan instead, it would have paid periodically USD700,000 (= USD40 million × [2.50% + 1.0%] × [180/360]).
Flows at maturity.
At the maturity of the swap (and after a successful exit from the LBO via a US IPO), the Canadian PE firm swaps back US dollars in exchange for Canadian dollars (USD × 1/S0).
In this specific example, it is worth noting that the exchange rate was assumed not to change.
A common use of currency swaps by investors is in transactions meant to earn extra yield by investing in a foreign bond market and swapping the proceeds into the domestic currency. Given that the investment is hedged against the risk of exchange rate fluctuations, this corresponds to a synthetic domestic yield, but the repackaging allows the investor to earn a higher yield compared with the yield from direct purchase of the domestic asset, because of the level of the basis on the cross-currency swap. For example, during periods when demand for US dollars is strong relative to demand for Japanese yen (JPY), the US–Japan interest rate differential implied by the currency markets may be significantly wider than the actual interest rate differential. During such a period, a US investor might choose among the following two options: (1) Invest in short-term US Treasury bonds, or (2) use a cross-currency swap to lend an equivalent amount of US dollars and buy yen; buy short-term Japanese government debt; each period pay yen and receive US dollars on the swap; and at maturity swap an equivalent amount of yen back into US dollars. When the basis is largely negative, due to relatively weak (strong) demand for yen (US dollars) from swap market participants, the borrowing costs in yen (US dollars) are low (high), making the return from lending US dollars via a cross-currency swap particularly attractive. By choosing Option 2, the investor can earn more than he could from the investment in short-term US Treasury debt.
The rates on the cross-currency basis swaps will depend on the demand for US dollar funding, because when the US dollar reference floating rate is elevated, the counterparty receiving US dollars at initiation of the swap will be willing to receive a lower interest rate on the non-dollar currency periodic payments. Exhibit 1 shows the levels of the basis for one-year cross-currency swaps from May 2016 to April 2018 in the Australian dollar (AUD), the Canadian dollar (CAD), the euro (EUR), and the British pound (GBP) versus USD Libor (quoted as six-month USD Libor versus six-month AUD bank bills, six-month CAD Libor, six-month Euribor, and six-month GBP Libor, respectively). Cross-currency basis spreads vary over time and are driven by credit and liquidity factors, and supply and demand for cross-currency financing. As noted previously, relatively strong demand for US dollar financing against the foreign currency would require the US dollar “borrower” in the swap to accept a lower rate on the periodic foreign currency cash flows it receives—for example, the foreign periodic reference rate less the basis. As shown in Exhibit 1, during the period covered this was the case for US dollar borrowers receiving periodic swap payments in all currencies shown except the Australian dollar.
Exhibit 1:
Historical Levels for One-Year Cross-Currency Swap Spreads (Basis) vs. Major Currencies (Six-Month Settlement)
Currency Forwards and Futures
Currency forwards and futures are actively used to manage currency risk. These two financial instruments are used to hedge against undesired moves in the exchange rate by buying or selling a specified amount of foreign currency, at a defined time in the future and at an agreed-on price at contract initiation. Futures contracts are standardized and best meet dealers’ and investors’ needs to manage their portfolios’ currency risk. Corporations often use customized forward contracts to manage the risk of cash flows in foreign currencies because they can be customized according to their needs.
For example, consider the general partner of a US-based venture capital (VC) firm that is calling down capital commitments for investment in “fintech” startups in Silicon Valley. It will receive in 30 days a payment of CAD50 million from a limited partner residing in Vancouver, British Columbia, and will immediately transfer the funds to its US dollar account. If the Canadian dollar were to depreciate versus the US dollar before the payment date, the US VC firm will receive fewer US dollars in exchange for the CAD50 million. To eliminate the foreign exchange risk associated with receiving this capital commitment, the firm can fix the price of the US dollars now via a forward contract in which it promises to sell CAD50 million for an agreed-on number of US dollars, based on the forward exchange rate, in 30 days.
EXAMPLE 8
Hedging Currency Risk with Futures
Consider the same US-based VC firm that is calling down capital commitments and will receive CAD50 million in 30 days. The general partner now decides to sell futures contracts to lock in the current USD/CAD rate. The hedge ratio is assumed to be equal to 1. The firm hedges its risk by selling Canadian dollar futures contracts with the closest expiry to the future Canadian dollar inflow.
Given a price for the Canadian dollar futures contract of USD/CAD 0.7838 (number of US dollars for 1 Canadian dollar) and a contract size of CAD100,000, determine how many Canadian dollar futures contracts the VC firm must sell to hedge its risk.
Solution:
To hedge the risk of the Canadian dollar depreciating against the US dollar, the VC firm must sell 500 futures contracts:
CAD50,000,000CAD100,000=500 contracts
When the futures contracts expire, the VC firm will receive (pay) any depreciation (appreciation) in the Canadian dollar versus the US dollar compared with the futures contract price of USD0.7838/CAD.7 If the changes in futures and spot prices are equal during the life of the futures contract, the hedge will be fully effective. A basis risk arises when the differential given by Futures pricet − Spot pricet is either positive or negative. In the absence of arbitrage, between the time when a hedging position is initiated and the time when it is liquidated, this spread may either widen or narrow to zero.
MANAGING EQUITY RISK
Learning Outcome
demonstrate how equity swaps, forwards, and futures can be used to modify a portfolio’s risk and return
Investors can achieve or modify their equity risk exposures using equity swaps and equity forwards and futures. The asset underlying these financial instruments could be an equity index, a single stock, or a basket of stocks.
Equity Swaps
An equity swap is a derivative contract in which two parties agree to exchange a series of cash flows whereby one party pays a variable series that will be determined by a single stock, a basket of stocks, or an equity index and the other party pays either (1) a variable series determined by a different equity or rate or (2) a fixed series. An equity swap is used to convert the returns from an equity investment into another series of returns, which either can be derived from another equity series or can be a fixed rate. There are three main types of equity swaps:
receive-equity return, pay-fixed;
receive-equity return, pay-floating; and
receive-equity return, pay-another equity return.
Because they are an OTC derivative instrument, each counterparty in the equity swap bears credit risk exposure to the other. For this reason, equity swaps are usually collateralized in order to reduce the credit risk exposure. At the same time, as equity swaps are created in the OTC market, they can be customized as desired by the counterparties.
A total return swap is a slightly modified equity swap; it also includes in the performance any dividends paid by the underlying stocks or index during the period until the swap maturity. The swap has a fixed tenor and may provide for one single payment at the end of the swap’s life, although more typically a series of periodic payments would be arranged instead. In another variation, at the time of each periodic payment, the notional amount could be reset or remain unchanged.
Equity swaps provide synthetic exposure to physical stocks. They are preferred by some investors over ownership of shares when access to a specific market is limited, when taxes are levied for owning physical stocks (e.g., stamp duty) but are not levied on swaps, the custodian fees are high, or the cost of monitoring the stock position is elevated (e.g., because of corporate actions). However, it is important to note that equity swaps require putting up collateral, are relatively illiquid contracts, and do not confer voting rights.
Example 9 shows how an equity swap might be used by an institutional investor with a portfolio indexed to the performance of the S&P 500 Index. He believes the stock market will decline over the next six months and would like to temporarily hedge part of the market exposure of his portfolio. He can do this by entering into a six-month equity swap with one payment at termination, exchanging the total return on the S&P 500 for a floating rate. We will consider two scenarios: In the first scenario, in six months the underlying portfolio is up 5%; in the second, it is down 5%.
EXAMPLE 9
Six-Month Equity Swap
An institutional investor holds a $100 million portfolio of US stocks indexed to the S&P 500. He expects the index will fall in the next six months and wants to reduce his market exposure by 30%. He enters into an equity swap with notional principal of $30 million whereby he agrees to pay the return on the index and to receive the floating reference interest rate, assumed to be 2.25%, minus 25 bps—so, 2.00% per annum.
Scenario 1:
In the first scenario, the stock market has increased by 5%. Thus, at swap settlement the institutional investor has an obligation to pay 5% × $30 million, or $1.5 million, and would receive 2% × 180/360 × $30 million, or $300,000. The two parties would net the payments and provide for a single payment of $1.2 million, which the institutional investor would pay. Because the portfolio has gained $5 million in this scenario, the profit and loss (P&L) on the combined position (including the original portfolio and the swap) is positive and equal to $3.8 million.
Scenario 1 Equity Portfolio Rises 5%
P&L on the stock portfolio:
+$5,000,000
Swap settlement:
Pay: $30 million × 5% =
−$1,500,000
Receive: $30 million × 2% × 180/360 =
+$300,000
Net payment on the swap:
−$1,200,000
P&L on the net position (70% of original exposure and 30% hedged): $5,000,000 − $1,200,000 =
+$3,800,000
Scenario 2:
In the second scenario, the stock market has decreased by 5%. So, it is slightly more complicated because the equity return that the institutional investor must pay is negative, which means he will receive money. He would receive $1.5 million because the S&P 500 had a negative performance in addition to receiving the $300,000. Because the portfolio has lost $5 million in this case, the P&L on the combined position is −$3.2 million. When the swap ends, the institutional investor returns to the same position in which he started, with the equity portfolio fully invested, and it is thereafter subject to full market risk once again.
Scenario 2 Equity Portfolio Declines 5%
P&L on the stock portfolio:
−$5,000,000
Swap settlement: Receive (Pay negative return): $30 million × 5% =
+$1,500,000
Receive: $30 million × 2% × 180/360 =
+$300,000
Net payment on the swap:
+$1,800,000
P&L on the net position (70% of original exposure and 30% hedged): −$5,000,000 + $1,800,000 =
−$3,200,000
To test the reasonableness of the result, a portfolio comprising 70% equities and 30% money market instruments assumed to earn 2% (1% over the six months) would achieve a return of 3.8% (= 0.7 × 5% + 0.3 × [2%/2]) or $3.8 million on $100 million in the bullish scenario (Scenario 1). In the bearish scenario (Scenario 2), the return would be −3.2% (= 0.7 × −5% + 0.3 × [2%/2]) or −$3.2 million on the initial $100 million portfolio. The total return swap effectively removes the risk associated with 30% of the equity portfolio allocation and converts it into money market equivalent returns.
Consider now a private high-net-worth investor who holds a large, concentrated position in a particular company’s stock that pays dividends on a regular basis. She expects that in the next six months the total return from the stock, including the dividends received, will be negative, so she wants to temporarily neutralize her long exposure. At the same time, she does not wish to lose ownership and her voting rights by selling the stock on an exchange.
This investor can enter into a total return swap requiring her to transfer the total performance of the stock (i.e., total return) to the counterparty of the swap, at prespecified dates for an agreed-on fee. Under the terms of the swap, she will pay to the counterparty the share price appreciation plus the dividends received over the life of the contract. If the stock price decreases, she will receive the share price depreciation but net of the dividends. At the same prespecified dates, the investor will receive in exchange from the counterparty an agreed-on floating-rate interest payment based on the swap notional.
Equity swaps that have a single stock as underlying can be cash settled or physically settled. If the swap is cash settled, on the termination date of the contract the equity swap receiver will receive (pay) the equity appreciation (depreciation) in cash. If the swap is physically settled, on the termination date the equity swap receiver will receive the quantity of single stock specified in the contract and pay the notional amount. Let us assume for example that a portfolio manager is the receiver in a six-month equity swap with notional principal of €4.5 million and no interim cash flows that requires physical settlement, at maturity, of 300,000 shares of the Italian insurer Generali. At maturity of the swap, the portfolio manager will receive 300,000 shares of Generali and will pay €4.5 million, which corresponds to a purchase price per share of €15 (= €4.5 million/300,000). He will also pay the interest on the swap based on the agreed-on rate. Now let us also assume that at the swap’s maturity the price of Generali is €16. This price implies a gain of €300,000 (= [€16 − €15] × 300,000) for the portfolio manager, assuming he sells the shares received in the swap at €16. If the same swap had cash settlement, instead of physical settlement, at maturity the portfolio manager would have received €300,000—given by €16 (the swap settlement price) less €15 (the agreed price on the swap) and multiplied by 300,000—against the payment of the interest on the swap.
Equity Forwards and Futures
Equity index futures are an indispensable tool for many investment managers: They are a low-cost instrument to implement tactical allocation decisions, achieve portfolio diversification, and attain international exposure. They are standardized contracts listed on an exchange, and when the underlying is a stock index, only cash settlement is available at contract expiration.
Single stock futures are also available to investors to acquire the desired exposure to a specific stock. This exposure is also achievable with equity forwards, which are OTC contracts that are used when the counterparties need a customized agreement. The underlying of a single stock futures contract is one specified stock, and the investor can receive or pay its performance. At expiration, the contract could require cash settlement or physical settlement using the stock.
In Example 9, rather than using an equity swap, the institutional investor could temporarily remove part of the market risk by selling S&P 500 Index futures. In the practical implementation of a stock index futures trade, we need to remember that the actual futures contract price is the quoted futures price times a designated multiplier. In determining the hedge ratio, the stock index futures price should be quoted on the same order of magnitude as the stock index.
For example, assume that a one-month futures contract on the S&P 500 is quoted at 2,700. Given the multiplier of $250, the actual futures price is equal to $675,000 (= 2,700 × $250). We also assume that the portfolio to be hedged carries average market risk, meaning a beta of 1.0.8 To hedge 30% of the $100 million portfolio, the portfolio manager would want to sell 44 S&P 500 futures contracts, determined as follows:
$30 million2,700($250)=44.444≈44 contracts
Suppose the institutional investor sold the 44 futures at 2,700 and at expiration the S&P 500 rises by 0.5%. The cash settlement of the contract is at 2,713.5. Because the futures position is short and the index rose, there is a “loss” of 13.5 index points—each point being worth $250—on 44 contracts, for a total cash outflow, paid by the institutional investor, of $148,500:
−13.5 points per contract × $250 per point × 44 contracts = −$148,500 (a loss).
If the S&P 500 Index rose by 0.5%, the 30% of the portfolio that has been hedged would also be expected to rise by the same amount, but there is a small difference due to rounding the number of futures contracts used for the hedge:
$30,000,000 × 0.5% = $150,000.
If instead the S&P 500 fell by 0.5%, the numbers would be the same, but the signs would change. The institutional investor would receive the “gain” because he had a short stock index futures position when the index fell, which would offset the loss on the hedged portion of his stock portfolio. In sum, the equity market risk is hedged away.
In the previous example, the beta of the portfolio was the same as the beta of the equity index futures. This situation usually does not occur, and in most hedging strategies, it is necessary to determine the exact “hedge ratio” in terms of the number of futures contracts. Consider that the investment manager wishes to change the beta of the equity portfolio, βS, to a target beta of βT. Because the value of the futures contract begins each day at zero, the dollar beta of the combination of stocks and futures, assuming the target beta is achieved, is βTS, where S is the market value of the stock portfolio.9 The number of futures contracts we shall use is Nf, which can be determined by setting the target dollar beta equal to the dollar beta of the stock portfolio (βSS) and the dollar beta of Nf futures (NfβfF), where βf is the beta of the futures and F is the value per futures contract:βTS = βSS + NfβfFWe then solve for Nf and obtain
𝑁𝑓=(𝛽𝑇−𝛽𝑆𝛽𝑓)(𝑆𝐹)10
Note that if the investor wants to increase the portfolio’s beta, βT will exceed βS and the sign of Nf will be positive, which means that she must buy futures. If she wants to decrease the beta, βT will be less than βS, the sign of Nf will be negative, and she must sell futures. This relationship should make sense: Selling futures will offset some of the risk of holding the stock, whereas buying futures will add risk.
In the special case in which the goal is to eliminate market risk, βT would be set to zero and the formula would reduce to
𝑁𝑓=−(𝛽𝑆𝛽𝑓)(𝑆𝐹)
In this case, the sign of Nf will always be negative, which makes sense, because in order to hedge away all the market risk, futures must be sold.
EXAMPLE 10
Increasing the Beta of a Portfolio with Futures
Paulo Bianchi is the manager of a fund that invests in UK defensive stocks, such as consumer staples producers and utilities. His firm’s market outlook for the next quarter has become more positive, so Bianchi decides to increase the beta on the £40 million portfolio he manages from its current level of βS = 0.85 to βT = 1.10 for the next three months. He will execute this increase in equity market risk exposure using futures on the FTSE 100 Index. The futures contract price is currently £7,300, the contract’s multiplier is £10 per index point (so each futures contract is worth £73,000), and its beta, βf, is 1.00.
At the end of the three-month period, the UK stock market has increased by 2%. The stock portfolio has increased in value to £40,680,000, calculated as £40,000,000 × [1 + (0.02 × 0.85)]. The FTSE 100 futures contract has risen to £74,460.
Determine the appropriate number of FTSE 100 Index futures Bianchi should buy to increase the portfolio’s beta to 1.10.
Solution to 1:
Using Equation 10 and the preceding data, the appropriate number of futures contracts to buy to increase the portfolio’s beta to 1.10 would be 137.
𝑁𝑓=(𝛽𝑇−𝛽𝑆𝛽𝑓)(𝑆𝐹)=(1.10−0.851.00)(£40,000,000£73,000)=136.99 (rounded to 137)
Demonstrate how the effective beta of the portfolio of stocks and the FTSE 100 Index futures matched Bianchi’s target beta of 1.10.
Solution to 2:
The profit on the futures contracts is 137 × (£74,460 − £73,000) = £200,020. Adding the profit from the futures to the value of the stock portfolio gives a total market value of £40,680,000 + £200,020 = £40,880,020. The rate of return for the combined position is
£40,880,020£40,000,000−1=0.0220, or 2.2%
Because the market went up by 2% and the overall gain was 2.2%, the effective beta of the portfolio was
0.02200.020=1.10
Thus, the effective beta matched the target beta of 1.10.
Cash Equitization
Cash securitization (also known as “cash equitization” or “cash overlay”) is a strategy designed to boost returns by finding ways to “equitize” unintended cash holdings. By purchasing futures contracts, fund managers attempt to replicate the performance of the underlying market in which the cash would have been invested. Given the liquidity of the futures market, doing so would be relatively easy. An alternative solution could be to purchase calls and sell puts on the underlying asset with the same exercise price and expiry date.
In this case, we have a cash holding, implying βS = 0, so the number of futures (with beta of βf) that would need to be purchased in a cash equitization transaction is given by
𝑁𝑓=(𝛽𝑇𝛽𝑓)(𝑆𝐹)11
EXAMPLE 11
Cash Equitization
Akari Fujiwara manages a large equity fund denominated in Japanese yen that is indexed to the Nikkei 225 stock index. She determines that the current level of excess cash that has built up in the portfolio amounts to JPY140 million. She decides to purchase futures contracts to replicate the return on her fund’s target index. Nikkei 225 index futures currently trade at a price of JPY23,000 per contract, the contract multiplier is JPY1,000 per index point (so each futures contract is worth JPY23 million), and the beta, βf, is 1.00.
Determine the appropriate number of futures Fujiwara must buy to equitize her portfolio’s excess cash position.
Solution:
Using Equation 11, which assumes βS = 0, the answer is found as follows:
𝑁𝑓=(𝛽𝑇𝛽𝑓)(𝑆𝐹)=(1.001.00)(JPY140,000,000JPY23,000,000)=6.087 (rounded to 6)
The appropriate number of futures to buy to equitize the portfolio’s excess cash position, based on the data provided, would be six contracts.
VOLATILITY DERIVATIVES: FUTURES AND OPTIONS
Learning Outcome
demonstrate the use of volatility derivatives and variance swaps
With the introduction of volatility futures and variance swaps, many investors now consider volatility an asset class in itself. In particular, long volatility exposure can be an effective hedge against a sell-off in a long equity portfolio, notably during periods of extreme market movements. Empirical studies have identified a negative correlation between volatility and stock index returns that becomes pronounced during stock market downturns. Importantly, variance swaps, which will be discussed in this section, have a valuable convexity feature—as realized volatility increases (decreases), the positive (negative) swap payoffs increase (decrease)—that makes them particularly attractive for hedging long equity portfolios. For example, some investors use strategies that systematically allocate to volatility futures or variance swaps to hedge the “tail” risk of their portfolios. Naturally, the counterparties are selling a kind of insurance; they expect such return tails will not materialize. The effectiveness of such hedges should be compared against more traditional “long volatility” hedging methods, such as implementing a rolling series of out-of-the-money put options or futures. The roll aspect affects portfolio returns, so the term structure should be carefully considered. For example, if futures prices are in backwardation (contango), then overall returns to an investor with a long position in the futures would be enhanced (diminished) owing to positive (negative) roll return. The results are necessarily reversed for the holder of the short futures position. In sum, all these derivatives strategies should be assessed on the basis of their ability to reduce portfolio risk and improve returns. In contrast, a common investment strategy implemented by opportunistic investors involves being systematically short volatility, thereby attempting to capture the risk premium embedded in option prices. This strategy is most profitable under stable market conditions, but it can lead to large losses if market volatility rises unexpectedly.
Volatility Futures and Options
The CBOE Volatility Index, known as the VIX or the “fear index,” is a measure of investors’ expectations of volatility in the S&P 500 over the next 30 days. It is calculated and published by the Chicago Board Options Exchange (CBOE) and is based on the prices of S&P 500 Index options. The CBOE began publishing real-time VIX data in 1993, and in 2004, VIX futures were introduced. Investors cannot invest directly in the VIX but instead must use VIX futures contracts that offer investors a pure play on the level of expected stock market volatility, regardless of the direction of the S&P 500. Volatility futures allow investors to implement their views depending on their expectations about the timing and magnitude of a change in implied volatility. For example, in order for a long VIX futures position to protect an equity portfolio during a downturn, the stock market’s implied volatility, as derived from S&P 500 Index options, must increase by more than the consensus expectation of implied volatility prior to the sell-off.
A family of volatility indexes has also been introduced for European equity markets, and they are designed to reflect market expectations of near-term to long-term volatility. The most well known of these is the VSTOXX index, based on real-time option prices on the EURO STOXX 50 index. The family of volatility indexes also includes the VDAX-NEW Index, based on DAX stock index options.
Next, we discuss various shapes of the VIX futures term structure. The CBOE Futures Exchange (CFE) lists nine standard (monthly) VIX index futures contracts and six weekly expirations in VIX futures. Each weekly and monthly contract settles 30 calendar days prior to the subsequent standard S&P 500 Index option’s expiration. The weekly futures have lower volumes and open interest than the monthly futures contracts have. Exhibit 2 presents the first six monthly VIX futures contracts at three different fixed points in time for all expires; these are not consecutive days but, rather, are at intervals of about two months apart.
Exhibit 2:
VIX Futures Contracts
CBOE VIX Futures Expiry
Day 1
Day 60
Day 120
April
16.68
33.46
9.77
May
17.00
19.85
14.05
June
17.00
19.10
14.55
July
17.35
18.50
15.25
August
17.51
18.75
15.60
September
17.80
18.90
16.10
Exhibit 3 shows the shape of the VIX futures term structure corresponding to the data in Exhibit 2. The vertical axis shows the futures prices, and the horizontal scale indicates the month of expiration.
Exhibit 3:
Shapes of the VIX Futures Term Structure
The shape of the VIX futures curve is always changing, reflecting the current volatility environment, investors’ expectations regarding the future level of volatility, and the buying and selling activity in VIX futures contracts by market participants. Depending on the mix of these factors, the VIX futures term structure can change from being positively sloped to flat or inverted in just a few months’ time.
Day 1 illustrates what happens when volatility is expected to remain stable over the near to long term: The term structure of VIX futures is flat. Day 60 shows the VIX futures in backwardation. This situation typically is a signal that investors expect more volatility in the short term and thus require higher prices for shorter-term contracts than for longer-term ones. In contrast, Day 120 is an example of the VIX futures being in contango. The curve is upward sloped, and it is steep for VIX buyers, with nearly 4.3 volatility points between the April and May expiries. Higher longer-term VIX futures prices are interpreted as an expectation that the VIX will rise because of increasing long-term volatility.
The VIX futures converge to the spot VIX as expiration approaches, so the two must be equal at expiration. When the VIX futures curve is in contango (backwardation) and assuming volatility expectations remain unchanged, the VIX futures price will get “pulled” closer to the VIX spot price, and they will decrease (increase) in price as they approach expiration. Traders calculate the daily roll as the difference between the front-month VIX futures price and the VIX spot price, divided by the number of business days until the VIX futures contract settles. Assuming that the basis declines linearly until settlement, when the term structure is in contango (backwardation), the trader who is long in back-month VIX futures would realize roll-down losses (profits).
Importantly, VIX futures may not reflect the index, especially when the VIX experiences large spikes, because longer-maturity futures contracts are less sensitive to short-term VIX movements. Furthermore, establishing long positions in VIX futures can be very expensive over time. When the short end of the VIX futures curve is much steeper than the long end of the curve, the carrying costs created from the contract roll down are elevated.
This phenomenon is particularly evident for investors who cannot invest directly in futures but must invest in volatility funds that attempt to track the VIX. These funds have attracted interest and substantial money flows because they are easily accessed in the form of exchange-traded products (ETPs) and, in particular, exchange-traded notes (ETNs) that provide exposure to short- and medium-term VIX futures. Some of these products also provide leveraged exposure. When using these investment products to hedge against a rise in the VIX, the VIX futures term structure should be taken into in consideration because volatility ETPs typically hold a mix of VIX futures that is adjusted daily to keep the average time to expiration of the portfolio constant. The daily rebalancing requires shorter-term futures to be sold and longer-dated futures to be purchased. When the VIX futures are in contango, the cost of rolling over hedges (i.e., negative carry) increases, thereby reducing profits and causing the ETP to underperform relative to the movement in the VIX. In contrast, “inverse” VIX ETPs offer investors the opportunity to profit from decreases in S&P 500 volatility. However, the purchase of these funds implies a directional positioning on volatility, and investors must accept the risk of large losses when volatility increases sharply.
In 2006, VIX options were introduced, providing an asymmetrical exposure to potential increases or decreases in anticipated volatility. VIX options are European style, and their prices depend on the prices of VIX futures with similar expirations because the market makers of VIX options typically hedge the risk of their option positions using VIX futures. To understand the use of VIX calls and puts, it is very important to recognize that the increases in the VIX (and VIX futures) are negatively correlated with the prices of equity assets. In particular, a trader or investor would purchase VIX call options when he expects that volatility will increase owing to a significant sell-off in the equity market. In contrast, VIX put options would be bought to profit from an expectation that volatility will decrease because of stable equity market conditions. Options on the VSTOXX index also exist, but they have lower volumes and open interest than those on the VIX.
VOLATILITY DERIVATIVES: VARIANCE SWAPS
Learning Outcome
demonstrate the use of volatility derivatives and variance swaps
Variance swaps are instruments used by investors for taking directional bets on implied versus realized volatility for speculative or hedging purposes. The term “variance swap” refers to the fact that these instruments have a payoff analogous to that of a swap. In a variance swap, the buyer of the contract will pay the difference between the fixed variance strike agreed on in the contract and the realized variance (annualized) on the underlying over the period specified and applied to a variance notional. In variance swaps, there is no exchange of cash at the contract inception or during the life of the swap. The payoff at expiration of a long variance position will be positive (negative) when realized variance is greater (less) than the swap’s variance strike. If the payment amount is positive (negative), the swap seller (buyer) pays the swap buyer (seller). The payoff at settlement is found as follows:
Settlement amountT = (Variance notional)(Realized variance − Variance strike)12
The realized variance is calculated as follows, where Ri = ln(Pi+1/Pi) and N is the number of days observed:
Realized variance=252×[∑𝑖=1𝑁−1𝑅𝑖2/(𝑁−1)]13
Since most market participants are accustomed to thinking in terms of volatility, variance swap traders typically agree on the following two things: (1) a variance swap trade size expressed in vega notional, NVega (not in variance notional), and (2) the strike (X), which represents the expected future variance of the underlying, expressed as volatility (not variance). This approach is intuitive because the vega notional represents the average profit and loss of the variance swap for a 1% change in volatility from the strike. For example, when the vega notional is $50,000, the profit and loss for one volatility point of difference between the realized volatility and the strike will be close to $50,000.
We must bear in mind that this is an approximation because the variance swap payoff is convex and the profit and loss is not linear for changes in the realized volatility. Specifically, to calculate the exact payoff, the variance strike is the strike squared and the variance notional, Nvariance, is defined and calculated as
Variance notional=Vega notional2×Strike price14
Thus, given the realized volatility (σ), we have the following equivalence:
Settlement amountT=𝑁𝑉𝑒𝑔𝑎(𝜎2−𝑋22×Strike price)=𝑁𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒(𝜎2−𝑋2)15
The strike on a variance swap is calculated on the basis of the implied volatility skew for a specific expiration, derived from calls and puts quoted in the market. As discussed previously, volatility skew is a plot of the differences in implied volatilities of a basket of options with the same maturity and underlying asset but with different strikes (and thus moneyness). As a rule of thumb, the strike of a variance swap typically corresponds to the implied volatility of the put that has 90% moneyness (calculated as the option’s strike divided by the current level of the underlying).
The mark-to-market valuation of a variance swap at time t (VarSwapt) will depend on realized volatility from the swap’s initiation to t, RealizedVol(0,t), and implied volatility at t, ImpliedVol(t,T), over the remaining life of the swap (T − t). PVt(T) is the present value at time t of $1 received at maturity T. The value of a variance swap at time t is given by the following formula:10
VarSwap𝑡=Variance notional×𝑃𝑉𝑡(𝑇)×{𝑡𝑇×[RealizedVol(0,𝑡)]2+𝑇−𝑡𝑇×[ImpliedVol(𝑡,𝑇)]2−Strike2}16
Importantly, the sensitivity of a variance swap to changes in implied volatility diminishes over time.
A feature of variance swaps that makes them particularly interesting to investors is that their payoffs are convex in volatility, as seen Exhibit 4. This convexity occurs because being long a variance swap is equivalent to be long a basket of options and short the underlying asset (typically by selling a futures contract). A long position in a variance swap is thus long gamma and has a convex payoff. This characteristic allows volatility sellers to sell variance swaps at a higher price than at-the-money options because the swap’s convex payoff profile is attractive to investors who desire a long volatility position as a tail risk hedge.
Exhibit 4:
The Payoff of a Variance Swap Is Convex in Volatility
EXAMPLE 12
Variance Swap Valuation and Settlement
Olivia Santos trades strategies that systematically sell volatility on the S&P 500 Index. She sells $50,000 vega notional of a one-year variance swap on the S&P 500 at a strike of 20% (quoted as annual volatility).
Now six months have passed, and the S&P 500 has experienced a realized volatility of 16% (annualized). On the same day, the fair strike of a new six-month variance swap on the S&P 500 is 19%.
Determine the following:
The current value of the variance swap sold by Santos (note that the annual interest rate is 2.5%)
Solution to 1:
Santos sold $50,000 vega notional of a one-year variance swap on the S&P 500 with a strike (in volatility terms) of 20%. The value of the variance swap sold by Santos is found using Equation 16:
VarSwap𝑡=Variance notional×𝑃𝑉𝑡(𝑇)×{𝑡𝑇×[RealizedVol(0,𝑡)]2+𝑇−𝑡𝑇×[ImpliedVol(𝑡,𝑇)]2−Strike2}
Values for the inputs are as follows:
Volatility strike on existing swap = 20.
Variance strike on existing swap = 202 = 400.
From Equation 14, Variance notional = Vega notional2×Strike=$50,0002×20=1,250.
RealizedVol(0,t)2 = 162 = 256.
ImpliedVol(t,T)2 = 192 = 361.
t = 6 months.
T = 12 months.
PVt(T) = 1/[1 + (2.5% × 6/12)] = 0.987654 (= Present Value Interest Factor for six months, where the annual rate is 2.5%).
The current value of the swap is
VarSwapt = 1,250 × (0.987654) × [(6/12) × 256 + (6/12) × 361 − 400]
= −$112,962.9263.
Given that Santos is short the variance swap, the mark-to-market value is positive for her, and it equals $112,963.
The settlement amount at expiration of the swap if the one-year realized volatility is 18%
Solution to 2:
The settlement amount is calculated using Equation 12 as follows:
SettlementT = Variance notional × (Realized variance − Variance strike) = 1,250 × (182 − 202)
= −$95,000
If the payment amount is positive (negative), the swap seller (buyer) pays the swap buyer (seller). So, in this case, Santos would receive $95,000 from the swap buyer.
USING DERIVATIVES IN ASSET ALLOCATION
Learning Outcome
demonstrate the use of derivatives to achieve targeted equity and interest rate risk exposures
Bernhard Steinbacher has a client with a holding of 100,000 shares in Tundra Corporation, currently trading for €14 per share. The client has owned the shares for many years and thus has a very low tax basis on this stock. Steinbacher wants to safeguard the value of the position since the client does not want to sell the shares. He cannot find exchange-traded options on the stock. Steinbacher wants to present a way in which the client could protect the investment portfolio from a decline in Tundra’s stock price.
Discuss a swap strategy that Steinbacher might recommend to his client.
Solution:
A possible solution is to enter into an equity swap trading the Tundra stock return for the floating reference interest rate. Given Tundra’s current share price of €14, the position is worth €1.4 million. Steinbacher can agree to exchange the total return on the shares (which includes the price performance and the dividends received) for the reference rate return on this sum of money. Now he needs to determine the time over which the protection is needed and must match the swap tenor to this period. After consulting with his client, Steinbacher decides on six months. The floating reference rate is 0.34%, expressed as an annual rate.
Scenario A:
Over the six months, Tundra pays a €0.10 dividend and the share price rises 1%.
The total return on the stock is(14×1.01)−14+0.1014=1.71%. For a six-month period, the reference rate return would be half the annual rate, or 0.17%. Tundra’s total return exceeds the six-month reference rate return: (1.71% − 0.17%) × €1.4 million = €21,620, which is a positive amount, so Steinbacher’s client would need to pay the swap counterparty.
Scenario B:
Over the six months, Tundra pays a €0.10 dividend and the share price falls 1%.
The total return on the stock is(14×0.99)−14+0.1014=−0.29%. Tundra’s total return is less than the six-month reference rate return: (−0.29% − 0.17%) × €1.4 million = −€6,380, which is a negative amount, so Steinbacher’s client would receive the negative return and the reference rate return from the swap counterparty (meaning the client will receive a positive cash inflow of €6,380).
Cash Equitization
Georgia McMillian manages a fund invested in UK stocks that is indexed to the FTSE 100 Index. The fund has £250 million of total assets under management, including £20 million of cash reserves invested at the three-month British pound floating rate of 0.63% (annualized). McMillian does not have an expectation on the direction of UK stocks over the next quarter. However, she is keen to minimize tracking error risk, so she implements a cash equitization strategy attempting to replicate the performance of the FTSE 100 on the cash reserves. Futures on the FTSE 100 settling in three months currently trade at a price of £7,900, the contract’s value is £10 per index point (so each futures contract is worth £79,000), and its beta, βf, is 1.0.
McMillian engages in a synthetic index strategy to gain exposure on a notional amount of £20 million to the FTSE 100 by purchasing equity index futures. The number of futures she must purchase is given by the following:𝑁𝑓=(𝛽𝑇𝛽𝑓)(𝑆𝐹)=(1.01.0)(20,000,00079,000)=253.16≈253where the beta of the futures contract, βf, and the target beta, βT, are both equal to 1.0.
Scenario: Three months later, the FTSE 100 Index has increased by 5%.
Three months later, the FTSE 100 has increased by 5%, and the original value of £230 million invested in UK stocks has increased to £241.5 million. The price of the FTSE 100 Index futures contract has increased to £8,282.5. Interest on the cash invested at the three-month floating rate amounts to £31,500 (£20,000,000 × 0.63% × 90/360). McMillian bought the futures at £7,900, and the cash settlement of the contract at is £8,282.5. So, there is a “gain” of 382.5 index points, each point being worth £10, on 253 contracts for a total cash inflow of £967,725 (382.5 points per contract × £10 per point × 253 contracts). Adding to the portfolio the profit from the futures and the cash reserves plus the interest earned on the cash gives a total market value for McMillian’s portfolio of £262,499,225 (= £241,500,000 + £20,000,000 + £967,725 + £31,500). The rate of return for the combined position is:
£262,499,225£250,000,000−1=0.05, or 5%
Importantly, without implementing this strategy, McMillian’s return would have been slightly over 4.6%, calculated as (£230 million/£250 million) × 5.0% + (£20 million/£250 million) × 0.63% × (90/360). So, she accomplished her goal of minimizing tracking error by following this strategy.
USING DERIVATIVES IN ASSET ALLOCATION
Learning Outcome
demonstrate the use of derivatives in asset allocation, rebalancing, and inferring market expectations
Changing Allocations between Asset Classes Using Futures
Mario Rossi manages a €500 million portfolio that is allocated 70% to stocks and 30% to bonds. Over the next three-month horizon, he is bearish on eurozone stocks, except for German shares, and is bullish on Italian bonds. So, Rossi wants to reduce the overall allocation to stocks by 10%, to 60%, and achieve the same weight (30%) in Italian stocks (which have a beta of 1.1 with respect to the FTSE MIB Index) and German stocks (which have a beta of 0.9 with respect to the DAX index). He also wants to increase the overall allocation to Italian government bonds (BTPs) by 10%, to 40%. The bond portion of his portfolio has a modified duration of 6.45. In summary, as shown in Exhibit 5, Rossi needs to remove €100 million of exposure to Italian stocks, add €50 million of exposure to German stocks, and add €50 million of exposure to Italian bonds in his portfolio.
Exhibit 5:
Summary of Rossi’s Original and New Asset Allocation
Stock Index
Original (€350 Million, 70%)
New (€300 Million, 60%)
Transaction
FTSE MIB
€250 million (50%)
€150 million (30%)
Sell €100 million
DAX
€100 million (20%)
€150 million (30%)
Buy €50 million
Bonds
Original (€150 Million, 30%)
New (€200 Million, 40%)
Transaction
Italian BTPs
€150 million (30%)
€200 million (40%)
Buy €50 million
Rossi uses stock index futures and bond futures to achieve this objective. Once the notional values to be traded are known, Rossi determines how many futures contracts should be purchased or sold to achieve the desired asset allocation. The FTSE MIB Index futures contract has a price of 23,100 and a multiplier of €5, for a value of €115,500. The DAX index futures contract has a price of 13,000 and a multiplier of €25, for a value of €325,000. Both futures contracts have a beta of 1. The BTP futures contract has a price of 132.50 and a contract size of €100,000. The cheapest-to-deliver bond has a price of €121; a modified duration of 8.19; a BPVCTD (from Equation 7) of €99.10, calculated as 8.19 × 0.0001 × [(121/100) × €100,000)]; and a conversion factor of 0.913292.
Determine how many stock index and bond futures contracts Rossi should use to implement the desired asset allocation and whether he should go long or short.
At the horizon date (three months later), the value of the Italian stock portfolio has fallen 5% whereas that of the German stock portfolio has increased 1%. The FTSE MIB futures price is 22,000, and the DAX futures price is 13,100. Determine the change in market value of the equity portfolio assuming the futures transactions specified in Part 1 have been carried out (note that you can ignore transaction costs).
At the horizon date, the Italian bond yield curve has a parallel shift downward of 25 bps. Determine the change in market value of the bond portfolio assuming the transactions specified in Part 1 have been carried out (note that you can ignore transactions costs).
Solution to 1:
The market value of the Italian stocks is 0.50(€500,000,000) = €250,000,000, and Rossi wants to reduce the exposure to this market by 0.20(€500,000,000) = €100,000,000. The market value of the German stocks is 0.20(€500,000,000) = €100,000,000, and he wants to increase the exposure to this market by 0.10(€500,000,000) = €50,000,000. He decides to sell enough futures contracts on the FTSE MIB to reduce the exposure to Italian stocks by €100 million and to purchase enough futures on the DAX index to increase the exposure to German stocks by €50 million.
The number of stock index futures, Nf, is𝑁𝑓=(𝛽𝑇−𝛽𝑆𝛽𝑓)(𝑆𝐹)where βT is the target beta of zero, βS is the stock beta of 1.1, βf is the futures beta of 1.0, S is the market value of the stocks involved in the transaction, and F is the value of the futures contract.
To achieve the desired reduction in exposure to Italian stocks, the market value of the stocks involved in the transaction will be S = €100,000,000. The Italian stocks’ beta (βS) is 1.1, and the target beta is βT = 0. The FTSE MIB Index futures have a contract value of €115,500 and a beta (βf) of 1.0:
𝑁𝑓=(𝛽𝑇−𝛽𝑆𝛽𝑓)(𝑆𝐹)=(0.0−1.11.0)(€100,000,000€115,500)=−952.38
Rossi sells 952 futures contracts (after rounding).
To achieve the desired exposure to German stocks (which have a beta of 0.9), the market value of the stocks involved in the transaction will be S = €50,000,000. In this case, βS = 0.0 because Rossi is starting with a notional “cash” position from the reduction in Italian stock exposure, and the target beta is now βT = 0.9. Given the value of the DAX index futures contract of €325,000 and a beta (βf) of 1.0, we obtain
𝑁𝑓=(𝛽𝑇−𝛽𝑆𝛽𝑓)(𝑆𝐹)=(0.9−0.01.0)(€50,000,000€325,000)=138.46
Rossi buys 138 futures contracts (after rounding).
The market value of the Italian bonds is 0.30(€500,000,000) = €150,000,000, and Rossi wants to increase the exposure to this market by 0.10(€500,000,000) = €50,000,000. He decides to purchase enough Euro-BTP futures so that €50 million exposure in Italian bonds is added to the portfolio. The target basis point value exposure (BPVT) is determined using Equation 4:
BPVT = MDURT × 0.01% × MVP = 6.45 × 0.0001 × €50,000,000 = €32,250
The cheapest-to-deliver bond is the BTP 4¾ 09/01/28, which has a conversion factor of 0.913292 and BPVCTD of €99.10. The number of Euro-BTP futures to buy to convert the €50 million in notional cash (BPVP = 0) to the desired exposure in Italian bonds is found using Equation 9:
𝐵𝑃𝑉𝐻𝑅=(𝐵𝑃𝑉𝑇−𝐵𝑃𝑉𝑃𝐵𝑃𝑉𝐶𝑇𝐷)×𝐶𝐹=[(€32,250−0)/€99.10]×0.913292=297.21
Rossi buys 297 Euro-BTP futures contracts (after rounding).
Solution to 2:
The value of the Italian stock portfolio decreases by €7,264,000. This outcome is the net effect of the following:
The original Italian stock portfolio decreases by €12,500,000 (= €250 million × −5%).
The short position in 952 FTSE MIB futures gains €5,236,000, calculated as− (22,000 − 23,100) × 952 × €5.
The value of the German stock portfolio increases by €1,345,000. This outcome is the net effect of the following:
The original German stock portfolio increases by €1,000,000 (= €100 million × 1%).
The long position in 138 DAX futures gains €345,000, calculated as (13,100 − 13,000) × 138 × €25.
The total value of the stock position has decreased by €5,919,000 or −1.691% (= −5,919,000/350,000,000).
Had Rossi sold the Italian stocks and then converted the proceeds into German stocks, the equity portfolio would have decreased by €6,000,000 or −1.71% (= −6,000,000/350,000,000). This outcome would have been the net effect of the following:
A decrease in the new Italian stock portfolio of €7,500,000 (= €150 million × −5%)
An increase in the new German stock portfolio of €1,500,000 (= €150 million × 1%)
Solution to 3:
The Italian bond yield curve had a parallel shift downward of 25 bps. The value of the Italian bond portfolio increases by €3,224,426. This outcome is the net effect of the following:
The €150 million portfolio’s BPVP, per Equation 5, is €96,750 (= 6.45 × 0.0001 × €150,000,000). Thus, for a 25 bp decrease in rates, the portfolio value increases by €2,418,750 (= €96,750 × 25).
The long position in 297 BTP futures has a BPVCTD of €99.10, and the CF is 0.913292. So, using Equation 6, BPVF is €108.51 (= €99.10/0.913292). Thus, for a 25 bp decrease in rates, the futures position increases in value by €805,676, calculated as follows: BPVF × Change in yield × Number of futures contracts = €108.51 × 25 × 297.
Had Rossi bought the Italian bonds, the new €200 million bond portfolio would have increased by €3,225,000. The portfolio has a BPVP of 6.45 × 0.0001 × €200,000,000 = €129,000. Thus, for a 25 bp decrease in rates, the portfolio value increases by €3,225,000 (= €129,000 × 25).
Rebalancing an Asset Allocation Using Futures
Yolanda Grant manages a portfolio with a target allocation of 40% in stocks and 60% in bonds. Over the last month, the value of the portfolio has increased from €100 million to €106 million, and Grant wants to rebalance it back to the target allocation (40% stocks/60% bonds). As shown in Exhibit 6, the current portfolio has €46 million (43.4%) in European stocks (with beta of 1.2 with respect to the EURO STOXX 50 index) and €60 million (56.6%) in German bunds. The bonds have a modified duration of 9.5.
Exhibit 6:
Summary of Grant’s Current and Rebalanced Allocation
Stocks
Current
Rebalanced
Transaction
European stocks
€46 million (43.4%)
€42.4 million (40%)
Sell €3.6 million
Bonds
Current
Rebalanced
Transaction
German bunds
€60 million (56.6%)
€63.6 million (60%)
Buy €3.6 million
Grant will use stock index futures and bond futures to achieve this objective. Once the notional values to be traded are known, she determines how many futures contracts should be purchased or sold to achieve the desired asset allocation.
The EURO STOXX 50 index futures contract has a price of 3,500 and a multiplier of €10, for a value of €35,000. The Euro-Bund futures contract has a contract size of €100,000. The cheapest-to-deliver bond has a modified duration of 8.623 and a price of 98.14, so (per Equation 7) its BPVCTD is €84.63, calculated as 8.623 × 0.0001 × [(98.14/100) × €100,000)]. Its conversion factor is 0.619489.
Determine how many stock index and bond futures contracts Grant should use to implement the desired asset allocation and whether she should go long or short.
Solution:
The market value of the European stocks is €46,000,000, and Grant wants to reduce the exposure to this market to 40% or 0.40(€106,000,000) = €42,400,000. She decides to sell enough futures contracts on the EURO STOXX 50 to reduce the allocation to European stocks by 0.034(€106,000,000) = €3,604,000.
To achieve the desired reduction in exposure to European stocks, the market value of the stocks involved in the transaction (S) will be €3,604,000. The European stocks’ beta (βS) is 1.2, and the target beta is βT = 0. The EURO STOXX 50 index futures have a contract value (F) of €35,000, with a beta (βf) of 1.0, so
𝑁𝑓=(𝛽𝑇−𝛽𝑆𝛽𝑓)(𝑆𝐹)=(0.0−1.21.0)(€3,604,000€35,000)=−123.57
Grant sells 124 futures contracts (after rounding).
The market value of the German bunds is €60,000,000, and Grant wants to increase the exposure to this market to 0.60(€106,000,000) = €63,600,000. She decides to purchase enough Euro-Bund futures so that €3,604,000 (= 0.034 × €106,000,000) of exposure to German bonds is added to the portfolio. The target basis point value (BPVT) is given by Equation 4, as follows:
BPVT = MDURT × 0.01% × MVP = 9.5 × 0.0001 ×€3,604,000 = €3,424
The cheapest-to-deliver bond has a BPVCTD of €84.63 and a conversion factor of 0.619489. The number of Euro-Bund futures to buy to convert the €3.6 million in notional cash (BPVP = 0) to the desired exposure in German bonds is found using Equation 9:
𝐵𝑃𝑉𝐻𝑅=(𝐵𝑃𝑉𝑇−𝐵𝑃𝑉𝑃𝐵𝑃𝑉𝐶𝑇𝐷)×𝐶𝐹=[(€3,424−0)/€84.63]×0.619489=25.06≈25 contracts
So, Grant should buy 25 Euro-Bund futures contracts.
Changing Allocations between Asset Classes Using Swaps
Tactical Money Management Inc. (TMM) is interested in changing the asset allocation on a $200 million segment of its portfolio. This money is invested 75% in US stocks and 25% in US bonds. Within the stock allocation, the funds are invested 60% in large cap, 30% in mid-cap, and 10% percent in small cap. Within the bond sector, the funds are invested 80% in US government bonds and 20% in investment-grade corporate bonds.
Given that it is bullish on equities, especially large-cap stocks, over the next year, TMM would like to change the overall allocation to 90% stocks and 10% bonds. Specifically, TMM would like to split the stock allocation into 65% large cap, 25% mid-cap, and 10% small cap. It also wants to change the bond allocation to 75% US government and 25% investment-grade corporate. The current position, the desired new position, and the necessary transactions to get from the current position to the new position are shown in Exhibit 7.
Exhibit 7:
Summary of TMM’s Current and New Asset Allocation
Stock
Current ($150 Million, 75%)
New ($180 Million, 90%)
Transaction
Large cap
$90 million (60%)
$117 million (65%)
Buy $27 million
Mid-cap
$45 million (30%)
$45 million (25%)
None
Small cap
$15 million (10%)
$18 million (10%)
Buy $3 million
Bonds
Current ($50 Million, 25%)
New ($20 Million, 10%)
Transaction
Government
$40 million (80%)
$15 million (75%)
Sell $25 million
Corporate
$10 million (20%)
$5 million (25%)
Sell $5 million
TMM knows these changes would entail a considerable amount of trading in stocks and bonds. So, TMM decides to execute a series of swaps that would enable it to change its position temporarily but more easily and less expensively than by executing the physical transactions. TMM engages Dynamic Dealers Inc. to perform the swaps.
TMM decides to increase the allocation in the large-cap sector by investing in the S&P 500 Index and to increase that in the small-cap sector by investing in the S&P SmallCap 600 Index (SPSC). To reduce the allocation in the overall fixed-income sector, TMM decides to replicate the performance of the Bloomberg Barclays US Treasury Index (BBT) for the government bond sector and the BofA Merrill Lynch US Corporate Index (BAMLC) for the corporate bond sector.
Discuss how TMM can use a combination of equity and fixed-income swaps to synthetically implement its desired asset allocation.
Solution:
To achieve the desired asset allocation, TMM takes the following exposures:
A long position of $27 million in the S&P 500
A long position of $3 million in the SPSC
A short position of $25 million in the BBT
A short position of $5 million in the BAMLC
The mid-cap exposure of $45 million does not change, so TMM does not need to incorporate a mid-cap index into the swap. TMM uses a combination of equity and fixed-income swaps to achieve its target allocation and structures the swap to have all payments occur on the same dates six months apart. TMM also decides that the swap should mature in one year. If it wishes to extend this period, TMM would need to renegotiate the swap at expiration. Likewise, TMM could decide to unwind the position before one year elapses, which it could do by executing a new swap with opposite payments for the remainder of the life of the original swap.
Every six months and at maturity, each of the equity swaps involves the settlement of the following cash flows:
Equity swap 1: TMM receives the total return of the S&P 500 and makes a floating payment tied to the market reference rate (MRR) minus the agreed-on spread, both on a notional principal of $27 million.
Equity swap 2: TMM receives the total return of the SPSC and makes a floating payment tied to the MRR minus the agreed-on spread, both on a notional principal of $3 million.
The fixed-income swaps will require the following cash flow settlements every six months and at maturity:
Fixed-income swap 1: TMM pays the total return of the BBT and receives floating payments tied to the MRR minus the agreed-on spread, both on notional principal of $25 million.
Fixed-income swap 2: TMM pays the total return of the BAMLC and receives floating payments tied to the MRR minus the agreed-on spread, both on a notional principal of $5 million.
It is important to recognize that this transaction will not perfectly replicate the performance of TMM’s equity and fixed-income portfolios, unless they are indexed to the indexes selected as underlying of the swaps. In addition, TMM could encounter a cash flow problem if its fixed-income payments exceed its equity receipts and its portfolio does not generate enough cash to fund its net obligation. The stock and bond portfolio will generate cash only from dividends and interest. Capital gains will not be received in cash unless a portion of the portfolio is sold. But avoiding selling a portion of the portfolio is the very reason why TMM wants to use swaps.
USING DERIVATIVES TO INFER MARKET EXPECTATIONS
Learning Outcome
demonstrate the use of derivatives in asset allocation, rebalancing, and inferring market expectations
As mentioned at the beginning of this reading, an important use of derivatives by market participants is for inferring market expectations. These expectations can be for changes in interest rates; for changes in prices for the whole economy (i.e., inflation), individual stocks, or other assets; or even for changes in key factors, such as implied volatility. Exhibit 8 provides a brief list of some of the myriad applications by which information embedded in derivatives prices is used to infer current market expectations. It is important to emphasize that these inferences relate to current expectations of future events—they do not foretell what will actually happen—which can change with the arrival of new information.
Exhibit 8:
Some Typical Applications of Derivatives for Inferring Market Expectations
Use Cases/Applications
Derivative Type
1. Inferring expectations for FOMC moves
Fed funds futures
2. Inferring expectations for inflation rates
CPI (inflation) swaps
3. Inferring expectations for market volatility
VIX futures
The first application in Exhibit 8, using fed funds futures to infer expectations of federal funds rate changes by the Federal Open Market Committee (FOMC), is likely the most common and well-publicized use of derivatives for inferring market expectations, so it is the focus of the following discussion.
Using Fed Funds Futures to Infer the Expected Average Federal Funds Rate
Market participants are interested in knowing the probabilities of various interest rate level outcomes, deriving from central banks’ future decisions, as implied by the pricing of financial instruments. This provides them with an indication about the extent to which markets are “pricing in” future monetary policy changes. Note that such implied probabilities represent the market’s view and may diverge from the guidance provided by central banks in their regular communications about the likely future course of monetary policy actions. Furthermore, especially for the longer-term horizon, these inferred probabilities usually do not have strong predictive power. Most information providers and the business media report current implied probability data for selected interest rates and historical analysis charts that show how the implied probabilities of policy rate settings have changed over time.
A commonly followed metric is the probability of a change in the federal funds rate at upcoming FOMC meetings that is implied by the prices of fed funds futures contracts. When the US central bank began its rate hiking cycle in 2015, it declared an intention to maintain a 25 bp “target range” (lower and upper bounds) for the federal funds rate. The Fed regularly communicates its “forward guidance” along with the so-called dot plot, which shows where each FOMC meeting participant believes the federal funds rate should be at the end of the year, for the next few years, and in the longer run.
To derive probabilities of potential upcoming Fed interest rate actions, market participants look at the pricing of fed funds futures, which are tied to the effective federal funds (FFE) rate—the rate actually transacted between depository institutions—not the Fed’s target federal funds rate. The underlying assumption is that the implied futures market rates are predicting the value of the monthly average effective federal funds rate. As shown in Exhibit 9, where the dots represent forecasts of the federal funds rate by each FOMC member, implied market expectations (dotted line) can diverge significantly from the Fed’s forward guidance (solid line, the median of the dots).
Exhibit 9:
Hypothetical Example of Market’s Implied Forecast vs. FOMC Forecast of Federal Funds Rate
Fed funds futures are traded on the Chicago Board of Trade, and the contract price is quoted as 100 minus the market’s expectation for the FFE rate, as follows:
Fed funds futures contract price = 100 − Expected FFE rate.17
At expiration, the contract is cash settled to the simple average (overnight) effective federal funds rate for the delivery month. The overnight rate is calculated and reported daily by the Federal Reserve Bank of New York.
To determine the probability of a change in the federal funds rate, the following formula is used, where the current federal funds rate is the midpoint of the current target range:
Effective federal funds rate implied by futures contract−Current federal funds rateFederal funds rate assuming a rate hike−Current federal funds rate18
Inferring Market Expectations
Andrew Okyung manages a portfolio of short-term floating-rate corporate debt, and he is interested in understanding current market expectations for any Fed rate actions at the upcoming FOMC meeting. He observes that the current price for the fed funds futures contract expiring after the next FOMC meeting is 97.90. The current federal funds rate target range is set between 1.75% and 2.00%.
Demonstrate how Okyung can use the information provided to determine the following:
The expected average FFE rate
The probability of a 25 bp interest rate hike at the next FOMC meeting
Solution to 1:
The FFE rate implied by the futures contract price is 2.10% (= 100 − 97.90). Okyung understands that this is the rate that market participants expect to be the average federal funds rate for that month.
Solution to 2:
Okyung knows that given that the FFE rate embedded in the fed funds futures price is 2.10%, there is a high probability that the FOMC will increase rates by 25 bps from its current target range of 1.75%–2.00% to the new target range of 2.00%–2.25%. Given the Fed’s declared incremental move size of 25 bps, he calculates the probability of a rate hike as2.100%−1.875%2.125%−1.875%=0.90, or 90%where 1.875% is the midpoint of the current target range (1.75%–2.00%) and 2.125% is the midpoint of the new target range (2.00%–2.25%) assuming a rate hike.
Exhibit 10 displays, as a hypothetical example, the trends in implied probabilities (y-axis) derived from fed funds futures prices of an FOMC rate action—either a 25 bp rate hike, a 25 bp rate cut, or no change—at the next FOMC meeting date. Note that the probability of a rate hike or cut is represented as the probability of a move from the current target range at the specified meeting date, and of course, the probabilities of all three actions at a particular meeting date sum to 1.
Exhibit 10:
Hypothetical Example of Trends of Probabilities for Federal Funds Rate Actions by the FOMC
Importantly, typical end-of-month (EOM) activity by large financial and banking institutions often induces “dips” in the FFE rate that create bias issues when using the rate as the basis for probability calculations of potential FOMC rate moves. For example, if such activity increased the price for the relevant fed funds futures contract to 98.05, then the FFE rate would decline to 1.95% (= 100 − 98.05). In this case, using the same equation as before, the probability of an FOMC rate hike decreases from 90% to just 30%:
1.950−1.8750.25=0.30, or 30%
To overcome this end-of-month bias, data providers have implemented various methods of “smoothing” the EOM dips. One prominent data provider uses a method that builds a forward rate structure based on where the market believes interest rates will settle in non-FOMC meeting months and then uses these forward rates to make appropriate adjustments. For each FOMC meeting month, it is assumed that an effective rate prevails until the meeting date, and then some rate prevails after the meeting, with the average effective rate over the month being implied by the futures price.
SUMMARY
This reading on swap, forward, and futures strategies shows a number of ways in which market participants might use these derivatives to enhance returns or to reduce risk to better meet portfolio objectives. Following are the key points.
Interest rate, currency, and equity swaps, forwards, and futures can be used to modify risk and return by altering the characteristics of the cash flows of an investment portfolio.
An interest rate swap is an OTC contract in which two parties agree to exchange cash flows on specified dates, one based on a floating interest rate and the other based on a fixed rate (swap rate), determined at swap initiation. Both rates are applied to the swap’s notional value to determine the size of the payments, which are typically netted. Interest rate swaps enable a party with a fixed (floating) risk or obligation to effectively convert it into a floating (fixed) one.
Investors can use short-dated interest rate futures and forward rate agreements or longer-dated fixed-income (bond) futures contracts to modify their portfolios’ interest rate risk exposure.
When hedging interest rate risk with bond futures, one must determine the basis point value of the portfolio to be hedged, the target basis point value, and the basis point value of the futures, which itself is determined by the basis point value of the cheapest-to-deliver bond and its conversion factor. The number of bond futures to buy or sell to reach the target basis point value is then determined by the basis point value hedge ratio:𝐵𝑃𝑉𝐻𝑅=(𝐵𝑃𝑉𝑇−𝐵𝑃𝑉𝑃𝐵𝑃𝑉𝐶𝑇𝐷)×𝐶𝐹.
Cross-currency basis swaps help parties in the swap to hedge against the risk of exchange rate fluctuations and to achieve better rate outcomes. Firms that need foreign-denominated cash can obtain funding in their local currency (likely at a more favorable rate) and then swap the local currency for the required foreign currency using a cross-currency basis swap.
Equity risk in a portfolio can be managed using equity swaps and total return swaps. There are three main types of equity swap: (1) receive-equity return, pay-fixed; (2) receive-equity return, pay-floating; and (3) receive-equity return, pay-another equity return. A total return swap is a modified equity swap; it also includes in the performance any dividends paid by the underlying stocks or index during the period until the swap maturity.
Equity risk in a portfolio can also be managed using equity futures and forwards. Equity futures are standardized, exchange-listed contracts, and when the underlying is a stock index, only cash settlement is available at contract expiration. The number of equity futures contracts to buy or sell is determined by𝑁𝑓=(𝛽𝑇−𝛽𝑆𝛽𝑓)(𝑆𝐹).
Cash equitization is a strategy designed to boost returns by finding ways to “equitize” unintended cash holdings. It is typically done using stock index futures and interest rate futures.
Derivatives on volatility include VIX futures and options and variance swaps. Importantly, VIX option prices are determined from VIX futures, and both instruments allow an investor to implement a view depending on her expectations about the timing and magnitude of a change in implied volatility.
In a variance swap, the buyer of the contract will pay the difference between the fixed variance strike specified in the contract and the realized variance (annualized) on the underlying over the period specified and applied to a variance notional. Thus, variance swaps allow directional bets on implied versus realized volatility.
Derivatives can be used to infer market participants’ current expectations for changes over the short term in inflation (e.g., CPI swaps) and market volatility (e.g., VIX futures). Another common application is using fed funds futures prices to derive the probability of a central bank move in the federal funds rate target at the FOMC’s next meeting.
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