11 June - YC Strategies skipped

INTRODUCTION

https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#read/study_task/2563473/introduction-30

The size and breadth of global fixed-income markets, as well as the term structure of interest rates within and across countries, lead investors to consider numerous factors when creating and managing a bond portfolio. While fixed-income index replication and bond portfolios that consider both an investor’s assets and liabilities were addressed earlier in the curriculum, we now turn our attention to active bond portfolio management. In contrast to a passive index strategy, active fixed-income management involves taking positions in primary risk factors that deviate from those of an index in order to generate excess return. Financial analysts who can successfully apply fixed-income concepts and tools to evaluate yield curve changes and position a portfolio based upon an interest rate view find this to be a valuable skill throughout their careers.

Prioritizing fixed-income risk factors is a key first step. In what follows, we focus on the yield curve, which represents the term structure of interest rates for government or benchmark securities, with the assumption that all promised principal and interest payments take place. Fixed-income securities, which trade at a spread above the benchmark to compensate investors for credit and liquidity risk, will be addressed later in the curriculum. The starting point for active portfolio managers is the current term structure of benchmark interest rates and an interest rate view established using macroeconomic variables introduced earlier. In what follows, we demonstrate how managers may position a fixed-income portfolio to capitalize on expectations regarding the level, slope, or shape (curvature) of yield curves using both long and short cash positions, derivatives, and leverage.

KEY YIELD CURVE AND FIXED-INCOME CONCEPTS FOR ACTIVE MANAGERS

https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#read/study_task/2563483/key-yield-curve-and-fixed-income-concepts-for-active-managers-1

Learning Outcome

• describe the factors affecting fixed-income portfolio returns due to a change in benchmark yields

The factors comprising an investor’s expected fixed-income portfolio returns introduced earlier in the curriculum are summarized in Equation 1:

E(R) ≈ Coupon income1

+/− Rolldown return

+/− E (Δ Price due to investor’s view of benchmark yields)

+/− E (Δ Price due to investor’s view of yield spreads)

+/− E (Δ Price due to investor’s view of currency value changes)

Sections 2 and 3 will focus on actively managing the first three components of Equation 1, and Section 4 will include changes in currency. Credit strategies driving yield spreads will be discussed in a later lesson. As active management hinges on an investor’s ability to identify actionable trades with specific securities, our review of yield curve and fixed-income concepts focuses on these practical considerations.

Yield Curve Dynamics

When someone refers to “the yield curve,” this implies that one yield curve for a given issuer applies to all investors. In fact, a yield curve is a stylized representation of the yields-to-maturity available to investors at various maturities for a specific issuer or group of issuers. Yield curve models make certain assumptions that may vary by investor or by the intended use of the curve, raising such issues as the following:

• Asynchronous observations of various maturities on the curve

• Maturity gaps that require interpolation and/or smoothing

• Observations that seem inconsistent with neighboring values

• Use of on-the-run bonds only versus all marketable bonds (i.e., including off-the-run bonds)

• Differences in accounting, regulatory, or tax treatment of certain bonds that may make them look like outliers

As an example, a yield curve of the most recently issued, or on-the-run, securities may differ significantly from one that includes off-the-run securities. Off-the-run bonds are typically less liquid than on-the-run bonds, and hence they have a lower price (higher yield-to-maturity). Inclusion of off-the-run bonds will tend to “pull” the yield curve higher.

This illustrates two key points about yield curves. First, although we often take reported yield curves as a “given,” they often do not consist of traded securities and must be derived from available bond yields-to-maturity using some type of model. This is particularly true for constant maturity yields, shown in some of the following exhibits. A constant maturity yield estimates, for example, what a hypothetical 5-year yield-to-maturity would be if a bond were available with exactly five years to maturity. While some derivatives reference the daily constant maturity yield, the current on-the-run 5-year Treasury issued before today has a maturity of less than five years. Estimating a constant maturity 5-year yield typically requires interpolating the yields-to-maturity on actively traded bonds with maturities near five years. Different models and assumptions can produce different yield curves. The difference between models becomes more pronounced as yields-to-maturity are converted to spot and forward rates (as spot and forward rate curves amplify yield curve steepness and curvature).

Second, a tradeoff exists between yield-to-maturity and liquidity. Active management strategies must assess this tradeoff when selecting bonds for the portfolio, especially if frequent trading is anticipated. While off-the-run bonds may earn a higher return if held to maturity, buying and selling them will likely involve increased trading costs (especially in a market crisis).

Primary yield curve risk factors are often categorized by three types: a change in (1) level (a parallel “shift” in the yield curve); (2) slope (a flattening or steepening “twist” of the yield curve); and (3) shape or curvature (or “butterfly movement”). Earlier in the curriculum, principal components analysis was used to decompose yield curve changes into these three separate factors. Level, slope, and curvature movements over time accounted for approximately 82%, 12%, and 4%, respectively, of US Treasury yield curve changes. Although based upon a specific historical period, the consistency of these results over time and across global markets underscores the importance of these factors in realizing excess portfolio returns under an active yield curve strategy.

The following exhibits provide historical context for the three yield curve factors using constant maturity US Treasury yields. Exhibit 1 shows US 10-year constant maturity yield levels.

Exhibit 1:

10-Year US Treasury Yield, 2007–2020 (%)

Source: US Federal Reserve.

During the period shown in Exhibit 1, 10-year US Treasury yields-to-maturity demonstrated significant volatility, falling to new lows in 2020 amid a flight to quality during the COVID-19 pandemic. Slower growth and accommodative monetary policy in the form of quantitative easing among global central banks since the 2008 global financial crisis years has driven government yields to zero and below. In 2020, negative yields were common on many Japanese, German, and Swiss government bonds, among others.

A change in yield level (or parallel shift) occurs when all yields-to-maturity represented on the curve change by the same number of basis points. Under this assumption, a portfolio manager might use a first-order duration statistic to approximate the impact of an expected yield curve change on portfolio value. This implies that yield curve changes occur only in parallel shifts, which is unreliable in cases where the yield curve’s slope and curvature also change. Larger yield curve changes necessitate the inclusion of second- order effects in order to better measure changes in portfolio value.

Yield curve slope is often defined as the difference in basis points between the yield-to-maturity on a long-maturity bond and the yield-to-maturity on a shorter-maturity bond. For example, as of July 2020, the slope as measured by the 2s30s spread, or the difference between the 30-year Treasury bond (30s) and the 2-year Treasury note (2s) yields-to-maturity (1.43% and 0.16%, respectively), was 127 bps. Exhibit 2 shows the 2s30s spread for US Treasury constant maturity yields. As this spread increases, or widens, the yield curve is said to steepen, while a decrease, or narrowing, is referred to as a flattening of the yield curve. In most instances, the spread is positive and the yield curve is upward-sloping. If the spread turns negative, as was the case just prior to the 2008 global financial crisis, the yield curve is described as “inverted.”

Exhibit 2:

2s30s US Yield Spread, 2007–2020 (%)

Source: US Federal Reserve.

Yield curve shape or curvature is the relationship between yields-to-maturity at the short end of the curve, at a midpoint along the curve (often referred to as the “belly” of the curve), and at the long end of the curve. A common measure of yield curve curvature is the butterfly spread:

$Butterfly spread=−(Short-term yield)+(2×Medium-term yield)−Long-term yield$Butterfly spread=−(Short-term yield) +(2×Medium-term yield)−Long-term yield2

The butterfly spread takes on larger positive values when the yield curve has more curvature. Exhibit 3 displays this measure of curvature for the US Treasury constant maturity yield curve using 2-year, 10-year, and 30-year tenors. Curvature indicates a difference between medium-term yields and a linear interpolation between short-term and long-term yields-to-maturity. A positive butterfly spread indicates a “humped” or concave shape to the midpoint of the curve, while a “saucer” or convex shape indicates the spread is negative. The butterfly spread changes when intermediate-term yield-to-maturity changes are of a different magnitude than those on the wings (the short- and long-end of the curve). Note that as in the case of yield curve slope, the butterfly spread was generally positive until 2020, except for the period just prior to the 2008 global financial crisis.

Exhibit 3:

US Butterfly Spread (2s/10s/30s), 2007–2020 (%)

Source: US Federal Reserve.

Duration and Convexity

As active managers position their portfolios to capitalize on expected changes in the level, slope, and curvature of the benchmark yield curve, the anticipated change in portfolio value due to yield-to-maturity changes is captured by the third term in Equation 1—namely, the expected change in price due to investor’s view of benchmark yields. The price/yield relationship for fixed-income bonds was established earlier in the curriculum as the combination of two factors: a negative, linear first-order factor (duration) and a usually positive, non-linear second-order factor (convexity), as shown in Exhibit 4.

Exhibit 4:

Price–Yield Relationship for a Fixed-Income Bond

The third term in Equation 1 (Δ Price due to investor’s view of benchmark yield), combines the duration and convexity effects in Equation 3 of the percentage change in the full price (%ΔPVFull) for a single bond as introduced earlier:

%∆PVFull ≈ −(ModDur × ΔYield) + [½ × Convexity × (ΔYield)2].3

Fixed-income portfolio managers often approximate changes in a bond portfolio’s present value (PV) by substituting market value (MV)-weighted averages for modified duration and convexity into Equation 3.

$AvgModDur=∑j=1JModDurj×(MVjMV)$AvgModDur=∑𝑗=1𝐽ModDur𝑗×(MV𝑗MV)4$AvgConvexity=∑j=1JConvexityj×(MVjMV)$AvgConvexity=∑𝑗=1𝐽Convexity𝑗×(MV𝑗MV)5

Active managers focus on the incremental effect on these summary statistics for a portfolio by adding or selling bonds in the portfolio or by buying and selling fixed-income derivatives. Duration is a first-order effect that attempts to capture a linear relationship between bond prices and yield-to-maturity. Convexity is a second-order effect that describes a bond’s price behavior for larger movements in yield-to-maturity. This additional term is a positive amount on a traditional (option-free) fixed-rate bond for either a yield increase or decrease, causing the yield/price relationship to deviate from a linear relationship. Because duration is a first-order effect, it follows that duration management—accounting for changes in yield curve level—will usually be a more important consideration for portfolio performance than convexity management. This is consistent with our previous discussion of the relative importance of the yield curve level, slope, and curvature. As we shall see later in this lesson, convexity management is more closely associated with yield curve slope and shape changes.

All else equal, positive convexity is a valuable feature in bonds. If a bond has higher positive convexity than an otherwise identical bond, then the bond price increases more if interest rates decrease (and decreases less if interest rates increase) than the duration estimate would suggest. Said another way, the expected price of a bond with positive convexity for a given rate change will be higher than the price change of an identical-duration, lower-convexity bond. This price behavior is valuable to investors; therefore, a bond with higher convexity might be expected to have a lower yield-to-maturity than a similar-duration bond with less convexity. All else equal, bonds with longer durations have higher convexity than bonds with shorter durations. Also, as noted earlier in the curriculum, convexity is affected by the dispersion of cash flows—that is, the variance of the times to receipt of cash flow. Higher cash flow dispersion leads to an increase in convexity. This is in contrast to Macaulay duration, which measures the weighted average of the times to cash flow receipt. Note that throughout this lesson, we will use “raw” versus scaled (or “raw” divided by 100) convexity figures often seen on trading platforms. We can see the convexity effect by comparing two bond portfolios:

EXAMPLE 1

US Treasury Securities Portfolio

Tenor	Coupon	Price	ModDur	Convexity
2y	0.250%	$100	1.994	5.0
5y	0.875%	$100	4.880	26.5
10y	2.000%	$100	9.023	90.8

Consider two $50 million portfolios: Portfolio A is fully invested in the 5-year Treasury bond, and Portfolio B is an investment split between the 2-year (58.94%) and the 10-year (41.06%) bonds. The Portfolio B weights were chosen to (approximately) match the 5-year bond duration of 4.88. How will the value of these portfolios change if all three Treasury yields-to-maturity immediately rise or fall by 50 bps?

Using Equation 3, we can derive the percentage value change for Portfolios A and B as well as the dollar value of each $50 million investment:

Portfolio	+ 50 bps % Δ Price	+ 50 bps Δ Price	− 50 bps % Δ Price	− 50 bps Δ Price
A	−2.407%	($1,203,438)	2.473%	$1,236,563
B	−2.390%	($1,194,883)	2.490%	$1,245,170

For example, for the case of a 50 bp increase in rates:

Portfolio 1	−2.407% = (−4.880 × 0.005) + [0.5 × 26.5 × (0.0052)]
Portfolio 2	−2.390% = 0.5894 × {[−1.994 × 0.005] + [0.5 × 5 × (0.0052)]} + 0.4106 × {[−9.023 × 0.005] + [0.5 × 90.8 × (0.0052)]}

Note that Portfolio B gains more ($8,607) than Portfolio A when rates fall 50 bps and loses less ($8,555) than Portfolio A when rates rise by 50 bps.

The first portfolio concentrated in a single intermediate maturity is often referred to as a bullet portfolio. The second portfolio, with similar duration but combining short- and long-term maturities, is a barbell portfolio. Although the bullet and barbell have the same duration, the barbell’s higher convexity (40.229 versus 26.5 for the bullet) results in a larger gain as yields-to-maturity fall and a smaller loss when yields-to-maturity rise. Convexity is therefore valuable when interest rate volatility is expected to rise. This dynamic tends to cause investors to bid up prices on more convex, longer-maturity bonds, which drives changes in yield curve shape. As a result, the long end of the curve may decline or even invert (or invert further), increasing the curvature of the yield curve.

EXAMPLE 2

Portfolio Convexity

1. Portfolio convexity is a second-order effect that causes the value of a portfolio to respond to a change in yields-to-maturity in a non-linear manner. Which of the following best describes the effect of positive portfolio convexity for a given change in yield-to-maturity?

   1. Convexity causes a greater increase in price for a decline in yields-to-maturity and a greater decrease in price when yields-to-maturity rise.

   2. Convexity causes a smaller increase in price for a decline in yields-to-maturity and a greater decrease in price when yields-to-maturity rise.

   3. Convexity causes a greater increase in price for a decline in yields-to-maturity and a smaller decrease in price when yields-to-maturity rise.

   Solution:

   The correct answer is c. Note that the convexity component of Equation 3 involves squaring the change in yield-to-maturity, or [½ × Convexity × (ΔYield)2], making the term positive as long as portfolio convexity is positive. This adds to the overall portfolio gain when yields-to-maturity decline and reduces the portfolio loss when yields-to-maturity rise.

YIELD CURVE STRATEGIES

https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#read/study_task/2563493/yield-curve-strategies-overview-1

Learning Outcomes

• formulate a portfolio positioning strategy given forward interest rates and an interest rate view that coincides with the market view

• formulate a portfolio positioning strategy given forward interest rates and an interest rate view that diverges from the market view in terms of rate level, slope, and shape

• formulate a portfolio positioning strategy based upon expected changes in interest rate volatility

• evaluate a portfolio’s sensitivity using key rate durations of the portfolio and its benchmark

Earlier in the curriculum, we established that yield curves are usually upward-sloping, with diminishing marginal yield-to-maturity increases at longer tenors—that is, flatter at longer maturities. As nominal yields-to-maturity incorporate an expected inflation premium, positively sloped yield curves are consistent with market expectations of rising or stable future inflation and relatively strong economic growth. Investor expectations of higher yields-to-maturity for assuming the increased interest rate risk of long-term bonds also contribute to this positive slope. Active managers often begin with growth and inflation forecasts, which they then translate into expected yield curve level, slope, and/or curvature changes. If their forecasts coincide with today’s yield curve, managers will choose active strategies that are consistent with a static or stable yield curve. If their forecasts differ from what today’s yield curve implies about these future yield curve characteristics, managers will position the portfolio to generate excess return based upon this divergent view, within the constraints of their investment mandate, using the cash and derivatives strategies we discuss next.

Static Yield Curve

A portfolio manager may believe that bonds are fairly priced and that the existing yield curve will remain unchanged over an investment horizon.

The two basic ways in which a manager may actively position a bond portfolio versus a benchmark index to generate excess return from a static or stable yield curve is to increase risk by adding either duration or leverage to the portfolio. If the yield curve is upward-sloping, longer duration exposure will result in a higher yield-to-maturity over time, while the “repo carry” trade (the difference between a higher-yielding instrument purchased and a lower-yielding (financing) instrument) will also generate excess returns.

Starting with cash-based instruments, “buy-and-hold” is an obvious strategy if the yield curve is upward-sloping. In an active context, this involves buying bonds with duration above the benchmark without active trading during a subsequent period. If the relationship between long- and short-term yields-to-maturity remains stable over this period, the manager is rewarded with higher return from the incremental duration. “Rolling down” the yield curve, a concept introduced previously, differs slightly from the “buy-and-hold” approach in terms of the investment time horizon and expected accumulation. The rolldown return component of Equation 1 (sometimes referred to as “carry-rolldown”) incorporates not only coupon income (adjusted over time for any price difference from par) but also additional return from the passage of time and the investor’s ability to sell the shorter-maturity bond in the future at a higher price (lower yield-to-maturity due to the upward-sloping yield curve) at the end of the investment horizon. If the yield curve is upward-sloping, buying bonds with a maturity beyond the investment horizon offers a total return (higher coupon plus price appreciation) greater than the purchase of a bond with maturity matching the investment horizon if the curve remains static. Finally, a common strategy known as a repurchase agreement or repo trade may be used in an expected stable rate environment to add leverage risk to the portfolio. The repo market involves buying a long-term security and financing it at a short-term rate below the long-term yield-to-maturity—that is, earning a positive “repo carry.” At the end of the trade, the bond is sold and the repo is unwound. These cash-based strategies are summarized in Exhibit 5 and Exhibit 6.

Exhibit 5:

Cash-Based Static Yield Curve Strategies

Strategy	Description	Income	Objective
Buy-and-hold	Constant without active trading	Coupon income	Add duration beyond target given static yield curve view
Rolling down the yield curve	Constant, with Δ Price as maturity shortens	Coupon income +/− Rolldown return	Add duration and increased return if future shorter-term yields are below current yield-to-maturity
Repo carry trade	Finance bond purchase in repo market	(Coupon income +/− Rolldown return)—Financing cost	Generate repo carry return if coupon plus rolldown exceeds financing cost

Exhibit 6:

Carry, Rolldown, and Buy-and-Hold Strategies under a Static Yield Curve

Excess return under these strategies depends upon stable rate levels and yield curve shape. Note that a more nuanced “buy-and-hold” strategy under this scenario could also involve less liquid and higher-yielding government bonds (such as off-the-run bonds). The lack of portfolio turnover may make the strategy seem passive, but in fact it may be quite aggressive as it introduces liquidity risk, a topic addressed in detail later in the curriculum. The ability to benefit from price appreciation by selling a shorter-dated bond at a premium when rolling down (or riding) the yield curve hinges on a reasonably static and upward-sloping yield curve. Not only will the repo carry be maintained under this yield curve scenario, but it also will generate excess return due to the reduced cash outlay versus a term bond purchase.

Active managers whose investment mandate extends to the use of synthetic means to increase risk by adding duration or leverage to the portfolio might consider using the derivatives-based strategies in Exhibit 7 to increase duration exposure beyond a benchmark target. Although the long futures example is similar to rolling down the yield curve, it relies solely on price appreciation rather than bond coupon income. The receive-fixed swap, on the other hand, is similar to the cash-based repo carry trade, but the investor receives the fixed swap rate and pays a market reference rate (MRR), which is often referred to as “swap carry.”

Exhibit 7:

Derivatives-Based Static Yield Curve Strategies

Strategy	Description	Targeted Return	Goal
Long futures position	Purchase contract for future bond delivery	(Δ Price / Δ Bond yield) − Margin cost	Synthetically increase duration (up-front margin and daily mark-to-market valuation)
Receive- fixed swap	Fixed-rate receiver on an interest rate swap	(Swap rate − MRR) + (Δ Swap mark-to-market / Δ Swap yield)	Synthetically increase portfolio duration (up-front / mark-to-market collateral) + / − Swap carry

As mentioned previously in the curriculum, global exchanges offer a wide range of derivatives contracts across swap, bond, and short-term market reference rates for different settlement dates, and over the counter (OTC) contracts may be uniquely tailored to end user needs. Our treatment here is limited to futures and swaps and will extend to options in a later section.

Although margining was historically limited to exchange-traded derivatives, the advent of derivatives central counterparty (CCP) clearing mandated by regulatory authorities following the 2008 global financial crisis to mitigate counterparty risk has given rise to similar cash flow implications for OTC derivatives. Active managers using both exchange-traded and OTC derivatives must therefore maintain sufficient cash or eligible collateral to fulfill margin or collateral requirements. They must also factor any resulting foregone portfolio return into their overall performance. That said, since the initial cash outlay for a derivative is limited to initial margin or collateral as opposed to the full price for a cash bond purchase, derivatives have a high degree of implicit leverage. That is, a small move in price/yield can have a very large effect on a derivative’s mark-to-market value (MTM) relative to the margin posted. Exhibit 8 shows these cash flow mechanics. This outsized price effect makes derivatives effective instruments for fixed-income portfolio management.

Exhibit 8:

Derivatives Cash Flow Impact for a Fixed-Income Portfolio

For example, bond futures involve a contract to take delivery of a bond on a specific future date. Changes in the futures contract value mirror those of the underlying bond’s price over time, allowing an investor to create an exposure profile similar to a long bond position by purchasing this contract with a fraction of the outlay of a cash bond purchase. While futures contracts are covered in detail elsewhere in the curriculum, for our purposes here it is important to establish the basis point value (BPV) of a futures contract. Most government bond futures are traded and settled using the least costly or cheapest-to-deliver (CTD) bond among those eligible for future delivery. For example, the CME Group’s Ultra 10-Year US Treasury Note Futures contract specifies delivery of an original 10-year issue Treasury security with not less than 9 years, five months and not more than 10 years to maturity with an assumed 6% yield-to-maturity and contract size of $100,000. The “duration” of the bond futures contract is assumed to match that of the CTD security. In order to determine the futures BPV, we use the following approximation introduced previously:

Futures BPV ≈ BPVCTD / CFCTD,6

where CFCTD is the conversion factor for the CTD security. For government bond futures with a fixed basket of underlying bonds, such as Australian Treasury bond futures, the futures BPV simply equals the BPV of an underlying basket of bonds.

The manager in Example 1 can replicate the 10-year Treasury exposure using futures by matching the BPV of the cash bond. As explained elsewhere, the BPV of the $20.53 million (or 41.06% × $50 million) 10-year Treasury position equals the modified duration (9.023) multiplied by the full price (also known as the money duration) times one basis point, or $18,524. If the CTD security under the Ultra 10-Year Futures contract is a Treasury bond also priced at par but with 9.5 years remaining to maturity, modified duration of 8.84, and a conversion factor of 0.684, then each $100,000 futures contract has a BPV of $129.24 ($88.40/0.684). The manager must therefore buy approximately 143 futures contracts ($18,524/$129.24) to replicate the exposure. Note that as shown in Exhibit 8, this will involve an outlay of initial margin and margin movement due to MTM changes rather than investment of full principal.

An interest rate swap involves the net exchange of fixed-for-floating payments, where the fixed rate (swap rate) is derived from short-term market reference rates for a given tenor. As shown in Exhibit 9, the swap contract may be seen as a combination of bonds, namely a fixed-rate bond versus a floating-rate bond of the same maturity.

Exhibit 9:

Swaps as a Duration Management Tool

Note the similarities between the “carry” trade in Exhibit 5 and the receive-fixed interest rate swap position on the right in Exhibit 9. The fixed-rate receiver is “long” a fixed-rate term bond and “short” a floating-rate bond, giving rise to an exposure profile that mimics a “long” cash bond position by increasing duration. A swap’s BPV may be estimated using Equation 7.

Swap BPV = ModDurSwap × Swap Notional/10,000.7

The difference between the receive-fixed swap and long fixed-rate bond positions is best understood via an example.

EXAMPLE 3

Calculating Bond versus Swap Returns

Say a UK-based manager seeks to extend duration beyond an index by adding 10-year exposure. The manager considers either buying and holding a 10-year, 2.25% semi-annual coupon UK government bond priced at ₤93.947, with a corresponding yield-to-maturity of 2.9535%, or entering a new 10-year, GBP receive-fixed interest rate swap at 2.8535% versus the six-month GBP MRR currently set at 0.5925%. The swap has a modified duration of 8.318. We compare the results of both strategies over a six-month time horizon for a ₤100 million par value during which both the bond yield-to-maturity and swap rates fall 50 bps. We ignore day count details in the calculation.

Position	Income	Price Appreciation/MTM	Gain in 6 Months
10y UK bond	₤1,125,000	₤4,337,779	₤5,462,778
10y GBP swap	₤1,130,500	₤4,234,260	₤5,364,760

The relevant return components from Equation 1 are income, namely coupon income for the bond versus “carry” for the swap, and E (Δ Price due to investor’s view of benchmark yield) in the form of price appreciation for the bond versus an MTM gain for the swap:

• 10-Year UK Government Bond:

  • Coupon income = ₤1,125,000, or (2.25%/2) × ₤100 million.

  • Price appreciation = ₤4,337,779. Using Excel, this is the difference between the 10-year, or [PV (0.029535/2, 20, 1.125, 100)], and the 9.5-year bond at the lower yield-to-maturity, or [PV (0.024535/2, 19, 1.125, 100)] × ₤1 million.

  We can separate bond price appreciation into two components:

  • Rolldown return: The difference between the 10-year and 9.5-year PV with no change in yield-to-maturity of ₤262,363, or [PV (0.029535/2, 20, 1.125, 100)] − [PV (0.024535/2, 19, 1.125, 100)] × ₤1 million].

  • (Δ Price due to investor’s view of benchmark yield): The difference in price for a 50 bp shift of the 9.5-year bond of ₤4,075,415, or [PV (0.029535/2, 19, 1.125, 100)] − [PV (0.024535/2, 19, 1.125, 100)] × ₤1 million.

• 10-Year GBP Swap:

  • Swap carry = ₤1,130,500, or [(2.8535% − 0.5925%)/2] × ₤100,000,000.

  • Swap MTM gain = ₤4,234,260. The swap MTM gain equals the difference between the fixed leg and floating leg, which is currently at par. The fixed leg equals the 9.5-year swap value given a 50 bp shift in the fixed swap rate, which is ₤104,234,260, or [PV(0.023535/2, 19, 2.8535/2, 100)] × ₤1 million, and the floating leg is priced at par and therefore equal to ₤100,000,000.

We can use Equation 7 to derive an approximate swap MTM change of ₤4,159,000 by multiplying swap BPV (8.318 × ₤100 million) by 50 bps. As in the case of a bond future, the cash outlay for the swap is limited to required collateral or margin for the transaction as opposed to the bond’s full cash price. Note that for the purposes of this example, we have ignored any interest on the difference between the bond investment and the cash outlay for the swap.

While these strategies are designed to gain from a static or stable interest rate term structure, we now turn to portfolio positioning in a changing yield curve environment.

EXAMPLE 4

Static Yield Curve Strategies under Curve Inversion

1. An investment manager who pursues the cash-based yield curve strategies described in Exhibit 5 faces an inverted yield curve (with a decline in long-term yields-to-maturity and a sharp increase in short-term yields-to-maturity) instead of a static yield curve post implementation. Which of the following is the least likely portfolio outcome under this scenario?

   1. The manager realizes a loss on a “buy-and-hold” position that extends duration beyond that of the index.

   2. The manager faces negative carry when financing a bond purchase in the repo market.

   3. The manager is able to reinvest coupon income from a yield curve rolldown strategy at a higher short-term yield-to-maturity.

   Solution:

   The correct answer is a. The fall in long-term yields-to-maturity will lead to price appreciation under the “buy-and-hold” strategy. The difference between long-term and short-term yields-to-maturity in b will fall, leading to negative carry if short-term yields-to-maturity rise sharply. As for c, higher short-term yields-to-maturity will enable the manager to reinvest bond coupon payments at a higher rate.

Dynamic Yield Curve

Exhibits 1 through 3 show that yield curves are dynamic over time, with significant changes in the level, slope, and curvature of rates across maturities. Unless otherwise specified, the sole focus here is on instantaneous yield-to-maturity changes affecting E(Δ Price due to investor’s view of benchmark yields), the third component of Equation 1.

Divergent Rate Level View

The principal components analysis cited earlier underscores that rate level changes are the key driver of changes in single bond or bond portfolio values. The first term in Equation 3 shows that bond value changes result from yield-to-maturity changes multiplied by a duration statistic. For active fixed-income managers with a divergent rate level view, positioning the portfolio to increase profit as yield levels fall or minimizing losses as yield levels rise is of primary importance. To be clear, a divergent rate level view implies an expectation of a parallel shift in the yield curve, as shown in Exhibit 10.

Exhibit 10:

Yield Level Changes

Exhibit 10 shows a general decline in bond yield levels, referred to as a bull market, since 2007. This trend began in late 1981 when the 10-year US Treasury yield-to-maturity peaked at nearly 16%, a consequence of contractionary US Federal Reserve monetary policy in which the short-term federal funds rate was raised to 20% to combat double-digit inflation. Extending duration beyond a target index over this period was a winning active strategy, despite occasional periods of yield increases. Exhibit 11 summarizes the major strategies an active manager might pursue if she expects lower yield levels and downside risks.

Exhibit 11:

Major Yield Curve Strategies to Increase Portfolio Duration

Strategy	Description	Expected Excess Return	Downside Risks
Cash bond purchase (“bullet”)	Extend duration with longer-dated bonds	Price appreciation as yield-to-maturity declines	Higher yield levels
Receive-fixed swap	Fixed-rate receiver on an interest rate swap	Swap MTM gain plus “carry” (fixed minus floating rate)	Higher swap yield levels and/or higher floating rates
Long futures position	Purchase contract for forward bond delivery	Futures MTM gain − Margin cost	Higher bond yields and/or higher margin cost

Assume the “index” portfolio equally weights the 2-, 5-, and 10-year Treasuries priced at par from Example 1, while a higher duration “active” portfolio is weighted 25% for 2- and 5-year Treasuries, respectively, and 50% in 10-year Treasuries. Average portfolio statistics are summarized here:

Portfolio	Coupon	Modified Duration	Convexity
Index	1.042%	5.299	40.8
Active (25/25/50)	1.281%	6.230	53.3

We can see from this table that the active portfolio has a blended coupon nearly 24 bps above that of the index.

We now turn to the impact of a parallel yield curve shift on the index versus active portfolios. Assuming an instantaneous 30 bp downward shift in yields-to-maturity, the index portfolio value would rise by approximately 1.608%, or (−5.299 × −0.003) + 0.5 × (40.8) × (−0.0032), versus an estimated 1.893% increase for the actively managed portfolio, a positive difference of nearly $285,000 for a $100 million portfolio.

EXAMPLE 5

Portfolio Impact of Higher Yield-to-Maturity Levels

1. Consider a $50 million Treasury portfolio equally weighted between 2-, 5-, and 10-year Treasuries using parameters from the prior example as the index, and an active portfolio with 20% each in 2- and 5-year Treasuries and the remaining 60% invested in 10-year Treasuries. Which of the following is closest to the active versus index portfolio value change due to a 40 bp rise in yields-to-maturity?

   1. Active portfolio declines by $181,197 more than the index portfolio

   2. Active portfolio declines by $289,915 more than the index portfolio

   3. Index portfolio declines by $289,915 more than the active portfolio

   Solution:

   The correct answer is b. First, we must establish average portfolio statistics for the 20/20/60 portfolio using a weighted average of duration (6.79 versus 5.299 for the index) and convexity (60.8 versus 40.8 for the index). Second, using these portfolio statistics, we must calculate %∆PVFull, as shown in Equation 3, for both the index and active portfolios, which are −2.087% for the index and −2.667% for the active portfolio, respectively. Finally, we multiply the difference of −0.58% by the $50 million notional to get −$289,915.

Receive-fixed swaps or long futures positions may be used in place of a cash bond strategy to take an active view on rates. Note that most fixed-income managers will tend to favor option-free over callable bonds if taking a divergent rate level view due to the greater liquidity of option-free bonds. An exception to this arises when investors formulate portfolio positioning strategies based upon expected changes in interest rate volatility, as we will discuss in detail later in this lesson.

As 2020 began, some analysts expected government yields-to-maturity to eventually rise following over a decade of quantitative easing after the 2008 global financial crisis. However, yields instead reached new lows during 2020 when the COVID-19 pandemic caused a sharp economic slowdown, prompting additional monetary and fiscal policy stimulus. If analysts expected a strong economic rebound to increase yield levels, they might seek to lessen the adverse impact of higher rate levels by reducing duration. Exhibit 12 outlines major strategies to achieve this goal.

Exhibit 12:

Major Yield Curve Strategies to Reduce Portfolio Duration

Strategy	Description	Expected Excess Return	Downside Risks
Cash bond sale (“bullet”)	Reduce duration with short sale/switch to shorter-dated bonds	Smaller price decline as yield-to-maturity increases	Lower yield levels
Pay-fixed (interest rate swap)	Fixed-rate payer on an interest rate swap	Swap MTM gain plus “swap carry” (MRR − Fixed swap rate)	Swap MTM loss amid lower swap yield levels and/or lower floating rates
Short futures position	Sell contract for forward bond delivery	Futures MTM gain − Margin cost	Futures MTM loss amid lower bond yields and/or higher margin cost

Returning to our “index” portfolio of equally weighted 2-, 5-, and 10-year Treasuries, we now consider an active portfolio positioned to reduce downside exposure to higher yields-to-maturity versus the index. In order to limit changes to the bond portfolio, the manager chooses a swap strategy instead.

EXAMPLE 6

Five-Year Pay-Fixed Swap Overlay

In this example, the manager enters into a pay-fixed swap overlay with a notional principal equal to one-half of the size of the total bond portfolio. We will focus solely on first-order effects of yield changes on price (ignoring coupon income and swap carry) to determine the active and index portfolio impact. As the pay-fixed swap is a “short” duration position, it is a negative contribution to portfolio duration and therefore subtracted from rather than added to the portfolio. Recall the $100 million “index” portfolio has a modified duration of 5.299, or (1.994 + 4.88 + 9.023)/3. If the manager enters a $50 million notional 5-year pay-fixed swap with an assumed modified duration of 4.32, the portfolio’s modified duration falls to 3.139, or [(5.299 × 100) − (4.32 × 50)]/100. Stated differently, the bond portfolio BPV falls from $52,990 to $31,390 with the swap. For a 25 bp yield increase, this $21,600 reduction in active portfolio BPV reduces the adverse impact of higher rates by approximately $540,000 versus the “index” portfolio.

One point worth noting related to short duration positions is that with the exception of distressed debt situations addressed later in the curriculum, the uncertain cost and availability of individual bonds to borrow and sell short leads many active managers to favor the use of derivatives over short sales to establish a short bond position. Derivatives also facilitate duration changes without interfering with other active bond strategies with a portfolio.

Portfolio managers frequently use average duration and yield level changes to estimate bond portfolio performance in broad terms. However, these approximations are only reasonable if we assume a parallel yield curve shift. As Exhibits 2 and 3 show, non-parallel changes, or shifts in the slope and/or shape of the yield curve, occur frequently and require closer examination of individual positions and rate changes across maturities.

Divergent Yield Curve Slope View

Exhibit 2 established that while a positively sloped yield curve prevails under most economic scenarios, this difference between long-term and short-term yields-to-maturity can vary significantly over time. Changes in monetary policy, as well as expectations for growth and inflation, affect yields differently across the term structure, resulting in an increase (steepening) or decrease (flattening) in this spread. Although the barbell strategy combining extreme maturities is often referred to in a long-only context as in Example 1, here we take a more generalized approach in which the short-term and long-term security positions within the barbell trade may move in opposite directions—that is, combining a “short” and a “long” position. This type of barbell is an effective tool employed by managers to position a bond portfolio for yield curve steepening or flattening changes, as shown in Exhibit 13.

Exhibit 13:

Barbell Strategy for a Yield Curve Slope Change

A manager could certainly use a bullet to increase or decrease exposure to a specific maturity in anticipation of a price change that changes yield curve slope, but a combination of positions in both short and long maturities with greater cash flow dispersion is particularly well-suited to position for yield curve slope changes or twists. Managers combine long or short positions in either maturity segment to take advantage of expected yield curve slope changes—which may be duration neutral, net long, or short duration depending upon how the curve is expected to steepen or flatten in the future. Also, in some instances, the investment policy statement may allow managers to use bonds, swaps, and/or futures to achieve this objective. Finally, while not all strategies shown are cash neutral, here we focus solely on portfolio value changes due to yield changes, ignoring any associated funding or other costs that might arise as a result.

Yield curve steepener strategies seek to gain from an increase in yield curve slope, or a greater difference between long-term and short-term yields-to-maturity. This may be achieved by combining a “long” shorter-dated bond position with a “short” longer-dated bond position. For example, assume an active manager seeks to benefit from yield curve steepening with a net zero duration by purchasing the 2-year Treasury and selling the 10-year Treasury securities from our earlier example, both of which are priced at par.

Tenor	Coupon	Position ($ MM)	Modified Duration	Convexity
Long 2y	0.25%	163.8	1.994	5.0
Short 10y	2.00%	−36.2	9.023	90.8

Note that here and throughout the lesson, negative portfolio positions reflect a “short” position. We can approximate the impact of parallel yield curve changes using portfolio duration and convexity. Portfolio duration is approximately zero, or [1.994 × 163.8/(163.8 − 36.2)] + [9.023 × −36.2/(163.8 − 36.2)], and portfolio convexity equals −19.34, or [5.0 × 163.8/(163.8 − 36.2)] + [90.8 × −36.2/(163.8 − 36.2). A 25 bp increase in both 2-year and 10-year Treasury yields-to-maturity therefore has no duration effect on the portfolio, although negative convexity leads to a 0.006%, or $7,712 decline in portfolio value, or $127,600,000 × 0.5 × −19.34 × 0.00252.

However, changes in the difference between short- and long-term yields-to-maturity are not captured by portfolio duration or convexity but rather require assessment of individual positions. For example, if yield curve slope increases from 175 bps to 225 bps due to a 25 bp decline in 2-year yields-to-maturity and a 25 bp rise in 10-year yields-to-maturity, the portfolio increases in value by $1,625,412 as follows:

2y: $819,102 = $163,800,000 × (−1.994 × −0.0025 + 0.5 × 5.0 × −0.00252)

10y: $806,310 = −$36,200,000 × (−9.023 × 0.0025 + 0.5 × 90.8 × 0.00252)

EXAMPLE 7

Barbell Performance under a Flattening Yield Curve

Consider a Treasury portfolio consisting of a $124.6 million long 2-year zero-coupon Treasury with an annualized 2% yield-to-maturity and a short $25.41 million 10-year zero-coupon bond with a 4% yield-to-maturity. Calculate the net portfolio duration and solve for the first-order change in portfolio value based upon modified duration assuming a 25 bp rise in 2-year yield-to-maturity and a 30 bp decline in 10-year yield-to-maturity.

First, recall from earlier in the curriculum that Macaulay duration (MacDur) is equal to maturity for zero-coupon bonds and modified duration (ModDur) is equal to MacDur/1+r, where r is the yield per period. We can therefore solve for the modified duration of the 2-year zero as 1.96 (= 2/1.02) and the 10-year zero as 9.62 (= 10/1.04), so net portfolio duration equals zero, or (124.6 - 25.41 × 1.96) + (-25.4/124.6 - 25.41 × 9.62).

We may show that the 2-year Treasury BPV is close to $24,430 (= 1.96 × 124,600,000/10,000) and the 10-year Treasury position BPV is also approximately $24,430 (= 9.61 × 25,410,000/10,000), but it is a short position. Therefore a 25 bp increase in 2-year yield-to-maturity decreases portfolio value by $610,750 (25 bps × $24,430), while a 30 bp decrease in the 10-year yield-to-maturity also decreases portfolio value (due to the short position) by an additional $732,900 (= 30 bps × $24,430), for a total approximate portfolio loss of $1,343,650.

The portfolio manager is indifferent as to whether the portfolio gain from a greater slope arises due to a greater change in value from short-term or long-term yield movements as the duration is matched between the two positions. Two variations of a steeper yield curve adapted from Smith (2014) are shown in Exhibit 14.

Exhibit 14:

Yield Curve Slope Changes—Steepening

In an earlier lesson on establishing a rate view, we highlighted a bull steepening scenario under which short-term yields-to-maturity fall by more than long-term yields-to-maturity if the monetary authority cuts benchmark rates to stimulate economic activity during a recession. Exhibit 15 shows the bull steepening that occurred in the UK gilt yield curve amid the 2008 global financial crisis. After reaching a cycle peak of 5.75% in July 2007, the Bank of England cut its monetary policy base rate six times, down to 2.00% in early December 2008, due to weakening economic conditions and financial market stress.

Exhibit 15:

UK Government Yields, 2007 versus 2008 (Year End)

Source: Bloomberg.

On the other hand, a bear steepening occurs when long-term yields-to-maturity rise more than short-term yields-to-maturity. This could result from a jump in long-term rates amid higher growth and inflation expectations while short-term rates remain unchanged. In this case, an analyst might expect the next central bank policy change to be a monetary tightening to curb inflation.

Bull or bear steepening expectations will change the strategy an active fixed-income manager might pursue, as seen in Exhibit 16.

Exhibit 16:

Yield Curve Steepener Strategies

Strategy	Description	Expected Excess Return	Downside Risks
Duration neutral	Net zero duration	Portfolio gain from yield curve slope increase	Yield curve flattening
Bear steepener	Net negative (“short”) duration	Portfolio gain from slope increase and/or rising yields	Yield curve flattening and/or lower yields
Bull steepener	Net positive (“long”) duration	Portfolio gain from slope increase and/or lower yields	Yield curve flattening and/or higher yields

For example, assume an active manager expects the next yield curve change to be a bull steepening and establishes the following portfolio using the same 2-year and 10-year Treasury securities as in our prior examples.

Tenor	Coupon	Position ($ MM)	Modified Duration	Convexity
Long 2y	0.25%	213.8	1.994	5.0
Short 10y	2.00%	−36.2	9.023	90.8

In contrast to the earlier duration-matched steepener, the bull steepener increases the 2-year long Treasury position by $50 million, introducing a net long duration position to capitalize on an anticipated greater decline in short-term yields-to-maturity. We can see this by solving for portfolio duration of 0.5613, or [1.994 × 213.8/(213.8 − 36.2)] + [9.023 × −36.2/(213.8 − 36.2)], which is equivalent to a portfolio BPV of approximately $9,969, or 0.5613 × [($213,800,000 − $36,200,000)/10,000]. We may use this portfolio BPV to estimate the approximate portfolio gain if the 2-year yield-to-maturity and the 10-year yield-to-maturity fall by 25 bps, which is equal to $249,225 (= 25 bps × $9,969).

Yield curve flattening involves an anticipated narrowing of the difference between long-term and short-term yields-to-maturity, two basic variations of which are shown in Exhibit 17 and are adapted from Smith (2014).

Exhibit 17:

Yield Curve Slope Changes—Flattening

A flatter yield curve may follow monetary policy actions due to changing growth and inflation expectations. For example, a bear flattening scenario might follow the bear steepening move seen in Exhibit 15 if policymakers respond to rising inflation expectations and higher long-term rates by raising short-term policy rates. It was established earlier in the curriculum that investors sell higher risk assets and buy default risk-free government bonds in a flight to quality during highly uncertain markets, a situation which often contributes to bull flattening as long-term rates fall more than short-term rates. Flattener strategies may use a barbell strategy, which reverses the exposure profile of a steepener—namely, a “short” short-term bond position and a “long” long-term bond position. The bull and bear variations of this strategy are summarized in Exhibit 18.

Exhibit 18:

Yield Curve Flattener Strategies

Strategy	Description	Expected Excess Return	Downside Risks
Duration neutral	Net zero duration position	Portfolio gain from yield curve slope decrease	Yield curve steepening
Bear flattener	Net negative duration position	Portfolio gain from slope decrease and/or rising yields	Yield curve steepening and/or lower yields
Bull flattener	Net positive duration position	Portfolio gain from slope decrease and/or lower yields	Yield curve steepening and/or higher yields

Say, for example, a French investor expects the government yield curve to flatten over the next six months following years of quantitative easing by the European Central Bank through 2019. Her lack of a view as to whether this will occur amid lower or higher rates causes her to choose a duration neutral flattener using available French government (OAT) zero-coupon securities. She decides to enter the following trade at the beginning of 2020:

Tenor	Yield	Price	Notional (€ MM)	Modified Duration	Position BPV	Convexity
Short 2y	−0.65%	€101.313	−83.24	2.013	(€16,975)	6.1
Long 10y	0.04%	€99.601	17.05	9.996	€16,977	110

Note that as the Excel PRICE function returns a #NUM! error value for bonds with negative yields-to-maturity, we calculate the 2-year OAT zero-coupon bond price of 101.313 using 100/(1 − 0.0065)2. The initial portfolio BPV close to zero tells us that parallel yield curve shifts will have little effect on portfolio value, while the short 2-year and long 10-year trades position the manager to profit from a decline in the current 69 bp spread between 2- and 10-year OAT yields-to-maturity. After six months, the portfolio looks as follows:

Tenor	Yield	Price	Notional (€ MM)	Modified Duration	Convexity
Short 1.5y	−0.63%	€100.95	−83.24	1.51	3.8
Long 9.5y	−0.20%	€101.92	17.05	9.52	100.2

At the end of six months (June 2020), the sharp decline in economic growth and inflation expectations due to the COVID-19 pandemic caused the OAT yield curve to flatten as the 10-year yield-to-maturity fell. The six-month barbell return of €695,332 is comprised of rolldown return and yield changes, calculated as follows:

Rolldown Return

Zero-coupon bonds usually accrete in value as time passes if rates remain constant and the yield-to-maturity is positive. However, under negative yields-to-maturity, amortization of the bond’s premium will typically result in a negative rolldown return. In our example, the investor is short the original 2-year zero and therefore realizes a positive rolldown return on the short position. Rolldown return on the barbell may be shown to be approximately €277,924, as follows:

“Short” 2-year: −€83.24 MM × ([1/(1 + −0.65%)1.5] − [1/(1 + −0.65%)2)]

“Long” 10-year: €17.05 MM × ([1/(1 + 0.04%)9.5] − [1/(1 + 0.04%)10)]

Δ Price Due to Benchmark Yield Changes

The yield difference falls from 69 bps to 43 bps, mostly due to a 24 bp decline in the 10-year yield-to-maturity. Note that the Excel DURATION and MDURATION functions also return a #NUM! error for negative yields-to-maturity. We may use either price changes, as shown next, or the modified duration and convexity statistics as of the end of the investment horizon, just shown, to calculate a return of €417,408 using Equation 3.

“Short” 2-year: -€83.24 MM × ([1/(1 + −0.63%)1.5] − [1/(1 + −0.65%)1.5])

“Long” 10-year: €17.05 MM × ([1/(1 − 0.20%)9.5] − 1/(1 + 0.04%)9.5])

As we have considered duration-neutral, long, and short duration strategies to position the portfolio for expected yield curve slope changes, average duration is clearly no longer a sufficient summary statistic. A barbell strategy has greater cash flow dispersion and is therefore more convex than a bullet strategy, implying that its value will decrease by less than a bullet if yields-to-maturity rise and increase by more than a bullet if yields-to-maturity fall. We therefore must consider portfolio convexity in addition to duration when weighing yield curve slope strategies under different scenarios.

Divergent Yield Curve Shape View

As described in Section 2.1, yield curve shape or curvature describes the relationship between short-, medium-, and long-term yields-to-maturity across the term structure. Recall from Equation 2 that we quantify the butterfly spread by subtracting both short- and long-term rates from twice the intermediate yield-to-maturity. Since the difference between short- and medium-term rates is typically greater than that between medium- and long-term rates, the butterfly spread is usually positive, as seen earlier in Exhibit 3.

What factors drive yield curve curvature changes as distinct from rate level or curve slope changes? The segmented markets hypothesis introduced previously offers one explanation: Different market participants face either regulatory or economic asset/liability management constraints that drive the supply and demand for fixed-income instruments within different segments of the term structure. For example, a potential factor driving the apparent butterfly spread volatility in Exhibit 3 is the active central bank purchases of Treasury securities at specific maturities under its quantitative easing policy.

The most common yield curve curvature strategy combines a long bullet with a short barbell portfolio (or vice versa) in what is referred to as a butterfly strategy to capitalize on expected yield curve shape changes. The short-term and long-term bond positions of the barbell form the “wings,” while the intermediate-term bullet bond position forms the “body” of the butterfly, as illustrated in Exhibit 19. Note that unlike the steepener and flattener cases, the investor is either “long” or “short” both a short-term and long-term bond and enters into an intermediate-term bullet trade in the opposite direction.

Exhibit 19:

Butterfly Strategy

For example, consider a situation in which an active manager expects the butterfly spread to rise due to lower 2- and 10-year yields-to-maturity and a higher 5-year Treasury yield-to-maturity. Using the same portfolio statistics as in prior examples with bonds priced at par, consider the following combined short (5-year) bullet and long (2-year and 10-year) barbell strategy.

Tenor	Coupon	Position ($ MM)	Modified Duration	Position BPV	Convexity
Long 2y	0.25%	110	1.994	$21,934	5.0
Short 5y	0.875%	−248.3	4.88	($121,170)	26.5
Long 10y	2.00%	110	9.023	$99,253	90.8

While the sum of portfolio positions (−$28.3 MM) shows that the investor has a net “short” bond position, we can verify the strategy is duration neutral by either adding up the position BPVs or calculating the portfolio duration, or [1.994 × (110/−28.3)] + [4.88 × (−248.3/−28.3)] + [9.023 × (110/−28.3)] to confirm that both are approximately zero. The portfolio convexity may be shown as −139.9, or [5.0 × (110/−28.3)] + [26.5 × (−248.3/−28.3)] + [90.8 × (110/−28.3)].

How does this portfolio perform if 2- and 10-year Treasury yields-to-maturity fall by 25 bps each and the 5-year yield-to-maturity rises by 50 bps? A duration-based estimate multiplying each position BPV by the respective yield change gives us an approximation of $9,088,175, or (+25 bps × $21,934) + −(50 bps × -$121,170) + (+25 bps × $99,253). A more precise answer of $9,038,877 incorporating convexity for each position may be derived using Equation 3. You might ask why the precise portfolio value change is below our approximation. The answer lies in the relative magnitude of yield changes across the curve. Since the 5-year yield-to-maturity is assumed to increase by 50 bps rather than 25 bps, the convexity impact of the short bullet position outweighs that of the long barbell. Although the portfolio is nearly immune to parallel yield curve changes with a BPV close to zero, the portfolio gain in our example coincides with an increase in the butterfly spread from −50 bps to +100 bps.

This example shows that an active manager’s specific view on how yield curve shape will change will dictate the details of the combined bullet and barbell strategy. Exhibit 20, adapted from Smith (2014), shows both the negative butterfly view just shown as well as a positive butterfly, which indicates a decrease in the butterfly spread due to an expected rise in short- and long-term yields-to-maturity combined with a lower medium-term yield-to-maturity. Note that a positive butterfly view indicates a decrease in butterfly spread due to a bond’s inverse price–yield relationship.

Exhibit 20:

Yield Curve Curvature Changes

Note that as in the case of yield curve slope strategies, the combination of a short bullet and long barbell increases portfolio convexity due to higher cash flow dispersion, making this a more meaningful portfolio risk measure for this strategy than average duration (which remains neutral in the Exhibit 20 example). Exhibit 21 summarizes the two butterfly strategies.

Exhibit 21:

Yield Curve Curvature Strategies

Expected Scenario	Investor’s Expectation	Active Position
Negative butterfly	Lower short- and long-term yields, Higher medium-term yields	Short bullet, Long barbell (long positions in short- and long-term bonds)
Positive butterfly	Higher short- and long-term yields, Lower medium-term yields	Long bullet, Short barbell (short positions in short- and long-term bonds)

Yield Curve Volatility Strategies

While the prior sections focused on strategies using option-free bonds and swaps and futures as opposed to bonds with embedded options and stand-alone option strategies, we now explicitly address the role of volatility in active fixed-income management.

Option-only strategies play a more modest role in overall yield curve management. In markets such as in the United States where a significant portion of outstanding fixed-income bonds, such as asset-backed securities, have embedded options, investors use cash bond positions with embedded options more frequently than stand-alone options to manage volatility. For example, as of 2019 approximately 30% of the Bloomberg Barclays US Aggregate Bond Index was comprised of securitized debt, which mostly includes bonds with embedded options. As outlined earlier, the purchase of a bond call (put) option offers an investor the right, but not the obligation, to buy (sell) an underlying bond at a pre-determined strike price. An active manager’s choice between purchasing or selling bonds with embedded call or put options versus an option-free bond with otherwise similar characteristics hinges upon expected changes in the option value and whether the investor is “short” volatility (i.e., has sold the right to call a bond at a fixed price to the issuer), as in the case of callable bonds, or “long” volatility (i.e., owns the right to sell the bond at a fixed price to the issuer), as for putable bonds. Exhibit 22 shows how callable and putable bond prices change versus option-free bonds as yields-to-maturity change.

Exhibit 22:

Callable and Putable versus Option-Free Bonds

EXAMPLE 8

Option-Free Bonds versus Callable/Putable Bonds

1. An investment manager is considering an incremental position in a callable, putable, or option-free bond with otherwise comparable characteristics. If she expects a downward parallel shift in the yield curve, it would be most profitable to be:

   1. long a callable bond.

   2. short a putable bond.

   3. long an option-free bond.

   Solution:

   “C” is correct. The value of a bond with an embedded option is equal to the sum of the value of an option-free bond plus the value to the embedded option. The bond investor can be either long or short the embedded option, depending on the type of bond. With a callable bond, the embedded call option is owned by the issuer of the bond, who can exercise this option if yields-to-maturity decrease (the bond investor is short the call option). With a putable bond, the embedded put option is owned by the bond investor, who can exercise the option if yields-to-maturity increase. For a decrease in yields-to-maturity—as given in the question—the value of the embedded call option increases and the value of the embedded put option decreases. This means that a long position in a callable bond (“A”) would underperform compared to a long position in an option-free bond. A short position in a putable bond (“B”) would underperform a long position in an option-free bond primarily because yields-to-maturity were declining, although the declining value of the embedded put option would mitigate some of the loss (the seller of the putable bond has “sold” the embedded put).

As mentioned earlier in the curriculum, effective duration and convexity are the relevant summary statistics when future bond cash flows are contingent upon interest rate changes.

$Effective Duration (EffDur)=(PV−)−(PV+)2×(ΔCurve)(PV0)$Effective  Duration (EffDur)=(PV−)−(PV+)2×(𝛥 Curve)(PV0)8$EffectiveConvexity (EffCon)=(PV−)+(PV+)−2(PV0)(ΔCurve)2×(PV0)$Effective  Convexity (EffCon)=(PV−)+(PV+)−2(PV0)(𝛥 Curve)2×(PV0)9

In Equation 8 and Equation 9, PV− and PV+ are the portfolio values from a decrease and increase in yield-to-maturity, respectively, PV0 is the original portfolio value, and ∆Curve is the change in the benchmark yield-to-maturity.

Although cash-based yield curve volatility strategies are limited to the availability of liquid callable or putable bonds, several stand-alone derivatives strategies involve the right, but not the obligation, to change portfolio duration and convexity based upon an interest rate-sensitive payoff profile.

Interest rate put and call options are generally based upon a bond’s price, not yield-to-maturity. Therefore, the purchase of a bond call option provides an investor the right, but not the obligation, to acquire an underlying bond at a pre-determined strike price. This purchased call option adds convexity to the portfolio and will be exercised if the bond price appreciates beyond the strike price (i.e., generally at a lower yield-to-maturity). On the other hand, a purchased bond put option benefits the owner if prices fall (i.e., yields-to-maturity rise) beyond the strike prior to expiration. Sale of a bond put (call) option limits an investor’s return to the up-front premium received in exchange for assuming the potential cost of exercise if bond prices fall below (rise above) the pre-determined strike. Note that the option seller must post margin based on exchange or counterparty requirements until expiration.

An interest rate swaption involves the right to enter into an interest rate swap at a specific strike price in the future. This instrument grants the contingent right to increase or decrease portfolio duration. For example, Exhibit 23 shows a purchased payer swaption, which a manager might purchase to benefit from higher rates using an option-based strategy.

Exhibit 23:

Purchased Payer Swaption

Options on bond futures contracts are liquid exchange-traded instruments frequently used by fixed-income market participants to buy or sell the right to enter into a futures position. Long option, swaption, and bond futures option strategies are summarized in Exhibit 24.

Exhibit 24:

Long Option, Swaption, and Bond Futures Option Strategies

Strategy	Description	Targeted Return	Portfolio Duration Impact
Long bond call option	Purchase right to take forward bond delivery	Max (Bond price at lower yield − Strike price, 0) − Call premium	Increase portfolio duration
Long bond put option	Purchase right to deliver bond in the future	Max (Strike price − Bond price at higher yield, 0) − Put premium	Decrease portfolio duration
Long payer swaption	Own the right to pay-fixed on an interest rate swap at a strike rate	Max (Strike rate − Swap rate, 0) − Swaption premium	Decrease in portfolio duration
Long receiver swaption	Own the right to receive-fixed on an interest rate swap at a strike rate	Max (Swap rate − Strike rate, 0) − Swaption premium	Increase in portfolio duration
Long call option on bond future	Own the right to take forward bond delivery at a strike price	Max (Bond futures price at lower yield − Strike price, 0) − Call premium	Increase in portfolio duration
Long put option on bond future	Own the right to deliver bond in the future at a strike price	Max (Strike price − Bond futures price at higher yield, 0) − Put premium	Decrease in portfolio duration

EXAMPLE 9

Choice of Option Strategy

1. A parallel upward shift in the yield curve is expected. Which of the following would be the best option strategy?

   1. Long a receiver swaption

   2. Short a payer swaption

   3. Long a put option on a bond futures contract

   Solution:

   C is correct. With an expected upward shift in the yield curve, the portfolio manager would want to reduce portfolio duration in anticipation of lower bond prices. A put option increases in value as the yield curve shifts upward, while the price of the underlying bond declines below the strike. A is incorrect because a receiver swaption is an option to receive-fixed in an interest rate swap. With fixed-rate bond prices expected to fall as rates rise, the portfolio manager would not want to exercise an option to receive a fixed strike rate, which is similar to owning a fixed-rate bond. B is incorrect because a payer swaption is an option to pay-fixed/receive-floating in an interest rate swap. A long, not a short, position in a payer swaption would benefit from higher rates.

   In an expected stable or static yield curve environment, an active manager may aim to “sell” volatility in the form of either owning callable bonds (which is an implicit “sale” of an option) or selling stand-alone options in order to earn premium income, if this is within the investment mandate. The active portfolio decision here depends upon the manager’s view as to whether future realized volatility will be greater or less than the implied volatility, as reflected by the price of a stand-alone option or a bond with embedded options. The manager will benefit if rates remain relatively constant and the bond is not called and/or the options sold expire worthless. Alternatively, if yield curve volatility is expected to increase, a manager may prefer to be long volatility in order to capitalize on large changes in level, yield curve slope, and/or shape using option-based contracts.

EXAMPLE 10

Option-Free versus Callable and Putable Bonds Amid Higher Yield Levels

1. Given a parallel shift upwards in the yield curve, what is the most likely ordering in terms of expected decline in value—from least to most—for otherwise comparable bonds? Assume that the embedded options are deep out-of-the-money.

   1. Callable bond, option-free bond, putable bond

   2. Putable bond, callable bond, option-free bond

   3. Putable bond, option-free bond, callable bond

   Solution:

   Answer: B is correct. The value of a bond with an embedded option may be considered as the value of an option-free bond plus the value of the embedded option. While the upward shift in the yield curve will cause the option-free component of each bond to depreciate in value, this change in yields-to-maturity will also affect the value of embedded options.

   For a putable bond, the bond investor has the option to “put” the bond back to the issuer if yields-to-maturity rise. The more rates rise, the more valuable this embedded option becomes. This increasing option value will partially offset the decline in value of the putable bond relative to the option-free bond. This can be seen in the lower panel of Exhibit 22: The dotted line for the putable bond has a flatter slope than the solid line for the option-free bond; its price will decrease more slowly as yields-to-maturity increase.

   For a callable bond, the bond issuer has an option to “call” the bond if yields-to-maturity decline; the more rates rise, the lower the call option value. Since the bond investor is short the embedded option and the value of the embedded option has fallen, this will partially offset the decline in the value of the callable bond relative to the option-free bond. The top panel of Exhibit 22 shows that the dotted line for the callable bond has a flatter slope than the solid line for the option-free bond.

   As rates continue to increase, the embedded option for the putable bond rises in value more quickly at the margin as it shifts toward becoming an in-the-money option. In contrast, the deep out-of-the-money embedded call option moves further out-of-the-money as rates increase and the marginal impact of further rate increases declines.

Key Rate Duration for a Portfolio

So far, we have evaluated changes in yield curve level, slope, and curvature using one, two, and three specific maturity points across the term structure of interest rates, respectively. The concept of key rate duration (or partial duration) introduced previously measures portfolio sensitivity over a set of maturities along the yield curve, with the sum of key rate durations being identical to the effective duration:

$KeyRateDurk=1PV×ΔPVΔrk$KeyRateDur𝑘=1PV×𝛥PV𝛥𝑟𝑘10$∑k=1nKeyRateDurk=EffDur$∑𝑘=1𝑛KeyRateDur𝑘=EffDur11where rk represents the kth key rate and PV is the portfolio value. In contrast to effective duration, key rate durations help identify “shaping risk” for a bond portfolio—that is, a portfolio’s sensitivity to changes in the shape of the benchmark yield curve. By breaking down a portfolio into its individual duration components by maturity, an active manager can pinpoint and quantify key exposures along the curve, as illustrated in the following simplified zero-coupon bond example.

Compare a passive zero-coupon US Treasury bond portfolio versus an actively managed portfolio:

“Index” Zero-Coupon Portfolio

Tenor	Coupon	Annualized Yield	Price (per $100)	Position ($ MM)	ModDur	KeyRateDur
2y	0.00%	1%	98.03	98.03	1.980	0.738
5y	0.00%	2%	90.57	90.57	4.902	1.688
10y	0.00%	3%	74.40	74.40	9.709	2.747

Assume the “index” portfolio is simply weighted by the price of the respective 2-, 5-, and 10-year bonds for a total portfolio value of $263 million, or $1 million × (98.03 + 90.57 + 74.4). We can calculate the portfolio modified duration as 5.173, or [1.98 × (98.03/263)] + [4.902 × (90.57/263)] + [9.709 × (74.40/263)]. Or, we could calculate each key rate duration by maturity, as in the far right column. For example, the 2-year key rate duration (KeyRateDur2) equals 0.738, or 1.98 × (98.03/263). Note that these three key rate duration values also sum to the portfolio value of 5.173.

“Active” Zero-Coupon Portfolio

Tenor	Coupon	Annualized Yield	Price (per $100)	Position ($ MM)	ModDur	KeyRateDur
2y	0.00%	1%	98.03	51.40	1.980	0.387
5y	0.00%	2%	90.57	−46.00	4.902	−0.857
10y	0.00%	3%	74.40	257.60	9.709	9.509

As in the case of the “index” portfolio, the “active” zero-coupon portfolio has a value of $263 million, or [$1 million × (51.4 − 46 + 257.6)], but the portfolio duration is greater at 9.039, or [1.98 × (51.4/263)] + [4.902 × (−46/263)] + [9.709 × (257.6/263)]. Note that the short 5-year active position has a negative key rate duration of −0.857, or 4.902 × (−46/263).

By now, you may have noticed that our active manager is positioned for the combination of a negative butterfly and a bull flattening at the long end of the yield curve. However, a comparison of the active versus index portfolio duration summary statistic does not tell the entire story. Instead, we can compare the key rate or partial durations for specific maturities across the index and active portfolios to better understand exposure differences:

Tenor	Active	Index	Difference
2y	0.39	0.74	−0.35
5y	−0.86	1.69	−2.55
10y	9.51	2.75	6.76
Portfolio	9.04	5.17	3.87

The key rate duration differences in this chart provide more detailed information regarding the exposure differences across maturities. For example, the negative differences for 2-year and 5-year maturities (−0.35 and −2.55, respectively) indicate that the active portfolio has lower exposure to short-term rates than the index portfolio. The large positive difference in the 10-year tenor shows that the active portfolio has far greater exposure to 10-year yield-to-maturity changes. This simple zero-coupon bond example may be extended to portfolios consisting of fixed-coupon bonds, swaps, and other rate-sensitive instruments that may be included in a fixed-income portfolio, as seen in the following example.

EXAMPLE 11

Key Rate Duration

1. A fixed-income manager is presented with the following key rate duration summary of his actively managed bond portfolio versus an equally weighted index portfolio across 5-, 10-, and 30-year maturities:

   Tenor	Active	Index	Difference
   5y	−1.188	1.633	−2.821
   10y	2.909	3.200	−0.291
   30y	11	8.067	2.933
   Portfolio	12.72	12.9	−0.179

   Assume the active manager has invested in the index bond portfolio and used only derivatives to create the active portfolio. Which of the following most likely represents the manager’s synthetic positions?

   1. Receive-fixed 5-year swap, short 10-year futures, and pay-fixed 30-year swap

   2. Pay-fixed 5-year swap, short 10-year futures, and receive-fixed 30-year swap

   3. Short 5-year futures, long 10-year futures, and receive-fixed 30-year swap

   Solution:

   Answer: B is correct. The key rate duration summary shows the investor to be net short 5- and 10-year key rate duration and long 30-year key rate duration versus the index. A combines synthetic long, short, and short positions in the 5-, 10-, and 30-year maturities, respectively. C combines short, long, and long positions across the curve. The combination of a pay-fixed (short duration) 5-year swap, a short 10-year futures position, and a receive-fixed (long duration) 30-year swap is, therefore, the best answer.

ACTIVE FIXED-INCOME MANAGEMENT ACROSS CURRENCIES

https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#read/study_task/2563503/active-fixed-income-management-across-currencies-1

ACTIVE FIXED-INCOME MANAGEMENT ACROSS CURRENCIES

Learning Outcome

• discuss yield curve strategies across currencies

The benefits of investing across borders to maximize return and diversify exposure is a consistent theme among portfolio managers. While both the tools as well as the strategic considerations of active versus passive currency risk management within an investment portfolio are addressed elsewhere, here we will primarily focus on extending our analysis of yield curve strategies from a single yield curve to multiple yield curves across currencies.

An earlier currency lesson noted that investors measure return in functional currency terms—that is, considering domestic currency returns on foreign currency assets, as shown in Equation 12 and Equation 13.

RDC and RFC are the domestic and foreign currency returns expressed as a percentage, RFX is the percentage change of the domestic versus foreign currency, while ωi is the respective portfolio weight of each foreign currency asset (in domestic currency terms) with the sum of ωi equal to 1. In the context of Equation 1, RDC simply combines the third factor, +/−E (Δ Price due to investor’s view of benchmark yield), and the fifth factor, +/−E (Δ Price due to investor’s view of currency value changes), factors in the expected fixed-income return model.

In a previous term structure lesson, we highlighted several macroeconomic factors that influence the bond term premium and required returns, such as inflation, economic growth, and monetary policy. Differences in these factors across countries are frequently reflected in the relative term structure of interest rates as well as in exchange rates.

For example, after a decade of economic expansion following the 2008 global financial crisis, the US Federal Reserve’s earlier reversal of quantitative easing versus the European Central Bank through 2019 led to significantly higher short-term government yields-to-maturity in the United States versus Europe.

Against this historical backdrop, assume a German fixed-income manager decides to buy short-term US Treasuries to take advantage of higher USD yields-to-maturity. At the end of March 2019, a USD Treasury zero-coupon bond maturing on 31 March 2021 had a price at 95.656, with an approximate yield-to-maturity of 2.25%. Based upon the then-current USD/EUR spot rate of 1.1218 (that is, $1.1218 = €1), the manager pays €85,270,102 (= $95,656,000/1.1218) for a $100 million face value Treasury security, as seen in Exhibit 25.

Exhibit 25:

USD/EUR Spot Trade and US Treasury Zero Purchase

As in the single currency yield curve case, the investor will benefit from bond price appreciation if the US Treasury yield-to-maturity falls during the holding period. In addition, since her domestic returns are measured in EUR, she will also benefit if the USD she receives upon sale of the bond or at maturity buy more EUR per USD in the future—that is, if USD/EUR decreases (i.e., USD appreciates versus EUR).

In fact, the flight to quality induced by the COVID-19 pandemic in early 2020 led to a sharp decline in US Treasury yields-to-maturity. Exhibit 26 shows how the relationship between US and German government rates changed between March 2019 and March 2020.

Exhibit 26:

US vs. German Government Yield Curves, 2019 and 2020

Source: Bloomberg.

As a result, one year after purchase (31 March 2020), the US Treasury zero-coupon bond maturing 31 March 2021 traded at a price of 100.028 and the USD/EUR spot was 1.1031.

Now we calculate the German investor’s 1-year domestic currency return from holding the $100 million par value US Treasury zero-coupon bond.

Equation 12 separates this return into two key components:

RFC: 4.57%, = ($100,028,000/$95,656,000) − 1, as the investor receives $100,028,000 upon sale of the US Treasury bond purchased a year earlier at $95,656,000.

RFX: 1.70%, = (1.1218/1.1031 − 1), as the investor converted €85,270,102 into USD to purchase the bond at 1.1218 and then converted USD proceeds back to EUR at 1.1031. The EUR depreciated (i.e., lower USD/EUR spot rate) over the 1-year period.

RDC may be shown to be 6.34%, solved either using Equation 12 or directly for the 1-year return on investment in EUR terms, = (€90,678,996/€ 85,270,102) −1.

In contrast to the unhedged 1-year example, let us now assume that the German manager fully hedges the foreign currency risk associated with the US Treasury bond purchase and holds it instead for two years, at which time she receives the bond’s face value of $100,000,000. Specifically, the manager enters a 2-year FX forward agreement at the time of bond purchase to sell the future $100,000,000 payment upon bond maturity and buy EUR at the then current 2-year USD/EUR forward rate of 1.1870, locking in a certain €84,245,998, = $100,000,000/1.1870, in two years’ time.

If fully hedged, the expected annualized return, RDC, in EUR terms on the 2-year US Treasury zero-coupon bond hedged EUR investment over two years is equal to −0.60%, = (€84,245,998/€85,270,102)0.5 − 1, which matches the 2-year annualized German government zero-coupon bond yield-to-maturity upon inception. This may also be calculated using Equation 12, with RFC = 2.25% and RFX = −2.785%, or (1.1218/1.1870)0.5 − 1.

The fully hedged investment example is a reminder from earlier lessons that covered interest rate parity establishes a fundamental no-arbitrage relationship between spot and forward rates for individual cash flows in T periods, as shown in Equation 14.

$F(DCFC,T)=S0(DC/FC)(1+rDC)T(1+rFC)T$𝐹(DCFC,𝑇)=𝑆0(DC/FC)(1+𝑟DC)𝑇(1+𝑟FC)𝑇14

F denotes the forward rate; S is the spot rate; and rDC and rFC reflect the respective domestic and foreign currency risk-free rates. If an investor uses a forward contract to fully hedge foreign currency cash flows, she should expect to earn the domestic risk-free rate, as seen in our example. Recall also that this implies in general that a higher-yielding currency will trade at a forward discount, while a lower-yielding currency will trade at a premium. This is consistent with USD/EUR spot versus forward exchange rates (1.1218 spot versus the 1.187 2-year forward rate) as well as the relationship between USD rates and EUR rates in 2019, as shown in Exhibit 26.

In contrast, uncovered interest rate parity suggests that over time, the returns on unhedged foreign currency exposure will be the same as on a domestic currency investment. Although forward FX rates should in theory be an unbiased predictor of future spot FX rates if uncovered interest rate parity holds, in practice investors sometimes seek to exploit a persistent divergence from interest rate parity conditions (known as the forward rate bias) by investing in higher-yielding currencies, which is in some cases enhanced by borrowing in lower-yielding currencies.

This demonstrates that active fixed-income strategies across currencies must factor in views on currency appreciation versus depreciation as well as yield curve changes across countries. Our investor’s USD versus EUR interest rate view in the previous example combined with an implicit view that USD/EUR would remain relatively stable led to the highest return in the unhedged case with a 1-year investment horizon. This stands in contrast to the relationship between USD/EUR spot and 2-year forward rates at the inception of the trade on 31 March 2019, when implied (annualized) EUR appreciation was 2.87%, = (1.187/1.1218)0.5 − 1.

The European fixed-income manager in our example might use leverage instead of cash by borrowing in euros when buying the 2-year US Treasury zero. This is an extension of the single currency repo carry trade shown in Exhibit 5, in which an investor borrows short-term in one currency and invests in another higher-yielding currency. This carry trade across currencies is a potential source of additional income subject to short-term availability if the positive interest rate differential persists for the life of the transaction. Given the preponderance of fixed-rate coupon versus zero-coupon bonds, our analysis turns next to these securities. As in the case of the fully hedged German investor in US Treasuries, we first establish the necessary building blocks to replicate a risk-free domestic currency return when investing in a foreign currency fixed-income coupon bond. We then consider how an active investor might deviate from this exposure profile to generate excess return.

Consider the example of a Japan-based investor who buys a fixed-rate USD coupon bond. In order to fully hedge JPY domestic currency cash flows for the foreign currency bond, as in the case of the earlier German investor, the investor must first sell Japanese yen (JPY) and purchase USD at the current spot rate to purchase the bond. At the end of each semi-annual interest period, the investor receives a USD coupon, which must be converted at the future JPY/USD spot rate (that is, the number of JPY required to buy one USD). At maturity, the investor receives the final semi-annual coupon and principal, which must be converted to JPY using the future JPY/USD spot rate to receive the final payment in domestic currency.

The fixed-rate foreign currency bond exposes the Japanese investor to a series of FX forward exposures that may be hedged upon purchase with a cross-currency swap, as seen in Exhibit 27 with the example of a par 10-year US Treasury bond with a 0.625% coupon issued in May 2020.

Exhibit 27:

Fixed-Fixed Cross-Currency Swap Diagram and Details

Trade Details	JPY/USD Fixed-Fixed Cross-Currency Swap
Start date	15 May 2020
Maturity date	15 May 2030
Fixed USD payer	JPY Investor
Fixed JPY payer	Swap counterparty
Initial exchange	JPY investor pays JPY10.706 billion and receives USD100 million as of 15 May 2020
Fixed USD rate	0.625% Semiannual, Act/Act
Fixed JPY rate	−0.726% Semiannual, Act/365
Final exchange	JPY investor pays USD100 million and receives JPY10.706 billion as of 15 May 2030

Note that the fixed-fixed cross-currency swap components, shown in Exhibit 28, are a combination of three distinct hedging transactions: a receive-fixed JPY interest rate swap, a USD-JPY cross-currency basis swap involving the exchange of floating JPY for floating USD payments, and a pay-fixed USD interest rate swap.

Exhibit 28:

Fixed-Fixed Cross-Currency Swap Components

Cross-Currency Basis and Covered Interest Rate Parity

The “basis” or spread, as shown in the cross-currency basis swap, is the difference between the USD interest rate and the synthetic USD interest rate derived from swapping JPY into US dollars. A positive (negative) currency basis means that the direct USD interest rate is higher (lower) than the synthetic USD interest rate. While covered interest rate parity suggests that cross-currency basis should be close to zero, Exhibit 29 shows that the JPY and EUR cross-currency basis was persistently negative following the 2008 global financial crisis.

Exhibit 29:

Five-Year JPY and EUR Cross-Currency Basis, 2006–2020

Cross-currency basis is widely seen as a barometer for global financial conditions. For example, greater credit and liquidity risk within the EU financial sector and the European Central Bank’s aggressive quantitative easing have been cited as causes of the wider USD/EUR cross-currency basis.

Du, Tepper, and Verdelhan (2018) investigate the persistent no-arbitrage violation of covered interest rate parity implied by wider cross-currency basis observed across G-10 countries and offer several explanations. First, higher financial intermediation costs since the 2008 global financial crisis, such as higher bank regulatory capital requirements, prevent market participants from taking advantage of basis arbitrage opportunities. Second, covered interest rate parity violations suggest international imbalances in the form of high demand for investments in high interest rate currencies and a large supply of savings in low interest rate currencies. These deviations are magnified by divergent monetary policies across jurisdictions.

The building blocks of the fixed-fixed cross-currency swap shown in Exhibit 28 offer an active fixed-income investor a simplified framework within which one can take interest rate or currency positions to deviate from a risk-free domestic currency return. For example, by foregoing the pay USD fixed swap, the JPY investor takes a USD rate view by earning the USD fixed coupon and paying USD floating while fully hedging the currency exposure via the cross-currency basis swap. Similar principles apply as in the single currency case—namely, to go long (or overweight) assets expected to appreciate and go short (or underweight) assets expected to decline in value or appreciate less. The overweight and underweight bond positions may now be denominated in different currencies, with the active strategy often using an underweight position in one currency to fund an overweight position in another. The resulting yield curve strategy faces three potential risks: (1) yield curve movements—level, slope, or curvature—in the overweight currency; (2) yield curve changes in the underweight currency; and (3) exchange rate changes.

Consider the following unhedged example of a higher- versus lower-yielding currency.

EXAMPLE 12

MXN Carry Trade

Consider the case of a portfolio manager examining a cross-currency carry trade between US dollar (USD) and Mexican peso (MXN) money market rates. The manager is contemplating borrowing in USD for one year and investing in 90-day Mexican treasury bills, rolling them over at maturity for the next 12 months. Assume that today’s 1-year USD interest rate is 1.85%, the 90-day MXN interest rate is currently 7.70% (annualized), and the MXN/USD spot exchange rate is 19.15 (that is, it takes 19.15 MXN to buy one USD).

If the manager expected that Mexican money market rates and the MXN/USD exchange rate would remain stable, the expected profit from this carry trade is:

(1 + 0.0770/4)4 − (1 + 0.0185) ≈ 6.08%.

However, money market and exchange rates are rarely stable; this trade is exposed to changes in both the 90-day MXN interest rate and the MXN/USD spot exchange rate. (The 1-year fixed-rate USD loan eliminates exposure to USD rate changes). Assume that 90-day MXN interest rates and exchange rates change as follows over the 12-month period.

Rate / Time	Today	90 Days	180 Days	270 Days	360 Days
90-day MXN rate	7.70%	7.85%	8.15%	8.20%	N/A
MXN/USD spot rate	19.15	18.05	19.05	18.80	19.65

Note that 90-day MXN yields-to-maturity rose and that MXN depreciated slightly versus USD over the 360-day period. If the manager had rolled over this trade for the full 12 months, the realized return would have been:

$(1+0.07704)(1+0.07854)(1+0.08154)(1+0.0824)×19.1519.65−(1+0.0185)≈3.61$1+0.077041+0.078541+0.081541+0.0824×19.1519.65−1+0.0185≈3.61%

While the cross-currency carry trade was ultimately profitable, it was exposed to risks over the horizon; moreover, despite the rise in 90-day MXN yields-to-maturity, a late-period MXN depreciation undercut the profitability of the trade. This underscores the fact that carry trades are unhedged and are most successful in stable (low volatility) markets: Unforeseen market volatility can quickly erase even the most attractive cross-currency carry opportunities. For example, in the first quarter of 2020 at the start of the COVID-19 pandemic, MXN depreciated against the USD by approximately 25% in just over a month.

While an endless number of unhedged strategies seeking to capitalize on a level, slope, or curvature view across currencies exist, Exhibit 30 summarizes several of these major strategies.

Exhibit 30:

Active Cross-Currency Strategies

Strategy	Purchase	Sell / Borrow	Expected Unhedged Return
Receive-fixed/pay-fixed	High-yielding fixed-income asset	Lower-yield fixed-rate loan	Carry (higher yield minus lower yield) assuming uncovered interest parity does not hold
Receive-fixed/pay-floating	High-yielding fixed-rate asset	Short-term, lower yield floating-rate loan rolled over until maturity	Carry (higher yield minus lower yield) plus long- versus short-term rate differential for lower-yielding currency
Receive-floating/pay-fixed	High-yield floating-rate asset	Lower-yield fixed-rate loan	Carry (higher floating yield minus lower fixed yield)
Receive-floating/pay-floating	High-yield floating-rate asset	Short-term, lower yield floating-rate loan rolled over until maturity	Carry (higher floating yield minus lower floating yield)

EXAMPLE 13

Bear Flattening Impact

1. A fixed-income manager is considering a foreign currency fixed-income investment in a relatively high-yielding market, where she expects bear flattening to occur in the near future and her lower-yielding domestic yield curve to remain stable and upward-sloping. Under this scenario, which of the following strategies will generate the largest carry benefit if her interest rate view is realized?

   1. Receive-fixed in foreign currency, pay-fixed in domestic currency

   2. Receive-fixed in foreign currency, pay-floating in domestic currency

   3. Receive-floating in foreign currency, pay-floating in domestic currency

   Solution:

   The correct answer is C. If the higher-yielding foreign currency experiences a bear flattening in the yield curve as the manager expects, then foreign currency short-term yields-to-maturity will increase by more than long-term yields-to-maturity; thus she will want receive-floating in foreign currency. Given the upward-sloping domestic yield curve, we would expect the carry difference between receiving foreign currency floating rates and paying domestic currency floating rates to be the highest.

A FRAMEWORK FOR EVALUATING YIELD CURVE STRATEGIES

https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#read/study_task/2563513/a-framework-for-evaluating-yield-curve-strategies-1

Learning Outcome

• evaluate the expected return and risks of a yield curve strategy

The factors affecting the expected return of a fixed-income portfolio were summarized in Equation 1. The key underlying assumption in this calculation is that the inputs rely on the fixed-income manager’s expectations under an active strategy. As we have seen earlier, unexpected changes to the level, slope, and shape of the yield curve as well as currency changes can impact a portfolio’s value in a number of ways—as quantified by the use of portfolio duration and convexity statistics in Equation 3 for a single currency and in Equation 13 for a multicurrency portfolio.

Practitioners frequently evaluate fixed-income portfolio risk using scenario analysis, which involves changing multiple assumptions at once to assess the overall impact of unexpected market changes on a portfolio’s value. Managers may use historical rate and currency changes or conduct specific stress tests using this analysis. For example, a leveraged investor might evaluate how much rates or currencies must move before she faces a collateral or margin call or is forced to unwind a position. Fixed-income portfolio models offer practitioners a variety of historical or user-defined scenarios. The following scenario analysis example shows how this may be done for the US Treasury portfolio seen earlier.

EXAMPLE 14

Scenario Analysis—US Treasury Securities Portfolio

Tenor	Coupon	Price	Modified Duration	Convexity
2y	0.25%	100	1.994	5.0
5y	0.875%	100	4.88	26.5
10y	2.00%	100	9.023	90.8

In Example 1, we compared two $50 million portfolios. Portfolio A is fully invested in the 5-year Treasury bond, while Portfolio B is split between 2-year (58.94%) and 10-year (41.06%) bonds to match a 5-year bond duration of 4.88. Rather than the earlier parallel yield curve shift, we now analyze two yield curve slope scenarios—namely, an immediate bear steepening and bull flattening of the US Treasury yield curve. The bear steepening scenario involves a 50 bp and 100 bp rise in 5- and 10-year yields-to-maturity, respectively, while the bull flattening is assumed to result from a 50 bp fall in 5-year rates and a 100 bp fall in 10-year rates. Using Equation 3, our scenario analysis looks as follows:

Scenario	Portfolio A % Δ Price	Portfolio A Δ Price	Portfolio B % Δ Price	Portfolio B Δ Price
Bear steepening	−2.407%	($1,203,437)	−3.518%	($1,759,216)
Bull flattening	2.473%	$1,236,563	3.891%	$1,945,628

We may conclude from our analysis that although Portfolios A and B have similar duration and therefore perform similarly if the yield curve experiences a parallel shift (except for the convexity difference) seen in Example 1, they perform very differently under various yield curve slope scenarios.

The fixed-income portfolio risk and return impact of rolldown return versus carry, changes in the level, slope, and shape of a single currency yield curve, and an extension to multiple currencies (where spot and forward FX rates are related to relative interest rates) are best illustrated with a pair of examples.

EXAMPLE 15

AUD Bullet versus Barbell

A US-based portfolio manager plans to invest in Australian zero-coupon bonds denominated in Australian dollars (AUD). He projects that over the next 12 months, the Australian zero-coupon yield curve will experience a downward parallel shift of 60 bps and that AUD will appreciate 0.25% against USD. The manager is weighing bullet and barbell strategies using the following data:

Statistic	Bullet	Barbell
Investment horizon (years)	1.0	1.0
Average bond price in portfolio (today)	98.00	98.00
Average portfolio bond price (in 1 year/stable yield curve)	99.75	100.00
Expected portfolio effective duration (in 1 year)	3.95	3.95
Expected portfolio convexity (in 1 year)	19.50	34.00
Expected change in AUD zero-coupon yields	−0.60%	−0.60%
Expected change in AUD versus USD	+0.25%	+0.25%

Solve for the expected return over the 1-year investment horizon for each portfolio using the step-by-step estimation approach in Equation 1.

Rolldown Return

The sum of coupon income (in %) and the price effect on bonds from “rolling down the yield curve.” Since both portfolios contain only zero-coupon bonds, there is no coupon income and we calculate the rolldown return using (PV1 − PV0) / PV0, where PV0 is today’s bond price and PV1 is the bond price in one year, assuming no shift in the yield curve.

1. Bullet: 1.7857% = (99.75 − 98.00) / 98.00

2. Barbell: 2.0408% = (100.000 − 98.00) / 98.00

E (Δ Price Due to Investor’s View of Benchmark Yield)

The effect of the interest rate view on expected portfolio return may be estimated using Equation 3, using effective duration and convexity in one year’s time to evaluate the expected 60 bp downward parallel yield curve shift:

1. Bullet: 2.4051% = (−3.95 × −0.0060) + [1/2 × 19.5 × (−0.0060)2]

2. Barbell: 2.4312% = (−3.95 × −0.0060) + [1/2 × 34.0 × (−0.0060)2]E(R) ≈ % Rolldown return + E (% Δ Price due to investor’s view of benchmark yield) + E (% Δ Price due to investor’s view of currency value changes)

In addition to rolldown return and expected price changes due to changes in yield-to-maturity, the expected 0.25% appreciation of AUD versus USD must be incorporated in order to arrive at the USD investor’s domestic currency return. Using Equation 12, RFC equals the sum of rolldown return and changes in price due to yield-to-maturity changes, while RFX is 0.25%. Expected returns are as follows:

E(R1) = 4.4513%, or [(1 + 0.017857 + 0.024051) × (1.0025)] − 1

E(R2) = 4.7332%, or [(1 + 0.020408 + 0.024312) × (1.0025)] − 1

Overall, the barbell outperforms the bullet by approximately 28 bps. Rolldown return contributes most of this outperformance. Rolldown return contributed approximately 25.5 bps of outperformance (i.e., 2.0408% − 1.7857%) for the barbell, and the greater convexity of the barbell portfolio contributed just over 2.6 bps of outperformance (i.e., 2.4312% − 2.4051%). Currency exposure had the same impact on both strategies. The strong rolldown contribution is likely driven by the stronger price appreciation (under the stable yield curve assumption) of longer-maturity zeros in the barbell portfolio relative to the price appreciation of the intermediate zeros in the bullet portfolio as the bonds ride the curve over the 1-year horizon to a shorter maturity.

EXAMPLE 16

US Treasury Bullet versus Barbell

Assume a 1-year investment horizon for a portfolio manager considering US Treasury market strategies. The manager is considering two strategies to capitalize on an expected rise in US Treasury security zero-coupon yield levels of 50 bps in the next 12 months:

1. A bullet portfolio fully invested in 5-year zero-coupon notes currently priced at 94.5392.

2. A barbell portfolio: 62.97% is invested in 2-year zero-coupon notes priced at 98.7816, and 37.03% is invested in 10-year zero-coupon bonds priced at 83.7906.

Further assumptions for evaluating these portfolios are shown here:

Statistic	Bullet	Barbell
Investment horizon (years)	1.0	1.0
Average bond price in portfolio (today)	94.5392	92.6437
Average portfolio bond price (in 1 year/stable yield curve)	96.0503	94.3525
Expected portfolio effective duration (in 1 year)	3.98	3.98
Expected portfolio convexity (in 1 year)	17.82	32.57
Expected change in US Treasury zero-coupon yields	0.50%	0.50%

Solve for the expected return over the 1-year investment horizon for each portfolio using the step-by-step estimation approach in Equation 1.

Rolldown Return

The sum of coupon income (in %) and the price effect on bonds from “rolling down the yield curve.” Since both portfolios contain only zero-coupon bonds, there is no coupon income and we calculate the rolldown return using (PV1 − PV0) / PV0, where PV0 is today’s bond price and PV1 is the bond price in one year, assuming no shift in the yield curve.

Bullet: (96.0503 − 94.5392) ÷ 94.5392 = 1.5984%

Barbell: (94.3525 − 92.6437) ÷ 92.6437 = 1.8445%

E (Δ Price Due to Investor’s View of Benchmark Yield)

The effect of the interest rate view on expected portfolio return may be estimated with Equation 3, using effective duration and convexity in one year’s time to evaluate the expected 50 bp upward parallel yield curve shift:

Bullet: −1.9677% = (−3.98 × 0.0050) + [1/2 × 17.82 × (0.0050)2]

Barbell: −1.9493% = (−3.98 × 0.0050) + [1/2 × 32.57 × (0.0050)2]

Expected total return in percentage terms for each portfolio is equal to:

E(R) = % Rolldown return + E (% Δ Price due to investor’s view of benchmark yield)

The total expected return over the 1-year investment horizon for the bullet portfolio is therefore −0.3693%, or 1.5984% − 1.9677%, and the expected return for the barbell portfolio is −0.1048%, or 1.8445% − 1.9493%.

If the manager’s expected market scenario materializes, the barbell portfolio outperforms the bullet portfolio by 26 bps. The higher barbell convexity contributed just under 2 bps of outperformance, whereas the rolldown return contributed nearly 25 bps. Stronger price appreciation (under the stable yield curve assumption) resulted from a greater rolldown effect from the 10-year zeros in the barbell versus the 5-year zeros over one year.