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https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#quiz/take/study_task/2562383
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https://study.cfainstitute.org/app/cfa-institute-program-level-iii-for-august-2024#quiz/take/study_task/2562383
Last updated
Q.
John Tomb is an investment advisor at an asset management firm. He is developing an asset allocation for James Youngmall, a client of the firm. Tomb considers two possible allocations for Youngmall. Allocation A consists of four asset classes: cash, US bonds, US equities, and global equities. Allocation B includes these same four asset classes, as well as global bonds.
Youngmall has a relatively low risk tolerance with a risk aversion coefficient (λ) of 7. Tomb runs mean–variance optimization (MVO) to maximize the following utility function to determine the preferred allocation for Youngmall:
The resulting MVO statistics for the two asset allocations are presented in Exhibit 1.
Exhibit 1:
MVO Portfolio Statistics
Allocation A
Allocation B
Expected return
6.7%
5.9%
Expected standard deviation
11.9%
10.7%
Determine which allocation in Exhibit 1 Tomb should recommend to Youngmall. Justify your response.
Determine which allocation in Exhibit 1 Tomb should recommend to Youngmall. (circle one)
Allocation A
Allocation B
Justify your response.
It should be E(Rm) - 0.5... instead of 0.005
Solution
Determine which allocation in Exhibit 1 Tomb should recommend to Youngmall. (circle one)
Allocation A
Allocation B
Justify your response.
Tomb should recommend Allocation B.
The expected utility of Allocation B is 1.89%, which is higher than Allocation A’s expected utility of 1.74%.
MVO provides a framework to determine how much to allocate to each asset class or to create the optimal asset mix. The given objective function is:��=�(��)−0.005���2
Using the given objective function and the expected returns and expected standard deviations for Allocations A and B, the expected utilities (certainty-equivalent returns) for the two allocations are calculated as:Allocation A: 6.7% – 0.005 (7) (11.9%)2 = 1.74%Allocation B: 5.9% – 0.005 (7) (10.7%)2 = 1.89%
Therefore, Tomb should recommend Allocation B because it results in higher expected utility than Allocation A.
Asset Class
Market Cap (trillions)
Beta
Expected Returns
MVO Asset Allocation
Cash
$4.2
0.0
2.0%
10%
US bonds
$26.8
0.5
4.5%
20%
US equities
$22.2
1.4
8.6%
35%
Global equities
$27.5
1.7
10.5%
20%
Global bonds
$27.1
0.6
4.7%
15%
Total
$107.8
Solution
Contrast, using the information provided above, the results of a reverse optimization approach with that of the MVO approach for each of the following:
The asset allocation mix
The asset allocation weights for the reverse optimization method are inputs into the optimization and are determined by the market capitalization weights of the global market portfolio.
The asset allocation weights for the MVO method are outputs of the optimization with the expected returns, covariances, and a risk aversion coefficient used as inputs.
The two methods result in significantly different asset allocation mixes.
In contrast to MVO, the reverse optimization method results in a higher percentage point allocation to global bonds, US bonds, and global equities as well as a lower percentage point allocation to cash and US equities.
The reverse optimization method takes the asset allocation weights as its inputs that are assumed to be optimal. These weights are calculated as the market capitalization weights of a global market portfolio. In contrast, the outputs of an MVO are the asset allocation weights, which are based on (1) expected returns and covariances that are forecasted using historical data and (2) a risk aversion coefficient. The two methods result in significantly different asset allocation mixes. In contrast to MVO, the reverse optimization method results in a 4.9, 5.5, and 10.1 higher percentage point allocation to US bonds, global equities, and global bonds, respectively, and a 6.1 and 14.4 lower percentage point allocation to cash and US equities, respectively.
The asset allocation under the two methods is as follows:
Asset Class
Market Cap (trillions)
Asset Allocation Weights
Reverse Optimization
MVO Approach
Difference
Cash
$4.2
3.9%
10%
–6.1%
US bonds
$26.8
24.9%
20%
4.9%
US equities
$22.2
20.6%
35%
–14.4%
Global equities
$27.5
25.5%
20%
5.5%
Global bonds
$27.1
25.1%
15%
10.1%
Total
$107.8
100.0%
100.0%
The values of the expected returns for US equities and global bonds
For the reverse optimization approach, the expected returns of asset classes are the outputs of optimization with the market capitalization weights, covariances, and the risk aversion coefficient used as inputs.
In contrast, for the MVO approach, the expected returns of asset classes are inputs to the optimization, with the expected returns generally estimated using historical data.
The computed values for the expected returns for global bonds and US equities using the reverse optimization method are 5.3% and 9.7%, respectively.
In contrast, the expected return estimates used in the MVO approach from Exhibit 1 for global bonds and US equities are 4.7% and 8.6%, respectively.
The output of the reverse optimization method are optimized returns which are viewed as unobserved equilibrium or imputed returns. The equilibrium returns are essentially long-run capital market returns provided by each asset class and are strongly linked to CAPM. In contrast, the expected returns in the MVO approach are generally forecasted based on historical data and are used as inputs along with covariances and the risk aversion coefficient in the optimization. The reverse-optimized returns are calculated using a CAPM approach. The return on an asset class using the CAPM approach is calculated as follows:Return on Asset Class = Risk-Free Rate + (Beta) (Market Risk Premium)
Therefore, the implied returns for global bonds and US equities are calculated as follows:Return on Global Bonds = 2.0% + (0.6) (5.5%) = 5.3%Return on US Equities = 2.0% + (1.4) (5.5%) = 9.7%
The implied equilibrium returns for global bonds and US equities are 5.3% and 9.7%, respectively. These implied returns are above the forecasted returns based on historical data (from Exhibit 1) used as inputs in the MVO approach for global bonds and US equities of 4.7% and 8.6%, respectively.
ADDRESSING THE CRITICISMS OF MEAN–VARIANCE OPTIMIZATION
Asset Class
Allocation 1
Allocation 2
Allocation 3
Cash
15%
5%
0%
Index-linked government bonds
70%
15%
85%
Corporate bonds
0%
30%
5%
Equities
15%
50%
10%
Portfolio Statistics
Expected return
3.4%
6.2%
3.6%
Expected standard deviation
7.0%
12.0%
8.5%
Determine which asset allocation in Exhibit 1 would be most appropriate for Johansson given her recommendation. (circle one)
Allocation 1
Allocation 2
Allocation 3
Justify your response.
Justify your response.
Allocation 3 is most appropriate.
To fully hedge the fund’s liabilities, 85% ($8.5 billion/$10.0 billion) of the fund’s assets would be linked to index-linked government bonds.
Residual $1.5 billion surplus would be invested into a return-seeking portfolio.
The pension fund currently has a surplus of $1.5 billion ($10.0 billion – $8.5 billion). To adopt a hedging/return-seeking portfolios approach, Johansson would first hedge the liabilities by allocating an amount equal to the present value of the fund’s liabilities, $8.5 billion, to a hedging portfolio. The hedging portfolio must include assets whose returns are driven by the same factors that drive the returns of the liabilities, which in this case are the index-linked government bonds.
So, Johansson should allocate 85% ($8.5 billion/$10.0 billion) of the fund’s assets to index-linked government bonds. The residual $1.5 billion surplus would then be invested into a return-seeking portfolio. Therefore, Allocation 3 would be the most appropriate asset allocation for the fund because it allocates 85% of the fund’s assets to index-linked government bonds and the remainder to a return-seeking portfolio consisting of corporate bonds and equities.
A
B
C
D
Portfolio Characteristics
Expected return
6.5%
7.9%
8.5%
8.8%
Expected volatility
6.0%
7.7%
8.8%
9.7%
Annualized Minimum Expectation Returns
Time Horizon
5 Years
Required Success
99%
0.3%
–0.1%
–0.7%
–1.3%
85%
3.7%
4.3%
4.4%
4.3%
75%
4.7%
5.6%
5.8%
5.9%
Required Success
99%
2.1%
2.2%
2.0%
1.7%
85%
4.5%
5.4%
5.6%
5.6%
75%
5.2%
6.3%
6.6%
6.7%
Required Success
99%
3.7%
4.3%
4.4%
4.3%
85%
5.3%
6.3%
6.7%
6.8%
75%
5.7%
6.9%
7.3%
7.5%
Select, for each of Armstrong’s three goals, which sub-portfolio module from Exhibit 1 Abbott should choose in constructing a portfolio. (circle one module for each goal)
Goal 1
Goal 2
Goal 3
Module A
Module A
Module A
Module B
Module B
Module B
Module C
Module C
Module C
Module D
Module D
Module D
Justify each selection.
Select, for each of Armstrong’s three goals, which sub-portfolio module from Exhibit 1 Abbott should choose in constructing a portfolio. (circle one module for each goal)
Goal 1
Goal 2
Goal 3
Module A
Module A
Module A
Module B
Module B
Module B
Module C
Module C
Module C
Module D
Module D
Module D
Justify each selection.
Module C should be chosen for Goal 1, Module B should be chosen for Goal 2, and Module D should be chosen for Goal 3.
The module that should be selected for each goal is the one that offers the highest return given the time horizon and required probability of success.
The module that should be selected for each goal is the one that offers the highest return given the time horizon and required probability of success. For Goal 1, which has a time horizon of five years and a required probability of success of 85%, Module C should be chosen because its 4.4% expected return is higher than the expected returns of all the other modules. Similarly, for Goal 2, which has a time horizon of 10 years and a required probability of success of 99%, Module B should be chosen because its 2.2% expected return is higher than the expected returns of all the other modules. Finally, for Goal 3, which has a time horizon of 25 years and a required probability of success of 75%, Module D should be chosen because its 7.5% expected return is higher than the expected returns of all the other modules.
Mike and Kerry Armstrong are a married couple who recently retired with total assets of $8 million. The Armstrongs meet with their financial advisor, Brent Abbott, to discuss three of their financial goals during their retirement.
Goal 1: An 85% chance of purchasing a vacation home for $5 million in five years.
Goal 2: A 99% chance of being able to maintain their current annual expenditures of $100,000 for the next 10 years, assuming annual inflation of 3% from Year 2 onward.
Goal 3: A 75% chance of being able to donate $10 million to charitable foundations in 25 years.
Abbott suggests using a goals-based approach to construct a portfolio. He develops a set of sub-portfolio modules, presented in Exhibit 1. Abbott suggests investing any excess capital in Module A.
Exhibit 1:
“Highest Probability- and Horizon-Adjusted Return” Sub-Portfolio Modules under Different Horizon and Probability Scenarios
A
B
C
D
Portfolio Characteristics
Expected return
6.5%
7.9%
8.5%
8.8%
Expected volatility
6.0%
7.7%
8.8%
9.7%
Annualized Minimum Expectation Returns
Time Horizon
5 Years
Required Success
99%
0.3%
–0.1%
–0.7%
–1.3%
85%
3.7%
4.3%
4.4%
4.3%
75%
4.7%
5.6%
5.8%
5.9%
Required Success
99%
2.1%
2.2%
2.0%
1.7%
85%
4.5%
5.4%
5.6%
5.6%
75%
5.2%
6.3%
6.6%
6.7%
Required Success
99%
3.7%
4.3%
4.4%
4.3%
85%
5.3%
6.3%
6.7%
6.8%
75%
5.7%
6.9%
7.3%
7.5%
QuestionQ.Construct the overall goals-based asset allocation for the Armstrongs given their three goals and Abbott’s suggestion for investing any excess capital. Show your calculations.
Construct the overall goals-based asset allocation for the Armstrongs given their three goals and Abbott’s suggestion for investing any excess capital. (insert the percentage of the total assets to be invested in each module)
Module A
Module B
Module C
Module D
Show your calculations.
Goals
1
2
3
Surplus
Horizon (years)
5
10
25
Probability of success
85%
99%
75%
Selected module
C
B
D
A
Discount rate
4.4%
2.2%
7.5%
Dollars invested (millions)
$4.03
$1.01
$1.64
$1.32
As a % of total
50.4%
12.7%
20.5%
16.4%
Supporting calculations:
For Goal 1, which has a time horizon of five years and a required probability of success of 85%, Module C should be chosen because its 4.4% expected return is higher than the expected returns of all the other modules. The present value of Goal 1 is calculated as follows:N = 5, FV = –5,000,000, I/Y = 4.4%; CPT PV = $4,031,508 (or $4.03 million)
So, approximately 50.4% of the total assets of $8 million (= $4.03 million/$8.00 million) should be allocated to Module C.
For Goal 2, which has a time horizon of 10 years and a required probability of success of 99%, Module B should be chosen because its 2.2% expected return is higher than the expected returns of all the other modules. The present value of Goal 2 is calculated as follows:PV=$100,000(1.022)1+$100,000(1.03)1(1.022)2+$100,000(1.03)2(1.022)3+⋯+$100,000(1.03)9(1.022)10PV=$100,000(1.022)1+$100,000(1.03)1(1.022)2+$100,000(1.03)2(1.022)3+⋯+$100,000(1.03)9(1.022)10
PV = $1,013,670 (or $1.01 million)
So, approximately 12.7% of the total assets of $8 million (= $1.01 million/$8.00 million) should be allocated to Module B.
For Goal 3, which has a time horizon of 25 years and a required probability of success of 75%, Module D should be chosen because its 7.5% expected return is higher than the expected returns of all the other modules. The present value of Goal 3 is calculated as follows:N = 25, FV = –10,000,000, I/Y = 7.5%; CPT PV = $1,639,791 (or $1.64 million)
So, approximately 20.5% of the total assets of $8 million (= $1.64 million/$8.00 million) should be allocated to Module D.
Finally, the surplus of $1,315,032 (= $8,000,000 – $4,031,508 – $1,013,670 – $1,639,791), representing 16.4% (= $1.32 million/$8.00 million), should be invested in Module A following Abbott’s suggestion.
Adviser’s Forecasts
Asset Allocation
Expected Return (%)
Standard Deviation of Returns (%)
A
10
12.0
B
8
8.0
C
6
2.0
Statement 1
An optimum risk budget minimizes total risk.
Statement 2
Risk budgeting decomposes total portfolio risk into its constituent parts.
Statement 3
An asset allocation is optimal from a risk-budgeting perspective when the ratio of excess return to marginal contribution to risk is different for all assets in the portfolio.
Characteristic 1
The factors commonly used in the factor-based approach generally have low correlations with the market and with each other.
Characteristic 2
The factors commonly used in the factor-based approach are typically different from the fundamental or structural factors used in multifactor models.
Solution
C is correct. The risk aversion coefficient (λ) for Mary Perkins is 8. The utility of each asset allocation is calculated as follows:
Asset Allocation A:
UA = 10.0% – 0.005(8)(12%)2
= 4.24%
Asset Allocation B:
UB = 8.0% – 0.005(8)(8%)2
= 5.44%
Asset Allocation C:
UC = 6.0% – 0.005(8)(2%)2
= 5.84%
Therefore, the preferred strategic allocation is Asset Allocation C, which generates the highest utility given Perkins’s level of risk aversion.
Adviser’s Forecasts
Asset Allocation
Expected Return (%)
Standard Deviation of Returns (%)
A
10
12.0
B
8
8.0
C
6
2.0
Statement 1
An optimum risk budget minimizes total risk.
Statement 2
Risk budgeting decomposes total portfolio risk into its constituent parts.
Statement 3
An asset allocation is optimal from a risk-budgeting perspective when the ratio of excess return to marginal contribution to risk is different for all assets in the portfolio.
Characteristic 1
The factors commonly used in the factor-based approach generally have low correlations with the market and with each other.
Characteristic 2
The factors commonly used in the factor-based approach are typically different from the fundamental or structural factors used in multifactor models.
Solution
C is correct. Less liquid asset classes—such as direct real estate, infrastructure, and private equity—represent unique challenges when applying many of the common asset allocation techniques. Common illiquid asset classes cannot be readily diversified to eliminate idiosyncratic risk, so representing overall asset class performance is problematic. Furthermore, there are far fewer indexes that attempt to represent aggregate performance for these less liquid asset classes than indexes of traditional highly liquid asset classes. Finally, the risk and return characteristics associated with actual investment vehicles—such as direct real estate funds, infrastructure funds, and private equity funds—are typically significantly different from the characteristics of the asset classes themselves.
Statement 1
An optimum risk budget minimizes total risk.
Statement 2
Risk budgeting decomposes total portfolio risk into its constituent parts.
Statement 3
An asset allocation is optimal from a risk-budgeting perspective when the ratio of excess return to marginal contribution to risk is different for all assets in the portfolio.
Q.
Which of Velky’s statements about risk budgeting is correct?
Solution
B is correct. The goal of risk budgeting is to maximize return per unit of risk. A risk budget identifies the total amount of risk and attributes risk to its constituent parts. An optimum risk budget allocates risk efficiently.
Characteristic 1
The factors commonly used in the factor-based approach generally have low correlations with the market and with each other.
Characteristic 2
The factors commonly used in the factor-based approach are typically different from the fundamental or structural factors used in multifactor models.
Solution
A is correct. The factors commonly used in the factor-based approach generally have low correlations with the market and with each other. This results from the fact that the factors typically represent what is referred to as a zero (dollar) investment or self-financing investment, in which the underperforming attribute is sold short to finance an offsetting long position in the better-performing attribute. Constructing factors in this manner removes most market exposure from the factors (because of the offsetting short and long positions); as a result, the factors generally have low correlations with the market and with one another. Also, the factors commonly used in the factor-based approach are typically similar to the fundamental or structural factors used in multifactor models.
Correct
Your result is correct.
Your answer is A.
Correct answer is A.
Megan Beade and Hanna Müller are senior analysts for a large, multi-divisional money management firm. Beade supports the institutional portfolio managers, and Müller does the same for the private wealth portfolio managers.
Beade reviews the asset allocation in Exhibit 1, derived from a mean–variance optimization (MVO) model for an institutional client, noting that details of the MVO are lacking.
Exhibit 1:
Asset Allocation and Market Weights (in percent)
Asset Classes
Asset Allocation
Investable Global Market Weights
Cash
0
—
US bonds
30
17
US TIPS
0
3
Non-US bonds
0
22
Emerging market equity
25
5
Non-US developed equity
20
29
US small- and mid-cap equity
25
4
US large-cap equity
0
20
The firm’s policy is to rebalance a portfolio when the asset class weight falls outside of a corridor around the target allocation. The width of each corridor is customized for each client and proportional to the target allocation. Beade recommends wider corridor widths for high-risk asset classes, narrower corridor widths for less liquid asset classes, and narrower corridor widths for taxable clients with high capital gains tax rates.
One client sponsors a defined benefit pension 2plan where the present value of the liabilities is $241 million and the market value of plan assets is $205 million. Beade expects interest rates to rise and both the present value of plan liabilities and the market value of plan assets to decrease by $25 million, changing the pension plan’s funding ratio.
Solution
A is correct. The original funding ratio is the market value of assets divided by the present value of liabilities. This plan’s ratio is $205 million/$241 million = 0.8506. When the assets and liabilities both decrease by $25 million, the funding ratio will decrease to $180 million/$216 million = 0.8333.
Beade uses a surplus optimization approach to liability-relative asset allocation based on the objective function
where E(Rs,m) is the expected surplus return for portfolio m, λ is the risk aversion coefficient, and σ2(Rs,m) is the variance of the surplus return. Beade establishes the expected surplus return and surplus variance for three different asset allocations, shown in Exhibit 2. Given λ = 1.50, she chooses the optimal asset mix.
Exhibit 2:
Expected Surplus Return and Volatility for Three Portfolios
Return
Standard Deviation
Portfolio 1
13.00%
24%
8.68
Portfolio 2
12.00%
18%
9.57
Portfolio 3
11.00%
19%
Portfolio
E(Rs,m)
σ2(Rs,m)
1
13.00
576
8.68
2
12.00
324
9.57
3
11.00
361
8.29
Client Haunani Kealoha has a large fixed obligation due in 10 years. Beade assesses that Kealoha has substantially more funds than are required to meet the fixed obligation. The client wants to earn a competitive risk-adjusted rate of return while maintaining a high level of certainty that there will be sufficient assets to meet the fixed obligation.
Solution
C is correct. The hedging/return-seeking portfolios approach is best for this client. Beade should construct two portfolios, one that includes riskless bonds that will pay off the fixed obligation in 10 years and the other a risky portfolio that earns a competitive risk-adjusted return. This approach is a simple two-step process of hedging the fixed obligation and then investing the balance of the assets in a return-seeking portfolio.
In the private wealth area, the firm has designed five subportfolios with differing asset allocations that are used to fund different client goals over a five-year horizon. Exhibit 3 shows the expected returns and volatilities of the subportfolios and the probabilities that the subportfolios will exceed an expected minimum return. Client Luis Rodríguez wants to satisfy two goals. Goal 1 requires a conservative portfolio providing the highest possible minimum return that will be met at least 95% of the time. Goal 2 requires a riskier portfolio that provides the highest minimum return that will be exceeded at least 85% of the time.
Exhibit 3:
Characteristics of Subportfolios
Subportfolio
A
B
C
D
E
Expected return, in percent
4.60
5.80
7.00
8.20
9.40
Expected volatility, in percent
3.46
5.51
8.08
10.80
13.59
Required Success Rate
Minimum Expected Return for Success Rate
99%
1.00
0.07
–1.40
–3.04
–4.74
95%
2.05
1.75
1.06
0.25
–0.60
90%
2.62
2.64
2.37
2.01
1.61
85%
3.00
3.25
3.26
3.19
3.10
75%
3.56
4.14
4.56
4.94
5.30
Müller uses a risk parity asset allocation approach with a client’s four–asset class portfolio. The expected return of the domestic bond asset class is the lowest of the asset classes, and the returns of the domestic bond asset class have the lowest covariance with other asset class returns. Müller estimates the weight that should be placed on domestic bonds.
Solution
C is correct. A risk parity asset allocation is based on the notion that each asset class should contribute equally to the total risk of the portfolio. Bonds have the lowest risk level and must contribute 25% of the portfolio’s total risk, so bonds must be overweighted (greater than 25%). The equal contribution of each asset class is calculated as:
where
wi = weight of asset i
Cov(ri,rp) = covariance of asset i with the portfolio
n = number of assets
= variance of the portfolio
In this example, there are four asset classes, and the variance of the total portfolio is assumed to be 25%; therefore, using a risk parity approach, the allocation to each asset class is expected to contribute (1/4 × 25%) = 6.25% of the total variance. Because bonds have the lowest covariance, they must have a higher relative weight to achieve the same contribution to risk as the other asset classes.
Müller and a client discuss other approaches to asset allocation that are not based on optimization models or goals-based models. Müller makes the following comments to the client:
Comment 1
An advantage of the “120 minus your age” heuristic over the 60/40 stock/bond heuristic is that it incorporates an age-based stock/bond allocation.
Comment 2
The Yale model emphasizes traditional investments and a commitment to active management.
Comment 3
A client’s asset allocation using the 1/N rule depends on the investment characteristics of each asset class.
Solution
A is correct. Comment 1 is correct because the “120 minus your age” rule reduces the equity allocation as the client ages, while the 60/40 rule makes no such adjustment. Comments 2 and 3 are not correct. The Yale model emphasizes investing in alternative assets (such as hedge funds, private equity, and real estate) as opposed to investing in traditional asset classes (such as stock and bonds). The 1/N rule allocates an equal weight to each asset without regard to its investment characteristics, treating all assets as indistinguishable in terms of mean returns, volatility, and correlations.
Preston Remington is the managing partner of Remington Wealth Partners. The firm manages high-net-worth private client investment portfolios using various asset allocation strategies. Analyst Hannah Montgomery assists Remington.
Remington and Montgomery’s first meeting of the day is with a new client, Spencer Shipman, who recently won $900,000 in the lottery. Shipman wants to fund a comfortable retirement. Earning a return on his investment portfolio that outpaces inflation over the long term is critical to him. He plans to withdraw $54,000 from the lottery winnings investment portfolio in one year to help fund the purchase of a vacation home and states that it is important that he be able to withdraw the $54,000 without reducing the initial $900,000 principal. Montgomery suggests they use a risk-adjusted expected return approach in selecting one of the portfolios provided in Exhibit 1.
Exhibit 1
Investment Portfolio One-Year Projections
Return
Standard Deviation
Portfolio 1
10.50%
20.0%
Portfolio 2
9.00
13.0
Portfolio 3
7.75
10.0
Solution
B is correct. Portfolio 2 has the highest probability of enabling Shipman to meet his goal for the vacation home. All three of the portfolios’ expected returns over the next year exceed the 6.0% (see calculations below) required return threshold to avoid reducing the portfolio. However, on a risk-adjusted basis, Portfolio 2 (probability ratio of 0.231) has a higher probability of meeting and surpassing the threshold than either Portfolio 1 (probability ratio of 0.175) or Portfolio 3 (probability ratio of 0.225).
Step 1
Calculate the required return threshold: 54,000 ÷ 900,000 = 0.06 = 6.0%.
Step 2
To decide which allocation is best for Shipman, calculate the probability ratio:[E(RP) – RL] ÷ σP, where
RP = The return for the portfolio
RL = The required return threshold
σP = The standard deviation of the portfolio
Portfolio 1: (10.50% – 6.0%) / 20.0% = 4.50% / 20.0% = 0.225.
Portfolio 2: (9.00% – 6.0%) / 13.0% = 3.00% / 13.0% = 0.231. (Highest)
Portfolio 3: (7.75% – 6.0%) / 10.0% = 1.75% / 10.0% = 0.175.
A is incorrect. Portfolio 1 was chosen because it has the highest projected return.
C is incorrect. Portfolio 3 was chosen because it has the lowest projected standard deviation.
Principles of Asset Allocation Learning Outcome
Recommend and justify an asset allocation using mean–variance optimization
Remington and Montgomery discuss the importance of strategic asset allocation with Shipman. Remington states that the firm’s practice is to establish targeted asset allocations and a corridor around the target. Movements of the asset allocations outside the corridor trigger a rebalancing of the portfolio. Remington explains that for a given asset class, the higher the transaction costs and the higher the correlation with the rest of the portfolio, the wider the rebalancing corridor. Montgomery adds that the higher the volatility of the rest of the portfolio, excluding the asset class being considered, the wider the corridor.
Solution
A is correct. The statement regarding volatility is the least accurate. The higher the volatility of the rest of the portfolio, excluding the asset class being considered, the more likely a large divergence from the strategic asset allocation becomes, which should point to a narrower optimal corridor, all else being equal.
B is incorrect. The higher the correlation of an asset class with the rest of the portfolio, the wider the optimal corridor. When asset classes move in sync, further divergence from target weights is less likely.
C is incorrect. The higher the transaction costs, the wider the optimal corridor. High transaction costs set a high hurdle for rebalancing benefits to overcome.
Principles of Asset Allocation Learning Outcome
Discuss factors affecting rebalancing policy
Remington and Montgomery next meet with client Katherine Winfield. The firm had established Winfield’s current asset allocation on the basis of reverse optimization using the investable global market portfolio weights with further adjustments to reflect Winfield’s views on expected returns.
Solution
B is correct. Winfield’s current asset allocation is most likely based on the Black–Litterman model. Black–Litterman starts with the excess returns produced from reverse optimization, which commonly uses the observed market-capitalization value of the assets or asset classes of the global opportunity set. It then alters the reverse-optimized expected returns that reflect an investor’s own distinctive views yet still behaves well in an optimizer.
A is incorrect. Asset allocations using mean–variance optimization tend to be concentrated in a subset of the available asset classes. Winfield’s portfolio will be allocated to all or most of the asset classes through the reverse-optimization process followed by adjustments reflecting the investor’s views.
C is incorrect. Reverse optimization takes as its inputs a set of asset allocation weights that are assumed to be optimal and, with covariances and the risk aversion coefficient, solves for expected returns. The starting weights are commonly the observed market-capitalization value of the assets or asset classes of the global opportunity set. The asset allocation using reverse optimization would not take into account the investor’s own views.
Principles of Asset Allocation Learning Outcome
Recommend and justify an asset allocation based on the global market portfolio
Remington and Montgomery discuss with Winfield some alternative asset allocation models that she may wish to consider, including resampled mean–variance optimization (resampling). Remington explains that resampling combines mean–variance optimization (MVO) with Monte Carlo simulation, leading to more diversified asset allocations. Montgomery comments that resampling, like other asset allocation models, is subject to criticisms, including that risker asset allocations tend to be under-diversified and the asset allocations inherit the estimation errors in the original inputs.
Solution
C is correct. Montgomery’s comment about the criticisms of resampling is incorrect regarding diversification of asset allocations. Risker asset allocations are over-diversified, not under-diversified. The comment is correct with regard to estimation errors because the asset allocations do inherit the estimation errors in the original inputs.
A and B are incorrect. Risker asset allocations are over-diversified, not under-diversified. However, the asset allocations do inherit the estimation errors in the original inputs.
Principles of Asset Allocation Learning Outcome
Discuss the use of Monte Carlo simulation and scenario analysis to evaluate the robustness of an asset allocation
Montgomery inquires whether asset allocation models based on heuristics or other techniques might be of interest to Winfield and makes the following comments:
The 60/40 stock/bond heuristic optimizes the growth benefits of equity and the risk reduction benefits of bonds.
The Norway model is a variation of the endowment model that actively invests in publicly traded securities while giving consideration to environmental, social, and governance issues.
The 1/N heuristic allocates assets equally across asset classes with regular rebalancing without regard to return, volatility, or correlation.
Solution
C is correct. The 1/N rule asset allocation heuristic involves equally weighting allocations to assets; 1/N of wealth is allocated to each of N assets available for investment at each rebalancing date. All assets are treated as indistinguishable in terms of mean returns, volatility, and correlations.
A is incorrect. It is not an optimization model. The 60/40 stock/bond heuristic allocates 60% of assets to equities, supplying a long-term growth foundation, and 40% to fixed income, supplying risk reduction benefits.
B is incorrect. The Norway model passively invests in publicly traded securities subject to environmental, social, and governance concerns. In comparison, the endowment model asset allocation emphasizes active management of large allocations to non-traditional investments, seeking to earn illiquidity premiums.
Principles of Asset Allocation Learning Outcome
Describe and evaluate heuristic and other approaches to asset allocation
Finally, Remington and Montgomery discuss Isabelle Sebastian. During a recent conversation, Sebastian, a long-term client with a $2,900,000 investment portfolio, reminded Remington that she will soon turn age 65 and wants to update her investment goals as follows:
Goal 1: Over the next 20 years, she needs to maintain her living expenditures, which are currently $120,000 per year (90% probability of success). Inflation is expected to average 2.5% annually over the time horizon, and withdrawals take place at the beginning of the year, starting immediately.
Goal 2: In 10 years, she wants to donate $1,500,000 in nominal terms to a charitable foundation (85% probability of success).
Exhibit 2 provides the details of the two sub-portfolios, including Sebastian’s allocation to the sub-portfolios and the probabilities that they will exceed the expected minimum return.
Exhibit 2
Investment Sub-Portfolios & Minimum Expected Return for Success Rate
Sub-Portfolio
BY
CZ
Expected return (%)
5.70
7.10
Expected volatility (%)
5.10
7.40
Current portfolio allocations (%)
40
60
Probability (%)
Minimum Expected Return (%)
Time horizon: 10 years
99
2.90
2.50
90
3.40
2.80
85
3.60
3.00
Time horizon: 20 years
95
5.10
5.40
90
5.20
5.70
85
5.60
5.90
Assume 0% correlation between the time horizon portfolios.
Solution
C is correct. Sebastian needs to adjust the sub-portfolio allocation to achieve her goals. By adjusting the allocations to 37% × $2,900,000 = $1,073,000 in BY and 63% × $2,900,000 = $1,827,000 in CZ, she will be able to achieve both of her goals based on the confidence intervals.
Goal 1: Sebastian needs to maintain her current living expenditure of $120,000 per year over 20 years with a 90% probability of success. Inflation is expected to average 2.5% annually over the time horizon.
Sub-portfolio CZ should be selected because it has a higher expected return (5.70%) at the 90% probability for the 20-year horizon. Although sub-portfolio CZ has an expected annual return of 7.10%, based on the 90% probability of success requirement, the discount factor is 5.70%.
Goal 1: k = 5.70%; g = 2.50%.
Determine the inflation-adjusted annual cash flow generated by sub-portfolio CZ:
$1,827,000×(0.057−0.025)[1−(1+0.0251+0.057)20](1.057)=$120,432.04>$120,000$1,827,000×0.057−0.0251−1+0.0251+0.057201.057=$120,432.04>$120,000
Goal 2: Sebastian wants to contribute $1,500,000 to a charitable foundation in 10 years with an 85% probability of success.
Sub-portfolio BY should be selected because it has a higher expected return (3.60%) at the 85% probability for the 10-year horizon. Although sub-portfolio BY has an expected annual return of 5.70%, based on the 85% probability of success requirement, the discount factor is 3.60%.
Goal 2: k = 3.60%.
Determine the amount needed today in sub-portfolio BY:
$1,500,000(1+0.036)10=$1,053,158.42<$1,073,000$1,500,0001+0.03610=$1,053,158.42<$1,073,000
A is incorrect: 40% × $2,900,000 = $1,160,000 in BY, and 60% × $2,900,000 = $1,740,000 in CZ.
Goal 1: k= 5.70%; g = 2.50%.
Determine the inflation-adjusted annual cash flow generated by sub-portfolio CZ:
$1,740,000×(0.057−0.025)[1−(1+0.0251+0.057)20](1.057)=$114,697.18<$120,000$1,740,000×0.057−0.0251−1+0.0251+0.057201.057=$114,697.18<$120,000
Goal 2: k = 3.60%.
Determine the amount needed today in sub-portfolio BY:
$1,500,000(1+0.036)10=$1,053,158.42<$1,160,000$1,500,0001+0.03610=$1,053,158.42<$1,160,000
Goal 1 is not realized because the inflation-adjusted annual payment is below $120,000.
Goal 2 is realized
B is incorrect: 43% × $2,900,000 = $1,247,000 in BY, and 57% × $2,900,000 = $1,653,000 in CZ.
Goal 1: k = 5.70%; g = 2.50%.
Determine the inflation-adjusted annual cash flow generated by sub-portfolio CZ:
$1,653,000×(0.057−0.025)[1−(1+0.0251+0.057)20](1.057)=$108,962.32<$120,000$1,653,000×0.057−0.0251−1+0.0251+0.057201.057=$108,962.32<$120,000
Goal 2: k = 3.60%.
Determine the amount needed today in sub-portfolio BY:
$1,500,000(1+0.036)10=$1,053,158.42<$1,247,000$1,500,0001+0.03610=$1,053,158.42<$1,247,000
Goal 1 is not realized because the inflation-adjusted annual payment is below $120,000.
Goal 2 is realized.
Principles of Asset Allocation Learning Outcome
Recommend and justify an asset allocation using a goals-based approach
Sabonete, S.A., is a multi-national consumer products company headquartered in Lisbon, Portugal, with annual revenue of approximately €2 billion. Over the past several years, the company’s growth strategy has centered on expanding market share in emerging markets. Over half of revenues are from emerging markets, and the remainder are from developed European economies. Oni Falana serves as the chief investment officer of Sabonete’s defined benefit pension plan (SPP).
Falana has engaged an outside consultant, Isabel Horvath, for assistance with asset allocation. Falana describes to Horvath the company’s key objectives with respect to the SPP:
Falana has engaged an outside consultant, Isabel Horvath, for assistance with asset allocation. Falana describes to Horvath the company’s key objectives with respect to the SPP:
· reach fully funded (100%) status in five years, at which point the liabilities will be fully hedged,
· minimize fluctuations in expected year-to-year required contributions, and
· minimize the administrative and investment costs associated with managing the fund.
Solution
C is correct. Based on the objectives described, the integrated asset–liability approach is the most appropriate asset allocation approach for the SPP, which is currently under-funded and seeks to achieve fully funded status in five years. Mean–variance optimization is an asset-only approach to asset allocation and fails to consider Sabonete’s goal of fully funding the liabilities. The basic two-portfolio approach assumes the pension plan has a surplus that can be allocated to a return-seeking portfolio.
A is incorrect. Mean–variance optimization is an asset-only approach to asset allocation and fails to consider Sabonete’s goal of fully funding the liabilities.
B is incorrect. The basic two-portfolio approach assumes the pension plan has a surplus that can be allocated to a return-seeking portfolio.
Principles of Asset Allocation Learning Outcome
Discuss approaches to liability-relative asset allocation
Falana provides Horvath with the following key facts and assumptions regarding Sabonete and the SPP:
· The fund is closed to new employees, but existing employees continue to accrue benefits under the original terms.
· The fund is 90% funded (€5 billion in assets and an accrued benefit obligation of €5.55 billion).
· A €75 million contribution will be made to the fund at the end of this quarter. Future contributions are likely to be substantially smaller.
· The average participant is 45 years old, and employee turnover has been low.
· The salary growth rate is 3.2%
· The liability discount rate, currently 3.5%, is benchmarked to 10-year AA corporate bonds.
· The risk-free rate is currently 3.0%, and short-term interest rates are expected to remain stable.[
The current asset allocation and other statistics (expected return, volatility, and correlation with global equities) of each asset class are indicated in Exhibit 1.
Exhibit 1 SPP Current Asset Allocation and Other Statistics by Asset Class
Statistics by Asset Class
Asset Class
Current Asset Allocation
Expected Return
Expected Volatility
Correlation with Global Equities
Global fixed income
30%
3.5%
3.5%
0.10
Global developed market equity
30%
6.0%
18.0%
1.00
Emerging market equity
25%
7.0%
22.0%
0.86
Real estate
10%
4.5%
10.0%
0.48
Private equity
5%
7.5%
25.0%
0.85
Hedge funds
0%
5%
6.5%
0.81
Solution
B is correct. A decline in short-term interest rates is least likely to affect SPP’s funded status, because liabilities are discounted using 10-year rates. In addition, it is highly likely that only a small proportion of the fixed-income assets would be invested in short-term assets given the long-term nature of the liabilities.
A is incorrect. A higher salary growth rate increases the projected liabilities and thus negatively affects the funded status of the SPP.
C is incorrect. Average participant age affects the date and timing of future benefit payments. If the average participant age increases, withdrawals from the plan will occur sooner, causing the present value of future withdrawals from the SPP to increase, resulting in a negative impact on its funded status.
Principles of Asset Allocation Learning Outcome
Describe and evaluate characteristics of liabilities that are relevant to asset allocation
key objectives with respect to the SPP:
· reach fully funded (100%) status in five years, at which point the liabilities will be fully hedged,
· minimize fluctuations in expected year-to-year required contributions, and
· minimize the administrative and investment costs associated with managing the fund.
Horvath starts by preparing an economic balance sheet for the SPP and makes the following notes:
· Emerging market equities have a 10% weight in the global market portfolio. However, given the firm’s familiarity with and the opportunities they perceive in emerging markets, the SPP has historically been over-weighted (25%) in this asset class.
· 60% of company revenue is from sales in emerging markets, of which half is attributable to sales in Africa. The revenue growth rate for Sabonete’s African business is high but very volatile. The firm’s revenues and profitability are quite sensitive to emerging markets.
The firm has significant investments in African real estate. It recently acquired several large parcels of land in Africa for €200 million and is planning to make a major investment in new manufacturing facilities to boost margins.
Based on these factors and the information from Exhibit 1, Horvath presents the current asset allocation of the plan along with two other options for consideration (see Exhibit 2).
Exhibit 2
Current and Proposed Asset Allocation Options
Asset Class
Current
Option 1
Option 2
Global fixed income
30%
25%
35%
Global developed market equity
30%
20%
30%
Emerging market equity
25%
20%
15%
Real estate
10%
10%
5%
Private equity
5%
10%
5%
Hedge funds
0%
15%
10%
Expected return
5.4%
5.5%
5.1%
Expected volatility
14%
13%
12%
Sharpe ratio
0.17
0.19
0.175
Recee Radell, an analyst in SPP’s pension office, makes the following statements regarding Option 1:
Statement 1
Based on its Sharpe ratio, Option 1 is most consistent with Sabonete’s objectives.
Statement 2
The 10% allocation to real estate is too high given the company’s recent real estate acquisitions in Africa.
Statement 3
The 20% allocation to emerging market equity is too high given the sensitivity of the firm’s revenues to emerging markets.
Solution
C is correct. Statement 3 is most appropriate. The 20% allocation to emerging market equity is too high given the company’s goals and objectives and the sensitivity of revenues to the African economy. A weak emerging market economic environment is likely to stress the pension fund’s investment in emerging market equity and its revenue from its emerging market business simultaneously. Thus, the high volatility of emerging market equity, its limited diversification potential relative to global equity, and the sensitivity of the firm’s revenues to emerging market economies make a large, over-weighted allocation to the asset class inconsistent with the firm’s objective of minimizing fluctuations in year-to-year required contributions.
A is incorrect. The Sharpe ratios for the current allocation, Option 1, and Option 2 are 0.17, 0.19, and 0.175, respectively, with Option 1 having the highest Sharpe ratio. The Sharpe ratio, while providing a means to rank choices on the basis of return per unit of volatility, does not capture other characteristics that are important to Sabonete, such as funded ratio, time horizon, and predictability of contributions.
B is incorrect. Sabonete’s land holdings outside of the pension fund are not considered a part of the extended balance sheet for the SPP and should not affect its asset allocation decisions.
Sabonete’s recent acquisition of land in Africa are outside of the pension fund and, therefore, should not be considered a part of the extended balance sheet for the SPP and should not affect its asset allocation decisions.
Overview of Asset Allocation Learning Outcome
Explain the use of risk factors in asset allocation and their relation to traditional asset class–based approaches
Falana plans to recommend Option 2, believing that the greater certainty of meeting the required year-to-year contributions is the more important objective. She asks Radell to recommend a rebalancing policy for Option 2. He proposes the ranges shown in Exhibit 3 and provides an estimate of the related transaction costs.
Exhibit 3
Rebalancing Ranges for Option 2
Asset Class
Asset Allocation Option 2
Lower Boundary
Upper Boundary
Transaction Costs
Global fixed income
35%
10%
35%
Low
Global developed market equity
30%
15%
35%
Low
Emerging market equity
15%
0%
30%
Moderate
Real estate
5%
0%
10%
High
Private equity
5%
0%
10%
High
Hedge funds
10%
0%
10%
High
Radell justifies his recommendation on the basis of the following statements:
· A wide rebalancing range for global fixed income is appropriate because of its low volatility, low transaction costs, and low correlation with other asset classes in the portfolio.
Solution
A is correct. Radell indicates that a wider rebalancing range for the global fixed-income fund is appropriate. This assertion is correct because this asset class exhibits low volatility relative to the rest of the portfolio (3.5%, Exhibit 1). The lower the volatility of an asset class relative to the rest of the portfolio, the wider the optimal rebalancing corridor.
B is incorrect. When an asset class exhibits a lower correlation with the rest of the portfolio, the rebalancing range should be narrower—not wider—to preserve its contribution to risk reduction.
C is incorrect. The low transactions costs associated with the global fixed-income asset class means that rebalancing will be less costly, justifying a narrower rebalancing corridor
Principles of Asset Allocation Learning Outcome
Discuss factors affecting rebalancing policy
The allocation for private equity is challenging because a low-cost passive investment vehicle does not exist and modeling with a private equity index captures only the return aspects of private equity without an appropriate representation of risk.
company’s key objectives with respect to the SPP:
· reach fully funded (100%) status in five years, at which point the liabilities will be fully hedged,
· minimize fluctuations in expected year-to-year required contributions, and
· minimize the administrative and investment costs associated with managing the fund.
Falana discusses the recommended asset allocation with the Pension Committee. A new committee member, James D’Alessandro, states that his preference would be Option 1 because the best pension funds have adopted the endowment model of asset allocation, which has a higher allocation to alternative investments.
Solution
A is correct. D’Alessandro suggests that the endowment model should be used by the pension plan. The endowment model is not well suited to meeting Sabonete’s objectives. The model seeks to earn illiquidity premiums, which require a long-time horizon to capture. Because Sabonete plans to hedge its liabilities in five years, its investment horizon is too short to capture the illiquidity premium that is a critical component of the endowment model.
B is incorrect. The endowment model can be expected to generate higher returns because of the illiquidity premium, which would more than offset the higher costs. However, Sabonete’s investment horizon is too short to capture the illiquidity premium inherent in the model. Further, the higher costs of the model are inconsistent with Sabonete’s desire to minimize the investment and administrative costs associated with the plan.
C is incorrect. Sabonete’s investment horizon is too short to capture the illiquidity premium that is a critical component of the endowment model. Further, the higher costs of the model are inconsistent with Sabonete’s objective to minimize the investment and administrative costs associated with the plan.
Principles of Asset Allocation Learning Outcome
Describe and evaluate heuristic and other approaches to asset allocation
Tina Swan Case Scenario
Tina Swan heads a consulting practice that advises large funds and high-net-worth individuals about portfolio asset allocation and portfolio performance. Swan and two junior advisers, Stephanie Gruber and Monica Morrison, are meeting with a client, XTR Funds (XTR), to select an appropriate mean–variance-optimized (MVO) portfolio combination that meets various restrictions. Three possible portfolios meet XTR’s criteria; each portfolio’s expected performance is higher than XTR’s target return of 5.7% (Exhibit 1). Consequently, each prospective portfolio can be combined with a risk-free security to generate XTR’s target return.
Exhibit 1
Mean–Variance-Optimized (MVO) Portfolios
MVO portfolio
Expected return (%)
Return standard deviation (%)
Portfolio weight in risk-free security (%)*
A
9.5
18.69
49.35
B
8.2
14.95
39.06
C
7.8
14.18
35.00
Risk-free security expected return: 1.8% XTR target return: 5.7%
* This is the proportional investment in the risk-free security that in combination with the associated MVO portfolio produces the target expected return of 5.7%.
(X)
(Y)
(Z) = (X) × [1 – (Y)]
= (5.7% – 1.8%) ÷ Z
MVO portfolio
Return standard deviation
Portfolio weight in risk-free security
Risk of MVO portfolio combined with risk-free security
Sharpe ratio
A
0.1869
0.4935
0.0947 = 0.1869 × [1 – 0.4935]
0.412 = (0.057 – 0.018) ÷ 0.0947
B
0.1495
0.3906
0.0911 = 0.1495 × [1 – 0.3906]
0.428 = (0.057 – 0.018) ÷ 0.0911
C
0.1418
0.3500
0.0922 = 0.1418 × [1 – 0.3500]
0.423 = (0.057 – 0.018) ÷ 0.0922
After meeting with XTR, Swan, Gruber, and Morrison discuss some of the criticisms of MVO portfolios.
Swan: MVO portfolios are diversified with respect to risk factors such as value, size, and quality.
Gruber: MVO portfolios are more sensitive to measurement errors in the expected return than to measurement errors in correlation and risk.
Morrison: Some of the issues with MVO can be corrected by using reverse optimization to solve for risk parameters based on inputs for expected return and correlation.
Solution
C is correct. Gruber is correct. MVO portfolios are more sensitive to measurement errors in the expected return than to measurement errors in correlation and risk.
A is incorrect. Reverse optimization uses inputs for risk and correlation (or covariance) to solve for expected return.
B is incorrect. MVO portfolios are based on market risk only.
Principles of Asset Allocation Learning Outcome
Recommend and justify an asset allocation using mean–variance optimization
Swan then asks Gruber about the issues associated with incorporating illiquid assets into a portfolio allocation. Gruber makes the following statements:
Illiquid assets offer both diversification benefits and an expected return discount relative to similar liquid assets as compensation for illiquidity.
Easily tracked indexes in asset classes similar to that of an illiquid asset often do not represent the non-idiosyncratic risk of the illiquid asset very accurately.
Unfortunately, there are no low-cost passive investment vehicles available to allow one to closely track the aggregate performance of less liquid asset classes.
Solution
B is correct. Statement 3 is the most accurate. Low-cost passive vehicles for tracking performance exist for publicly traded liquid assets but do not exist for illiquid assets.
A is incorrect. Statement 1 is incorrect because an illiquid asset usually has a premium associated with its return due to the illiquidity.
C is incorrect. Statement 2 is incorrect because easily tracked indexes for an asset class usually do not capture the idiosyncratic risk component of less liquid assets.
Principles of Asset Allocation Learning Outcome
Discuss asset class liquidity considerations in asset allocation
Morrison presents a portfolio allocation for another client, Terry Williams, based on the information in her current Investment Policy Statement (IPS) using an asset-only asset allocation (Exhibit 2).
Exhibit 2
Optimized Output for Asset Classes within a Portfolio
Asset class
Expected return (%)
Portfolio weight (%)
Beta relative to overall portfolio
A
10.2
63.34
1.091
B
8.4
10.00
0.857
C
8.9
20.00
0.922
D
6.3
6.66
0.584
Risk-free security expected return: 1.80% Portfolio expected return: 9.50% Standard deviation of the portfolio return: 21.63%
Solution
C is correct. The marginal contribution to total risk (MCTR) is the beta relative to the portfolio multiplied by the standard deviation of the portfolio: 1.091 × 0.2163 = 0.2360 = 23.6%.
B is incorrect because 14.9% is the weighted MCTR, which is the portfolio weight multiplied by the MCTR: 0.6334 × 0.2360 = 0.14948.
A is incorrect because 13.7% is the portfolio weight multiplied by the portfolio standard deviation: 0.6334 × 0.2163 = 0.1370.
Solution
A is correct. Because the portfolio is optimal, the ratio of the excess return to the marginal contribution to total risk (MCTR) is equal to the Sharpe ratio for the portfolio:Sharpe ratio = where
kp = portfolio expected return
RF = risk-free rate
σp = standard deviation of portfolio return
B is incorrect because 0.439 is the Sharpe ratio calculated with the portfolio expected return (kp) rather than the return premium (kp – RF): 0.095/0.2163 = 0.439.
C is incorrect because 0.077 is the Treynor measure of the portfolio (the portfolio beta is equal to 1.0): (0.095 – 0.018)/1.0 = 0.077.
Principles of Asset Allocation Learning Outcome
Explain absolute and relative risk budgets and their use in determining and implementing an asset allocation
Finally, the team discusses a third client, Roberta Harmon. Harmon just made a cash deposit of $679,000 with a 10-year goal of achieving $775,000. She expressed the desire to meet this goal with 90% certainty without any excess risk. Her IPS allows for the use of equities and fixed-income securities. Exhibit 3 summarizes her investment goal and the characteristics of the investments contained in her IPS.
Exhibit 3
Summary of Harmon’s Investment Goal and Investment Characteristics
Investment Goal
Current balance
$679,000
Investment goal
$775,000
Investment horizon
10 years
Investment Characteristics
Expected annual return (%)
Annual volatility (%)
Annual return with 90% success rate (%)
Equity portfolio
5.75
10.00
1.70
Fixed-income portfolio
1.95
2.25
1.04
The following comments are made:
Gruber: It would be safest for Harmon to invest 100% in fixed income to meet her goal.
Morrison: I disagree. To meet her goal, Harmon’s portfolio will have to contain some equity investments, but no further optimization will be required.
Swan: I agree that the portfolio will require equity exposure, but optimization will also be required.
Solution
A is correct. Swan’s comment is most accurate. The annual return required to achieve Harmon’s goal is 1.33%:
($775,000$679,000)1/10−1=1.33%$775,000$679,0001/10−1=1.33%
The fixed-income portfolio can provide only a 1.04% return with 90% certainty, so some equity exposure is required to increase the portfolio return. However, various combinations of the fixed-income and equity portfolios will need to be determined to find the risk/return results for alternative weightings. The correlation between the two portfolios will have a strong influence on the degree of risk reduction that can be achieved. The optimized portfolio will be the one that achieves the 1.33% return with 90% certainty and has the lowest level of risk.
B is incorrect. Further optimization will be needed depending on the correlation between the equity portfolio and the fixed-income portfolio.
C is incorrect. The fixed-income portfolio alone will not generate the required annual return of 1.33% with 90% certainty.
Principles of Asset Allocation Learning Outcome
Recommend and justify an asset allocation using a goals-based approach
Monte Carlo simulation can accommodate many future possible scenarios, such as portfolio rebalancing costs and non-normal distributions (i.e., distributions that require more than expected return and volatility as parameters).
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