Book 5. Risk and Investment Management

FRM Part 2

IM 8. Portfolio Performance Evaluation

Presented by: Sudhanshu

Module 1. Time-Weighted and Dollar-Weighted Returns

Module 2. Risk-Adjusted Performance Measures

Module 3. Alpha, Hedge Funds, Dynamic Risk, Market Risk and Style

Module 1. Time-Weighted and Dollar-Weighted Returns

Topic 1. Time-Weighted and Dollar-Weighted Returns

Topic 1. Time-Weighted Vs Dollar-Weighted Returns

  • Definition: The dollar-weighted rate of return is the Internal Rate of Return (IRR) on a portfolio.

  • It accounts for the timing and size of all cash flows (inflows and outflows).

  • Key Concept: It gives more weight to periods where larger sums of money are invested.

  • Appropriate Use: Best for evaluating the performance of an investor who has control over the timing of their deposits and withdrawals. It reflects the return on the investor's actual invested capital.

  • Calculating the Dollar-Weighted Rate of Return (Example)
  • Scenario:

    • t = 0: Buy 1 share for $100.

    • t = 1: Buy 1 more share for $120. (Received a $2 dividend from the first share).

    • t = 2: Sell both shares for $130 each. (Received a $2 dividend per share).

  • Cash Flows:

    • CF0​: -$100 (Initial Outlay)

    • CF1​: $2 (Dividend) - $120 (Purchase) = -$118

    • CF2​: ($130 * 2) (Sale Proceeds) + ($2 * 2) (Dividends) = +$264

  • Formula: Set the present value of inflows equal to the present value of outflows and solve for r:

 

  • Result: The IRR, or DWRR, is 13.86%.
  • Time-Weighted Rate of Return (TWRR)

    • Definition: The time-weighted rate of return measures compound growth. It is the geometric mean of the holding period returns (HPRs) for each subperiod.

    • Key Concept: It completely removes the effect of cash inflows and outflows.

    • Appropriate Use: This is the preferred method for evaluating a portfolio manager's skill because their performance should not be judged by cash flow decisions made by the client, which are outside of their control.

\$ 100+\frac{\$ 118}{(1+r)}=\frac{\$ 264}{(1+r)^2}
  • Example: Calculating the Time-Weighted Rate of Return
    • Scenario: Same as the DWRR example.
  • Calculation:

    • Step 1: Break the investment period into subperiods based on cash flows.

      • Subperiod 1: From t=0 to t=1.

      • Subperiod 2: From t=1 to t=2.

    • Step 2: Calculate the Holding Period Return (HPR) for each subperiod.

      •  

      •  

      •  

    • Step 3: Calculate the geometric mean of the HPRs.

      • ​

      •  

  • When DWRR > TWRR

    • The DWRR will be higher than the TWRR if the investor adds funds before a period of good performance.

    • This suggests the investor had superior market timing ability.

H P R_1=\frac{(\text { Ending Price+Dividends })}{\text { Beginning Price }}-1=\frac{(\$ 120+\$ 2)}{\$ 100}-1=22 \%
H P R_2=\frac{(\text { Ending Price }+ \text { Dividends })}{\text { Beginning Price }}-1=\frac{(\$ 260+\$ 4)}{\$ 240}-1=10 \%
T W R R=\sqrt{\left(1+H P R_1\right) \times\left(1+H P R_2\right)}-1
T W R R=\sqrt{(1.22) \times(1.10)}-1=15.84 \%
  • When DWRR < TWRR

    • The DWRR will be lower than the TWRR if the investor adds funds before a period of poor performance.

    • The TWRR, in this case, would show the true performance of the manager without being penalized for the investor's poor timing.

  • Key Differences and Relationship

 

 

 

 

 

 

 

 

 

 

Practice Questions: Q1

Q1. Assume you purchase a share of stock for $ 50 at time t=0 and another share at $ 65 at time t=1, and at the end of Year 1 and Year 2, the stock paid a $ 2.00 dividend. Also at the end of Year 2, you sold both shares for $ 70 each.
The dollar-weighted rate of return on the investment is:

A. 10.77 %.

B. 15.45 %.

C. 15.79 %.

D. 18.02 %.

Practice Questions: Q1 Answer

Explanation: D is correct.

One way to do this problem is to set up the cash flows so that the PV of inflows = PV of outflows and then to plug in each of the multiple choices.

 

 


Alternatively, on your financial calculator, solve for IRR:

 

 

 

 

 

 

 

 

 

50+65 /(1+\mathrm{IRR})=2 /(1+\mathrm{IRR})+144 /(1+\mathrm{IRR})^2 \rightarrow \mathrm{IRR}= 18.02 \%
-50-\frac{65-2}{1+\operatorname{IRR}}+\frac{2(70+2)}{(1+\operatorname{IRR})^2}=0

Practice Questions: Q2

Q2.Assume you purchase a share of stock for $ 50 at time t=0 and another share at $ 65 at time t=1, and at the end of Year 1 and Year 2, the stock paid a $ 2.00 dividend. Also at the end of Year 2, you sold both shares for $ 70 each.

The time-weighted rate of return on the investment is:

A. 18.04 %.

B. 18.27 %.

C. 20.13 %.

D. 21.83 %.

Practice Questions: Q2 Answer

Explanation: D is correct.

 

 

 

\begin{aligned} & \mathrm{HPR}_1=(65+2) / 50-1=34 \%, \mathrm{HPR}_2=(140+4) / 130-1=10.77 \% \\ & \text { time-weighted return }=[(1.34)(1.1077)]^{0.5}-1=21.83 \% \end{aligned}

Module 2. Risk-Adjusted Perfomance Measures

Topic 1. Universe Comparisons

Topic 2. Sharpe Ratio

Topic 3. Treynor Measure

Topic 4. Jensen's Alpha

Topic 5. Information Ratio

Topic 6. M-Squared Measure

Topic 1. Universe Comparisons

  • Concept: A method for comparing portfolios by first grouping them into categories based on their investment style (e.g., small-cap growth, large-cap value).

  • Process: Portfolios are then ranked based on their return within their specific style universe.

  • Purpose: This approach makes rankings more meaningful by standardizing the comparison based on investment style.

  • Limitation: It may not fully account for all risk differences if there is still variation in risk profiles among the funds within a single style.

Topic 2. The Sharpe Ratio

  • Definition: Measures the amount of excess return (above the risk-free rate) earned per unit of total risk.

  • Formula: 


    ​​

  • Variables:

    •       Sharpe Ratio

    •     ​: Average account return

    •    
      ​      : Average risk-free return

    •     : Standard deviation of account returns (total risk)

  • Key Insight: The Sharpe Ratio evaluates a portfolio's performance in terms of both overall return and diversification, as it uses total risk as the denominator.

S_A=\frac{\bar{R}_A-\bar{R}_F}{\sigma_A}
S_A:
\bar{R}_F
\bar{R}_A
\sigma_A

Topic 3. The Treynor Measure

  • Definition: Similar to the Sharpe Ratio, but measures the excess return per unit of systematic risk (beta).

  • Formula: 


    ​​

  • Variables:

    • ​    : Treynor Measure

    •       ​: Average account return

    •        ​: Average risk-free return

    •       : Average beta (systematic risk)

  • Key Insight: For a well-diversified portfolio, the Sharpe and Treynor rankings will be similar. For a less-diversified portfolio, a higher Treynor ranking compared to the Sharpe ranking suggests the presence of significant unsystematic risk.

T_A=\frac{\bar{R}_A-\bar{R}_F}{\beta_A}
T_A
\bar{R}_F
\bar{R}_A
\beta_A

Practice Questions: Q1

Q1. The following information is available for Funds ABC, RST, JKL, and XYZ:

 

 

 

 

 

 

The average risk-free rate was 5%. Rank the funds from best to worst according to their Treynor measure.

A. JKL, RST, ABC, XYZ.

B. JKL, RST, XYZ, ABC.

C. RST, JKL, ABC, XYZ.

D. XYZ, ABC, RST, JKL.

Practice Questions: Q1 Answer

Explanation: A is correct.

Treynor measures:

 

 

 

 

 

 

The following table summarizes the results:
 

 

 

 

 

 

 

\begin{aligned} & \mathrm{T}_{\mathrm{ABC}}=\frac{0.15-0.05}{1.25}=0.08=8 \\ & \mathrm{~T}_{\mathrm{RST}}=\frac{0.18-0.05}{1.00}=0.13=13 \\ & \mathrm{~T}_{J \mathrm{KL}}=\frac{0.25-0.05}{1.20}=0.1667=16.7 \\ & \mathrm{~T}_{\mathrm{XYZ}}=\frac{0.11-0.05}{1.36}=0.0441=4.4 \end{aligned}

Topic 4. Jensen's Alpha

  • Definition: The difference between a portfolio's actual return and its expected return, as predicted by the Capital Asset Pricing Model (CAPM). It is a direct measure of a manager's performance.

  • Formula: 

    αA​=RA​−E(RA​)

    where:

    E(RA​)=RF​+βA​[E(RM​)−RF​]

  • Variables:

    • αA​: Jensen's Alpha

    • RA​: Actual return on the account

    • E(RA​): Expected return based on CAPM

    • RF​: Risk-free rate

    • E(RM​): Expected market return

    • βA​: Average beta

  • Key Insight: A positive and statistically significant alpha indicates a superior manager who has outperformed the market on a risk-adjusted basis. Like the Treynor measure, it only considers systematic risk.

Practice Questions: Q2

Q2. The following data has been collected to appraise Funds A, B, C, and D:

 

 

 

 

 

The risk-free rate of return for the relevant period was 4%. Calculate and rank the funds from best to worst using Jensen’s alpha.

A. B, D, A, C.

B. A, C, D, B.

C. C, A, D, B.

D. C, D, A, B.

Practice Questions: Q2 Answer

Explanation: C is correct.

CAPM returns:

 

 

 

 

 

 

 

 

 

 

 

 

 

\begin{aligned} & R_A=4+0.91(8.6-4)=8.19 \% \\ & R_B=4+0.84(8.6-4)=7.86 \% \\ & R_C=4+1.02(8.6-4)=8.69 \% \\ & R_D=4+1.34(8.6-4)=10.16 \% \end{aligned}

Topic 5. The Information Ratio

  • Definition: Measures a portfolio's surplus return relative to a benchmark, divided by the standard deviation of that surplus return.

  • Formula: 


    ​

  • Variables:

    • IRA​: Information Ratio

    •        : Average account return

    •             ​: Average benchmark return

    • σA−B​: Standard deviation of the excess returns (difference between account and benchmark returns)

  • Key Insight: A higher Information Ratio indicates better performance, as it shows the amount of "risk" (variability of surplus returns) taken to achieve returns above the benchmark.

IR_A=\frac{\bar{R}_A-\bar{R}_B}{\sigma_{A-B}}
\bar{R}_A
\bar{R}_B

Topic 6. The M-Squared Measure

  • Definition: A performance measure that compares the managed portfolio's return against the market return after adjusting for differences in total risk (standard deviation).

  • Concept: A hypothetical portfolio (P′) is created by combining the managed portfolio (P) and a risk-free asset to match the standard deviation of the market portfolio.

  • Calculation: The M2 measure is the difference in returns between this hypothetical portfolio (P′) and the market portfolio.

  • Example from PDF:

    • Portfolio P Mean Return = 10%

    • Market Mean Return = 12%

    • Risk-free Rate = 4%

    • Hypothetical Portfolio P′ Return = 7%

    • M2=RP′​−RM​=0.07−0.12=−0.05

  • Key Insight: The M2 measure will produce the same rankings as the Sharpe Ratio, as both are based on total risk.

Module 3. Alpha, Hedge Funds, Dynamic Risk, Market Timing and Style

Topic 1.  Statistical Significance of Alpha Returns

Topic 2. Measuring Hedge Fund Performance

Topic 3. Performance Evaluation With Dynamic Risk Levels

Topic 4. Measuring Market Timing Ability

Topic 5. Style Analysis

Topic 1. Statistical Significance of Alpha Returns

  • Purpose: To determine if a positive alpha is a result of a manager's skill or merely a random outcome.

  • Methodology: The significance of alpha is tested using regression analysis. The alpha is the intercept of a regression of the portfolio's excess returns against the market's excess returns.

  • Key Concept: A statistically significant alpha (indicated by a high t-statistic) suggests that the manager has indeed outperformed the market on a risk-adjusted basis. A positive alpha that is not statistically significant may just be due to luck.

Topic 2. Measuring Hedge Fund Performance

  • Challenge: Evaluating hedge fund performance is complicated due to their unique trading practices.

  • Illiquidity: Many hedge fund investments are illiquid.

    • This can lead to the "smoothing" of returns, where reported returns are less volatile than the underlying assets.

    • This smoothing effect makes standard deviation an unreliable measure of risk.

  • Non-linear Returns: Hedge funds often use options and other derivatives.

    • The returns from these instruments have a non-linear relationship with the market, meaning a simple beta (linear relationship) may be an inaccurate measure of risk.

Topic 3. Performance Evaluation With Dynamic Risk Levels

  • Issue: A portfolio manager's risk level (beta) may not be constant over time.

    • Managers may shift their exposure to the market based on their beliefs about future market movements, a practice known as market timing.

  • Solution: To properly evaluate performance, it is necessary to use models that allow for a dynamic or time-varying beta. These models can help to separate a manager's stock-picking ability from their market timing ability.

Topic 4. Measuring Market Timing Ability

  • Concept: A manager's ability to increase the portfolio's beta (risk) in anticipation of a rising market and decrease it in anticipation of a falling market.

  • Two Common Models:

    • Treynor-Mazuy Model: 

      RP​−RF​=α+β(RM​−RF​)+γ(RM​−RF​)2+ϵ

      A positive and significant γ indicates successful market timing.

    • Henriksson-Merton Model: 

      RP​−RF​=α+β(RM​−RF​)+γ(RM​−RF​)×D+ϵ

      A positive and significant γ indicates successful market timing. Here, D is a dummy variable that is 1 if RM​−RF​>0 and 0 otherwise.

Topic 5. Style Analysis

  • Purpose: To conduct performance attribution by breaking down a portfolio's returns into a set of distinct asset classes or investment styles.

  • Methodology: A regression model is used where the portfolio's returns are the dependent variable. The independent variables are the returns of various style indices (e.g., small-cap, large-cap, value, growth).

  • Key Insight: The coefficients of the regression represent the percentage of the portfolio invested in each style. The model provides insights into the manager's investment strategy and the sources of the portfolio's returns.

Practice Questions: Q1

Q1. Sharpe’s style analysis, used to evaluate an active portfolio manager’s performance, measures performance relative to:

A. a passive benchmark of the same style.

B. broad-based market indices.

C. the performance of an equity index fund.

D. an average of similar actively managed investment funds.

Practice Questions: Q1 Answer

Explanation: A is correct.

Sharpe’s style analysis measures performance relative to a passive benchmark of the same style.

IM 8. Portfolio Performance Evaluation

By Prateek Yadav

IM 8. Portfolio Performance Evaluation

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