Interpret Portfolio Performance - Series 65

1. What is the formula for time-weighted return (TWR) across $n$ subperiods?

   $\text{TWR}=\left(\prod_{t=1}^{n}(1+r_t)\right)-1$. Compounds subperiod returns to measure performance independent of external cash flows.

2. Calculate Sharpe ratio: $r_p=0.10$, $r_f=0.02$, $\sigma_p=0.16$. What is Sharpe?

   $0.50$. Divides the excess return over the risk-free rate by the portfolio's standard deviation to measure risk-adjusted performance.

3. Identify the required condition on portfolio weights $w_i$ in $r_p=\sum w_i r_i$.

   $\sum_{i=1}^{n} w_i=1$. Ensures the weights fully allocate the portfolio's value without over- or under-representation.

4. What is the formula for a portfolio's weighted average return using weights $w_i$?

   $r_p=\sum_{i=1}^{n} w_i r_i$. Aggregates individual asset returns proportionally to their weights in the portfolio.

5. Calculate HPR: begin $100$, end $110$, income $2$. What is $\text{HPR}$?

   $0.12$. Incorporates both capital appreciation and income divided by the beginning value to determine total return.

6. What is the Treynor ratio formula using portfolio beta $\beta_p$?

   $\text{Treynor}=\frac{r_p-r_f}{\beta_p}$. Assesses reward per unit of systematic risk by dividing excess return by the portfolio's beta.

7. What is the formula for beta using covariance and market variance?

   $\beta_p=\frac{\text{Cov}(r_p,r_m)}{\text{Var}(r_m)}$. Measures systematic risk as the covariance of portfolio and market returns divided by market variance.

8. Which mean return is always less than or equal to the arithmetic mean when returns vary?

   Geometric mean return. Accounts for compounding effects, yielding a value less than or equal to the arithmetic mean unless all returns are identical.

9. Calculate a 2-asset return: $w_1=0.60,r_1=0.10;\;w_2=0.40,r_2=-0.05$. What is $r_p$?

   $0.04$. Sums the products of each asset's weight and return to derive the overall portfolio performance.

10. What is tracking error in benchmarked performance measurement?

    Standard deviation of $(r_p-r_b)$. Quantifies the volatility of a portfolio's returns relative to its benchmark, measuring deviation risk.

11. What is the formula for an arithmetic mean return across $n$ periods?

    $\bar{r}=\frac{\sum_{t=1}^{n} r_t}{n}$. Provides a simple average of periodic returns by summing them and dividing by the number of periods.

12. What is the Sharpe ratio formula using portfolio return $r_p$, risk-free rate $r_f$, and $\sigma_p$?

    $\text{Sharpe}=\frac{r_p-r_f}{\sigma_p}$. Evaluates risk-adjusted performance by dividing excess return over the risk-free rate by portfolio standard deviation.

13. Which return measure is most sensitive to the timing and size of investor cash flows?

    Money-weighted return (IRR). Incorporates the impact of cash flow timing and magnitude directly into the return calculation.

14. What is the formula for alpha relative to a benchmark using CAPM terms?

    $\alpha=r_p-\left[r_f+\beta_p(r_m-r_f)\right]$. Quantifies abnormal return by subtracting the CAPM-expected return from the actual portfolio return.

15. What is the formula for the geometric mean return across $n$ periods?

    $\left(\prod_{t=1}^{n}(1+r_t)\right)^{\frac{1}{n}}-1$. Represents the compound growth rate by taking the nth root of the product of one plus each periodic return, then subtracting one.

16. What is the formula for money-weighted return (MWR), also called the internal rate of return?

    $\sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}=0$. Solves for the rate that makes the present value of all cash flows, including initial and final values, equal to zero.

17. What is the approximate real return when nominal return is $r_n$ and inflation is $i$?

    $r_{real}\approx r_n-i$. Offers a simplified estimate of purchasing power return by subtracting inflation from nominal return.

18. What is the primary purpose of time-weighted return (TWR) in performance reporting?

    To remove the effect of external cash flows. Isolates the manager's performance by eliminating distortions from investor-driven cash inflows or outflows.

19. Calculate TWR for two subperiod returns $+10\%$ then $-10\%$. What is TWR?

    $-0.01$. Compounds the subperiod returns by multiplying one plus each return and subtracting one from the product.

20. What is the formula for the real return given nominal return $r_n$ and inflation $i$?

    $r_{real}=\frac{1+r_n}{1+i}-1$. Adjusts nominal return for inflation by dividing one plus nominal by one plus inflation rate, then subtracting one.

21. What is active return relative to a benchmark with return $r_b$?

    $r_p-r_b$. Represents the excess return of the portfolio over its benchmark, indicating outperformance or underperformance.

22. Calculate approximate real return: nominal $8\%$, inflation $3\%$. What is $r_{real}$ approximately?

    $0.05$. Subtracts the inflation rate from the nominal return to estimate the inflation-adjusted return.

23. What is the formula for the holding period return (HPR) including income?

    $\text{HPR}=\frac{\text{Ending value}-\text{Beginning value}+\text{Income}}{\text{Beginning value}}$. Captures total return by including capital gain or loss and any income received relative to the initial investment.

24. What is the information ratio formula using active return and tracking error?

    $\text{IR}=\frac{r_p-r_b}{\sigma_{p-b}}$. Gauges active management skill by dividing active return by the standard deviation of return differences from the benchmark.

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