Risk Based Measures of Performance Appraisal

Finance academics and professionals have developed several returns-based measures to assess the value of active management. These concepts lend themselves well to testing and show up frequently on exams. Important measures which will be reviewed and/or introduced in this Los summary include the following:

• Sharpe ratio.
• Treynor ratio.
• Information ratio.
• Appraisal ratio.
• Sortino ratio.
• Capture ratios.

Sharpe Ratio – Review

The Sharpe ratio measures the additional return for bearing risk above the risk-free rate, stated per unit of return volatility. In performance appraisal, this additional return is often known as excess return. This use contrasts with how “excess return” is used in return performance attribution—that is, as a return beyond a benchmark's return. The Sharpe ratio is commonly used on an ex-post basis to evaluate historical risk-adjusted returns.

$$ \frac { (R_p – R_{fr}) }{ \sigma_a} $$

Where:

\(R_p\) = Return on portfolio or security.

\(R_{fr}\) = Risk-free rate.

\(\sigma_a\) = Standard deviation of returns.

One drawback of the Sharpe Ratio is that it does not distinguish between upside and downside volatility. Investment returns may be variable but only to the upside as the security or portfolio performs well. However because these returns are choppy, or a little less predictable, the Sharpe Ratio will fall as the standard deviation rises.

Treynor Ratio – Review

The Treynor Ratio begins exactly the same as the Sharpe Ratio. The ratio differs however in the denominator when the risk metric changes from standard deviation to Beta. This is useful when the performance of a portfolio is in someway comparable to a benchmark:

$$ \frac { (R_p – R_{fr}) }{ B_a } $$

Where:

\(R_p\) = Return on portfolio or security.

\(R_{fr}\) = Risk-free rate.

\(B_a\) = Beta.

The Information Ratio

$$ IR = \frac {[E(r_p) – E(r_b)] }{ [\sigma_{(p – b)}] }$$

Where:

\(E(r_p)\) = Expected return on the portfolio.

\(E(r_b)\) = Expected return on the benchmark.

\(\sigma_{(p – b)}\) = Portfolio tracking risk.

The Appraisal Ratio

The appraisal ratio (AR) is a returns-based measure, like the IR. It is the annualized alpha divided by the annualized residual risk. The difference being that with the appraisal ratio, both the alpha and the residual risk are computed from a factor regression. The AR can be computed using any factor model appropriate for the portfolio. The appraisal ratio measures the reward of active management relative to the risk of active management (alpha from a factor model):

$$ AR =\frac { \alpha }{ \sigma_\epsilon} $$

Where:

\(\alpha\) = Reward from active management.

\(\sigma_\epsilon\) = Risk of active management.

where \(\sigma_\epsilon\) is the standard deviation of \(\epsilon_t\), commonly denoted as the “standard error of regression,” in rea-world application this number would be output from common statistical software.

The Sortino Ratio

The Sortino Ratio is a modification of the Sharpe Ratio, in that it seeks to solve the dilemma of punishing investors for ‘good’ volatility. In place of the risk-free rate, the Sortino Ratio uses a target rate of return, and in the denominator uses downside risk or a measure of downside volatility:

$$ \frac {(R_p – R_t) }{ \sigma_d} $$

Where:

\(R_p\) = Return on portfolio or security.

\(R_t\) = Target rate of return.

\(\sigma_d\) = Target semi deviation.

Sortino Ratio – Calculation Example

The formula for target semi deviation is:

$$
\text{Target semi deviation} = \text{min}(r_a – r_T,0)^2 $$

Where:

\(r_a\) = Actual return.

\(r_T\) = Target return.

This means that the smaller of zero, or a negative number will be chosen. For example, if the target return was 4%, and actual portfolio returns in a given year were 6%:

Then;

$$ \text{Target semi deviation}= \text{min}(r_a – r_T,0)^2 = (6\%-4\%,0)^2 = 0 $$

If, on the other hand, returns were -1%

Then;

$$ \text{Target semi deviation}= \text{min}(r_a – r_T,0)^2= (-1\%-4\%,0)^2 = 0.0025 $$

The second scenario would add to the value of the numerator, and thus lower the Sortino Ratio.

The numerator in the Sortino Ratio is the average portfolio returns over the period minus the minimum target return.

Capture Ratios and Drawdown

Capture

Capture ratios help investors understand the upside and downside participation of an investment manager during different market periods. Managers who's portfolios gain at a rate as fast or faster than the benchmark (100% +) are said to outperform. Conversely, managers who's portfolios decline faster than the benchmark are said to be underperforming.

The expression for upside capture capture is:

$$ UC_{(m,B,t)} =\frac {R_{(m,t)}}{R_{(B,t)}} \ \ \text{ if } R_{(B,t)} \ge 0 $$

The expression for downside capture is:

$$ DC_{(m,B,t)} =\frac { R_{(m,t)}}{R_{(B,t)}} \ \ \text{ if } R_{(B,t)} \lt 0 $$

Where:

\(UC_{(m,B,t)}\) = Upside capture for manager m relative to benchmark B for time t.

\(DC_{(m,B,t)}\) = Downside capture for manager m relative to benchmark B for time t.

\(R_{(m,t)}\) = Return of manager m for time t

\(R_{(B,t)}\) = return of benchmark B for time t.

The upside/downside capture (CR), is the upside capture divided by the downside capture. It measures the lack of symmetry of return. A capture ratio greater than 1 means positive asymmetry, whereas a capture ratio lower than 1 indicates negative asymmetry.

Drawdown

Drawdown is measured as the cumulative highest to lowest value point during a continuous period. Drawdown duration is the total time from the start of the drawdown until the cumulative drawdown recovers to zero (much like a high-water mark), which can be segmented into the drawdown phase (start to trough) and the recovery phase (lowest-to-zero cumulative return).

$$
\text{Maximum } DD_{(m,t)} = \text{min} \left( \left[\frac {V_{(m,t)} – V_{(m,t^\ast)}}{V_{(m,t^\ast)}} \right], 0 \right) $$

Where:

\(V_{(m,t)}\) = Portfolio value of manager m at time t.

\(V_{(m,t^\ast)}\) = Peak portfolio value of manager m \(t \gt t^\ast\).

In order to visualize how a low-point would be calculated, a true drawdown, it could look something like this:

$$ \frac { (\text{Lowest portfolio value} – \text{Highest portfolio value})}{\text{Highest portfolio value}} $$

Limitations to Appraisal Based Ratios

Ratios for appraisal must take into account the assumptions of each ratio and be appropriate for the investment process, investor risk tolerance, and investor time horizon.

In spite of the fact that appraisal ratios can be used to identify manager skill (rather than luck), these ratios are often based on investment return data, which can be limited and subject to error.

Question

Portfolio Delta delivered 12.0% annual returns on average over the past 60 months. Its average annual volatility as measured by standard deviation was 14.0%, and its downside volatility as measured by target semi-standard deviation was 8.0%. Assuming the risk-free rate is 3%, and the target rate of return is 5.0% per year, the Sortino ratio of Portfolio Delta is closest to:

1. 0.50.
2. 1.13.
3. 0.88.

Solution

The correct answer is C.

$$
\frac { (R_p – R_t) }{ \sigma_d }$$

Where:

\(R_p\) = Return on portfolio or security.

\(R_t\) = Target rate of return.

\(\sigma_d\) = Target semi deviation.

Therefore,

$$ \frac { (12 – 5) }{ 8} = 0.875 $$

A is incorrect. It uses standard deviation instead of downside deviation.

B is incorrect. It uses the risk-free rate rather than target return.

Reading 12: Portfolio Performance Evaluation

Los 12 (o) Describe limitations of appraisal measures and related metrics