5. Measures of risk and performance

Question 1

1. What are the two main differences between the formula for variance and the formula for semivariance?

Answer 1

• The semivariance uses a formula otherwise identical to the variance formula except that it only includes the negative deviations in the numerator and a smaller number of observations in the denominator.

Question 2

2. What is the main difference between the formula for semistandard deviation and target semistandard deviation?

Answer 2

• Target semivariance is similar to semivariance except that target semivariance substitutes the investor’s target rate of return in place of the asset’s mean return.

Question 3

3. Define tracking error and average tracking error

Answer 3

• Tracking error indicates the dispersion of the returns of an investment relative to a benchmark return, where a benchmark return is the contemporaneous realized return on an index or peer group of comparable risk.
• Average tracking error simply refers to the average difference between an investment’s return relative to its benchmark. In other words, it is the numerator of the information ratio.

Question 4

4. What is the difference between value at risk and conditional value-at-risk?

Answer 4

• Value at risk (VaR or VAR) is the loss figure associated with a particular percentile of a cumulative loss function. In other words, VaR is the maximum loss over a specified time period within a specified probability.
• Conditional value-at-risk (CVaR), also known as expected tail loss, is the expected loss of the investor given that the VaR has been equaled or exceeded. CVaR will exceed VaR (if the overall maximum potential loss exceeds the VaR).

Question 5

5. Name the two primary approaches for estimating the volatility used in computing value-at-risk.

Answer 5

• Estimate the standard deviation (volatility) as being equal to the asset’s historical standard deviation of returns
• Estimate volatility based on the implied volatilities from option prices

Question 6

6. What are the steps involved in directly estimating VaR from historical data rather than through a parametric technique?

Answer 6

• Collect the percentage price changes
• Rank the gains/losses from the highest to the lowest
• Select the outcome (loss) reflecting the quantile specified by the VaR (e.g., for a VaR based on
95% confidence pick the observation with a loss larger than 95% of the other outcomes).

Question 7

7. When is Monte Carlo analysis most appropriate as an estimation technique?

Answer 7

• It is best used in difficult problems where it is not practical to find expected values and standard deviations using mathematical solutions.

Question 8

8. What is the difference between the formulas for the Sharpe and Treynor ratios?

Answer 8

• The Treynor ratio differs from the Sharpe ratio by the use of systematic risk rather than total risk in the denominator.

Question 9

9. Define Return on VaR.

Answer 9

• Return on VaR (RoVaR) is simply the expected or average return of an asset divided by a specified VaR (expressing VaR as a positive number):

Question 10

10. Describe the intuition of Jensen’s alpha.

Answer 10

• Jensen’s alpha is a direct measure of the absolute amount by which an asset is estimated to outperform, if positive, the return on efficiently priced assets of equal systematic risk in a single- factor market model.

Question 11

drawdown

Answer 11

is defined as the maximum loss in the value of an
asset over a specified time interval and is usually expressed in
percentage-return form rather than currency.

Question 12

Information ratio

Answer 12

has a numerator formed by the
difference between the average return of a portfolio (or other
asset) and its benchmark, and a denominator equal to its
tracking error: Information Ratio = [E(Rp) − RBenchmark]∕TE
where E(Rp) is the expected or mean return for portfolio p,
RBenchmark is the expected or mean return of the benchmark,
and TE is the tracking error of the portfolio relative to its

Question 13

M2 approach

Answer 13

or M-squared approach, expresses the
excess return of an investment after its risk has been
normalized to equal the risk of the market portfolio.

Question 14

Maximum drawdown

Answer 14

is defined as the largest decline over

any time interval within the entire observation period.

Question 15

parametric VaR

Answer 15

A VaR computation assuming normality and using the

statistics of the normal distribution is known as this.

Question 16

shortfall risk

Answer 16

is simply the probability that the return will be

less than the investor’s target rate of return.

Question 17

Sortino ratio

Answer 17

subtracts a benchmark return, rather than
the riskless rate, from the asset’s return in its numerator and
uses downside standard deviation as the measure of risk in its
denominator: Sortino Ratio = [E(Rp) − RTarget ]∕TSSD
where E(Rp) is the expected return, or mean return in practice,
for portfolio p; RTarget is the user’s target rate of return; and
TSSD is the target semistandard deviation (or downside
deviation).

Question 18

target semivariance

Answer 18

is similar to semivariance except that
target semivariance substitutes the investor’s target rate of
return in place of the mean return.

Question 19

well-diversified portfolio

Answer 19

is traditionally interpreted as any portfolio containing only
trivial amounts of diversifiable risk.

Question 20

Let’s return to the example of JAC Fund’s $1
million holding of the ETF with an expected return of zero. Estimating roughly
that the daily standard deviation of the ETF is 1.35%, for a 99% confidence
interval, the 10-day VaR is found through substituting the known values into the
equation:

Answer 20

Parametric VaR = N x SD x square root (days) x value

In order to solve this application we need to use equation 5.4. In this case, with a
99% confidence interval the z-score is 2.33. Following equation 5.4, we multiply
2.33 by .1.35% by the square root of 10 (the days in the period) to get a product
of 9.94%. 9.94% represents the percentage change in the value. To complete
this solution we multiply 9.94% by $1,000,000 for an answer of $99,469.44. The
z-score is a value that is assumed to be provided rather than being memorized or
calculated.

Step One: Press 2.33 → x → 0.0135

Step Two: Press x → 10 →

Step Three: Press x → 1,000,000

Step Four: Press =

Answer: 99469.44

Question 21

Consider a portfolio that earns 10% per year and

has an annual standard deviation of 20% when the risk-free rate is 3%.

Answer 21

The Sharpe ratio in this application is calculated by finding the difference between
10% and 3% (the portfolio return and the risk free rate otherwise known as excess
return), then dividing by 20% for a quotient of 0.35.

Step One: Press 0.1 → - → 0.03
Step Two: Press ÷ → 0.2
Step Three: Press =
Answer: 0.35

Question 22

Consider a portfolio that earns 10% per year and
has an annual standard deviation of 20% when the risk-free rate is 3%.

Ignoring compounding for simplicity, and
assuming statistically independent returns through time, the Sharpe ratios based
on semiannual returns and quarterly returns are, using the same annual values
as illustrated earlier, as follows:

Answer 22

Semiannual Sharpe Ratio
Step One: Press 0.1 → - → 0.03
Step Two: Press ÷ → 2
Step Three: Press = “0.035”
Step Four: Press 0.5 →
Step Five: Press x → 0.2
Step Six: Press = “0.1414”
Step Seven: Press 0.035 → ÷ → 0.1414
Answer: 0.247

Quarterly Sharpe Ratio
Step One: Press 0.1 → - → 0.03
Step Two: Press ÷ → 4
Step Three: Press = “0.0175”
Step Four: Press 0.25 →
Step Five: Press x → 0.2
Step Six: Press = “0.1”
Step Seven: Press 0.0175 → ÷ → 0.1
Answer: 0.175

Question 23

Consider a portfolio that earns 10% per year and
has a beta with respect to the market portfolio of 1.5 when the risk-free rate is
3%.

Answer 23

The Treynor ratio is (10% – 3%)/1.5, or 0.0467 (4.67%).

Question 24

Consider a portfolio that earns 10% per year
when the investor’s target rate of return is 8% per year. The semistandard
deviation based on returns relative to the target is 16% annualized.

Answer 24

The Sortino

ratio would be (10% – 8%)/16%, or 0.125.

Question 25

If a portfolio consistently outperformed its
benchmark by 4% per year, but its performance relative to that benchmark
typically deviated from that 4% mean with an annualized standard deviation of
10%, then its information ratio would be?

Answer 25

To calculate the information ratio we need to divide 4% (the amount that the
portfolio outperformed the benchmark per year) by 10% (the annual standard
deviation of returns of the portfolio) for an answer of 0.40.

Question 26

A portfolio is expected to earn 7% annualized
return when the riskless rate is 4% and the expected return of the market is 8%.
If the beta of the portfolio is 0.5, the alpha of the portfolio is?

Answer 26

Alpha(p) = E(Rp) - R(f) - Beta(p)(E(Rm) - R(f))

In order to solve this application, we need to apply equation 5.15. First,
subtracted 0.08 (the expected return of the market) by 0.04 (the riskless rate).
Multiply the difference of 0.4 by 0.5 (the portfolio’s beta) for a product of 0.02.
Subtract 0.07 by 0.04 for a difference of 0.03. Subtract 0.03 by 0.02 for a
difference of 0.01 or 1% (Jensen’s alpha).

Question 27

Consider a portfolio with M2 = 4%. The portfolio
is expected to earn 10%, while the riskless rate is only 2%. What is the ratio of
the volatility of the market to the volatility of the portfolio?

Answer 27

M(^2) = R(f) + (SDm / SDp) (E(Rp) - R(f))

Inserting the given
rates generates 4% = 2% + [(ratio of volatilities) x 8%]. The ratio of the volatility
of the market to the volatility of the portfolio must be 25%.