Thread starter
#1

*Learning objectives: Define the properties of a coherent risk measure and explain the meaning of each property. Explain why VaR is not a coherent risk measure.*

**Questions:**

809.1. A credit portfolio contains five identical bonds. Each bond has a face value of $1,000 and default probability of 80 basis points (0.80%). The bonds are completely independent: their pairwise default correlations are zero. If any bond defaults, we will assume it has a 50.0% recovery rate. With respect to any of the individual bonds, consider the following definitions:

- The worst expected loss with 99.0% confidence is zero, because the default probability is less than the 1.0% significance level. In this way, the relative VaR is zero.
- The unexpected loss is given by (loss quantile) - EL; in this case, 0 - (0.0080 * $1,000 * 0.50) = -$4.00. In this way, the 99.0% credit value at risk (CVaR) is -$4.00.

**nearest**to the portfolio's 99.0% CVaR?

a. -$20.00

b. Zero

c. $480.00

d. $4,960.00

809.2. Peter has $10,000 to invest over a one-year horizon and considers three choices:

- A fund with an expected return of 9.0% and volatility of 30.0% (both per annum)
- A fund with an expected return of 15.0% and volatility of 30.0% (both per annum)
- One-year at-the-money (ATM) call options on a stock with a volatility of 30.0%

I. The 95.0% absolute value at risk (aVaR) is lowest for the second equity fund; i.e., -0.150 + 1.65*0.40 < -0.090 + 1.65*0.40

II. Although the underlying stock volatility is 30.0% in each case, the options effectively leverage the position and represent the highest risk (in fact, the entire investment can be lost)

Which coherence properties are

**NEAREST**to being demonstrated, respectively, by these two observations?

a. Subadditivity, Monotonicity

b. Monotonicity, Positive homogeneity

c. Translation invariance, Subadditivity

d. Positive homogeneity, Translation invariance

809.3. Robert's all-equity portfolio returned +15.0% last year. He outperformed his benchmark, which returned only +9.0%. Consequently, he asserts that he outperformed his benchmark by +6.0%. His colleague Barbara observes that Robert's portfolio experienced a volatility of 32.0% per annum, but the benchmark's volatility was only 20.0%. She introduces him to the M-squared (M^2) measure. The M^2 measure imagines that Robert's portfolio is mixed with cash (i.e., a risk-free position) to the extent that its volatility would match its benchmark. In this case, if 37.50% were allocated to risk-free cash, then the imagined portfolio would have a volatility equal to the benchmark: 62.50% * 32.0% + 37.50%*0 = 20.0%. In this case, if the risk-free rate is assumed to be 4.0%, then the return of the imagined portfolio would be 15.0%*62.50% + 4.0%*37.50% = 10.8750%. As this compares to the benchmark's return of 9.0%, the M^2 performance measure is given by 10.875% - 9.0% = +1.8750%. According to Barbara, this is a better measure of Robert's risk-adjusted performance relative to his benchmark.

By imagining Robert's all-equity portfolio as re-mixed with 37.50% cash, the re-mixed portfolio's risk, as measured by volatility, is reduced from 32.0% to 20.0%. Which coherence property is

**NEAREST**to being demonstrated by this observation?

a. Monotonicity

b. Subadditivity

c. Positive homogeneity

d. Translation invariance

**Answers here:**

## Stay connected