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P1.T4.809. Coherent risk measures (Dowd Ch.2)

Nicole Seaman

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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.


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.
The estimation of the portfolio CVaR illustrates VaR's lack of subadditivity. Which of the following is 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%
To keep it simple, let's assume he will invest all of his funds into one of the strategies (he will not allocate among the strategies) so that he has only three choices. From the perspective of potential returns (upside), clearly, the highest potential is offered by the call options. However, from the perspective solely of risk, he makes the following two observations:

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

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