Week 4 - Market Risk

Question 1

14.16 Consider a position consisting of an $300,000 investment in gold and a $500,000 investment in silver. Suppose that the daily volatilities of these two assets are 1.8% and 1.2%, respectively, and that the coefficient of correlation between their returns is 0.6. What are the 10-day 97.5% VaR and ES for the portfolio? By how much does diversification reduce the VaR and ES?

Answer 1

Step 1:
Calculate the portfolio volatility as

Portfolio Volatility = √[(300)² × (0.018)² + (500)² × (0.012)² + 2 × (0.375) × (0.625) × (0.018) × (0.012) × (0.6)] = 10.2

N-1(97.5%) = N-1(0.975) = N(1-0.975) = N(0.25) = 1.96

Step 2:

This gives the one-day VaR:
VaR1 = 1.96 ∗ σP = 19.99

The 10-day VaR?
VaR10 =√10VaR1 = √1019.99 = 63.22 or 63,220$

To calculate how much diversification reduce the VaR:

VaRs10 = 0.012 ∗ 500 ∗ √10 ∗ 1.96 = 37188

VaRg10 = 0.018 ∗ 300 ∗ √10 ∗ 1.96 = 33470

Diversification benefit:
VaRs10 + VaRg10 − VaRP10 = 7,438$

The Expected Shortfall:

ES = μ + σ*φ(Φ−1(α)) / 1−α
where φ(x)=1 / √ 2π * e ^(-χ^ 2 / 2)

Esport = 75.41 ⇒ ESp = $75410

The benefit of diversification:
ESg = 39.92 ⇒ ESg = $39920

ESs = 0.012∗500 10∗ √2π ∗ 0.025 = 44.36 ⇒ ESg = $44360

ESg +ESs −ESP = 39920+44360−75410 = 8870

Question 2

14.20 A company has a long position in a two-year bond and a three-year bond, as well as a short position in a five-year bond. Each bond has a principal of $100 million and pays a 5% coupon annually. Calculate the company’s exposure to the one- year, two-year, three-year, four-year, and five-year rates. Use the data in Tables 14.7 and 14.8 to calculate a 20-day 95% VaR on the assumption that rate changes are explained by (a) one factor, (b) two factors, and (c) three factors. Assume that the zero-coupon yield curve is flat at 5%.

Answer 2

step 1:

Make a table with the cash flows,
Calculate the PV (discount the CF’s)

For 1b.p. Δy calculate ΔB using:

ΔΒ = -D * B * Δy

where D = Σti * (PV / B)

Add the the effect of 1bp change in yield on the PV of cashflow to the table.

step 2:

Calculate the sensitivity to the first PCA factor as:

ΔΒi * PC1 + ΔΒi+1 * PC2 …..

−0.00047∗0.216−0.0190∗0.331−0.0258∗0.372+0.0016∗0.392+0.0409∗0.404 = −0.00116

Calculate the sensitivity to the second and third PCA factors:

Sensitivity to the second factor = 0.01588
Sensitivity to the third factor
= −0.01363

Assuming one factor, the standard deviation of the one-day change in portfolio is:

Sensitivity to the first factor × Standard deviation of first factor score (s1 * σ1)

0.00116 ∗ 17.55 = 0.0204

Calculate the 20-day 95% VaR
* Z-score for 95% is 1.645
* one-day 95% VaR
* 20-day VaR:

VaR1 = 0.0204 ∗ 1.645 = 0.033558
VaR20 = VaR1 ∗ √20 = 0.15

Assuming two factor, the standard deviation of the one-day change in portfolio is:

√(s21 ∗ sd21 + s2 ∗ sd2) = 0.0785

VaR1 = 0.0785 ∗ 1.645 =…
VaR20 = VaR1 ∗√ 20 = 0.577

Assuming three factor, the standard deviation of the one-day change in portfolio is:

√s21 ∗sd21 +s2 ∗sd2 +s23 ∗sd23 = 0.0834

VaR1 = 0.0834 ∗ 1.645
VaR20 = VaR1 ∗ √20 = 0.614

Question 3

If a portfolio has 2 assets X and Y how do you calculate the diversification benefit in VaR?

Answer 3

First you calculate the VaR for X, Y and then, the diversification benefit form is:

VaR(X) - VaR(Y) - VaR(port)

Question 4

A company has a position in bonds worth $6 million. The modified duration of the portfolio is5.2 years. Assume that only parallel shifts in the yield curve can take place and that the standard deviation of the daily yield change (when yield is measured in percent) is 0.09. Use the duration model to estimate the 20-day 90% VaR for the portfolio. Explain carefully the weaknesses of this approach to calculating VaR. Explain two alternatives that give more accuracy.

Answer 4

The change in the value of the portfolio for a small change Δy in the yield is approximately −DBΔy where D is the duration and B is the value of the portfolio. It follows that the standard deviation of the daily change in the value of the bond portfolio equals DBΔy where Δy is the standard deviation of the daily change in the yield. In this case D = 5.2, B = 6,000, 000, and Δy = 0.0009 so that the standard deviation of the daily change in the value of the bond portfolio is:

5.4 * 6,000,000 * 0.0009 = 28,080

The 20-day VaR 90% of the portfolio is 1.282 * 28,080 * √20 = 160,990 $

This approach assumes that only parallel shifts in the term structure can take place. Equivalently it assumes that all rates are perfectly correlated or that only one factor drives term structure movements. Alternative more accurate approaches described in the chapter are (a) cash flow mapping and (b) a principal components analysis.

Question 5

Using the (Merton) Black and Scholes model the expected loss on the debt is:

Answer 5

D - E0,

E0 = V0N(d1) − De−rTN(d2)

Question 6

Using the (Merton) Black and Scholes model the probability of default is:

Answer 6

Prob of default = N(-d2)