RMBI4210 — Statistical Methods for Risk Management Homework Problems
Course instructor: Prof. Y.K. Kwok
1. Let c be the coupon rate per period and y be the yield per period. There are m periods per year (say, m = 4 for quarterly coupon payments), and let n be the number of periods remaining until maturity. Show that the duration D is given by
D = 1 + y − 1 + y + n(c − y) . my mc[(1+y)n −1]+my
Here, the yield per year λ equals my. Show that, as T → ∞, we obtain D → m1 + λ1 .
Remark
The above analytic results are revealed in the following numerical example. Consider the
duration calculated for various bonds as shown in the following table, where λ = 0.05 and
m=2.WeobtainD→1+ 1 =20.5. 2 0.05
Duration of a Bond Yielding 5% as Function of Maturity and Coupon Rate Coupon rate
Years to maturity 1
2
5
10
25
50
100
∞
1% 0.997 1.984 4.875 9.416 20.164 26.666 22.572 20.500
2% 5% 0.995 0.988 1.969 1.928 4.763 4.485 8.950 7.989
17.715 14.536 22.284 18.765 21.200 20.363 20.500 20.500
10% 0.977 1.868 4.156 7.107
12.754 17.384 20.067 20.500
The table shows that duration does not increase appreciably with maturity. In fact, with fixed yield, duration increases only to a finite limit as maturity is increased.
2. The return-to-maturity expectations hypothesis states that the return generated by hold- ing a bond for term t to T will equal the expected return generated by continually rolling over a bond whose term is a period evenly divisible into T − t. Explain why the above relationship can be expressed formally as
1 B(t,T)
where B(t, T ) is the time-t price of a discount bond maturing at T and rt is the one-period spot rate at time t. The operator Et indicates that expectation is taken based on the information available at the current time t.
Remark
=E[(1+r)(1+r ̃ )···(1+r ̃ )], t t t+1 T−1
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Suppose the investor starts with one dollar at time t and invests in a discount bond maturing one year later the “deterministic” return is
1+rt = 1 , B(t, t + 1)
where B(t, t+1) is known at time t. At time t+1, the investor uses the proceed 1 B(t, t + 1)
to invest in a discount bond maturing one year later. The bond price is B ̃(t + 1, t + 2), which is not known at time t. The return over [t + 1,t + 2] is
1+r ̃= 1 ,
t+1
B ̃(t + 1, t + 2)
where “tilde” quantities represent stochastic quantities. At time t + 2, the investor again
invests in 1 units of discount bond maturing one year later. After B(t, t + 1)B ̃(t + 1, t + 2)
T − t years, the random return at time T is
(1+r)(1+r ̃ )···(1+r ̃ )= 1 .
t t+1 T−1
B(t,t + 1)B ̃(t + 1,t + 2)···B ̃(T − 1,T) This strategy is like investing in a money market account with annual rolling over.
3. Let Bt be the time-t value of the bond maturity T years later and paying annual coupon amount c. Let i be the interest rate, assuming to be constant throughout the life of the
bond. Recall that
Show that
c[ 1 ] BT
Bt = i 1− (1+i)T + (1+i)T .
∆Bt =Bt+1−Bt =i− c.
Bt Bt Bt
In the continuous time limit, show that the governing equation for the bond value function
B(t) is given by
dB(t) = i(t)B(t) − c(t), t < T, dt
with the terminal condition: B(T) = BT. There are two factors that affect the bond value B(t), one is the instantaneous short rate i(t) and the other is the coupon rate c(t). The second term on the right hand side exhibits a negative sign in the coupon effect since the bond value decreases in value after paying a coupon amount c(t)dt over (t, t + dt). Deduce the closed form solution for B(t).
4. Show that all curves rH = rH(i) for various horizons H(H = 1,2,...,∞) go through the point (i0, i0). In other words, show that (i0, i0) is a fixed point for all curves rH (i).
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5. Looking at the figure below, the quadratic approximation curve of the bond’s value lies between the bond’s value curve and the tangent line to the left of the tangency point and outside (above) these lines to the right of the tangency point. The fact that a quadratic (and hence “better”) approximation behaves like this is not intuitive: we would tend to think that a “better approximation” would always lie between the exact value curve and its linear approximation. How can you explain this apparently non-intuitive result?
Figure 1: *
Linear and quadratic approximations of a bond’s value.
6. Consider the following two bonds:
Maturity Coupon rate Par value
Bond A 15 years 10% $1000
Bond B 11 years 5% $1000
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(a) The current yield to maturity is taken to be 12%. Determine the convexity of each bond.
(b) Suppose you have a defensive strategy, and that you want to immunize the investor. What is each bond’s rate of return at horizon H = D if interest rates keep jumping from 12% to either 10% or 14%?
(c) By examining the rates of return of the two bonds under an increase or decrease of interest rates, and different choices of horizon, which bond would you choose?
7. An investor is considering the purchase of 10-year U.S. Treasury bonds and plans to hold them to maturity. Federal taxes on coupons must be paid during the year they are received, and tax must also be paid on the capital gain realized at maturity (defined as the difference between face value and original price). This investor’s federal tax bracket rate is r = 30%, as it is for most individuals. There are two bonds with par 100 that meet the investor’s requirements. Bond 1 is a 10-year, 10% bond with a price (in decimal form) of P1 = 92.21. Bond 2 is a 10-year, 7% bond with a price of P2 = 75.84. Based on the price information contained in these two bonds, the investor would like to compute the theoretical price of a hypothetical 10-year zero-coupon bond that had no coupon payments and required tax payment only at maturity equal in amount to 30% of the realized capital gain (the par value minus the original price). This theoretical price should be such that the price of this bond and those of bonds 1 and 2 are mutually consistent (in mathematical term, they are equal in value) on an after-tax basis. Find this theoretical price, and show that it does not depend on the tax rate t. Assume all cash flows occur at the end of each year.
8. Orange County managed an investment pool into which several municipalities made short- term investments. A total of $7.5 billion was invested in this pool, and this money was used to purchase securities. Using these securities as collateral, the pool borrowed $12.5 billion from Wall Street brokerages, and these funds were used to purchase additional se- curities. The $20 billion total was invested primarily in long-term fixed-income securities to obtain a higher yield than the short-term alternatives. Furthermore, as interest rates slowly declined, as they did in 1992-1994, an even greater return was obtained. Things fell apart in 1994, when interest rates rose sharply.
Hypothetically, assume that initially the duration of the invested portfolio was 10 years, the short-term rate was 6%, the average coupon interest on the portfolio was 8.5% of face value, the cost of Wall Street money was 7%, and short-term interest rates were falling at 21% per year.
(a) What was the rate of return that the pool investors obtained during this early period? Does it compare favorably with 6% that these investors would have obtained by investing normally in short-term securities?
(b) When interest rates had fallen two percentage points and began increasing at 2% per year, what rate of return was obtained by the pool?
Additional assumptions made in the calculations:
(a) Assume the bond portfolio is restructured annually to maintain a duration of 10 years.
(b) Assume the value of money borrowed is maintained at $12.5 billion every year.
(c) Assume Orange County makes interest on borrowed fund at the rate which prevailed at the beginning of the given year.
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Hints
• In the first year, the coupon rate was 8.5%. With a duration of 10 years, the change in portfolio value due to change in interest rate is given by
−duration · P · change in i. 1+i
The coupon rate dropped by 0.5% per year until down to 6.5%. This is the coupon rate applied on the bond fund in the fifth year. The interest cost of borrowing of 12.5 billion per annum with cost of fund 7% is given by 12.5×(0.07).
• At the beginning of the fifth year, the coupon rate became 6.5%. The change in cost of fund is 2%, so the cost of fund applicable for the whole fifth year is 5%. The cost of fund then increased by 2% to 7% at the end of the fifth year (applicable for the whole 6th year).
9. The
risk-free bond is 50 basis points. The recovery rate is 30%. Estimate the average default intensity per year over the 3-year period. Next, suppose that the spread between the yield on a 5-year bond issued by the same company and the yield on a similar risk-free bond is 60 basis points. Assume the same recovery rate of 30%. Estimate the average default intensity per year over the 5-year period. What do your results indicate about the average default intensity in years 4 and 5?
10. A company has issued 3-year and 5-year bonds with a coupon of 4% per annum payable annually. The yields on the bonds (expressed with continuous compounding) are 4.5% and 4.75%, respectively. Risk-free rates are 3.5% with continuous compounding for all maturities. The recovery rate is 40%. Defaults can take place halfway through each year. The default rates per year are Q1 for years 1 to 3 and Q2 for years 4 and 5. Estimate Q1 and Q2.
11. Suppose that the risk-free zero-curve is flat at 6% per annum with continuous compound- ing. Consider a four-year plain vanilla credit default swap with annual payments on an underlying risky bond. Suppose that the recovery rate is 20% and the compensation payment is (1− recovery rate) times notional. The forward probabilities of default of the bond during the first year, the second year, the third year, and the fourth year are assumed to be 0.01, 0.015, 0.02 and 0.025, respectively. Assume that the credit premium is paid by the Protection Seller at the end of each year (if the bond survives until then), and accrual premium from the last premium payment date to the time of default is paid when the bond defaults. If default does occur, it would take place either in mid-year or the end of the year. What is the credit default swap spread? What would be the credit default spread if the instrument were a binary credit default swap?
Hint :
The probability of survival until the end of the second year = 100% − (1% + 1.5%) = 97.5%, and the probability of survival until Year 1.5 is 100% − (1.0% + 0.5 × 1.5%) = 98.25%, and similar calculations for other survival probabilities can be performed. There are 4 swap premium payments if the bond survives throughout the life of the CDS. However, there are 8 possible dates at which the bond may default.
spread between the yield on a 3-year corporate bond and the yield on a similar
12. Suppose that:
(a) The yield on a 5-year risk-free bond is 7%.
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(b) The yield on a 5-year corporate bond issued by company X is 9.5%.
(c) A 5-year credit default swap providing insurance against the default of company X
costs premium rate of 150 basis points per year.
What arbitrage opportunity is there in this situation? What arbitrage opportunity would there be if the credit default spread were 300 basis points instead of 150 basis points? Give two reasons why arbitrage opportunities such as those you identify may not be extracted fully by an arbitrageur.
13. Both the Barings’ fall and Daiwa’s huge loss involve rouge traders (Nicholas Leeson and Toshihide Igushi, respectively). Search the relevant web sites to obtain the basic facts about these two cases of poor risk management. Comment on the similarities and differ- ences in these two cases in terms of (i) lack of controls within the institutions, (ii) how the market events triggered these huge losses.
14. A bank’s position in options on the dollar-euro exchange rate has a delta of 30,000 and a gamma of −80,000. Explain how these numbers can be interpreted. The exchange rate (dollars per euro) is 0.90. What position would you take to make the position delta neutral? After a short period of time, the exchange rate moves to 0.93. Estimate the new delta. What additional trade is necessary to keep the position delta neutral? Assuming the bank did set up a delta-neutral position originally, has it gained or lost money from the exchange-rate movement?
15. A non-dividend-paying stock has a current price of $100 per share. You have just sold a six-month European call option contract on 100 shares of this stock at a strike price of $101 per share. You want to implement a dynamic delta hedging scheme to hedge the risk of having sold the option. The option has a delta of 0.50. You believe that delta would fall to 0.44 if the stock price falls to $99 per share. Identify what action you should take now (i.e., when you have just written the option contract) to make your position delta neutral. After the option is written, if the stock price falls to $99 per share, identify what action should be taken at that time (after initiation time) to rebalance your delta-hedged position.
16. Suppose an existing short option position is delta-neutral, but has a gamma of −600. Also assume that there exists a traded option with a delta of 0.75 and a gamma of 1.50. In order to maintain the position gamma-neutral and delta-neutral, which of the following is the appropriate hedging strategy to implement?
(a) Buy 400 options and sell 300 shares of the underlying asset. (b) Buy 300 options and sell 400 shares of the underlying asset. (c) Sell 400 options and buy 300 shares of the underlying asset. (d) Sell 300 options and buy 400 shares of the underlying asset.
17. A market risk manager at Marcat Securities, is analyzing the risk of its S&P 500 index options trading desk. His risk report shows the desk is net long gamma and short vega. Which of the following portfolios of options shows exposures consistent with this report?
(a) The desk has substantial long-expiry long call positions and substantial short-expiry short put positions.
(b) The desk has substantial long-expiry long put positions and substantial long-expiry short call positions.
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(c) The desk has substantial long-expiry long call positions and substantial short-expiry short call positions.
(d) The desk has substantial short-expiry long call positions and substantial long-expiry short call positions.
18. You are implementing a portfolio insurance strategy using index futures designed to pro- tect the value of a portfolio of stocks. Expecting the value of your stock portfolio to decrease, which trade would you make to protect your portfolio?
19. A bronze producer will sell 1,000 mt (metric tons) of bronze in three months at the prevailing market price at that time. The standard deviation of the price of bronze over a three-month period is 2.6%. The company decides to use three-month futures on copper to hedge. The copper futures contract is for 25 mt of copper. The standard deviation of the futures price is 3.2%. The correlation between three-month changes in the futures price and the price of bronze is 0.77. To hedge its price exposure, how many futures contracts should the company buy/sell?
(a) Sell 38 futures (b) Buy 25 futures (c) Buy 63 futures
(d) Sell 25 futures
20. Suppose that each of two investments has a 0.9% chance of a loss of $10 million, a 99.1% of a loss of $1, and 0% chance of a profit. The investments are independent of each other.
(a) What is the VaR for one of the investments when the confidence level is 99%?
(b) What is the expected shortfall for one of the investments when the confidence level is 99%?
(c) What is the VaR for a portfolio consisting of the two investments when the confidence level is 99%?
(d) What is the expected shortfall for a portfolio consisting of the two investments when the confidence level is 99%?
(e) Show that in this example VaR does not satisfy the subadditivity condition whereas expected shortfall does.
(f) What is the difference between expected shortfall and VaR? What is the theoretical advantage of expected shortfall over VaR?
21. Suppose that daily gains and losses are normally distributed with standard deviation of $5 million.
(a) Estimate the minimum regulatory capital the bank is required to hold (assume a multiplicative factor of 4.0).
(b) Estimate the economic capital using a one-year time horizon and a 99.9% confidence limit, assuming that there is a correlation of 0.05 between gains or losses on successive days.
22. An option on the Bovespa stock index is struck on 3, 000 Brazilian Real (BRL). The delta of the option is 0.6, and the annual volatility of the index is 24%. Using delta-normal assumptions, what is the 10-day VaR at the 95% confidence level? Assume 260 days per year.
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23. Suppose that the economic capital estimate for two business units are as follows:
Type of risk
Business Unit Business Unit
12 Market risk 10 50 Credit risk 30 30 Operational risk 50 10
The correlation between market risk and credit risk in the same business unit is 0.3. The correlation between credit risk in one business unit and credit risk in the other is 0.7. The correlation between market risk in one business unit and market risk in the other is 0.2. All other correlations are zero. Calculate the total economic capital. How much should be allocated to each business unit?
24. Suppose we choose the mixture distribution in the Bernuolli mixture model to be the beta distribution whose density function is given by
1
f(p)=β(a,b)p (1−p) , a,b>0, 0
This models the arrival of default of firm i due either to its idiosyncratic shock or the
macro-economic shock. The survival function of firm i is given by Si(t) = P[τi > t] = e−(λi+λ)t, i = 1,2,…,m.
(a) Find the joint survival function as defined by
S(t1,t2,…,tm) = P[τ1 > t1,τ2 > t2,…,τm > tm].
(b) The exponential survival copula associated with the survival function S(t1, t2, . . . , tm) can be found via
Cτ(u ,u ,…,u ) = S(S−1(u ),S−1(u ),…,S−1(u )). 12m1122mm
Fixing some i, j ∈ {1, 2, . . . , m} with i ̸= j, show that the two-dimensional marginal copula is given by
Cτ(ui,uj) = Cτ(1,…,1,ui,1,…,1,uj,1,…,1) = min(uj u1−θi , ui u1−θj ).
Find θi and θj in terms of λi, λj and λ. 10
ij