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ORBS7220/MATH4837 Risk and Portfolio Management
Assignment 2
Instructions: Layout the intermediate steps systematically. Give exact answers or round them to 4 decimals unless specified otherwise. For percentages (e.g. return and volatility), show 4 decimals after conversion to percentages, e.g. ^σ = 12.3456%, not 0.1235.
1. [20 marks] A three-year bond with a yield of 6% p.a. (continuously compounded) paysa 4% coupon at the end of each year. The face value is $100. (a) What is the bond's yield duration?
(b) Use the duration to estimate the bond's price if the yield decreases 0.1%.
(c) Recalculate the bond's price on the basis of a 5.9% per annum yield (continuously compounded). What is the error of the prediction in part (b)?
(d) Suppose that a second bond with a market price of $105 and a duration of 2.5 is used to hedge against interest rate risk. How much face value of the second bond should one short for each $100 face value of the first bond?
2. [30 marks] The values of a stock index over four days are given in the table below.
Day |
Index |
0 |
20436 |
1 |
20794 |
2 |
21059 |
3 |
20751 |
The volatility on Day 2 is estimated with ^(σ)2 = ju1 j .
(a) Estimate the daily volatility ^σ3 on Day 3 using the EWMA model with λ = 0.95.
(b) Estimate the daily volatility ^σ3 on Day 3 using the GARCH(1,1) model with α = 0.01, β = 0.95, γ = 0.04, and σL = 0.01.
(c) Calculate the log-likelihood for part (b).
(d) Estimate the daily volatility ^σ3 on Day 3 using the GARCH(1,1) model with α = 0.02, β = 0.96, γ = 0.02, and σL = 0.02.
(e) Calculate the log-likelihood for part (d).
(f) Which set of parameters, part (b) or (d), is better?
3. [20 marks] A bank has two $10m one-year loans
Outcome |
Probability |
Neither loan defaults |
95% |
Loan 1 defaults, Loan 2 does not default |
2.5% |
Loan 2 defaults, Loan 1 does not default |
2.5% |
Both loans default |
0% |
(a) Calculate the one-year 97% VaR for the portfolio of two loans.
(b) Calculate the one-year 97% ES for the portfolio of two loans.
4. [30 marks] Now is the end of Day 3. The stock price, daily return, and daily volatility over 4 days are depicted in the table. An investor has purchased 1000 shares.
Day |
Stock price |
daily return |
daily volatility |
0 |
5.0 |
— |
— |
1 |
4.8 |
-4.0000% |
0.1% |
2 |
4.6 |
-4.1667% |
0.4% |
3 |
5.1 |
10.8698% |
0.2% |
4 |
|
|
0.2% |
(a) Use historical simulation to estimate the volatility-adjusted daily losses (under Scenario 1 to 3).
(b) Using the results in part (a), calculate the one-day 50% VaR on Day 4.
(c) Using the results in part (a), calculate the one-day 50% ES on Day 4.