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answer the whole question QUESTION 1 1. As an investment manager working for the...
May 17, 2024
answer the whole question
Solution by Steps
step 1
Calculate the duration of the 2-year bond using the Macaulay duration formula: D=1.5%×1+1.5%×(1+2.25%)−1×2(1+2.25%)−1×1.5%+(1+2.25%)−2×1.5% D = \frac{1.5\% \times 1 + 1.5\% \times (1 + 2.25\%)^{-1} \times 2}{(1 + 2.25\%)^{-1} \times 1.5\% + (1 + 2.25\%)^{-2} \times 1.5\%}
step 2
Calculate the duration of the 3.5-year bond using the Macaulay duration formula: D=2.5%×1+2.5%×(1+2.25%)−1×2+2.5%×(1+2.25%)−2×3.5(1+2.25%)−1×2.5%+(1+2.25%)−2×2.5%+(1+2.25%)−3.5×2.5% D = \frac{2.5\% \times 1 + 2.5\% \times (1 + 2.25\%)^{-1} \times 2 + 2.5\% \times (1 + 2.25\%)^{-2} \times 3.5}{(1 + 2.25\%)^{-1} \times 2.5\% + (1 + 2.25\%)^{-2} \times 2.5\% + (1 + 2.25\%)^{-3.5} \times 2.5\%}
step 3
Calculate the market value of each bond: Market Value of 2-year bond=$120 million \text{Market Value of 2-year bond} = \$120 \text{ million} Market Value of 3.5-year bond=$150 million \text{Market Value of 3.5-year bond} = \$150 \text{ million}
step 4
Calculate the portfolio duration using the weighted average duration formula: Dp=120×D2-year+150×D3.5-year120+150 D_p = \frac{120 \times D_{2\text{-year}} + 150 \times D_{3.5\text{-year}}}{120 + 150}
step 5
Compare the calculated portfolio duration with the staff's estimate of 5 years
Answer
The staff's estimate of 5 years for portfolio duration is incorrect.
Part (b)
step 1
Set up the equation for the portfolio duration to match the pension liability duration of 3 years: 3=w1×D2-year+w2×D3.5-year 3 = w_1 \times D_{2\text{-year}} + w_2 \times D_{3.5\text{-year}} where w1+w2=1 w_1 + w_2 = 1
step 2
Solve for w1 w_1 and w2 w_2 : w1=3−D3.5-yearD2-year−D3.5-year w_1 = \frac{3 - D_{3.5\text{-year}}}{D_{2\text{-year}} - D_{3.5\text{-year}}} w2=1−w1 w_2 = 1 - w_1
Answer
The proportions of the portfolio should be w1 w_1 in 2-year bonds and w2 w_2 in 3.5-year bonds to achieve a portfolio duration of 3 years.
Part (c)
step 1
Calculate the total market value of the portfolio: Total Market Value=$120 million+$150 million=$270 million \text{Total Market Value} = \$120 \text{ million} + \$150 \text{ million} = \$270 \text{ million}
step 2
Compare the total market value with the present value of the pension liability (\$180 million)
step 3
If there is excess money, calculate the adjustment needed: Excess=$270 million−$180 million=$90 million \text{Excess} = \$270 \text{ million} - \$180 \text{ million} = \$90 \text{ million} Adjust investments in the two bonds proportionally to maintain the duration of 3 years. \text{Adjust investments in the two bonds proportionally to maintain the duration of 3 years.}
Answer
There is too much money invested in the bond portfolio. Adjust the investments to maintain the duration of 3 years and reinvest the excess \$90 million more profitably.
Key Concept
Duration matching and portfolio adjustment
Explanation
The key concept is to match the duration of the bond portfolio with the pension liability to minimize interest rate risk.
answer the whole question
Solution by Steps
step 1
To hedge the interest-rate risk using Eurodollar futures contracts, you would take a short position in the futures contracts. This means you would sell Eurodollar futures contracts to lock in the borrowing rate
step 2
If you decide to use an FRA (Forward Rate Agreement) instead, you would enter into a long 6×96 \times 9 FRA. This means you agree to borrow at a fixed rate starting in six months for a period of three months
Answer
Short 10 Eurodollar futures contracts or Long 6×96 \times 9 FRA
Part (a)(ii)
step 1
The number of Eurodollar futures contracts needed can be calculated using the formula: N=L×(T/360)P×Δ N = \frac{L \times (T/360)}{P \times \Delta} where L L is the principal amount, T T is the duration in days, P P is the price of the futures contract, and Δ \Delta is the change in the futures price per basis point change in interest rate
step 2
Given: - Principal L=$10,000,000 L = \$10,000,000 - Duration T=90 T = 90 days - Price P=1,000,000 P = 1,000,000 - Change in futures price Δ=0.01 \Delta = 0.01 The number of contracts needed is: N=10,000,000×(90/360)1,000,000×0.01=9.901 N = \frac{10,000,000 \times (90/360)}{1,000,000 \times 0.01} = 9.901
Answer
9.901 contracts
Part (a)(iii)
step 1
For the FRA, the payoff is calculated as: Payoff=L×(RLIBOR−RFRA1+RLIBOR×(T/360)) \text{Payoff} = L \times \left( \frac{R_{\text{LIBOR}} - R_{\text{FRA}}}{1 + R_{\text{LIBOR}} \times (T/360)} \right) where RLIBOR R_{\text{LIBOR}} is the LIBOR rate at maturity, RFRA R_{\text{FRA}} is the locked-in rate, and T T is the duration in days
step 2
For the Eurodollar futures, the payoff is: Payoff=L×(RLIBOR−RFutures)×(T/360) \text{Payoff} = L \times (R_{\text{LIBOR}} - R_{\text{Futures}}) \times (T/360) where RFutures R_{\text{Futures}} is the locked-in rate
step 3
Given: - RLIBOR=5% R_{\text{LIBOR}} = 5\% - RFRA=4% R_{\text{FRA}} = 4\% - T=90 T = 90 days For FRA: Payoff=10,000,000×(0.05−0.041+0.05×(90/360))=24,691 \text{Payoff} = 10,000,000 \times \left( \frac{0.05 - 0.04}{1 + 0.05 \times (90/360)} \right) = 24,691 For Eurodollar futures: Payoff=10,000,000×(0.05−0.04)×(90/360)=24,753 \text{Payoff} = 10,000,000 \times (0.05 - 0.04) \times (90/360) = 24,753
step 4
Given: - RLIBOR=3% R_{\text{LIBOR}} = 3\% For FRA: Payoff=10,000,000×(0.03−0.041+0.03×(90/360))=−24,814 \text{Payoff} = 10,000,000 \times \left( \frac{0.03 - 0.04}{1 + 0.03 \times (90/360)} \right) = -24,814 For Eurodollar futures: Payoff=10,000,000×(0.03−0.04)×(90/360)=−24,753 \text{Payoff} = 10,000,000 \times (0.03 - 0.04) \times (90/360) = -24,753
Answer
LIBOR = 5%: FRA = 24,691; Eurodollar futures = 24,753
LIBOR = 3%: FRA = -24,814; Eurodollar futures = -24,753
Part (b)
step 1
Duration-based hedging involves matching the duration of the hedging instrument with the duration of the exposure. The hedge ratio is calculated to minimize the interest rate risk by equating the weighted average duration of the assets and liabilities
step 2
The computation of the hedge ratio in duration-based hedging differs from the minimum-variance hedge computation. In duration-based hedging, the focus is on matching durations, while in minimum-variance hedging, the focus is on minimizing the variance of the portfolio's value
Answer
Duration-based hedging involves matching durations, while minimum-variance hedging focuses on minimizing portfolio variance.
Key Concept
Hedging strategies using Eurodollar futures and FRA
Explanation
The key concept involves understanding how to hedge interest-rate risk using different financial instruments and calculating the necessary contracts and payoffs.
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