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Answer: 6.950%
For a par swap, the present value of the net cash flows must be zero. ### Known present values From the table, the present values of the first three net cash flows are: - `-0.693` - `-0.488` - `-0.191` Sum: \[ -0.693 - 0.488 - 0.191 = -1.372 \] So the final cash flow at 2.0 years must have a present value of `+1.372`. ### Discount factor at 2.0 years Using the 24-month OIS zero rate of 3.60% with continuous compounding: \[ DF_{2.0} = e^{-0.036 \times 2} = e^{-0.072} \approx 0.93053 \] ### Solve for the final forward rate Let the unknown forward LIBOR for 1.5 to 2.0 years be `x`. The final net cash flow is: \[ \frac{x - 4.00\%}{2} \times 100 = 50(x - 0.04) \] Set its present value equal to 1.372: \[ 50(x - 0.04) \times 0.93053 = 1.372 \] \[ x - 0.04 \approx 0.02948 \] \[ x \approx 0.06948 = 6.95\% \] **Correct answer: D**
Author: Manit Arora
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Q-722.3. Suppose that the 6-month, 12-month, 18-month, and 24-month overnight indexed swap (OIS) zero rates with continuous compounding are 2.00%, 2.40%, 3.00%, and 3.60%, respectively. Suppose further that the six-month LIBOR rate is 2.60% with semi-annual compounding. The forward LIBOR rate for the period between 6 and 12 months is 3.00%, with semi-annual compounding. The forward LIBOR rate for the period between 12 and 18 months is 3.60%, with semi-annual compounding. (Please Note: this question is inspired by Hull's Example 7.2 in 10th Edition).
| Par | $100.00 |
|---|---|
| 2-year swap rate | 4.00% |
| Period | OIS Zero rates (CC) | Forward LIBOR (s.a.) | Cash Flow | |
|---|---|---|---|---|
| FV | PV | |||
| 0.50 | 2.00% | 2.600% | ($0.700) | ($0.693) |
| 1.00 | 2.40% | 3.000% | ($0.500) | ($0.488) |
| 1.50 | 3.00% | 3.600% | ($0.200) | ($0.191) |
| 2.00 | 3.60% | ??? |
Finally and importantly, assume the two-year swap rate is 4.00%. Conditional on the realization of the LIBOR forward rates, the future cash flow in six months is, therefore (2.60\% - 4.00\%)/2 * \`100.0 = -\ and its present value is about -\`0.70 * \exp(-0.020 * 0.50) = -\; that is, we are using the OIS zero rates as the risk-free rate for discounting purposes.
Which is nearest to an estimate for the forward LIBOR rate for the 18- to 24-month period, F(1.5, 2.0)?
A
3.880%
B
4.503%
C
5.747%
D
6.950%
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