VIPSolutions‘The current prices of zero-coupon bonds are in the following table. If the Expectations Hypothesis holds, what is the market's expectation of 2-year spot rate in 3 years from today, Eq[R(3,5)]2 mani ci PY Maturity Date Price 1 year 096 2years 091 dyears 085 dyears 079 Syears 072 In order to hedge their interest rate risk for borrowing beginning 3 months from now, ExxonMobil’s treasury department entered into a 3x5 Forward Rate Agreement (FRA) with Barclays with a notional of $100mm where LIBOR was the reference rate and the forward rate agreement rate was 7%. If in 3 months, 2-month LIBOR decreased to 4.5%, what transfer would take place at the settlement date? I»
Question:
‘The current prices of zero-coupon bonds are in the following table. If the Expectations
Hypothesis holds, what is the market's expectation of 2-year spot rate in 3 years from
today, Eq[R(3,5)]2
mani ci PY
Maturity Date Price
1 year 096
2years 091
dyears 085
dyears 079
Syears 072
In order to hedge their interest rate risk for borrowing beginning 3 months from now,
ExxonMobil’s treasury department entered into a 3x5 Forward Rate Agreement (FRA)
with Barclays with a notional of $100mm where LIBOR was the reference rate and the
forward rate agreement rate was 7%. If in 3 months, 2-month LIBOR decreased to 4.5%,
what transfer would take place at the settlement date? I»
This is AI generated Response.
Asked by: Sampangi Swetha
Created at: 2025-06-28 10:02:01
Asked by: Sampangi Swetha
Created at: 2025-06-28 10:02:01
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---
### **Question 1: Expectation of 2-year spot rate in 3 years**
#### **Given Table:**
| Maturity Date | Price |
|:-------------:|:-----:|
| 1 year | 0.96 |
| 2 years | 0.91 |
| 3 years | 0.85 |
| 4 years | 0.79 |
| 5 years | 0.72 |
We are to find the market's expectation of the **2-year spot rate in 3 years** (i.e., the forward rate from year 3 to year 5).
#### **Step 1: Find the forward rate \( f_{3,5} \)**
The formula for the forward rate from \( t_1 \) to \( t_2 \) is:
\[
f_{t_1, t_2} = \left( \frac{P(0,t_1)}{P(0,t_2)} \right)^{1/(t_2-t_1)} - 1
\]
Where \( P(0,t) \) is the price of a zero-coupon bond maturing at \( t \) years.
So for 3 to 5 years:
\[
f_{3,5} = \left( \frac{P(0,3)}{P(0,5)} \right)^{1/2} - 1
\]
Given:
- \( P(0,3) = 0.85 \)
- \( P(0,5) = 0.72 \)
\[
f_{3,5} = \left( \frac{0.85}{0.72} \right)^{1/2} - 1
\]
\[
f_{3,5} = (1.1806)^{0.5} - 1
\]
\[
f_{3,5} = 1.0865 - 1 = 0.0865 \text{ or } 8.65\%
\]
**Final Answer:**
\[
\boxed{8.65\%}
\]
---
### **Question 2: 3x5 FRA Settlement Value**
#### **Given:**
- Notional: $100,000,000
- FRA Rate (\( K \)): 7%
- Actual 2-month LIBOR after 3 months (\( R \)): 4.5%
- FRA Period: 2 months (from 3 months to 5 months)
- Settlement in 3 months (so discounting is needed for the settlement amount)
#### **FRA Settlement Formula:**
The value of the FRA to the borrower at settlement (time \( t \)) is:
\[
\text{Settlement Amount} = \frac{(R - K) \times \text{Notional} \times \frac{\Delta t}{360}}{1 + R \times \frac{\Delta t}{360}}
\]
Where:
- \( R \) = Actual LIBOR (4.5% or 0.045)
- \( K \) = FRA Rate (7% or 0.07)
- \( \Delta t \) = 60 days (2 months), or \( \frac{2}{12} = 0.1667 \) years (but conventionally 60/360 = 1/6)
Plug in the values:
\[
\text{Settlement Amount} = \frac{(0.045 - 0.07) \times 100,000,000 \times \frac{60}{360}}{1 + 0.045 \times \frac{60}{360}}
\]
\[
= \frac{(-0.025) \times 100,000,000 \times \frac{1}{6}}{1 + 0.0075}
\]
\[
= \frac{(-0.025) \times 16,666,667}{1.0075}
\]
\[
= \frac{-416,666.67}{1.0075}
\]
\[
= -413,567.16
\]
**Interpretation:**
- Since the actual LIBOR is less than the FRA rate, **ExxonMobil receives $413,567.16 from Barclays** at settlement.
---
## **Summary:**
1. **Forward 2-year spot rate in 3 years:** **8.65%**
2. **FRA settlement amount:** **ExxonMobil receives $413,567.16 from Barclays**
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