An investor wants to take advantage of the 5-year spot rate, currently at a level of 4.0%. Unfortunately, the investor just invested all of his funds in a 2-year bond with a yield of 3.2%. The investor contacts his broker, who tells him that in two years he can purchase a 3-year bond and end up with the same return currently offered on the 5-year bond. What 3-year forward rate beginning two years from now will allow the investor to earn a return equivalent to the 5-year spot rate?
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A. B. C.B
To determine the 3-year forward rate beginning two years from now that will allow the investor to earn a return equivalent to the 5-year spot rate, we need to use the concept of forward rates and spot rates.
First, let's understand the terms involved:
Spot Rate: It represents the interest rate on a bond that is paid at a specific point in time in the future, known as the spot date. In this case, the 5-year spot rate is given as 4.0%.
Forward Rate: It represents the expected interest rate on a bond that will start at some future date. In this case, we need to find the 3-year forward rate beginning two years from now.
To solve this problem, we'll use the concept of the yield on a bond portfolio. The yield on the bond portfolio can be calculated as the geometric average of the spot rates over the investment horizon.
Let's calculate the yield on the bond portfolio for the 5-year spot rate:
Yield on 5-year bond = (1 + Spot Rate1) × (1 + Spot Rate2) × (1 + Spot Rate3) × (1 + Spot Rate4) × (1 + Spot Rate5) - 1
Given that the 5-year spot rate is 4.0%, we can calculate the yield on the 5-year bond:
Yield on 5-year bond = (1 + 0.04)^5 - 1 = 0.21665 or 21.665%
Next, let's determine the yield on the 2-year bond the investor currently holds. The yield on the 2-year bond is given as 3.2%. We can calculate it as:
Yield on 2-year bond = (1 + 0.032)^2 - 1 = 0.0656 or 6.56%
Now, we need to find the forward rate that will make the investor's return equivalent to the 5-year spot rate. Since the investor plans to purchase a 3-year bond in two years, we'll calculate the yield on this future bond.
Let's assume the forward rate beginning two years from now is X. Using the yield on a bond portfolio formula, we can calculate the yield on the bond portfolio for the 3-year bond:
Yield on 3-year bond = (1 + 0.032)^2 × (1 + X) - 1
The investor wants the yield on the 3-year bond to be equal to the yield on the 5-year bond, which is 21.665%. So, we have the equation:
(1 + 0.032)^2 × (1 + X) - 1 = 0.21665
Simplifying the equation:
(1.0656)^2 × (1 + X) - 1 = 0.21665
1.136717 × (1 + X) - 1 = 0.21665
1.136717 × (1 + X) = 1.21665
1 + X = 1.21665 / 1.136717
1 + X = 1.071246
X = 1.071246 - 1
X = 0.071246 or 7.1246%
Therefore, the 3-year forward rate beginning two years from now that will allow the investor to earn a return equivalent to the 5-year spot rate is approximately 7.1246%.
Answer: None of the provided options (A, B, C) is correct.
Please note that this is a fictional exam question, and the calculations provided here are for educational purposes only.