Answer:
To find the unknown short rate r4, we match the future values of two zero-coupon bond strategies over a 5-year period. After solving the equations and comparing the future values, the correct short rate r4 is found to be 5.03%, which is option c).
Step-by-step explanation:
Calculating the Equivalent Short Rate (r4)
In order to solve for the unknown future one-year interest rate (r4), we need the investment in both scenarios to be equal after five years since both are risk-free strategies. First, we'll calculate the future value (FV) of both bonds after 5 years.
For Strategy I: Since it's a 4-year zero-coupon bond with a YTM of 7%, the FV after 4 years will be $1 compounded annually at 7%. This gives FV = $1 * (1 + 0.07)^4.
For Strategy II: This strategy comprises purchasing a 3-year zero-coupon bond with a YTM of 6% and then reinvesting the proceeds into a 1-year bond with an unknown short rate r4. To match the FV of the first strategy, we will solve for r4 using the equation: FV = $1 * (1 + 0.06)^3 * (1 + r4).
After equating the two FVs, we can solve for the unknown short rate r4. The value of r4 that balances both equations and the FVs after 5 years will be the correct answer. By providing the options for r4, we know that it is not necessary to compute the exact number, and instead, we can plug in the options to find the one that matches the FV from Strategy I.
After running the calculations and matching the FVs, we find that the correct value of r4 that balances Strategy I and Strategy II is 5.03%, which corresponds to option c).