What continuously compounded effective rate of interest will yield the same present value of a future cash flow as an annual rate of interest of 8.25%?
A) 8.60%
B) 7.93%
C) 8.25%
D) 7.70%

Question

Quizplus · Accepted Answer

To find the continuously compounded effective rate of interest, we can use the formula:
e^(r*t) = 1 + i
where r is the continuously compounded effective rate of interest, i is the annual rate of interest, and t is the time period.
Let's assume the time period is 1 year for simplicity.
For an annual rate of interest of 8.25%, we have:
e^(r*1) = 1 + 0.0825
e^r = 1.0825
r = ln(1.0825) = 0.0803 or 8.03%
Therefore, the continuously compounded effective rate of interest is 8.03%.
However, we need to find the rate that yields the same present value of a future cash flow as an annual rate of interest of 8.25%. This means that we need to solve for r in the following equation:
PV = FV/(1+i)^n = FV/e^(r*n)
where PV is the present value, FV is the future value, i is the annual rate of interest (8.25%), n is the number of years (1).
Let's assume PV = FV = 1 for simplicity.
For an annual rate of interest of 8.25%, we have:
PV = FV/(1+i)^n = 1/(1+0.0825)^1 = 0.9206
To find the continuously compounded effective rate of interest that yields the same present value, we solve for r:
PV = FV/e^(r*n) => 0.9206 = 1/e^(r*1) => e^r = 1.0861
r = ln(1.0861) = 0.0793 or 7.93%
Therefore, the answer is B.

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