At an effective annual interest rate 𝑖, the present value of a perpetuity paying 10 at the end of each 3 year period with first payment at the end of year 6, is 32. At the same effective annual rate a perpetuity -immediate paying 4 at the end of each 4-months period is X. Calculate X.

To solve this problem, we need to use the formula for the present value of a perpetuity. The formula is:

PV = PMT / i

where:

• PV is the present value
• PMT is the payment per period
• i is the interest rate per period

First, we need to find the interest rate i. We know that the present value of a perpetuity paying 10 at the end of each 3 year period with first payment at the end of year 6 is 32. So we can set up the equation:

32 = 10 / i

Solving for i, we get i = 10 / 32 = 0.3125 or 31.25% per 3 years.

However, we need the effective annual interest rate. To convert the 3-year interest rate to an annual interest rate, we use the formula:

(1 + i)^(1/n) - 1

where n is the number of periods per year. In this case, n = 1/3 because there is one third of a year in 3 years. So the annual interest rate is:

(1 + 0.3125)^(1/3) - 1 = 0.0954 or 9.54%

Now we can find the present value of a perpetuity-immediate paying 4 at the end of each 4-months period. There are 3 periods in a year (12 months / 4 months = 3), so the payment per period is 4 * 3 = 12. The interest rate per period is 0.0954 / 3 = 0.0318. So the present value is:

X = 12 / 0.0318 = 377.36

So the present value of a perpetuity-immediate paying 4 at the end of each 4-months period is approximately 377.36.