Question

Calculus is needed.

Perpetuity X has level payments of $220 at the end of each year. Perpetuity Y also has end-of-year payments but they begin at $11 and increase by $11 each year. Find the rate of interest which will make the difference in present values between these two perpetuities a maximum. (Round your answer to two decimal places.)

Answer 1

Getting 220 each year starting at the end of year 1 generates a present value with first term 220/(1+i) and ratio 1/(1+i), which generates sum a/(1-r) (a the first term, r the ratio) =

220/(1+I)/(1-1/(1+i)) = 220/i

Getting 11 the first year and then increasing by 11 each year is equivalent to getting 11 each year, then 11 each year starting the second year, then 11 each year starting the third year ...

Thus, this is 11/i with ratio 1/(1+i) = 11/i/(1 - 1/(1+i)) = 11(1+i)/i^2

Thus, we maximize 220/i - 11(1+i)/i^2

The first derivative is -220/i^2 - 11/i^2 + 22(1+i)/i^3 = 1/i^3(22 - 209i)

Then, this is 0 when 22 - 209i = 0, or i = 22/209 = 0.105263 = 10.5263%

As f'(x) = 1/i^3(22 - 209i) = 22/i^3 - 209/i^2, f''(x) = -66/i^4 + 418/i^3 = 1/i^4(418i - 66)

As i = 0.1052, (418i - 66) = -22,

so f''(x) = -22/i^4 < 0, so this is a maximum.