Question

1 a) The present value of an annuity lasting 72 interest periods

and paying $ 2,000 at the beginning of each “k” interest periods is

$ 9,538.60. Given that the effective interest rate per interest period

is 3.6575%, find k.

1 b) Find an expression in terms of i⁽⁴⁾ for the accumulated value at the

end of the twenty-one years of an annuity that pays $ 400 at the

beginning of each four-month period for 21 years. Your

expression should NOT be a sum of terms. Your answer should be a

nice formula.

1 c) Jason purchases a deferred perpetuity for $ 27,040. The

perpetuity has quarterly payments of $ 1,500. Express the waiting

time until the first payment as a function of the annual effective

interest rate i.

Answer 1

You have asked multiple unrelated questions in the same post. I have addressed the first one. Please post the balance questions separately.

r = interest rate per period = 3.6575%

R = interest rate per k interest periods

Let's call k interest period as one relevant period

Hence, (1 + R) = (1 + r)^k

And number of relevant periods, N = n / k = 72 / k

Hence, PV of period end annuity A

If the payments are made at the beginning of the period, then PV of period beginning annuity A

We have (1 + R) = (1 + r)^k

Hence, (1 + R)^-N = [(1 + r)^k]^-N

We have N = n / k = 72 / k

Hence, (1 + R)^-N = [(1 + r)^k]^-72/k = (1 + r)^-72

Hence, PV = 9,538.60

Hence, (1 + R) / R = 9,538.60 / 1,839.42 = 5.19

Hence, R = 1 / (5.19 - 1) = 0.2389

Hence, 1 + R = 1.2389 = (1 + r)^k = (1 + 0.036575)^k = 1.036575^k

Hence, k = ln(1.2389) / ln (1.036575) = 5.96 = 6

Hence, the desired value of k = 6