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I've been working at this question for some time now and I'm quite stuck. Some help would be greatly appreciated. I can figure out recurring payments by themselves, but I'm drawing a blank when it comes to annuities and varying interest rates.

A woman has reached her retirement age of 65 on October 15, 2015. She invests $300,000 and buys an annuity with monthly payments, first payment due on November 15, 2015 and the final payment due on July 15, 2039. What size monthly payment does she receive if the interest rate is j(12) = 6% for the 1st 5 years and j(12) = 3.9% thereafter?

Again, thanks for your help.

closed as off-topic by Pete B., Dheer, Dilip Sarwate, Nathan L, Michael Oct 28 '17 at 17:59

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Taking first a simple case, based on the example here: Calculating the Present Value of an Ordinary Annuity.

enter image description here

If the first two periods had interest rate 10% the calculation would be

pv = 1000 (1/1.1^1 +
           1/1.1^2 +
           1/(1.1^2*1.05^1) +
           1/(1.1^2*1.05^2) +
           1/(1.1^2*1.05^3)) = 3986.16

or

enter image description here

where

m = 2
n = 5 - m
c = 1000
r1 = 0.10
r2 = 0.05

By induction

enter image description here

Check

pv = ((1 + r1)^-m (1 + r2)^-n (-c r1 + 
     c (1 + r2)^n (r1 + (-1 + (1 + r1)^m) r2)))/(r1 r2) = 3986.16

Now applying this formula to the OP's case. Oct 2015 to July 2039 is 285 months.

m = 60
n = 285 - m = 225
r1 = 0.06/12
r2 = 0.039/12
pv = 300000

pv = ((1 + r1)^-m (1 + r2)^-n (-c r1 + 
     c (1 + r2)^n (r1 + (-1 + (1 + r1)^m) r2)))/(r1 r2)

∴ c = (pv r1 (1 + r1)^m r2 (1 + r2)^n)/(-r1 +
      (1 + r2)^n (r1 + (-1 + (1 + r1)^m) r2))

∴ c = 1765.57

Assuming the interest rates are nominal compounded monthly, the monthly payment is $1765.57

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Another way of looking at the transaction to arrive at the correct answer of @Chris Degnen:

Consider that the original investment of $300,000 is divided into two amounts.

The first, A1, is used to fund a five-year ordinary annuity, making monthly payments of R, at an interest rate of 6% per year, compounded monthly.

So:

A1 = R x (1-1.005^-60) / 0.005 = 51.72556075 x R

The balance of the $300000, A2, is left to grow for 60 periods at 0.5% per period, so that it becomes A3:

A3 = A2 x 1.005^60 = 1.348850153 x A2

This amount A3 is then used to fund a 225 month ordinary annuity of R per month at 3.9% compounded monthly:

A3 = R x (1-1.00325^-225) / 0.00325 = 159.4221506 x R

So, from these two results:

A2 = A3 / 1.348850153 = 159.4221506 x R / 1.348850153 = 118.1911499 x R

So:

A1 + A2 = 300000 = 51.72556075 x R + 118.1911499 x R = 169.9167107 x R

So that:

R = 300000/169.9167107 = 1765.5709

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