Finding monthly payment for ordinary simple annuity with varying interest rates?
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 0,000 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.
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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 0,000 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 0000, 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
Taking first a simple case, based on the example here: Calculating the Present Value of an Ordinary Annuity.
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
where
m = 2
n = 5 - m
c = 1000
r1 = 0.10
r2 = 0.05
By induction
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 65.57
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