Perpetuities with differing payment and compounding periods in practice

actuarial-sciencefinance

I've been studying annuities and perpetuities recently and have tried to solve a simple problem using them.

I want to find the lump sum I'd need now so that I could withdraw $\$10$ each month forever. The interest rate is compounded only once per year and say the rate is $11\%$. My understanding is that you'd just find the present value of the discounted infinite stream of payments after converting the $11\%$ to an effective rate that's compounded monthly i.e. $(1+0.11)^{1/12}-1\approx 0.873\%$ which would give: $$PV=\frac{C}{j}=\frac{10}{(1+0.11)^{1/12}-1}\approx \$1144.87$$

My question is how does this work practically? For example if we had the money in an account that earned interest at the rate 11% rate per year, how does the interest work when we start taking the money out? The assumption for the perpetuity is that the first payment is due after a month so the money will have earned some interest: $\$1144.87\times (1+0.00873)$, then we'd take the $10 payment away. If the interest were actually compounded monthly then everything works out: the earned interest perfectly covers the payment, but this only works if the interest is paid back into the account each month. If it's not then the account balance dwindles down until the annual interest payment is paid in (and the interest payment we'd get is a function of the intermediate decreasing balances).

From playing around on Google Sheets and running through a year of payments that earn interest on the intermediate balances it seems we still earn enough interest for the process to continue forever, but it actually gives slightly more than $1144.87 after the interest is paid.

I understand how I could analytically solve this problem, I'm more trying to find out where my intuition for using present values/perpetuities is off. Is there something I'm misunderstanding here?

Edit: Just to clarify things following on from @Kurt G. I must have made a mistake before when I said 'more'.

To make things clearer I'll restate the problem hopefully in a better way. I want to find the account balance $P$ such that if I deposit it in an account that earns interest of 11% per year (compounded annually and paid once at the end of the year) I'll be able to take $\$C$ out at the beginning of each month forever.

One way to do this is like how you've done in your answer. If you took a year's 12 payments as a lump sum at the beginning of every year you'd need $P = 12 C/0.11+12C$. The problem with doing this is that we're not taking into account the interest that could be earnt on the $\$12C$ we take, since we only spend $\$C$ each month the rest could be earning interest. So I thought by doing it the way I did that would take that into account.

From doing it analytically by equating the interest earned on the intermediate balances and a year's worth of payments we get ($j$ is the effective monthly compounding rate, i.e. 0.873% here)
$$(P-A)j + (P-2A)j + \dots + (P-12A)j = \sum_{i=1}^{12} (P-i C)j = 12 C$$
which then gives
$$P=\frac{2C+13Cj}{2j}=\frac{C}{j}+\frac{13C}{2}$$
which is the value of an ordinary perpetuity plus another term (in general if $k$ is the number of payments we want to take a year it'd be $C/j + (k+1)C/2$).

Using $C=\$10$, $j=0.00873$, we get

  • $\$1210.909091$ using $P = 12 C/0.11+12C$.
  • $\$1209.872927$ using the formula I derived.

$\$1209.872927$ seems to work (the link to the Sheets is here) as we each year we always have the same starting balance.
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I think where I might have been going wrong is how interest works in annuities/perpetuities. I'm guessing when I converted the interest from 11% to a monthly equivalent we're then assuming that the interest gets paid out each month (at a lower rate to account for compounding) but that's not the case for this problem. So the only way to do it then would be either deriving it like above or overestimating it by lumping the payments into one, but I'm not entirely sure.

Sorry this is so long!

Best Answer

Please clarify how your annuity works. I find it hard to believe that from an amount of $\$1144.87$ (or "slightly more") one can withdraw perpetually $\$10$ every month even if the remaining amount earns interest of $11\%$ each year. The following simple python code shows that time is up after approximately 28 years. Since the interest of $11\%$ is added only once pear year we can for simplicity assume that we withraw $\$120$ once every year:

T = 100 # years
A = 1144.87
r = 0.11
for y in range(T):
    print( "{:3d} {:8.2f}".format( y, A ) )
    A -= 120
    A *= (1+r)
    if A < 0:
       print( "{:3d} {:8.2f}".format( y+1, A ) )
       break

Output:

  0  1144.87
  1  1137.61
  2  1129.54
  3  1120.59
  4  1110.66
  5  1099.63
  6  1087.39
  7  1073.80
  8  1058.72
  9  1041.98
 10  1023.40
 11  1002.77
 12   979.87
 13   954.46
 14   926.25
 15   894.94
 16   860.18
 17   821.60
 18   778.78
 19   731.24
 20   678.48
 21   619.91
 22   554.90
 23   482.74
 24   402.65
 25   313.74
 26   215.05
 27   105.50
 28   -16.09
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