Comparing the present value of total payment today and partial payments over 3 months

Question

I'm trying to buy something. I can either pay $2495 in a lump sum payment, or $997 over 3 x monthly installments.

Let's say I can invest in the stock market and get an 8% return on my money.

What's the present value of using the payment plan?

I got $3394.83 but I'm skeptical whether that's right because the 3 payments got me a bit confused. If it was 3 payments over 3 years, it would be easier.

• First $997 payment = 2.67%
• Second payment = 5.4%
• Third payment = 8.22%

Please show how you got your answer if you can.

Answer 1

Its kind of a dumb question because no one believes that you can earn 8% in the short term in the market, but for arguments sake the math is painfully easy. Keep in mind I am an engineer not a finance guy.

So the first payment will earn you one month at 8%, the second, two. In effect three months at 8% on 997. You can do it that way because the payments are equal:

997 * (.08 /12) *3 = earnings ~= 20

So with the second method you pay:

997 * 3 - 20 = 2971

Answer 2

I got $3394.83

The first problem with this is that it is backwards. The NPV (Net Present Value) of three future payments of $997 has to be less than the nominal value. The nominal value is simple: $2991.

First step, convert the 8% annual return from the stock market to a monthly return. Everyone else assumed that the 8% is a monthly return, but that is clearly absurd. The correct way to do this would be to solve for m in

(1 + m/100)^12 = 1 + 8/100
m = 100 * (1.08)^(1/12) - 100
m = .64%

But we often approximate this by dividing 8% by 12, which would be .67%. Either way, you divide each payment by the number of months of compounding.

$997 / (1 + m/100)
$997 / (1 + m/100)^2
$997 / (1 + m/100)^3

Sum those up using m equal to about .64% (I left the calculated value in memory and used that rather than the rounded value) and you get about $2952.92 which is smaller than $2991.

Obviously $2952.92 is much larger than $2495 and you should not do this. If the three payments were $842.39 instead, then it would about break even.

Note that this neglects risk. In a three month period, the stock market is as likely to fall short of an annualized 8% return as to beat it. This would make more sense if your alternative was to pay off some of your mortgage immediately and take the payments or yp pay a lump sum now and increase future mortgage payments. Then your return would be safer.

Someone noted in a comment that we would normally base the NPV on the interest rate of the payments. That's for calculating the NPV to the one making the loan. Here, we want to calculate the NPV for the borrower. So the question is what the borrower would do with the money if making payments and not the lump sum.

The question assumes that the borrower would invest in the stock market, which is a risky option and not normally advisable. I suggest a mortgage based alternative. If the borrower is going to stuff the money under the mattress until needed, then the answer is simple. The nominal value of $2991 is also the NPV, as mattresses don't pay interest. Similarly, many banks don't pay interest on checking these days. So for someone facing a real decision like this, I'd almost always recommend paying the lump sum and getting it over with. Even if the payments are "same as cash" with no premium charged.

Answer 3

What's the present value of using the payment plan?

In all common sense the present value of a loan is the value that you can pay in the present to avoid taking a loan, which in this case is the lump sum payment of $2495. That rather supposes the question is a trick, providing irrelevant information about the stock market.

However, if some strange interpretation is required which ignores the lump sum and wants to know how much you need in the present to pay the loan while being able to make 8% on the stock market that can be done.

I will initially assume that since the lender's APR works out about 9.6% per month that the 8% from the stock market is also per month, but will also calculate for 8% annual effective and an 8% annual nominal rate.

The calculation

If you have $x in hand (present value) and it is exactly enough to take the loan while investing in the stock market, the value in successive months is $x plus the market return less the loan payment. In the third month the loan is paid down so the balance is zero. I.e.

v1 = 1.08 x - 997
v2 = 1.08 v1 - 997
v3 = 1.08 v2 - 997 = 0

∴ v3 = 1.08 (1.08 (1.08 x - 997) - 997) - 997 = 0

∴ x = 2569.37

So the present value of using the payment plan while investing is $2569.37.

You would need $2569.37 to cover the loan while investing, which is more than the $2495 lump sum payment requires. Therefore, it would be advisable to make the lump sum payment because it is less expensive: If you have $2569.37 in hand it would be best to pay the lump sum and invest the remaining $74.37 in the stock market. Otherwise you invest $2569.37 (initially), pay the loan and end up with $0 in three months.

One might ask, what rate of return would the stock market need to yield to make it worth taking the loan?

The APR proposed by the loan can be calculated.

The present value of a loan is equal to the sum of the payments discounted to present value. I.e. with

s = present value of loan
n = number of periods
d = periodic payment
r = periodic interest rate

∴ by induction

So by comparing the $2495 lump sum payment with $997 over 3 x monthly instalments the interest rate implied by the loan can be found.

s = (d - d (1 + r)^-n) / r

∴ 2495 = (997 - 997 (1 + r)^-3) / r

Solving for r

r = 0.0964431 = 9.64431 % per period (month)

∴ APR = 9.64431 * 12 = 115.732 % nominal compounded monthly

  or (1 + 0.0964431)^12 - 1 = 201.879 % effective annual interest

If you could obtain 9.64431% per month on the stock market the $x cash in hand required would be calculated by

v1 = 1.0964431 x - 997
v2 = 1.0964431 v1 - 997
v3 = 1.0964431 v2 - 997

∴ v3 = 1.0964431 (1.0964431 (1.0964431 x - 997) - 997) - 997 = 0

∴ x = 2495

This is equal to the lump sum payment, so the calculated interest is comparable to the stock market rate of return. If you could gain more than 9.64431% per month on the stock market it would be better to invest and take the loan.

Recurrence Form

Solving the recurrence form shows the calculation is equivalent to the loan formula, e.g.

v1 = 1.08 pv - 997
v2 = 1.08 v1 - 997
v3 = 1.08 v2 - 997

becomes v[m + 1] = (1 + y) v[m] - p where v[0] = pv

where

m is the month number
v[m] is the value in month m
y is the stock market yield
p is the payment amount
pv is the present value

∴ v[m] = (p + (1 + y)^m (pv y - p)) / y

In the final month v[final] = 0, i.e. when m = 3

(p + (1 + y)^m (pv y - p)) / y = 0

∴ pv = (p - p (1 + y)^-m) / y

Compare with the earlier loan formula: s = (d - d (1 + r)^-n) / r

They are exactly equivalent, which is quite interesting, (because it wasn't immediately obvious to me that what the lender charges is the mirror opposite of what you gain by investing).

The present value can be now be calculated using the formula.

Still assuming the 8% stock market return is per month.

m = 3
p = 997
y = 0.08

pv = (p - p (1 + y)^-m) / y

∴ pv = (997 - 997 (1 + 0.08)^-3) / 0.08 = 2569.37

If the stock market yield is 8% per annum effective rate

monthly yield, y = (1 + 0.08)^(1/12) - 1 = 0.00643403

∴ pv = (997 - 997 (1 + 0.00643403)^-3) / 0.00643403 = 2952.92

and if it is given as a nominal annual yield, 8% compounded monthly

monthly yield, y = 0.08 / 12 = 0.00666667

∴ pv = (997 - 997 (1 + 0.00666667)^-3) / 0.00666667 = 2951.56