# Which equation is used for calculation interest rate of lump sum?

I was watching CFA level1 videos and found this odd question.

Elmer has won his \$4 million state lottery and has been offered 20 annual payments of \$200,000 each beginning today or a single payment of \$2,267,000. What is the annual discount rate used to calculate the lump-sum payout amount?

``````I know how to type in calculator by
Begin Mode, N=20, FV=0, PMT=-200,000, PV=2,267,000
ANS: I/Y = 7%
``````

However, I am interested in mathematical calculation. I have tried EMI, i=N√FVPV−1, etc. equations. Doesn't see one of them working. I am wondering which equation is used for calculation interest rate of lump sum? Why following formula doesn't work? -200,000 * (I/Y**20) = 2,267,000

If you accepted the \$200,000 per year you could deposit each payment at zero interest and end up with \$4m after 20 years. If you accepted the single payment of \$2,267,000 you would have to deposit it in a bank at 2.87987 % per annum to end up with \$4m after 20 years.

``````4000000 = 2267000 (1 + r)^20 ∴ r = 2.87987 %
``````

The OP only asks for the single payment calculation. Nevertheless, here is the rest.

The advantage of annual payments would remain up until the bank deposit rate reached slightly over 7% at which point the advantage turns to the single payment. I.e. future value of an annuity due for the annual payments:

Future value `s` equals the sum of the appreciated payments `a`. Formula is by induction.

Choosing `r` slightly over 7% for an example

``````  r = 0.070008
a = 200000
n = 20
∴ s = (a (1 + r) ((1 + r)^n - 1))/r = \$8,773,867.36
``````

vs. single payment

``````2267000 (1 + 0.070008)^20 = \$8,773,886.56
``````

In this case the future value from the single payment is greater.

With `r > 7.00077 %` is it advantageous to take the single payment. The plot below illustrates the cross-over point with the exact intercept found by solver.

Note the answer as stated by the OP is 7%. Solution via solver The mathematical calculation is an iterative root-finding algorithm. There's not a known closed form equation for the interest rate that results in a specific present value, except for very simple cases like a single cash flow. So the calculator tries various interest rates until it finds one that gives the correct present value of the coupon stream.

Why following formula doesn't work? -200,000 * (I/Y**20) = 2,267,000

It looks like you're trying to calculate the future value of a cash flow, but are comparing it to the present value. The idea of present value is "At what interest rate could I invest 2,267,000 now, and 200k per year, and end up with the same amount in the end".

• Thank you. That make senses. No wonder I couldn't find any formulas. Appreciate it!!! Aug 21 at 15:21

FV = PV (1+r)^n

• Indeed `4000000 = 2267000 (1 + r)^20 ∴ r = 2.87987 %` Aug 23 at 18:46