Question

1.

Suppose you won $2 million in a lottery and it is to be paid in 25 equal annual installments starting today. An alternative to the installment plan is to receive a lump sum payment today. If the applicable discount rate was 7% per year, how large would the lump sum payment have to be? Round to the nearest cent.

2.

If you invest $86 per month (starting next month) every month for 38 years, and you can earn 10% per year (compounded monthly), how much will you have at the end of 38 years? Round to the nearest cent.

Answer 1

1.
If $2 million is to be paid in 25 equal annual instalment, each instalment would be $80,000 [being $2 million / 25 months].
The present value of annuity for 24 years [I take 24 instead of 25 because the first instalment will be given NOW, and hence no need to apply discount] at 7% interest rate is 11.4693 from the present value of annuity table. Therefore, the lumpsum would be $997,544 [being ($80,000 x 11.4693) + $80,000].

It can also be found out using excel formula =PV(7%,24,-80000)+80000.

It can also be calculated by normal calculator operations as usual.

2.
As usual, this can be calculated in various ways. One way to look at it is, we are to calculate future value of annuity at 0.8333% for 456 periods investing $86 each period. [0.8333% is the interest for 1 month when annual rate is 10%, i.e., 10%/12 = 0.8333%; 456 periods are 38 years x 12 months].

In Excel, you can use the formula =FV(0.8333%,456,-86). It will yiield the result as $443,738.57. Note that if we use more decimals in the interest rate as 0.8333333333%, the result will vary but ultimately rest at $443,789.27.

You can also calculate it from this link: https://financial-calculators.com/future-value-of-an-annuity-calculator There you will keep initial amount as 0, and 456 as periods, with cash flow frequency as 'monthly', and compounding monthly, annual intereest rate at 10% and $86 as periodic amount.

There are also statistical formula to calculate this.