A sponsor offers you $2,000,000 today or $2,100,000 next year. The bigger number is not always the better deal. By the end of this week you can prove which one wins.
Learning objectives. After this week you can (1) compute simple and compound interest; (2) find a future value and a present value; and (3) turn a nominal rate into a real rate.
Remember this: a dollar today is worth more than a dollar tomorrow, because today's dollar can earn interest.
1) Simple interest
Simple Interest = Principal × Rate × Time
$10,000 at 5% for one year earns 10,000 × 0.05 × 1 = $500. Simple interest is paid on the original principal only.
2) Compound interest and future value
Future Value = PV × (1 + r)n
With compound interest you also earn interest on past interest. $1,000 at 10% for two years grows to 1,000 × 1.10 × 1.10 = $1,210, more than the $1,200 simple interest gives. Compounding more often at the same yearly rate gives a slightly higher future value.
3) Present value: running it backward
Present Value = FV ÷ (1 + r)n
Discounting asks what a future sum is worth today. $1,210 two years away at 10% is worth 1,210 ÷ 1.21 = $1,000 now. A higher rate, or a longer wait, makes the present value smaller.
The sponsor offer. At 8%, $2,100,000 in one year is worth 2,100,000 ÷ 1.08 ≈ $1,944,444 today. The $2,000,000 now is the better deal.
4) Real vs. nominal rates
Real Rate ≈ Nominal Rate − Inflation
An 8% loan during 3% inflation has a real rate of about 5%. The real rate is what the money actually costs once rising prices are taken out.
Quick self-check
(a) $2,000 at 10% simple interest for one year: how much interest? (b) $1,000 at 5% for one year: future value? (c) Nominal 7%, inflation 2%: real rate? (Answers: $200; $1,050; about 5%.)
Common mistakes
- Adding the years instead of compounding. Two years at 10% is × 1.21, not × 1.20.
- Comparing sums from different dates. Discount the later one to today first.
- Judging a loan by its sticker rate. Subtract inflation to see the real cost.
This week in the arena
Every dollar you spend this week is a dollar you can't hold for a later race. Take your biggest budget line and price it as if you had borrowed it at 8%. Then ask what the same cash would grow to if you held it two race weeks at 1% a week instead.