Time value of money
Why a dollar today is worth more than a dollar later: compounding, present value and the discount rate.
A dollar today
A dollar today beats a dollar next year, because today’s dollar can be invested and start earning. Future dollars are also exposed to inflation and to the risk of never arriving.
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Economists call this the time value of money, and interest is its price: lenders charge it because they give up the use of their money until it comes back.
Inflation eats into future dollars. At 3% a year, after ten years $100 buys only what about $74 buys today.
Compounding forward
Money growing at a rate r for n years becomes PV × (1 + r)ⁿ. At 10%, 1,000 becomes 1,100 after one year and 1,210 after two, because the second year also earns on the first year’s interest.
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Compound interest earns interest on interest; simple interest pays only on the original amount. Over a few years the difference is small, over decades it’s enormous: 1,000 at 10% simple interest for 30 years becomes 4,000; compounded, about 17,450.
The n counts periods. With monthly compounding, the yearly rate is split into twelve monthly rates and n counts months, which earns a little more. That’s the difference between an interest rate and its APY (annual percentage yield).
Discounting back
Discounting runs compounding in reverse: it tells you what a future amount is worth today. The rate you use, the discount rate, reflects risk. The riskier the cash, the higher the rate and the less it’s worth now.
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The discount rate is the return you could earn elsewhere for similar risk. Investors often start from the risk-free rate, what US Treasury bonds pay, and add a premium for the extra risk.
Discounting shrinks distant money fast. At 10%, 1,000 arriving in 10 years is worth about 386 today; arriving in 30 years, about 57.
Key terms
- Time value of money
- A dollar today is worth more than a dollar later.
- Present value
- Future amount ÷ (1 + r)ⁿ. What a future cash flow is worth today.
- Discount rate
- The rate used to discount; higher for riskier cash flows.
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