Basic, lesson 1 of 4
Time value of money
A dollar today is worth more than a dollar next year, because today's dollar can be invested. Every valuation in banking is built on that one sentence.
8 min, 3 checks
1Why a dollar today beats a dollar later
Offer someone $100 today or $100 in a year and they take it today. Not because they are impatient, but because $100 today can sit in a savings account or a Treasury bill and become more than $100 by next year.
That gap is the time value of money. The rate you could earn on the money in the meantime is what turns today's dollar into tomorrow's, and it is what lets you compare cash that arrives at different times.
Finance cares about this because almost every decision is a trade between money now and money later. A company builds a factory today to earn cash for twenty years. An investor buys a company today to get dividends and a sale price later. To judge either, you have to put all that cash on the same clock.
2Future value and compounding
Future value answers: if I invest this today at a given rate, what will it be worth later? Each year the balance grows by the rate, and next year's growth is earned on the bigger balance. That is compounding: interest on interest.
FV = PV x (1 + r)^n
PV is the amount today, r is the rate per period, n is the number of periods.
Worked example
$100 at 8% for five years
- 1Year 1: 100 x 1.08 = 108.00
- 2Year 2: 108.00 x 1.08 = 116.64
- 3Keep going, or do it in one step: 100 x 1.08^5
- 41.08^5 = 1.4693
$146.93. Simple interest would give $140; the extra $6.93 is interest earned on interest.
Check 1
You invest $1,000 at 6% a year for 10 years, compounded annually. What is it worth at the end, to the nearest dollar?
A handy shortcut is the rule of 72: divide 72 by the rate in percent to get the rough years to double. At 8%, money doubles in about 9 years (the exact answer is 9.01).
3Present value: running the clock backward
Present value flips the question: what is a future amount worth today? You divide by the growth factor instead of multiplying. This is called discounting, and the rate you use is the discount rate.
PV = FV / (1 + r)^n
Worked example
$1,000 three years from now at 10%
- 1Growth factor: 1.10^3 = 1.331
- 2PV = 1,000 / 1.331
$751.31. Someone who can earn 10% elsewhere is indifferent between $751.31 today and $1,000 in three years.
Check 2
Holding the future amount fixed, what happens to present value when the discount rate rises?
4Compounding frequency
Rates are often quoted per year but compounded more often. A 12% rate compounded monthly means 1% a month, and twelve rounds of 1% growth is more than 12%.
| Compounding | Rate per period | Effective annual rate |
|---|---|---|
| Annual | 12% | 12.00% |
| Quarterly | 3% | 12.55% |
| Monthly | 1% | 12.68% |
Check 3
A card charges 1% a month. What is the effective annual rate, in percent, to one decimal?
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