The Time Value of Money

Would you rather have $100 now or $100 a year from now? Now, obviously, and not just because you are impatient. If you take the $100 today and invest it at, say, 6%, you will have $106 next year. So a dollar today is genuinely worth more than a dollar later: it has time to earn. This is the time value of money, and it is the reason every future cash flow has to be shrunk down, discounted, before you can compare it to money in hand.

Compounding: money grows on money

Leave money invested and it earns interest. Leave it longer and it earns interest on the interest, this snowballing is called compounding, and it is what makes the growth curve bend upward instead of rising in a straight line. Starting from a present value $PV$, after $t$ years at rate $r$ per year the future value is

$$FV = PV\,(1 + r)^t.$$

Here $PV$ is the money today, $r$ is the annual interest rate (as a decimal), and $t$ is the number of years. The exponent $t$ is where the magic lives: because it multiplies growth on top of previous growth, doubling the time far more than doubles the money.

Below is that race, live. The green curve is compound growth; the amber dashed line is simple interest (the same rate, but paid only on the original principal). Push the rate and the years and watch the two lines pull apart — the shaded gap is the "interest on interest," and it's almost the entire story at long horizons.

Discounting: running the clock backward

Pricing works in reverse. If you are promised a future cash flow, what is it worth today? You undo the compounding by dividing instead of multiplying:

$$PV = \frac{FV}{(1 + r)^t}.$$

This is discounting, and $r$ here is the discount rate. The further away the money (larger $t$) or the higher the rate, the more you shrink it. This one operation is the engine under bond prices, expected cash-flow valuation, and even the $Ke^{-rT}$ term in put-call parity, where a strike paid at expiry is discounted back to today.

A quick mental shortcut worth memorizing is the Rule of 72: money doubles in roughly $72 / (\text{rate in \%})$ years. At 6%, that is $72/6 = 12$ years to double, a fast sanity check you can do in your head.

Worked example

You invest $1,000 for 10 years at 6% per year. Compare compounding against plain (simple) interest.

$$FV_{\text{compound}} = 1000\,(1.06)^{10} = 1000 \times 1.7908 = 1{,}790.85.$$

Simple interest, which pays 6% only on the original $1,000 each year, gives $1000 + 1000(0.06)(10) = 1{,}600$. Compounding earns an extra $1790.85 - 1600 = 190.85$ purely from interest-on-interest, and that gap grows explosively over longer horizons.

Now run it backward. What is $1,000 received in 10 years worth today, at the same 6%?

$$PV = \frac{1000}{(1.06)^{10}} = \frac{1000}{1.7908} = 558.39.$$

So a promise of $1,000 a decade out is worth only about $558 in today's money, even with no risk at all. Distance in time alone chops nearly half its value.

Common pitfalls

• Ignoring compounding frequency. 6% compounded monthly beats 6% compounded annually, because interest starts earning sooner. In the limit of continuous compounding, $FV = PV\,e^{rt}$, the form quants default to.
• Confusing rate and return. A stated (nominal) 12% rate compounded monthly is really $(1 + 0.12/12)^{12} - 1 = 12.68\%$ effective. Always compare effective annual rates, not the headline number.
• Forgetting inflation. These formulas grow nominal dollars. If prices rise 3% a year, a 6% return is only about 3% richer in real purchasing power. Discount by the real rate when you care about what the money can buy.
• Discounting risky and safe flows the same way. A guaranteed dollar and a risky dollar should not use the same rate. Riskier cash flows demand a higher discount rate, which is the bridge to the risk-return tradeoff.