The Time Value of Money

Would we rather have a hundred dollars today or a hundred dollars a year from now? Today. The hundred we hold now can be invested and turned into more than a hundred by next year, and even if it is spent rather than invested, money now gives us options that money later does not.

That observation (a dollar today is worth more than a dollar tomorrow) is the premise behind nearly all of finance. It is the reason interest exists, the reason bonds are priced the way they are, and the foundation of every valuation model that follows. The last post reduced every financial asset to three questions: how much money is coming back, when it shows up, and what the odds are that it shows up at all. This post builds the machinery that puts a price on timing.

Measuring Returns

Before we can compound a return we need a way to measure one. The gross return on an investment is its payoff divided by its price:

R = \frac{payoff}{price}

Suppose we buy a stock for 100 and sell it for 110. The gross return is 1.10, or 110%. More often the quoted figure is the net return, which strips the original investment back out:

r = \frac{payoff}{price} - 1

On the same trade our net return is 0.10, or 10%. Someone who says they made 10% on a trade means the net return.

For stocks the payoff arrives from two sources. Capital gain is the price appreciation, and dividend yield is the income the company pays out while the position is held. Together:

r = \frac{P _{t + 1} + D _{t + 1}}{P _{t}} - 1

Take a stock bought at 50 that falls to 46 and pays a 3 dividend before it is sold. The return is $(46 + 3) /50 - 1 = - 2%$ . The dividend softened the blow without erasing the loss.

Returns are often quoted in basis points, where one basis point is one hundredth of a percent (0.01%). A return of 5% is 500 basis points, and financial news reporting that the Fed raised rates by 25 bps is reporting a quarter of a percent.

Future Value and Compound Interest

With a return defined, the next question is how a balance grows over time. We invest an amount $X$ at a net return of $r$ per period, and after one period we hold:

F V = X \times (1 + r)

After two periods, we earn the second period’s return on the new, larger balance:

F V = X \times (1 + r) \times (1 + r) = X \times (1 + r)^{2}

And after $n$ periods in general:

F V = X \times (1 + r)^{n}

This is compound interest. We earn returns on the original investment and on the returns already collected. A 1,000 balance at 10% grows by 100 in the first year. In the second it grows by 110, which is 10% of 1,100. In the third, 121, and the increments keep growing.

Simple interest pays only on the original principal. The same 1,000 grows by 100 every year, forever, with no acceleration at all.

The difference between the two starts small and gets enormous over time. Drag the rate and years sliders below and watch the gap open up.

Scenario Principal: $1,000 Annual Rate: 8% Years: 20 $0 $1.0k $2.0k $3.0k $4.0k $5.0k 05101520 Years Compound $4,661 Simple $2,600 Interest on interest $2,061

That widening space between the solid line and the dashed line is interest earned on interest, and the readout beneath the chart reports it directly. At 8% over 20 years our 1,000 becomes about 4,661 with compounding but only 2,600 with simple interest. The extra 2,061 is money that the money earned, and over 40 years the gap grows far wider.

Compounding is why starting early matters so much for retirement savings. Ten extra years of it does more work than a large increase in contributions later.

Present Value and Discounting

Future value pushes a balance forward. Present value runs the same machinery in reverse, asking what a payment arriving later is worth today.

If we can earn a return of $r_{d}$ per period elsewhere (the discount rate), a payment of $X$ arriving in $n$ periods is worth:

P V = \frac{X}{( 1 + r _{d} ) ^{n}}

Consider a promise of 100 in three years, against a comparable alternative earning 5% annually. That promise is worth $100/1.0 5^{3} = 86.38$ today, because 86.38 invested at 5% grows to 100 over three years. No more and no less.

The discount rate is an opportunity cost, the return available on a similarly risky alternative. A higher discount rate marks either more impatience or better options elsewhere, and either way it makes future cash flows worth less today.

A security paying multiple cash flows at different times is handled one payment at a time. We discount each separately and add the results:

P V = t = 1 \sum n \frac{X _{t}}{( 1 + r _{d} ) ^{t}}

Take a bond paying 50 next year, 50 the year after, and 1,050 in year three. We discount each of the three payments, sum them, and the total is the bond’s fair price. This is the core of fixed-income pricing, and the next post builds on it directly.

The same formula explains where price movements come from. A price is a discounted stream of future cash flows, so it can move for two reasons: the market revised its expectations of those cash flows (“cash flow news”), or it revised the rate at which they are discounted (“discount-rate news”). There is no third source. A stock that drops 15% after an earnings call has had its cash flow estimates cut, or its required return raised. Usually both.

APR and the Effective Annual Rate

Compounding and discounting both take a rate per period as an input, and the rate quoted in the market is not always the rate earned.

Financial institutions quote an annual percentage rate (APR), calculated as:

A PR = m \times r_{E}

where $m$ is the number of compounding periods per year and $r_{E}$ is the effective rate per period. A bank that pays 0.5% per month advertises a 6% APR ( $12 \times 0.5%$ ).

APR ignores compounding. Interest credited at 0.5% this month earns interest in every month after it, so the rate earned over the year, the effective annual rate (EAR), is:

E A R = (1 + \frac{A PR}{m})^{m} - 1

Compounding a 6% APR monthly, we get $E A R = (1.005)^{12} - 1 = 6.168%$ . The difference is not huge, and it is money the quoted rate never mentions.

The gap between APR and EAR grows with the compounding frequency, since more periods mean more opportunities for interest to compound on itself. Step through the frequency buttons below, from annual to continuous, and watch the EAR climb while the APR stays fixed.

APR: 10% Annual m = 1 Semi-annual m = 2 Quarterly m = 4 Monthly m = 12 Daily m = 365 Continuous m = \infty APR 10.00% EAR 10.47% Rate per period 0.833% Effective Annual Rate 10.471% Extra over APR +0.471% Frequency Periods / yr EAR Annual 1 10.000% Semi-annual 2 10.250% Quarterly 4 10.381% Monthly 12 10.471% Daily 365 10.516% Continuous \infty 10.517%

The EAR keeps climbing as frequency rises, but it converges to a limit. That limit is continuous compounding, where interest compounds at every instant:

E A R = e^{A PR} - 1

where $e \approx 2.718$ is Euler’s number. At 10% APR, continuous compounding gives an EAR of about 10.52%. Most of the compounding benefit arrives early, since the jump from annual to monthly is much bigger than the jump from daily to continuous.

APR survives as a historical accident. Before calculators, converting between compounding frequencies required serious arithmetic, and bankers needed a quick “good enough” way to compare rates quoted on different schedules. APR was that shortcut, and it stuck. A phone can compute an EAR in milliseconds now, but regulation still requires APR disclosure, and whenever interest compounds more than once a year, APR understates what is paid on a loan or earned on a deposit.

Annuities

A great many financial instruments (loans, bond coupons, lease payments) involve the same fixed payment repeating on a schedule for a set number of periods. That pattern is an annuity, and it shows up often enough to be worth a shortcut.

We could always discount each payment individually with the summation formula from above. For 100 per year over 5 years at a 6% discount rate, that gives:

P V = \frac{100}{1.06} + \frac{100}{1.0 6 ^{2}} + \frac{100}{1.0 6 ^{3}} + \frac{100}{1.0 6 ^{4}} + \frac{100}{1.0 6 ^{5}} = 421.24

Writing out every term gets tedious fast, and a clean closed form exists. The present value of $n$ equal payments of $C$ , discounted at rate $r_{d}$ per period, is:

P V = C \times \frac{1 - ( 1 + r _{d} ) ^{- n}}{r _{d}}

The fraction $\frac{1 - ( 1 + r _{d} ) ^{- n}}{r _{d}}$ is called the annuity factor, a multiplier giving how much present value each dollar of periodic payment buys. At 6% for 5 years the factor is about 4.212, so each dollar of annual payment is worth 4.21 today. We multiply by 100 and recover 421.24, the same figure as before.

The closed form comes from a trick worth seeing, because an annuity is the difference between two perpetuities.

Take a perpetuity that pays $C$ per period forever, starting one period from now. Its value is $C / r_{d}$ . Now take a second perpetuity that also pays $C$ per period forever but does not start until period $n + 1$ . It is the same perpetuity discounted back $n$ periods, so we value it at $(C / r_{d}) \times (1 + r_{d})^{- n}$ today. We subtract the second from the first, everything from period $n + 1$ onward cancels, and we are left with the payments in periods 1 through $n$ . That is the annuity.

We buy an infinite stream of payments and sell back the tail we do not want.

This formula is going to be our workhorse when we price bonds in the next post. A bond’s coupon stream is an annuity: same payment, same interval, known number of periods.

Perpetuities and Growing Perpetuities

The annuity formula also tells us what happens when the payments never stop. We let $n$ go to infinity, the term $(1 + r_{d})^{- n}$ goes to zero, and the formula collapses to something simple. A perpetuity, a fixed payment $X$ every period forever, is worth:

P V = \frac{X}{r _{d}}

An infinite stream of cash flows has a finite price. That is discounting doing its work: payments arriving 100 years from now contribute almost nothing to the total. A perpetuity paying 50 per year, discounted at 5%, is worth $50/0.05 = 1, 000$ .

Now we add growth. A growing perpetuity pays $D_{1}$ in the first period, then $D_{1} (1 + g)$ , then $D_{1} (1 + g)^{2}$ , and so on forever. As long as the growth rate $g$ is less than the discount rate $r_{d}$ , this also converges:

P V = \frac{D _{1}}{r _{d} - g}

This is the Gordon growth formula, one of the most important equations in equity valuation, and we lean on it directly when we get to pricing stocks. If dividends grow at 3% and investors demand 8%, the stock is worth $D_{1} /0.05$ . The closer $g$ gets to $r_{d}$ , the higher the price goes, and if $g$ ever reaches $r_{d}$ the value goes to infinity, which is the model announcing that it has broken down (nothing is worth infinity).

What’s Next

We now have the pricing machinery to value any stream of cash flows: discount each one, add them up. The next post applies it directly to bonds, the simplest asset to price because their cash flows (the coupons and the face value) are spelled out in the contract rather than forecast. We’ll build a bond pricing calculator, work through the inverse relationship between price and yield, and read what the yield curve says about where the economy is headed.