The time value of money

Interest is the price of time

If I offer you $100 today or$100 in a year, you take it today — even ignoring inflation and the risk I might not pay. Money in hand can be put to work: lent out, invested in T-bills, used now instead of later. The interest rate is the market price of that flexibility. It's what someone must pay you to give up your money for a period of time.

This one idea underlies almost everything in finance: bond pricing, options pricing, why the Sharpe ratio subtracts a risk-free rate, why a "guaranteed $1M in 40 years" pension promise is worth far less than$1M today.

Future value: compounding forward

Two abbreviations you'll see everywhere in finance: PV is present value — money in hand today. FV is future value — what that money becomes later.

Invest $PV$ today at rate $r$ per year, reinvesting all interest. After $n$ years:

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

In words: future value equals present value times "one plus r, raised to the power n." The raised $n$ is an exponent: it means multiply $(1+r)$ by itself $n$ times — once per year. At 8% for 3 years, that's $1.08 \times 1.08 \times 1.08$, not $1.08 \times 3$.

That exponent is the whole story. Ten thousand dollars at 8%:

By year 30 the compound value is roughly 3× what linear intuition suggests. This is why fees, small edges, and small return differences matter so much over long horizons — they compound too.

Present value: discounting backward

Run the same machine in reverse. A cash flow — any payment you'll receive — of size $C$ arriving in $n$ years is worth today:

$PV = \frac{C}{(1+r)^n}$

In words: the value today equals the future cash amount $C$, divided by "one plus r to the power n." It's the future-value formula solved backwards — instead of multiplying by the growth factor, you divide by it.

This is called discounting, and $r$ here is the discount rate — the rate you use to shrink future money back to today's terms. A dollar tomorrow is worth less than a dollar today, because you could have invested today's dollar and had more than a dollar tomorrow. Concretely: at $r = 8\%$, $100 arriving in 9 years is worth about **$50 today** — because $50 invested at 8$100 in 9 years ($1.08^9 \approx 1.999$; read that as "1.08 to the power 9 is roughly 2").

Every asset price is, at some level, a discounted stream of expected future cash flows. When interest rates rise, discount rates rise, and the present value of far-future cash flows falls. That's why long-duration assets — assets whose cash flows arrive far in the future, like long bonds and high-growth tech stocks whose profits are decades away — get hit hardest when rates go up.

Compounding frequency, briefly

$(1+r)^n$ assumes interest compounds once a year. Compound more often and you earn slightly more. Example: 8% compounded monthly gives $(1 + 0.08/12)^{12} - 1 \approx 8.30\%$ effective — read it as "one plus the yearly rate split into 12 monthly pieces, compounded 12 times, minus the 1 you started with." Push the frequency to the limit — compounding every instant — and you get continuous compounding:

$FV = PV \, e^{rt}$

In words: future value equals present value times "e to the power r times t," where $t$ is time in years. The letter $e$ is not a variable — it's a fixed mathematical constant, roughly 2.718 (like $\pi$), that shows up whenever growth happens continuously.

At 8%, $e^{0.08} \approx 1.0833$ vs. 1.08 for annual compounding — a small difference. But the continuous form is mathematically cleaner, which is why derivatives pricing (Hull's world) and log returns (which add across time exactly the way $e^{rt}$ exponents add) live in continuous-compounding land. For now just recognize $e^{rt}$ when you see it.

The risk-free rate: the baseline everything is measured against

The risk-free rate, written $r_f$ ("r sub f"), is what you earn for giving up money over time while taking (essentially) zero default risk — risk that the borrower fails to pay you back. In practice it's the yield — the annualized return — on short-term US Treasury bills (T-bills: short-term IOUs from the US government). It is pure time value: no risk premium at all.

This is the baseline hurdle for every investment. Earning 5% when T-bills pay 5% means your risk earned you nothing. That's why:

• The Sharpe ratio (A2-04) is $(E[r] - r_f)/\sigma$ — read: expected return minus the risk-free rate, divided by sigma (the spread of returns). Only return above the risk-free baseline counts as compensation for risk.
• CAPM and factor models explain excess returns $r - r_f$ (return minus the risk-free rate), not raw returns.
• A backtest run during a 5%-rate era that "makes 6%" is far less impressive than the same 6% during a 0%-rate era.

When you see $r_f$ subtracted anywhere in this curriculum, it's this lesson operating: strip out the price of time first, then judge what the risk bought you.

The rule of 72

Quick mental math: money doubles in approximately $72 / r$ years, with $r$ in percent. In words: divide 72 by the interest rate to get the doubling time. At 8%: $72/8 = 9$ years. (The exact answer is $\ln 2 / \ln 1.08 \approx 9.01$, where $\ln$ is the natural logarithm — the inverse of the $e^x$ function from above. You don't need to compute it; the point is the shortcut is remarkably accurate for rates between ~4% and ~12%.)

The power move is chaining doublings. A 30-year horizon at 8% is $30/9 \approx 3.3$ doublings: $2^{3.3} \approx 10\times$ — two multiplied by itself 3.3 times is about ten. That single estimate — "8% for 30 years is about 10×" — reframes retirement math, fee drag, and why starting to invest 9 years earlier roughly doubles the final outcome for the same contributions.

Try it

Explain it in your own words

Your generative activity: explain to a friend with no finance background why $100 arriving in 9 years is worth only about$50 today when rates are 8% — without using the word "formula." If you can make the reinvestment argument in plain speech, you own this concept.