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

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

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

The 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 $C$ arriving in $n$ years is worth today:

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

This is called discounting, and $r$ here is the discount rate. 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 ($1.08^9 \approx 1.999$).

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 — which is why long-duration assets (long bonds, 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: 8% compounded monthly gives $(1 + 0.08/12)^{12} - 1 \approx 8.30\%$ effective. Push the frequency to the limit and you get continuous compounding:

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

At 8%, $e^{0.08} \approx 1.0833$ vs. 1.08 — 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 $r_f$ is what you earn for giving up money over time while taking (essentially) zero default risk — in practice, the yield on short-term US Treasury bills. 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 $(E[r] - r_f)/\sigma$ — only return above the risk-free baseline counts as compensation for risk.
• CAPM and factor models explain excess returns $r - r_f$, 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. At 8%: $72/8 = 9$ years (exact answer: $\ln 2 / \ln 1.08 \approx 9.01$ — the rule 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: $2^{3.3} \approx 10\times$. 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.