Interest, compounding and discounting

Almost every pricing formula in finance compares money at different dates. Before options, portfolios or bonds, we need one tool: a way to move an amount of money forward or backward in time. This lesson builds that tool.

The time value of money

If you have 100 today, you can put it in a bank account (or buy a government bill) and have more than 100 in a year, with essentially no risk. So 100 today is worth more than 100 promised in a year. The extra amount you are paid for waiting is interest, and the rate at which it accrues is the interest rate $r$, quoted per year.

Throughout these lessons $r$ means the risk-free rate: the rate on a loan that is certain to be repaid. Risky lending earns more, but that extra is compensation for risk, not for time.

Simple interest

With simple interest you earn $r$ on the original amount only. After $T$ years an initial amount $B_0$ has grown to

Growth is linear in time. This is how some short-term money market instruments are quoted, but it ignores the fact that interest already earned could itself earn interest.

Compound interest

With annual compounding, the interest earned in year one is added to the balance, and in year two you earn interest on the larger amount. After $T$ whole years:

If interest is credited $m$ times per year, each period pays $r/m$ and there are $mT$ periods:

A snowball is a fair picture: the bigger it gets, the more snow it picks up per turn. More frequent compounding gives slightly more growth, because interest starts earning interest sooner.

Continuous compounding

Let the number of compounding periods $m$ grow without limit. The expression above converges to an exponential:

Continuous compounding is the convention used in option pricing (Black–Scholes, binomial trees with small steps) because exponentials are easy to multiply and differentiate: growing for $T_1$ and then for $T_2$ is the same as growing for $T_1 + T_2$, since $e^{rT_1}e^{rT_2} = e^{r(T_1+T_2)}$.

The figure compares the three conventions for the same quoted rate. At low rates and short horizons they are almost identical; the gap opens up as $r$ and $T$ grow.

Discounting

Discounting is compounding run backwards. If a payment $C$ arrives at time $T$, its present value is the amount you would need to invest today to have exactly $C$ at $T$:

The multiplier $e^{-rT}$ is called the discount factor. It is always between 0 and 1 for a positive rate, and it shrinks the further away the payment is.

Example: with $r = 5\%$ and continuous compounding, 100 paid in two years is worth $100\,e^{-0.1} \approx 90.48$ today. Investing 90.48 at 5% for two years gives back exactly 100.

Converting between conventions

The same growth can be quoted in different conventions. A continuously compounded rate $r_c$ and an annually compounded rate $r_a$ describe the same growth when

So 5% continuous is the same as about 5.13% annual. When you compare rates, always check which convention they use.

Summary

• Money today is worth more than the same amount later, because it can earn the risk-free rate $r$.
• Compounding moves money forward: $B_T = B_0(1+r)^T$, or $B_0e^{rT}$ in continuous time.
• Discounting moves money back: $\text{PV} = Ce^{-rT}$. The factor $e^{-rT}$ is the discount factor.

Every later lesson uses the discount factor $e^{-rT}$. Whenever a formula compares a future payment with a price today, this factor is how the two are put on the same footing.