Discounting: compounding conventions and present value

Everything in derivatives sits on top of one idea: a pound today is worth more than a pound later, and the exchange rate between them is the discount factor. The conventions are fiddly and the interview tests whether you can move between them without thinking.

Convention	Value of 1 after $T$ years	Used in
Simple	$1 + rT$	Money-market instruments under a year
Compound, $m$ times a year	$(1 + r/m)^{mT}$	Bonds, deposits, quoted yields
Continuous	$e^{rT}$	Option pricing, and anywhere calculus is involved

Table 2.1 · The three conventions. Continuous compounding is the limit of the second row as

m \to \infty

. It is used in derivatives not because anyone compounds continuously but because

e^{rT}

differentiates cleanly and the exponents add.

Equation 2.2

Converting between them

Two rates are equivalent when they produce the same value after a year. The continuously compounded rate is always the *lowest* of the equivalent set, because it compounds most often.

\left(1 + \frac{r_m}{m}\right)^{m} = e^{r_c} \quad \Rightarrow \quad r_c = m\ln\!\left(1 + \frac{r_m}{m}\right)

$r_m$: Nominal rate quoted with $m$ compoundings a year.
$r_c$: The continuously compounded equivalent.

Figure 2.3 · A future pound shrinks when brought back to today. At a 5% annual rate, £1 in ten years is worth about £0.614 today. The curve is

1/(1.05)^T

; each extra year multiplies the present value by another 1/1.05.

Derivation 2.4

Perpetuities and annuities

Both are geometric series, and the annuity is just the difference of two perpetuities.

$PV_{\text{perp}} = \sum_{t=1}^{\infty} \frac{C}{(1+r)^t} = \frac{C}{r}$
A geometric series with ratio $1/(1+r)$ .
$PV_{\text{annuity}} = PV_{\text{perp}} - \frac{PV_{\text{perp}}\text{ starting at } n}{(1+r)^n}$
An $n$ -year annuity is a perpetuity minus a perpetuity deferred $n$ years.
$= \frac{C}{r}\left(1 - \frac{1}{(1+r)^n}\right)$

PV_{\text{annuity}} = \frac{C}{r}\left(1 - (1+r)^{-n}\right)

Proposition 2.5

The doubling rules

Money doubles in roughly $72/r$ years when $r$ is quoted as a percentage. The exact continuous figure is $\ln 2 / r \approx 69.3/r$ , and 72 is preferred because it divides neatly and because annual compounding pushes the answer slightly above the continuous one in the range people care about.

Holds when

Rule of 72 is most accurate around $6$ – $10\%$ ; below about $3\%$ use 70.
Tripling is the rule of 114, from $\ln 3 \approx 1.10$ .
For a fall rather than a growth, the same arithmetic gives the halving time.

Discounting: compounding conventions and present value

After this lesson you should be able to

Converting between them

Perpetuities and annuities

The doubling rules

The rest of this lesson is in Premium