Interest, present value and break-even

Simple and compound interest

Simple interest is paid on the original amount only. At a rate $r$ a year, each year adds $P \times r$, so after $n$ years the balance is $P(1 + nr)$.

Compound interest is added to the balance and from then on earns interest itself. Each year multiplies the balance by $1 + r$, so after $n$ years

$$\text{FV} = \text{PV} \times (1 + r)^n,$$

where PV is the amount today and FV the future value. This is the multiplier from percentages and successive changes, with the same change applied $n$ times.

€1,000 at 10% a year for three years:

Year	Simple	Compound
1	1,100	1,100
2	1,200	1,210
3	1,300	1,331

The gap of €31 is interest on interest: 10% of the first year's €100 in year 2, then 10% of the €210 earned so far in year 3. After the first year compounding always gives more, and the gap widens with time.

Read which kind the question describes: "compounded", "reinvested" or "added to the balance" mean compound interest; "simple interest" or interest "paid out each year" mean simple. If a yearly rate is credited in parts, use the rate per period and the number of periods: 10% a year credited every half-year is 5% per half-year, so two years multiply by $1.05^4$, not $1.1^2$.

Powers by hand

Calculators are not permitted during the test, so keep a few powers ready and build the others:

• $1.1^2 = 1.21$, $1.1^3 = 1.331$, $1.2^2 = 1.44$, $1.2^3 = 1.728$, $1.05^2 = 1.1025$.
• Square any multiplier with $(1 + r)^2 = 1 + 2r + r^2$: $1.06^2 = 1 + 0.12 + 0.0036 = 1.1236$.
• For small rates, $(1 + r)^n \approx 1 + nr + \frac{n(n - 1)}{2}r^2$. For $1.04^3$ this gives $1 + 0.12 + 3 \times 0.0016 = 1.1248$; the exact value is 1.124864.
• To estimate how long money takes to double, divide 72 by the rate in percent: at 8% a year, about 9 years.

Present value: compounding backwards

A euro today is worth more than a euro in a year, because today's euro can be invested. The present value of an amount CF due in $n$ years is the sum that, invested now at rate $r$, grows into CF:

$$\text{PV} = \frac{\text{CF}}{(1 + r)^n}.$$

€1,331 due in three years at 10% is worth $\frac{1331}{1.331} = 1\,000$ euros today. Discounting divides by the growth factor, so a higher rate or a longer wait gives a smaller present value. With several payments, discount each one separately and add: €110 in one year and €121 in two years, at 10%, are worth $\frac{110}{1.1} + \frac{121}{1.21} = 100 + 100 = 200$ euros today. The rate may be called a discount rate, a required return or a cost of capital; it plays the same role each time.

The yield of a zero-coupon bond

A zero-coupon bond pays one amount, its face value, at maturity and nothing before. Bought below face value, it grows into the face value, and the yearly rate that does this is its yield $y$:

$$\text{price} \times (1 + y)^n = \text{face value}, \qquad (1 + y)^n = \frac{\text{face value}}{\text{price}}.$$

So divide face value by price, then take the $n$-th root. By hand, look for a ratio that is a power you know: a ratio of 1.21 over two years means $1 + y = 1.1$, a yield of 10%. A higher price means a lower yield, because less growth is needed to reach the same face value.

Perpetuities

A perpetuity pays the same amount $C$ every year forever, starting one year from now. Its present value is

$$\text{PV} = \frac{C}{r}.$$

The reason: deposit $\frac{C}{r}$ at rate $r$, and each year's interest is exactly $C$. You can pay out $C$ every year and never touch the deposit. €600 a year forever, at 4%, is worth $\frac{600}{0.04} = 15\,000$ euros.

A growing perpetuity pays $D_1$ in one year, and each later payment is larger than the one before by a rate $g$, forever. If $r > g$, its value today is

$$P = \frac{D_1}{r - g}.$$

The same deposit argument works: a deposit $P$ earns $rP$, and paying out $D_1 = (r - g)P$ leaves $gP$, so the deposit, and with it the next payment, grows by $g$ each year. This is the usual way to value a share whose dividends are expected to grow at a steady rate. Two cautions: $D_1$ is next year's payment, so a payment just made must first be multiplied by $1 + g$; and the formula means nothing when $g \ge r$.

Break-even and contribution margin

Each unit sold brings in its price $p$ and costs its variable cost $v$ (materials, delivery, commission). The difference $p - v$ is the contribution margin: what each unit contributes towards the fixed costs $F$, such as rent and salaries, which do not change with output. For $Q$ units,

$$\text{profit} = (p - v)Q - F.$$

Profit is zero at the break-even quantity

$$Q = \frac{F}{p - v},$$

and a target profit $T$ needs $Q = \frac{F + T}{p - v}$. With fixed costs of €48,000, a price of €25 and a variable cost of €13, each unit contributes €12, so break-even is $\frac{48\,000}{12} = 4\,000$ units. Units come whole, so round a fractional answer up: with fixed costs of €50,000 the division gives about 4,166.7, and 4,166 units fall just short.