Theory

Compounding, annuities and amortization

How a capital grows under simple, compound and continuous compounding, how amounts are brought to the same date with present value and equivalent rates, how annuities and amortization schedules are valued, and how an investment is assessed with NPV and IRR.

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Simple interest and compound interest
Present value and discount

Simple interest and compound interest

1. Principal, interest and accumulated value

Whoever lends a principal $C$ today receives after a time $t$ , measured in years, a larger sum, the accumulated value $M$ . The difference $I = M - C$ is the interest, that is, the compensation for giving up the use of the money. The annual interest rate $i$ is the interest earned in one year by a principal of $1$ euro, and it is written as a decimal number or as a percentage; for example $i = 0.05$ is the same as $i = 5\%$ . The accumulation factor is the ratio $\dfrac{M}{C}$ , that is, the value after the time $t$ of $1$ euro invested today; the law that expresses it as a function of $t$ is called the compounding regime. The two regimes used in practice differ in how they treat the interest already earned.

2. Simple interest

Under simple interest the interest is proportional to the principal and to the time, $I = C\,i\,t$ , so the accumulated value is

M = C\,(1 + i\,t), \qquad t \geq 0 .

The interest earned does not itself earn interest, therefore every year the principal receives the same interest $C\,i$ and the accumulated value is a linear function of time, with slope $C\,i$ . It is used for transactions lasting less than a year. A principal of $2000$ euros at $4\%$ for $3$ years earns interest of $2000 \cdot 0.04 \cdot 3 = 240$ euros, so the accumulated value is $2240$ euros; for $9$ months one sets $t = \dfrac{9}{12} = \dfrac{3}{4}$ , and the interest is $60$ euros.

3. Compound interest

Under compound interest the interest earned at the end of each year is added to the principal and earns interest in the following years. At the end of each year the accumulated value is therefore multiplied by $1 + i$ , and after $n$ years it equals $C\,(1 + i)^{n}$ , that is, the term of a geometric progression with common ratio $1 + i$ . The formula extends to every time $t \geq 0$ :

M = C\,(1 + i)^{t} .

The accumulated value is an exponential function of time. With $C = 2000$ euros and $i = 4\%$ , after $3$ years one has $M = 2000 \cdot 1.04^{3} = 2249.73$ euros, that is, $9.73$ euros more than under simple interest, and the difference is due to the interest earned by the interest of the first two years.

M = 1000 \cdot 1.1^{t}

M = 1000\,(1 + 0.1\,t)

Accumulated value of

1000

euros at

10%

per year under compound and simple interest: the two accumulated values coincide for

t = 1

4. Comparison of the two regimes

Let $i > 0$ . Then

\begin{cases} (1 + i)^{t} < 1 + i\,t & \text{if } 0 < t < 1 , \\[4pt] (1 + i)^{t} > 1 + i\,t & \text{if } t > 1 , \end{cases}

and the two accumulation factors coincide for $t = 0$ and for $t = 1$ . The result follows from the convexity of the exponential function, covered in the topic on curve sketching and optimization; indeed the graph of $(1 + i)^{t}$ lies below the line through the points with abscissas $0$ and $1$ between these two points, and above it outside. For durations shorter than a year simple interest therefore gives the larger accumulated value. With $1000$ euros at $10\%$ for six months one obtains $1050$ euros under simple interest and $1000 \cdot \sqrt{1.1} \approx 1048.81$ euros under compound interest.

Present value and discount

1. Present value

The present value $V$ of an amount $M$ available in $t$ years is the principal that, invested today, has accumulated value $M$ after the time $t$ . It is obtained by solving the accumulated value formula for $C$ :

V = M\,(1 + i)^{-t} \quad \text{(compound interest)}, \qquad V = \frac{M}{1 + i\,t} \quad \text{(simple interest)} .

The passage from $M$ to $V$ is called discounting, and the number $v = \dfrac{1}{1 + i}$ is the annual discount factor, with which $V = M\,v^{t}$ . Present value is used to compare amounts available at different dates, by bringing all of them back to today. At $3\%$ compound, $5000$ euros available in $4$ years are worth $5000 \cdot 1.03^{-4} \approx 4442.44$ euros today. Since $0 < v < 1$ , the present value decreases as the due date moves further away and tends to $0$ as $t \to +\infty$ .

V = 1000 \cdot 1.05^{-t}

V = 1000 \cdot 1.1^{-t}

Present value of

1000

euros available in

t

years, at

5%

and at

10%

: a higher rate corresponds to a lower present value

2. Discount and discount rate

The discount is the difference $D = M - V$ between the amount at maturity and its present value; it is the interest seen by whoever pays in advance. The annual discount rate $d$ is the discount on $1$ euro due in one year, so

d = 1 - v = 1 - \frac{1}{1 + i} = \frac{i}{1 + i} .

Under commercial discount, used for short transactions, the discount is proportional to the amount and to the time, $D = M\,d\,t$ , so

V = M\,(1 - d\,t), \qquad 0 \leq t < \frac{1}{d} .

The restriction on $t$ keeps the present value positive. A bill of exchange of $5000$ euros maturing in $9$ months, discounted today at the commercial rate $d = 4\%$ , gives a present value of $5000 \left( 1 - 0.04 \cdot \dfrac{3}{4} \right) = 4850$ euros. With rational discount, that is, discounting under simple interest, and the rate $i = 4\%$ one would instead obtain $\dfrac{5000}{1.03} \approx 4854.37$ euros.

3. Unknown time and rate

Under compound interest, given $C$ , $M$ with $M > C > 0$ and the rate $i > 0$ , the time needed for $C$ to become $M$ is

t = \frac{\ln \dfrac{M}{C}}{\ln(1 + i)} ;

given $C$ , $M$ and $t > 0$ , the rate that links them is

i = \left( \frac{M}{C} \right)^{\frac{1}{t}} - 1 .

The formulas follow from $(1 + i)^{t} = \dfrac{M}{C}$ , by taking the logarithm of both sides in the first case and the $t$ -th root in the second, as in the topic on real functions and economic models. For example $1000$ euros at $4\%$ become $1500$ euros after $t = \dfrac{\ln 1.5}{\ln 1.04} \approx 10.34$ years, and a principal that grows from $1000$ to $1210$ euros in $2$ years has been invested at the rate $\sqrt{1.21} - 1 = 10\%$ .

The theory continues with 6 more sections (equivalent rates and nominal rate, continuous compounding, financial equivalence and average due date…), plus all worked examples, the solutions and the PDF.

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