Interest
1 Interest
In finance you never do anything for free and you would always prefer to have money now rather than later. For example, if you are loaning someone money, you would expect to be compensated for the fact that you are giving up the opportunity to use that money for some other purpose. This is the time value of money and the compensation is called interest.
Interest is the cost of borrowing money or the return on investment for lending money. It is typically expressed as a percentage of the principal amount over a specified period of time.
Suppose we loan initial amount (the principal) \(P\) to a borrower to hold for some period of time. To compensate us for the use of our money, the borrower agrees to pay us interest at an agreed rate \(r\). Their interest accrues when the amount of interest earned on an investment is added to the principal. The total amount owed at the end of the loan period is called the future value \(F\) of the loan.
How much interest is earned depends on the rate, number of repayments \(m\) (more typically determined from loan duration \(T\) and repayment frequency \(T/n\)), and whether the interest is simple or compound.
1.1 Simple vs Compound Interest
In simple interest, the interest is calculated only on the principal amount
\[ F_s = P(1 + rm). \]
In compound interest, the interest is calculated on the accrued total (principal + interest)
\[ F_c = P(1 + r)^m. \]
Suppose we loan \(P=\$100\) at a monthly interest rate of \(r=1.5\%\) for \(n=12\) months. The future value of the loan under simple and compound interest is:
\[ \begin{aligned} F_s & = 100(1 + 0.015 \cdot 12) = 100(1 + 0.18) = 100(1.18) = \$118 \\ F_c & = 100(1 + 0.015)^{12} \approx 100(1.195618) \approx \$119.56. \end{aligned} \]
1.2 Annualization
In finance, interest rates are often expressed on an annual basis, even if the interest is compounded more frequently.
The nominal rate \(r^{(m)}\) is a stated annual rate that is compounded \(m\) times per year. It is quoted as a yearly figure, but interest is actually credited every \(1/m\) of a year at the periodic rate
\[ \text{periodic rate} = \frac{r^{(m)}}{m}. \]
The effective annual rate \(r\) is the single once-a-year rate that produces the same growth as the nominal rate over one year. Since \(1\) dollar grows to \(\left(1+\frac{r^{(m)}}{m}\right)^m\) after one year with \(m\) compounding periods,
\[ 1 + r = \left(1 + \frac{r^{(m)}}{m}\right)^m \quad\Longleftrightarrow\quad r = \left(1 + \frac{r^{(m)}}{m}\right)^m - 1. \]
In North American consumer finance, the effective annual rate is called APY (annual percentage yield) for savings. The nominal rate on loans is called APR (annual percentage rate), which is usually just the periodic rate times the number of periods, so it understates the true cost relative to the effective rate. The process of converting a nomial rate to an effective annual rate is called annualization. The main purpose of annualization is to provide a standardized way to compare interest rates across different financial products and investment opportunities. It allows investors and borrowers to make informed decisions by understanding the true cost or return of a financial product over a one-year period.
Suppose we take out a loan with a nominal interest rate of \(r^{(12)} = 6\%\) compounded monthly. The effective annual rate is given by
\[ r = \left(1 + \frac{0.06}{12}\right)^{ 12} - 1 \approx 0.061678 - 1 \approx 0.061678 \approx 6.17\%. \]
1.3 Continuous Compounding
Continuous compounding is the limit of the nominal-rate setup as the number of compounding periods per year \(m \to \infty\). Interest is effectively credited over infinitesimally small periods, and the effective annual rate approaches an upper limit of \(e^r - 1\).
Start with \(A = P\left(1 + \frac{r}{m}\right)^{mt}\) and take logs:
\[ \ln\frac{A}{P} = mt\,\ln\!\left(1+\frac{r}{m}\right) \;\xrightarrow{\,m\to\infty\,}\; rt, \]
since \(\ln(1+x) \sim x\) as \(x \to 0\). Exponentiating gives
\[ A = P\,e^{rt}. \]
One can reach the same result by applying L’Hôpital’s rule to the logarithm of the growth factor. There is no \(m\) in the formula: a common error is to carry \(n\) into the continuous formula, but \(e^{rt}\) already accounts for infinitely frequent compounding.
In actuarial notation, the continuously compounded rate is called the force of interest \(\delta\). The accumulation function is \(a(t) = e^{\int_0^t \delta(s)\,ds}\), which reduces to \(e^{\delta t}\) when \(\delta\) is constant. Equivalently, \(\delta\) is the logarithmic derivative of the accumulation function:
\[ \delta(t) = \frac{a'(t)}{a(t)}. \]
With a constant force of interest,
\[ 1 + i = e^{\delta}, \qquad \delta = \ln(1+i), \qquad \delta = \lim_{m\to\infty} i^{(m)}.\]
The force of interest is below the effective annual rate but above the effective discount rate. In terms of the nominal rates from before, \(i^{(m)}\) decreases toward \(\delta\) as \(m\) grows:
\[i > i^{(2)} > i^{(4)} > i^{(12)} > \cdots > \delta.\]
Converting a continuous rate to an effective annual rate (APY) is just \(e^{r}-1\). At \(r = 6\%\) this gives about \(6.184\%\). Using the 12% nominal example from earlier: calcfyi
| Compounding | Effective annual rate |
|---|---|
| Annual (\(m=1\)) | \(12.000\%\) |
| Monthly (\(m=12\)) | \(12.683\%\) |
| Continuous (\(m\to\infty\)) | \(e^{0.12}-1 \approx 12.750\%\) |
Most of the gain from more frequent compounding comes by \(m=12\). Daily and continuous compounding differ by under a dollar in one worked example, so continuous compounding is used mainly for mathematical convenience. calcfyi
Investing $1000 at 8% for 5 years gives \(1000\,e^{0.4} \approx \$1491.82\) under continuous compounding, versus $1469.33 with annual compounding. calcfyi
To find a doubling time, solve \(e^{\delta t} = 2\):
\[t = \frac{\ln 2}{\delta}.\]