Monthly Rates, etc.
Kerry Back
Monthly rates
- If we earn \(r_m\) each month, then $1 will grow in a year to $\((1+r_m)^{12}\).
- So, the annual return (gain per $1) is \(r_a = (1+r_m)^{12}-1\).
- For example, 1% per month corresponds to 12.68% per year.
- We can reverse this to compute the monthly return from the annual return as \(r_m = (1+r_a)^{1/12} - 1\).
- 12% per year corresponds to 0.95% per month.
Bank rates
- If a bank says it will loan you money at 12% per year with monthly payments, it will charge you 1% per month.
- Bank formula is \(r_m = r_a/12\).
Frequent compounding at bank rates
- 12% per year compounded monthly at the bank formula implies an annual rate of \(1.01^{12}-1 = 0.1268\)
- 12% per year compounded weekly at the bank formula implies an annual rate of \((1+0.12/52)^{52}-1 = 0.1273\)
- 12% per year compounded daily at the bank formula implies an annual rate of \((1+0.12/365)^{365} = 0.12747\)
- More frequent compounding \(\rightarrow\) higher effective annual rate
Continuous compounding
A useful fact: as \(n \rightarrow \infty\), \[(1+r/n)^n \rightarrow e^r\]
- We say that an annual rate of \(r\) compounded continuously produces an effective annual rate of \(e^r-1\).
- For example, 12% compounded continuously implies an effective annual rate of \(e^{0.12}-1 = 0.12750\).
- Given an effective annual rate of \(r_a\), the corresponding continuously compounded rate is \(\log (1+r_a)\).