2-4 How Interest Is Paid and Quoted

semianually = twice a year

If the interest is 5% compounded semiannually, it is equivalent to 10.25% compounded anually. The effective annual interest rate on the bond is 10.25%. -> 100$ x 1.05 x 1.05 = 110,25$.

APR = annual percentage rate

If for example could be that a loan can have an APR of 12% with interest to be paid monthly. By this the bank means that each month you need to pay one-twelth of the annual rate. So each month you will have to pay 12% / 12 = 1% interest a month. The bank is quoting a rate of 12%, but the actual annual interest rate on the loan is ${1.01}^{12}-1=12,68$.

Always distinguish between the quoted annual interest rate and the effective annual rate. This is because the annual rate is often calculated as the total annual payment divided by the number of payments in the year. When interest is paid once a year, the quoted rate = the effective rate. In the case of interest being paid more frequently -> effective interest rate > quoted interest rate.

If you invest $1 at a rate of r per year compounded m times a year. Your investment at the end of the year will be worth $ [1 + (r / m)] ^ m $ and the effective interest rate is $ [1 + (r / m) ] ^ m - 1 $.

🔁 Continious Compounding

There is no limit to how frequently interest could be paid. It could be weekly (m = 52) or even daily (m = 365). There is no limit to how frequently interest can be paid.

When m approaches $\infty$ the interest rate $[1+(r/m){]}^m$ approaches $e^r$(the base for natural logarithms). So $1 invested at a continiously compounded rate of r will grow to $e^r$ by the end of the first year. By the end of t years it will grow to $e^{r\ast t}$.

Example: 17% continuosly compounded = $e^{0.17}=1.185=18,5$. The PV(present value) of the continuous cash flow stream is 100/.17 = 588,24$.

Investors are willing to pay more for the continuous cash payments because the cash starts to flow instantly.