3 Nominal Interest and Discount Rates
Quoted annual rates of interest frequently do not refer to the annual effective rate. ## Nominal Interest Rates A nominal annual rate of interest compounded or convertible m times per year refers to an interest compounding period of \(\frac{1}{m}\) years. \[\text{interest rate for $\frac{1}{m}$ year period} = \frac{\text{quoted nominal annual interest rate}}{m}\] We denote the nominal annual interest rate compounding m-times per year to be \(i^{(m)}\). The compound rate per period is \(\frac{i^{(m)}}{m}\), and the compound frequency is \(m\). ### PV and AV Using a Nominal Interest Rate The accumulated value is the present value times an accumulation factor: \[AV_t = PV_0 \bigg(1 + \frac{i^{(m)}}{m}\bigg)^{mt}\] Example: Tom is trying to decide between two banks in which to open an account. Bank A offers an annual rate of 15.25% with interest compounded semiannually, and Bank B offers an annual rate of 15% with interest compounded monthly. Which bank will give Tom higher annual effective growth?
Solution: Bank A pays an effective 6-month interest rate of \(\frac{1}{2}15.25\% = 7.625\%\) . In one year (two effective interest periods) a deposit of amount 1 will grow to \((1.07625)^2 =1.158314\) in Bank A. Bank B pays an effective monthly interest rate of \(\frac{1}{12}(15\%) =1.25\%\). In one year (12 effective interest periods) a deposit of amount 1 will grow to \((1.0125)^12 =1.160755\) in Bank B. Thus, Bank B has an equivalent annual effective rate that is almost 2.5% higher than that of Bank A.
3.0.1 Equivalent Nominal Interest Rates
If \(i^{(m)}\) is equivalent to \(i^{(p)}\) then the accumulation factor for one unit of time is: \[ \bigg(1 + \frac{i^{(m)}}{m}\bigg)^m = \bigg(1 + \frac{i^{(p)}}{p}\bigg)^p\] Setting \(p=1\), we have the following relationship: \[1 + i = \bigg(1 + \frac{i^{(m)}}{m}\bigg)^m\] Solving for the nominal interest rate, we have: \[i^{(m)} = \big[(1+i)^{1/m} - 1\big] \times m\] When the interest rates are annual rates, the nominal interest rate is known as the annual percentage rate (APR) and the annual effective interest rate is known as the annual percentage yield (APY). \[\begin{align} i^{(m)} &= \text{APR} = \text{nominal} \\ i &= \text{APY} = \text{effective} \end{align}\] Example: Suppose the annual effective interest rate is 12%. Find the equivalent nominal interest rate for \(m =1,2,3,4,6,8,12,52,365, \infty\).
Solution: Using the relationship above, we have the following table
| m | \((1+i)^{1/m} - 1\) | \(i^{(m)} = m[(1+i)^{1/m} - 1]\) |
|---|---|---|
| 1 | .1200 | .12 |
| 2 | .0583 | .1156 |
| 3 | .0385 | .1155 |
| 4 | .0287 | .1149 |
| 6 | .0191 | .1144 |
| 8 | .0143 | .1141 |
| 12 | .0095 | .1139 |
| 52 | .00218 | .1135 |
| 365 | .000311 | .113346 |
| \(\infty\) | \(\lim m[(1 + i)^{1/ m} - 1] = \ln(1+i) = .113329\) |
The limiting case \(m\to\infty\) is called continuous compounding and is related ot the notions of force of interest and instantaneous growth rate of an investment. ## Nominal Interest Rates ### Definition - Nominal Annual Rate of Discount A nominal annual rate of discount compounded or convertible m times per year refers to a discount compounding period of \(\frac{1}{m}\) years. \[\text{discount rate for $\frac{1}{m}$ year period} = \frac{\text{quoted nominal annual discount rate}}{m}\] We denote the nominal annual discount rate compounding m-times per year to be \(d^{(m)}\). The periodic effective discount rate is \(\frac{d^{(m)}}{m}\), and the frequency of discount is \(m\). ### PV and AV Using a Nominal Discount Rate The present value is the accumulated value times a discount factor: \[ PV_0= AV_t \bigg(1 - \frac{d^{(m)}}{m}\bigg)^{mt}\] ### Equivalent Nominal Interest Rates If \(d^{(m)}\) is equivalent to \(d^{(p)}\) then the discount factor for one unit of time is: \[ \bigg(1 - \frac{d^{(m)}}{m}\bigg)^m = \bigg(1 - \frac{i^{(p)}}{p}\bigg)^p\] Setting \(p=1\), we have the following relationship: \[v = 1 - d = \bigg(1 - \frac{d^{(m)}}{m}\bigg)^{m}\] Solving for the nominal discount rate, we have: \[d^{(m)} = \big[1 - (1-d)^{1/m}\big] \times m = \big[1-v^{1/m}\big]\times m\] Example: Suppose that the annual effective rate of discount is \(d=.107143\). Find the equivalent nominal annual discount rates \(d^{(m)}\) for \(m=1,2,3, 4,6,8,12,52,365,\infty\).
Solution:
| m | \(1-(1-d)^{1/m}\) | \(d^{(m)} = m[1-(1-d)^{1/m}]\) |
|---|---|---|
| 1 | .107143 | .107143 |
| 2 | .0551 | .1102 |
| 3 | .0371 | .1112 |
| 4 | .0279 | .1117 |
| 6 | .0187 | .1123 |
| 8 | .0141 | .1125 |
| 12 | .0094 | .1128 |
| 52 | .0022 | .1132 |
| 365 | .0003 | .11331 |
| \(\infty\) | \(\lim m[1-(1 - d)^{1/ m}] = -\ln(1-d) = .113329\) |
3.1 Equivalent Nominal Interest and Discount Rates
Setting the accumulated value of \(1\) under compound interest equal to the accumulated value under compound discount, we have \[\bigg(1+\frac{i^{(m)}}{m}\bigg)^{mt} = \bigg(1-\frac{d^{(p)}}{p}\bigg)^{-pt}\] We notice, as with \(i\) and \(d\), the relationship between \(i^{(m)}\) and \(d^{(p)}\) does not depend on \(t\).
3.1.1 Converting Between Different Nominal Rates
The nominal interest rates and the nominal discount rates are related as follows: \[1+i=\bigg(1+\frac{i^{(m)}}{m}\bigg)^{m} = \bigg(1-\frac{d^{(p)}}{p}\bigg)^{-p} = 1-d\] If a nominal interest rate is convertible at the same frequency a s a nominal discount rate, then we can simplify the expression above to: \[1 + \frac{i^{(m)}}{m} = \bigg(1 - \frac{d^{(m)}}{m}\bigg)^{-1}\] Note that this expression is analogous to \(1+i = (1-d)^{-1}\), derived in the previous section. The expression is also equivalent to: \[\frac{d^{(m)}}{m} = \frac{[i^{(m)}/m]}{1+[i^{(m)}/m]}, \qquad \frac{i^{(m)}}{m} = \frac{[d^{(m)}/m]}{1-[d^{(m)}/m]}\] ### Properties For \(m > n\), we have - \(i^{(m)} < i^{(n)}\) (\(\{i^{(n)}\}\) is a decreasing sequence) - \(d^{(m)} > d^{(n)}\) (\(\{d^{(n)}\}\) is an increasing sequence) - \(i^{(m)} \searrow i^{(\infty)} = d^{(\infty)} \nwarrow d^{(m)}\) - \(i^{(\infty)} = d^{(\infty)} = \ln(1+i) = -\ln(1-d)\); and this equals to the force of interest