We can imagine dividing the year into smaller and smaller periods, and thereby apply compounding monthly, weekly, daily, or even every minute or second. This leads
Interest rate (%)
Nominal 1.00 5.00 10 00 20.00 30 00 50 00 75 00 100 00 Effective 1 01 5 13 1052 22.14 34.99 64 87 11170 171.83
The nominal interest rales in the top row correspond, under continuous compounding. to the effective ratea shown in the second row The increase due to compounding is quite dramatic at large nominal rates to the idea of continuous compounding, We can determine the effect of continuous compounding by considering the limit of ordinary compounding as the number m of periods in a year goes to infinity, To determine the yearly effect of this continuous compounding we use the fact that lim [1 + (r/m)]m = er oo where e — 2.7818,... is the base of the natural logarithm, The effective rate of interest /•' is the value satisfying 1 4- r' = er, If the nominal interest rate is 8% per year, then with continuous compounding the growth would be ¿j08 = 1.08.3.3, and hence the effective interest rate is 8.33%, (Recall that quarterly compounding produces an effective rate of 8,24% ) Table 2,1 shows the effect of continuous compounding for various nominal rates Note that as the nominal rate increases, the compounding effect becomes more dramatic
We can also calculate how much an account will have grown after any arbitrary length of time, We denote time by the variable /, measured in years. Thus / = 1 corresponds to 1 year, and t = „25 corresponds to 3 months. Select a time t and divide the year into a (large) number m of small periods, each of length 1 fm. Then i ~ kjm for some k, meaning that k periods approximately coincides with the time t. If m is very large, this approximation can be made very accurate Therefore k ~ mt Using the general formula for compounding, we know that the growth factor for k periods is
[| + (,-/„,)]* = [| + (, fm)]»» — j[l + (, /m)f J'
where that last expression is valid in the limit as w goes to infinity, corresponding to continuous compounding. Hence continuous compounding leads to the familiar exponential growth curve Such a curve is shown in Figure 2 2 for a 10% nominal interest ¡ate
We have examined how a single investment (say a bank deposit) grows over time due to interest compounding It should be clear that exactly the same thing happens to debt If I borrow money from the bank at an interest rate r and make no payments to the bank, then my debt increases according to the same formulas. Specifically, if my debt is compounded monthly, then after k months my debt will have grown by a factor of [1 +0 /12)]*
FIGURE 2.2 Exponential growth curve; continuous compound growth,. Under continuous compounding at 10%, the value of $1 doubles in about 7 years In 20 years it grows by a factor of about 8
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