Financial: Compound Interest

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The idea of compound interest is that we invest money at some annual rate. However, at regular intervals the amount of interest earned is computed and paid back into the account holding our original investment and all of the previously earned interest. By doing this we earn interest on the interest that was previously paid. Let us look at the example in Table 1.

Let us look at what is happening with compound interest. This time we will look at it using variables instead of numbers. We will say that PVi is the present value at the start of period i. Thus, PV1 is the present value at the very start. The interest rate will be r, given as an annual rate in decimal form. The variable t will hold the number of compounding periods in a year. Thus, if we are compounding every 3 months we would make t=4. We will say that FVi is the future value at the end of period i. Thus, FV1 is the future value at the end of the first period. The relationship is
FV1 = PV1(1 + r/t)
Of course, the future value at the end of period 1 is exactly the present value at the start of period 2. Therefore,
PV2 = PV1(1 + r/t)
To find FV2 we compute
FV2 = PV2(1 + r/t)
but we know that PV2=PV1(1 + r/t) so we can rewrite that as
FV2 = (PV1(1 + r/t))*(1 + r/t)
which we could simplify to
FV2 = PV1(1 + r/t)2
Of course, the future value at the end of period 2 is exactly the present value at the start of period 3. Therefore,
PV3 = PV1(1 + r/t)2
To find FV3 we compute
FV3 = PV3(1 + r/t)
but we know that PV3=PV1(1 + r/t)2 so we can rewrite that as
FV3 = (PV1(1 + r/t)2)*(1 + r/t)
which we could simplify to
FV3 = PV1(1 + r/t)3
Of course, the future value at the end of period 3 is exactly the present value at the start of period 4. Therefore,
PV4 = PV1(1 + r/t)3
To find FV4 we compute
FV4 = PV4(1 + r/t)
but we know that PV4=PV1(1 + r/t)3 so we can rewrite that as
FV4 = (PV1(1 + r/t)3)*(1 + r/t)
which we could simplify to
FV4 = PV1(1 + r/t)4


At this point the pattern should be quite clear. The equation for the future value at the end of n periods is
FVn = PV1(1 + r/t)n


The formula
FVn = PV1(1 + r/t)n
is great if we are asked to find the future value when we know the present value, the annual rate, the number of periods in a year, and the number of compounding periods.

We could solve that equation for PV1 to get
PV1 = FVn/(1 + r/t)n
which would be helpful if we want to find the present value that will give us a specific future value at a given rate, frequency (i.e., times per year), and number of compounding periods.

Or, we could solve the equation for r to get
r = ((FVn/PV1 )1/n - 1)*t
which would be helpful if we wanted to find the rate that is required to grow a known present value to a desired future value when we know that we will be compounding a certain number of times per year and we want this to be done in a given number of compounding periods.

Or, we could solve the equation for n. This, however, requires the use of either one of the logarithm functions, ln() or log(). The result is
n = ln(FVn/PV1) / ln(1 + r/t) = log(FVn/PV1) / log(1 + r/t)
which would be helpful if we wanted to find the number of compounding periods required to grow a known present value to a desired future value when we know that we will be compounding a certain number of times per year and we know the annual interest rate.

Figure 01
Find the future value when we invest $2500 at 2.7% the compounded monthly for 5 years (i.e., 60 months). Use the formula
FVn = PV1(1 + r/t)n
Figure 01 shows evaluating the right side of the formula for those values to get an answer of $2860.91.
Figure 02
Find the present value that will give us $4500 at the end of 7 years if we invest that present value at 3.75% compounded daily. [Note that for these kinds of calculations it is common to use 365 as the number of days in a year.] Use the formula
PV1 = FVn/(1 + r/t)n
Figure 02 shows evaluating the right side of the formula for those values to get an answer of $3461.12.
Figure 03
Find the rate that is required to grow $4250 to a 5900 over 8 years compounded monthly. Use the formula
r = ((FVn/PV1 )1/n - 1)*t
Figure 03 shows evaluating the right side of the formula for those values to get an answer of 0.041074 or about 4.1%.
Figure 04
Find the number of compounding periods required to grow $4250 to $5900 when we know that we will be compounding monthly at 4% annual interest. Use either of the two versions of the formula, one using the natural logarithm, ln() and the other using the base-10 logarithm, log()
n = ln(FVn/PV1) / ln(1 + r/t) = log(FVn/PV1) / log(1 + r/t)
Figure 03 shows evaluating the two versions of the formula for those values to get an answer of 98.5739, which means 8 years and just over 2 and a jhalf months.


Just as we saw for simple interest, the four Figures above can be used as models for the four different types of compound interestproblems. However, to do this we need to recall and fill in appropriately the four different equations that we described above. A more elegant method is to create a program that will ask the user for the approarite data and then evaluate the appropriate expression to get the desired answer. Here is such a program, named COMPOUND:
Then we can use that program to solve the problems that we saw in the figures above.
Figure 05
Once the program starts we see the menu created by line 10 in the program.
Figure 06
Choosing the first option takes us to line 11 and from there to line 12 to display "FIND FUTURE", and then the calculator prompts us with the contents of Str1, leaving us at Figure 02.
Figure 07
Responding to the prompt with 2500 and then the key, the program stores the 2500 in the variable P and then goes on to ask for the rate. In Figure 07 we see the status of the program after we have given it all the requested information. Press to have the calculator accept the last value and move on to display the answer.
Figure 08
Figure 08 shows that answer. After verifying that this is the same result that we had in Figure 1, we press and the calculator returns to the menu, shown in Figure 09.
Figure 09
Now we can select the second option.
Figure 10
That brings us to Figure 10 where we have entered all the values in the second problem from Figure 02. Again, press to continue.
Figure 11
The answer shown in Figure 11 is the same as the one we found in Figure 02.
Figure 12
Without showing the return to the menu and then moving forward, Figure 12 shows the values and answer for the third type of problem.
Figure 13
Figure 13 shows the values and the answer for the fourth kind of problem.
Figure 14
And, Figure 14 shows the result of selecting the QUIT option in the menu.


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©Roger M. Palay     Saline, MI 48176     March, 2017