Compounding interest

[gemini070900]

We approximate the smaller difference between regular compounding and continuous compounding . With a nominal annual rate r over t years, continuous compounding yields ert , while compounding m times per year yields (1+r/m)mt, for a difference of D=ert-(1+r/m)mt that can be approximated by

D=r2 te rt
----------
2m +4r/3 + r2t/2

For a $ 1000.00 initial investment, calculate both the true difference and the approximate difference between continuous compounding and :

A) quarterly compounding at a nominal rate of 4 % for 10 yrs

B) daily compounding at a nominal rate of 18 % for 10 yrs

 

[Denis]

A) quarterly compounding at a nominal rate of 4 % for 10 yrs
B) daily compounding at a nominal rate of 18 % for 10 yrs

A: (1 + .04/4)^(10*4)

B: (1 + .18/365)^(10*365) ; may be 360 days in some cases...

 

[mmm4444bot]

continuous compounding yields ert , while compounding m times per year yields (1+r/m)^(mt), for a difference of D = ert - (1 + r/m)^(mt) that can be approximated by

Please note my changes (in red) to your typing above. It is very important to use the caret symbol ^ (shifted 6, on most keyboards) to indicate exponents. If an exponent is some algebraic expression, we need to put parentheses around it.

D=r2 te rt
     ----------
      2m +4r/3 + r2t/2

What is that supposed to be?

I mean, it's a formula for D, but it looks garbled!

Let's not use the same symbol to represent two different definitions. Otherwise, were we to look at the symbol D, how would we know whether it's the true difference or the approximate difference? We need to pick a different symbol for the approximate difference. How about a lower-case d, instead.

Let D = the true difference

Let d = the approximate difference

So, here is what I think your typing above means for the approximate difference formula:

\(\displaystyle d \ = \ \frac{r^2 \cdot t \cdot e^{rt}}{2m + \frac{4r}{3} + \frac{r^2 t}{2}}\)

?

For a $ 1000.00 initial investment, calculate both the true difference and the approximate difference between continuous compounding and :

A) quarterly compounding at a nominal rate of 4 % for 10 yrs

B) daily compounding at a nominal rate of 18 % for 10 yrs

I'll set up (A).

The formula for the true difference between continous compounding and quarterly compounding:

D = e^(r*t) - (1 + r/m)^(m*t)

To evaluate D, we substitute the Real numbers r = 0.04, t = 10, m = 4 into the formula.

D = e^(0.04*10) - (1 + 0.04/4)^(4*10)

Now, use a calculator, to get a decimal number for D.

The formula for the approximate difference between continous compounding and quarterly compounding:

d = [r^2 * t * e^(r*t)]/[2*m + 4*r/3 + r^(2*t)/2]

d = [0.04^2 * t * e^(0.04*10)]/[2*4 + (4*0.04)/3 + 0.04^(2*10)/2]

Now, use a calculator, to get a decimal number for d.

I just compared the values for D and d, and I see that d is a good approximation for D; their first three significant digits are equal.

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