Okay, so once again I've been having troubles with a couple of questions and need help! They need to be solved using either the steps of TVM (total value of

Quote:

Okay, so once again I've been having troubles with a couple of questions and need help! They need to be solved using either the steps of TVM (total value of money calculator) or using the equations A=P(1+i)^n. Here are those questions:

1. Barb invests $15,000 for 7 years and it doubles in value. What interest rate, compounded quarterly was the money invested at?

2. How much longer will it take $25,000 to amount to $100,000 if it is invested at 5% compounded annually compared to 8% quarterly?

1) 30000 = 15000 (1+r/4)^(7*4)

2 = (1+r/4)^28

2^(1/28) = 1 + r/4

2^(1/28)-1 = r/4

r = 4(2^(1/28) - 1) =10.026%

2)

let the 25000 be invested for n years at 5%
100000 = 25000 (1+0.05)^n

4 = (1.05)^n

have to take logs to solve this one

log(4) = n log(1.05)

n = log(4)/log(1.05) = 28.4 yrs so you would need 29 years to get to 100k

now let it be invested for k quarter years at 8% compounded quarterly

100000 = 25000 (1 + 0.08/4)^k

4 = 1.02^k

k = log(4)/log(1.02) = 70 quarter years = 17.5 yrs

the difference is 29 - 17.5 = 11.5 yrs

I am wanting to expand my answer to this question so how would someone explain and solve the questions I posted above keeping in mind these: compare simple and compound interest, relate compound interest to exponential growth.

Would you still solve the questions the same way and if so how would you include the comparisons?

Compound interest just takes a given annual rate, divides it by some number of periods per year K, and applies that new interest rate K times a year.

The formula for the future value, given rate r, over N years, compounded K times a year is

FV = PV*(1+r/K) ^KN

To compare, let's say you have 5% annual interest

Compounded annually you would have, over say 30 years

FV = PV*(1+0.05) ³⁰

FV/PV = 4.322

Compounded quarterly you would have

FV = PV*(1+0.05/4) ^4*30 = PV*(1+0.0125) ¹²⁰

FV/PV = 4.440

For equal rates and years and present values compounding yields a higher future value.

Now how does this relate to exponential growth?

If you recall exponential growth is described by

f(t) = f0*e ^g*t where f0 is the initial value of f and g describes the growth or decay rate. Positive g is growth, negative g is decay. A higher value of g leads to faster growth or decay.

now rewrite your compound interest formula as a function of t years

FV(t) = PV*(1+r/K) ^K*t

remembering that b ⁿ = e ^n*log(b) we get

FV(t) = PV*e ^{log(1+r/K)*K*t}

i.e. FV(t) is exponential growth with

f0 = PV
g = log(1+r/K)*K