Question

Nandana invests $500 at the start of each year for 20 years in a bank account paying interest at the effective annual rate i. She takes the interest paid at the end of each year and invests it in a different account paying an effective annual rate i/2. The effective annual rate she earns on her combined investments is 6%.

a) How much money does she have at the end of 20 years? (Total of both accounts.)

b) What is i?

please show full work done with all formulas

Answer 1

a. given effective annual rate on combined investment is 6%.

using FV = PV(1+r)^n = 500*(1+.06)^21 = 1699.782. since the invesment is at the start of year.

b. the formula used above has the assumptions that the interest earned is reinvested at the same rate. if the reinvestment rate for interest earned is different from investment interest rate, then the problem would be like the following.

at 20th year, investment of year 1 would be 500 + (500*i*(1+i/2))^19,

at 20th year, cumulative investment of year 1 and 2 would be 500*2 + (500*2*i*(1+i/2))^18

so the sum of all this investments would be like

500*n + sum of ((20-n)*(1+i/2)^n*i) where n variies from 1 0 to 20

this equals to the above answer 1699.782

solving for the problem would be i.

the numbers are in combination of ap * gp, hence formula needs to be derived.