Question

Karim deposits $100 every two years for 40 years into an account that earns an effective annual interest rate i. The accumulated value after 20 years is X. The accumulated value after 40 years is Y.

a) i = 2%. Find X. Find Y.

b) i is unknown, but Y = 4X. Find i. Find X.

Answer 1

The amount deposited every two years (C) = $100

Number of years = 40

The effective annual rate of interest = i

Since the amount is deposited after every two years

finding the effective two-year rate of interest (r)= (1+i)² -1

a. If i= 2%

effective two-year rate of interest = (1+i)² -1

effective two-year rate of interest (r) = (1.02)² -1 = 1.0404 -1 = 0.0404 = 4.04%

Accumulated value after 20 years = X

n = 10 ( 2year period)

Amount X after 20 years = (C/r)[(1+r)ⁿ -1]

Amount X after 20 years =(100/0.0404)[(1.0404)¹⁰ -1]

Amount X after 20 years = 2475.24*0.4859 = $1202.84

Accumulated value after 40 years = Y

n = 20 ( 2year period)

Amount X after 40 years = (C/r)[(1+r)ⁿ -1]

Amount X after 40 years =(100/0.0404)[(1.0404)²⁰ -1]

Amount X after 40 years = 2475.24*1.2080 = $2990.09

b.

If Y = 4X

from above answer

X = (C/r)[(1+r)¹⁰ -1]

Y = (C/r)[(1+r)²⁰ -1]

putting value of X and Y

(C/r)[(1+r)²⁰ -1] = 4*(C/r)[(1+r)¹⁰ -1]

(1+r)²⁰ -1 = 4*[(1+r)¹⁰ -1]

suppose (1+r)¹⁰ = P

P² -1 = 4P -4

P² -4P +3 =0

(P-3)(P-1) = 0

P= 3 or P=1

if P= 3

(1+r)¹⁰ = 3

r = 1.1161-1 = 0.1161 = 11.61%

if P= 1

(1+r)¹⁰ = 1

r = 0

taking r = 11.61%

r= (1+i)² -1

0.1161 +1 = (1+i)²

i = 1.0564 -1 = 5.64%