Question

You deposit $500 at the beginning of each month into your saving account every month. After five years (60 deposits total), your account value is $50,000. Assuming monthly compounding, what is your monthly rate that the bank provides?

a.   1.22%

b.   1.14

c.    1.43%

d.   1.57%

Answer 1

Amount to be invest at the BEGINNING of each year = FV of Annuity = P*[{(1+i)^n}-1]/i and FV of Single Deposit = P*[(1+i)^n]

Note: In above formula, P is the Annuity amount starting from 1 YEAR FROM NOW. Therefore, FV of Annuity starting from TODAY will be, FV of [FV of Annuity of next 6 deposits] after 1 year + FV of Today’s Deposit

Where, P = 500, FV = 50000, n = 60-1 = 59

Therefore,

50000 = [500*[{(1+i)^59}-1]/i]*[1+i] + [500*(1+i)^60]

50000/500 = [{{(1+i)^59}-1]/i}*{1+i}]+[(1+i)^60]

100 = [{{(1+i)^59}-1]/i}*{1+i}]+[(1+i)^60]

By Trial & Error,

Taking i = 1.22% = 0.0122

[{{(1+0.0122)^59}-1]/0.0122}*{1+0.0122}]+[(1+0.0122)^60] = 86.70868+2.07 = 88.7787

Taking i = 1.43% = 0.0143

[{{(1+0.0143)^59}-1]/0.0143}*{1+0.0143}]+[(1+0.0143)^60] = 92.9965+2.344= 95.34

Taking i = 1.57% = 0.0157

[{{(1+0.0157)^59}-1]/0.0157}*{1+0.0157}]+[(1+0.0157)^60] = 97.497+2.546 = 100.04

Therefore, Monthly Rate = i = 1.57%