Question

Nataya makes monthly deposits into Fund A at the beginning of each month for 25 years. Her first deposit is
$40, and each subsequent deposit increases by $10. Aarif contributes $250,000 once, today, into Fund B. The
nominal rate that both of these funds earn is 6% per year. However, Fund A compounds interest monthly
and Fund B compounds interest quarterly. What is the difference in the balance between these two funds at
the end of 25 years?

Answer 1

Fund A:
Stream can be broken into two streams
First stream is constant payment of 40 per month i.e., annuity due of 40

Present Value=40/(6%/12)*(1-1/(1+6%/12)^(12*25))*(1+6%/12)=6239.32

Second stream is arithmetic gradient annuity of 10 per month

The above formula is for ordinary annuity but we have annuity due hence multiplied by (1+periodic rate)

Present Value=10/(6%/12*(1+6%/12)^(12*25))*(((1+6%/12)^(12*25)-1)/(6%/12)-12*25)*(1+6%/12)=176914.49

Total Present Value=6239.32+176914.49=183153.81

Total Future Value=Present value*(1+periodic rate)^number of periods=183153.81*(1+6%/12)^(12*25)=817776.23

Future Value in Fund B=Future value of lump sum=Present Value*(1+r/m)^(m*n)=250000*(1+6%/4)^(4*25)=1108011.412

Difference in the balance=1108011.412-817776.23=290235.182