Question

Problem 2 An investment is under consideration. If the total annual payments to the investment of $10,000/year is made uniformly over the year and for 10 years, compare the accumulated interest of the investment at the end of the 10 years if the payments are (1) made at the end of each year with discrete yearly interest compounding, (2) made at the end of each week with weekly discrete compounded interest and (3) made continuously with continuous interest compounding. The nominal interest rate, r, is 8%. For (3), the annual payments are assumed to be uniform and continuous throughout the year.

Answer 1

Total Investment after n payments have been made:

B_n=A(1+n)ⁿ+{(P/i)*[(1 + i)ⁿ – 1]}

Where B_n= balance at the end of period

A= Initial Payment at the beginning

P = Principal,

i = nominal annual interest rate in percentage terms, and

n = number of compounding periods.

1).   Payments are made at the end of each year with yearly interest compounding.

P= $10000

i = 8%

A= 0 ( as payments are made at the end of year 1)

n= 10

B_n= 0 + [10000/0.08]*[(1 + 0.08)¹⁰– 1]

= 125000*1.1589

= $ 144862.5

Total annual payments= $ 10000*10: = $100000

Accumulated Interest = Accumulated amount of interest & principal - total annual payments

= $ 144862.5 - $ 100000

= $ 44862.5

2).   Payments are made at the end of each week with weekly interest compounding.

P= $10000/52: P= $ 192.308

i = 8%/52 : i=0.0015

A= 0 ( as payments are made at the end of year 1)

n= 52 weeks* 10 years: n= 520

B_n= 0 + [192.308/0.0015]*[(1 + 0.0015)⁵²⁰– 1]

= 128205.33*1.1801

= $ 151295

Total annual payments= $ 192.308*52*10: = $100000

Accumulated Interest = Accumulated amount of interest & principal - total annual payments

= $ 151295 - $ 100000

= $ 51295

3). Payments are made Continously Throughout the year.

Formula= CF*( e^rt-1)/(e^r-1)

CF= Cash flows

R= rate

T= time

= 10000* ( e^0.08*10-1)/( e^0.08​​​​​​-1)

= 10000* ( 2.22554-1)/(1.08329-1)

= $ 147141.31

Total annual payments= $ 10000*10: = $100000

Accumulated Interest = Accumulated amount of interest & principal - total annual payments

= $ 147141.31 - $ 100000

= $ 47141

So, Weekly compounding interest pays the highest accumulated interest.