Nonannual compounding:

Question

A) You plan to make five deposits of $1000 each, one every six months , with the first payment being made In 6 months. You will then make no more deposits. If the bank pays 4% nominal interest, compounded semiannually, how much will be in your account after 3 years?

B) One year from today you must make a payment of $10,000. To prepare for this Payment, you plan to make two equal quarterly deposits (At the end of the quarters 1 and 2) in a bank that pays 4% nominal interest compounded quarterly. How large must each of the two payments be?

Answer 1

A) At the end of the 5 Payments in 3 years:

1. Payment 1 has been sitting in the bank for 5*6 months (3 years is 36 months, and you first deposited after six months, so it has been 30 months) = 1000*(1 + .04/2)^5 = 1000*(1.02)^5
2. Payment 2 has been siting in the bank for 4*6 months = 1000*(1.02)^4
3. Payment 3 has been sitting in the bank for 3*6 months = 1000*(1.02)^3
4. Payment 4 has been sitting in the bank for 2*6 months = 1000*(1.02)^2
5. Payment 5 has been sitting in the bank for 6 months = 1000*(1.02)
6. Total in your account after 3 years = 1000*(1.02^5 + 1.02^4 + 1.02^3 + 1.02^2 + 1.02) = $5,308.12

After 3 years, your account will have $5,308.12

B)

X*(1 + .04/4)^3 + X*(1 +04/4)^2 = $10000

X*(1.01)^3 + X*(1.01)^2 = 10000

X*(1.01^3 + 1.01^2) = 10000

X = 10000 / (1.01^3 + 1.01^2)

X = $4877.09

Each of the two payments must be $4,877.09.

Answer 2

A) Interest calculation

6 mos

Deposit: $1,000.00

Balance: $1,000.00 $1,000.00 x 0.02 = $20.00

12 mos

Interest: $20.00

Deposit: $1,000.00

Balance: $2,020.00 $2,020.00 x 0.02 = $40.40

18 mos

Interest: $40.40

Deposit: $1,000.00

Balance: $3,060.40 $3,060.40 x 0.02 = $61.21

24 mos

Interest: $61.21

Deposit: $1,000.00

Balance: $4,121.61 $4,121.61 x 0.02 = $82.43

30 mos

Interest: $82.43

Deposit: $1,000.00

Balance:$5,204.04

This could have been obtained also by using the ordinary annuity table for five periods @ 2.0%. From this point forward, the amount is a single sum for five periods at 2.0%. The factor is (1.02)⁵ = 1.10408.

$5,204.04 x 1.10408 = $5,745.68.

B) The sum that must be in place after the second payment is $10,000.00/(1.01)^{2 =} $10,000.00/1.0201 = $9,802.95. This amount is equal to two payments plus interest on the first: (P x 1.01) + P = 2.01P.

2.01P = $9,802.95. Solving for P: P = $9,802.95/2.01 = $4,877.09