A series of cash flows are established on an account as listed below:

Question

A series of cash flows are established on an account as listed below:

1. Payment of P375 at the end of the 2nd, 5th, and 7th year at an interest rate of 9% compounded semi-annually.
2. Payment of P220 at the end of 2nd, and 6th year at an interest rate of 7.5% compounded quarterly.
3. Withdrawal of P640 at the end of 3rd, and 5th year at an interest rate of 5% compounded semi-quarterly.

Determine the following:

1. Total present worth of all cash flows. Is the sum a positive net cash flow or a negative net cash flow? 

2. Future worth of all the payments after 10 years? 

3. To balance the cash flows, what amount should be added at the end of the 4th year at an interest rate of 10% compounded annually? 

Answer 1

Let

i = interest rate

r = effective annual rate

n = number of compounding periods per year

r = ( 1 + i /n )ⁿ - 1

For interest rate of 9%, compounded semi-annually,

i = 0.09

n = 2 (because compounded twice per year)

Therefore, effective annual rate = ( 1 + 0.09 /2 ) ² - 1 = 0.092 or 9.2 %

Net present worth for payment of 375 at the end of 2nd year = 375 / (1 + 0.092)²

Net present worth for payment of 375 at the end of 5th year = 375 / (1 + 0.092)⁵

Similarly, for interest rate of 7.5%, compounded quarterly, effective annual rate = (1 + 0.075 /4 )⁴ - 1 = 0.077 or 7.7 %

Similarly, for interest rate of 5%, compounded semi-quarterly, effective annual rate = (1 + 0.05 /8 )⁸ - 1 = 0.051 or 5.1%

Using this logic, net present worth can be calculated for payments and withdrawals

Net present worth = 375 / (1 + 0.092)² + 375 / (1 + 0.092)⁵ + 375 / (1 + 0.092)⁷

+ 220 / (1 + 0.077)² + 220 / (1 + 0.077)⁶ – 640 / (1 + 0.051)³ – 640 / (1 + 0.051)⁵