Question

1. 18 pts] For this question, suppose the market interest rate is 4 percent, and round all answers to the nearest $1. a) What is the present discounted value of a perpetuity paying $100 per year, every year, with the first payment coming one year from today? b) What is the present discounted value of a perpetuity paying $100 per year, every year, with the first payment coming 6 years from today? c) What is the present discounted value of a financial instrument that pays you $100 per year, forever, starting next year, with the exception of year 6? d) (Challenging.) What is the present discounted value of a financial instrument that pays you $100 every other year, forever, starting after 2 years? (Le. you get paid after waiting 2 years, 4 years, 6, years, etc.)

Answer 1

1. Market rate of interest = r = 4%

Cash Inflow = 100 every year.

Therefore, Present discounted value of a perpetuity (does not have any maturity date and continues to give out payments) is :-

PV = 100/(1+r) + 100/(1+r)² + 100/(1+r)³ + 100/(1+r)⁴ + 100/(1+r)⁵ + ………………

Where payments are received at first after one year from today and hence each round of cash inflow is discounted by the market rate of interest raised to the number of years gone.

i.e., 100/(1+r) = the value of $ 100 received next year in today’s market; or how one needs to invest today to receive 100 next year.

= > PV = 100{ 1/(1+r) + 1/(1+r)² + 1/(1+r)³ + 1/(1+r)⁴ + 1/(1+r)⁵ + ………………}

This is a infinite GP series:-

= > PV =100/{ ( 1/(1+r))/1- ( 1/(1+r))} = 100/r =100/(4/100) = 25*100= 2500

2. Now the Present discounted value if the first round of payment comes 6 years later is nothing but the PV obtained in the previous answer – first six round of payments received

= > PV₊₆ = PV - 100/(1+r) + 100/(1+r)² + 100/(1+r)³ + 100/(1+r)⁴ + 100/(1+r)⁵

             = PV - 100{ 1/(1+r) + 1/(1+r)² + 1/(1+r)³ + 1/(1+r)⁴ + 1/(1+r)⁵}

             = PV – 100 { (1/(1+r)(1 - ( 1/(1+r)⁵ )/ 1- ( 1/(1+r)}

          = 2500 – 100 {4.4518}

          =2500 – 445.18

              = 2055

3. Now the Present discounted value if the payment comes 1 year later except the 6^th year is nothing but the PV obtained in the a)’s answer minus the sixth payment PV

PV_-6 = 2500 - 100/(1+r)⁶

        = 2500 -100/(1+0.04)⁶

       = 2421

4. Clearly,

PV = 100/(1+r)² + 100/(1+r)⁴ + 100/(1+r)⁶ + 100/(1+r)⁸ + ………………

= > PV = 100/(1+r)² { 1 + 1/(1+r)² + 1/(1+r)⁴ + ………….}

= > PV = 100/(1+r)² { 1/(1- (1/1+r))² = 100/(( 1+r)² – 1)

= > PV = 100/ (r² + 2r)

= > PV = 100/ 0.0816

= > PV = 1225.49 = 1226