Question

9. a) What must be the interest rate in order for an investment of $1,000 to produce proceeds of $2,000 in 20 years?

b) If a cash flow of $2,000 in 20 years has a price today of $1,000, what must be the discount rate (i.e., the “implied” rate)?

10. An asset promises to pay $50,000 in five years and $100,000 in ten years. What is its price if the 5-year rate of discount is 10% and the 10-year rate of discount is 5%?

Answer 1

9 a) Here, P=$ 1000, A= $ 2000, T =20 years

where P is the principal (or amount invested), A is the amount received (i.e. the proceeds) and T is the time

let the rate of interest be r

therefore by the formula of compound interest,

A = P(1+r/100)^T => 2000= 1000(1+r/100)²⁰ => (1+r/100)²⁰ =2000/1000 => (1+r/100) = (2)^1/20 => 1+r/100 = 1.0353

=>r/100 = 0.0353 => r=3.53

Thus at 3.53% pa interest rate, an investment of $1000 will produce proceeds of $ 2000 in 20 years

9 b)

Here, P=$ 1000, A= $ 2000, T =20 years

where P is the price today, A is the cash flow and T is the time

let the rate of interest be r

therefore by the formula of compound interest,

P = A/(1+r/100)^T => 1000= 2000/(1+r/100)²⁰ => (1+r/100)²⁰ =2000/1000 => (1+r/100) = (2)^1/20 => 1+r/100 = 1.0353

=>r/100 = 0.0353 => r=3.53

Thus, If a cash flow of $2,000 in 20 years has a price today of $1,000, the discount rate is 3.53% pa

9 c)

Based on the formula for pricing of an asset,

PV=FV/(1+r/100)ⁿ

Where PV is the present value of the asset, FV is the future value of the asset, r is the rate of discount in percentage and n is the time period

thus considering the fact that the asset is going to pay $50000 in 5 years at a discount rate of 10%,

PV = 50000/(1+0.05)⁵ = 39176.31 i.e. $ 39176.31

and considering the fact that the asset is going to pay $100000 in 10 years at a discount rate of 5%

PV=100000/(1+0.1)¹⁰= 38554.33 i.e. $ 38554.33

Thus the asset price should be somewhere between $38554.33 and $ 39176.31