Question

2. Consider the following cash flows. All market interest rates are 12%. Year 0 1 160 2 170 3 180 4 230 Cash Flow a. What price would you pay for these cash flows? What total wealth do you expect after 1% years if you sell the rights to the remaining cash flows? Assume interest rates remain constant. (5 points) b. What is the duration of these cash flows? (5 points) C. Immediately after buying these cash flows, all market interest rates drop to 11%. What is the impact on your total wealth after 1.5 years? (5 points)

Answer 1

We need to find the present value of the cashflows by discounting the cash flows with the given interest rate of 12%

formula to calculate PV = C1/(1+r) + C2/(1+r)² + C3/(1+r)³ + C4/(1+r)⁴  

in our case PV = 160/(1+12%) + 170/(1+12%)² + 180/(1+12%)³ + 230/(1+12%)⁴ = $552.6697

Price to pay for these cash flows = $552.6697

After 1 1/2 years, we will again calculate the present value of the remaining cash flows. In this case the first cash flow of $170 will occur after 1/2 year, second cash flow of $180 will occur after 1.5 yrs, and the third after 2.5 years.

formula to calculate PV = C2/(1+r)^0.5 + C3/(1+r)^1.5 + C4/(1+r)^2.5

substituting the remaining cash flow and time period in the formula above, we get

PV = 170/(1+12%)^0.5 + 180/(1+12%)^1.5 + 230/(1+12%)^2.5 = 160.6349+151.5607+173.2508 = $485.7494

Total wealth after 1 1/2 years = $485.7494

Sup part b:

Duration it is the measure of how long, on an average the holder of the bond has to wait before he/she receives his/her payments on the bond.

We will calculate Macaulay‟s duration which is the weighted average of the times when the payments are made.

Formula for duration = =

{"=1t*C/(1+r) PV of cash flows

=(1*C1/(1+r)¹ + 2*C2/(1+r)² + 3*C3/(1+r)³ + 4*C4/(1+r)⁴ ) / PV of cash flows

= (1*160/(1+12%)¹ + 2*170/(1+12%)² + 3*180/(1+12%)³ + 4*230/(1+12%)⁴ ) / 552.6697

= (142.8571429 + 271.0459184 + 384.3613338 + 584.6766321) / 552.6697

= 1382.941027 / 552.6697

= 2.502292084 = 2.5023

Duration of the cash flows = 2.5023

Sub-part c:

We will use the same approach as sub part 2 and replace the interest rate from 12% to 11%

formula to calculate PV = C2/(1+r)^0.5 + C3/(1+r)^1.5 + C4/(1+r)^2.5

substituting the remaining cash flow and time period in the formula above, we get

PV = 170/(1+11%)^0.5 + 180/(1+11%)^1.5 + 230/(1+11%)^2.5

= 161.3568593 + 153.9175128  + 177.1823221 = 492.4567

Impact on the total wealth after 1.5 years = Value of wealth at 11% - Value of wealth at 12 % -

= 492.4567 - 485.7494 = $6.7073

Impact : The wealth increases by $6.7073