A (yearly) cash flow stream is x=(-40,10,10,10,10,10,10). The spot rates are (yearly in percentage) s=(5.0,5.3,5.6,5.8,6.0,6.1).
Use the discount factors to determine the (net) present value of the stream.
(I got a little confused here. Should I
1. use the running present value to solve and get the PV, or
2. use the formula PV=x0 + d1x1 +...+ dnxn)?
Please help. Thank Q!
To calculate the present value (PV) of the cash flow stream, you can use the formula PV = x0 + d1x1 + d2x2 + ... + dnxn, where x0, x1, ..., xn represent the cash flows at each period, and d1, d2, ..., dn represent the discount factors.
Let's break down the calculation step by step:
1. Multiply each cash flow by its corresponding discount factor: Multiply -40 by the discount factor for year 0 (which is 1, as there is no discount for the initial cash flow). Then, multiply each cash flow of 10 by its corresponding discount factor for its respective year.
2. Add up the results from step 1: Sum all the values obtained in step 1 to calculate the present value.
Let's calculate the present value using the given information:
x0 = -40 (cash flow at year 0)
x1 = 10 (cash flow at year 1)
x2 = 10 (cash flow at year 2)
x3 = 10 (cash flow at year 3)
x4 = 10 (cash flow at year 4)
x5 = 10 (cash flow at year 5)
x6 = 10 (cash flow at year 6)
s0 (spot rate for year 0) is 5.0% = 0.05, and similarly for the other spot rates:
s1 = 0.053
s2 = 0.056
s3 = 0.058
s4 = 0.06
s5 = 0.061
Now, let's calculate the discount factors (d) using the spot rates:
d0 = 1 (no discount for year 0)
d1 = 1/(1+s1) = 1/(1+0.053) = 0.94883
d2 = 1/(1+s2) = 1/(1+0.056) = 0.94456
d3 = 1/(1+s3) = 1/(1+0.058) = 0.93947
d4 = 1/(1+s4) = 1/(1+0.06) = 0.93458
d5 = 1/(1+s5) = 1/(1+0.061) = 0.92988
Now, calculate the present value:
PV = x0 + d1x1 + d2x2 + d3x3 + d4x4 + d5x5 + d6x6
= -40 + 0.94883(10) + 0.94456(10) + 0.93947(10) + 0.93458(10) + 0.92988(10) + 0.92988(10)
= -40 + 9.4883 + 9.4456 + 9.3947 + 9.3458 + 9.2988 + 9.2988
= -40 + 56.372
Therefore, the (net) present value of the cash flow stream is 16.372.
So to answer your initial question, you should use the formula PV = x0 + d1x1 + d2x2 + ... + dnxn to calculate the present value.