How could you use Equation 5.2 to find the PV of an uneven stream of cash flows?
ID: 2773762 • Letter: H
Question
How could you use Equation 5.2 to find the PV of an uneven stream of cash flows? What s the present value of a 5-year ordinary annuity of $100 plus an additional $500 at the end of Year 5 if the interest rate is 6%? Vs/hat is the PV 'if the $A00 payments occur in Years 1 through 10 and the $500 comes at the end of year 10? ($794.87; $1,015.21) What's the present value of the following uneven cash flow stream- $0 at Time 0, $ 100 in Year 1 (or at Time 1), $200 in Year 2, $0 in Year 3, and $400 in Year 4 if the interest rate is 8%? ($558.07) Would a typical common stock provide cash flows more like an annuity or more like an uneven cash flow stream? Explain.Explanation / Answer
PV of n uneven cash stream formula is as follows
For the 1st part
r = 6% = 0.06
C1 = Cash Flow at the end of 1st year = 100
C2= Cash Flow at the end of 2nd year = 100
C3= Cash Flow at the end of 3rd year = 100
C4= Cash Flow at the end of 4th year = 100
C5= Cash Flow at the end of 5th year = 100 + 500 (additional) = 600
Hence NPV = C1/(1+r)1+ C2/(1+r)2 + C3 / (1+r)3 + C4 / (1+r)4 + C5/(1+r)5
= 100/1.06 + 100/(1+0.06)2 + 100 /(1+0.06)3 + 100/(1+0.06)4+600/(1+0.06)5
= 100/1.06 + 100 /1.1236 + 100 /1.191 + 100 /1.2624 + 600/1.338
= 94.34+89.00+83.96+79.21+448.43
= 794.94
0.07 difference from the given answer due to rounding off
If additional $ 500 i received in year 10 , then NPV is as follows
C1 = Cash Flow at the end of 1st year = 100
C2= Cash Flow at the end of 2nd year = 100
C3= Cash Flow at the end of 3rd year = 100
C4= Cash Flow at the end of 4th year = 100
C5= Cash Flow at the end of 5th year = 100
C6 = Cash Flow at the end of 6th year = 100
C7= Cash Flow at the end of 7th year = 100
C8= Cash Flow at the end of 8th year = 100
C9= Cash Flow at the end of 9th year = 100
C10= Cash Flow at the end of 10th year = 100 + 500 (additional) = 600
Hence NPV = C1/(1+r)1+ C2/(1+r)2 + C3 / (1+r)3 + C4 / (1+r)4 + C5/(1+r)5 + C6/(1+r)6+ C7/(1+r)7+ C8/ (1+r)8+ C9/ (1+r)9 + C10/(1+r)10
= 100/1.06 + 100/(1+0.06)2 + 100 /(1+0.06)3 + 100/(1+0.06)4+100/(1+0.06)5+100/(1+0.06)6+ 100 /(1+0.06)7+ 100/(1+0.06)8+100/(1+0.06)9 + 100/(1+0.06)10
= 100/1.06 + 100 /1.1236 + 100 /1.191 + 100 /1.2624 + 100/1.338 +100/1.4185 +100/1.5036 +100/1.5938 + 100/1.6895 + 600/1.7908
= 94.34+89.00+83.96+79.21+74.74+70.50+66.51+62.74+59.19+335.04
= $ 1015.23
Difference of 0.02 due to rounding off
2nd part
NPV of uneven cash flow :
r = 8% = 0.08
C0 = Cash Flow at the end of 0 year = 0
C1 = Cash Flow at the end of 1st year = 100
C2= Cash Flow at the end of 2nd year = 200
C3= Cash Flow at the end of 3rd year = 0
C4= Cash Flow at the end of 4th year = 400
Hence NPV = C0/(1+r)0 + C1/(1+r)1+ C2/(1+r)2 + C3 / (1+r)3 + C4 / (1+r)4
= 0 + 100/1.08 +200/(1.08)2 + 0 + 400/(1+0.8)4
= 100/1.08 +200/1.1664+400/1.3605
= 92.59 + 171.47 + 294.01
= $ 558.07
A typical common stock provides cash flow as uneven cash flow stream because the dividend payout is not fixed . It depends on many factors like profitability of a company . dividend payout ratio - which may differ year to year etc. Also common stock are to be paid the last after payment to debt owners and preferred stock owners. This is true in the case of both normal circumstances and liquidation scenario
Hence as dividend payout ratio is not fixed and the value f common stock is also not fixed , the cash flow received from common stock is i=uneven cash flow and not annuity
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