Question

Present value Mixed streams Consider the mixed streams of cash flows shown in the following​ table

A B

1   $-15,000 $5,000
2   $12,500 $7,500
3   $10,000 $10,000
4   $7,500 $12,500
5   $5,000 $-15,000
Totals   $20,000   $20,000

a.  Find the present value of each stream using a 6​% discount rate.

b.  Compare the calculated present values and discuss them in light of the undiscounted cash flows totaling $20,000 in each case. Is there some discount rate at which the present values of the two streams would be​ equal?

Answer 1

PV of A:

Year	CF	Discount Factor	Discounted CF
1	$-15,000.00	1/(1+0.12)^1=	0.892857143	0.892857142857143*-15000=	$ -13,392.86
2	$ 12,500.00	1/(1+0.12)^2=	0.797193878	0.79719387755102*12500=	$     9,964.92
3	$ 10,000.00	1/(1+0.12)^3=	0.711780248	0.711780247813411*10000=	$     7,117.80
4	$    7,500.00	1/(1+0.12)^4=	0.635518078	0.635518078404831*7500=	$     4,766.39
5	$    5,000.00	1/(1+0.12)^5=	0.567426856	0.567426855718599*5000=	$     2,837.13
				NPV = Sum of all Discounted CF	$   11,293.39

PV of B

Year	CF	Discount Factor	Discounted CF
1	$    5,000.00	1/(1+0.12)^1=	0.892857143	0.892857142857143*5000=	4,464.29
2	$    7,500.00	1/(1+0.12)^2=	0.797193878	0.79719387755102*7500=	5,978.95
3	$ 10,000.00	1/(1+0.12)^3=	0.711780248	0.711780247813411*10000=	7,117.80
4	$ 12,500.00	1/(1+0.12)^4=	0.635518078	0.635518078404831*12500=	7,943.98
5	$-15,000.00	1/(1+0.12)^5=	0.567426856	0.567426855718599*-15000=	-8,511.40
				NPV = Sum of all Discounted CF	16,993.62

To find the rate at which the NPV of the two projects will be equal, we need to calculate the IRR for the incremental CFs.

IRR is the rate at which NPV = 0
IRR can be calculated using either a financial calculator or excel or through hit and trial:
At IRR NPV of incremental CFs will equal 0 which implies that the NPV of both the streams is equal

Incremental CFs are

Year	CF
1	-15000-5000 = $-20,000.00
2	12500-7500 = $    5,000.00
3	10000-10000 = $                 -
4	7500 - 12500 = $   -5,000.00
5	5000-(-15000) = $ 20,000.00

As there are multiple sign changes there is no IRR for the project. So there is no rate at which the NPV of the two streams can be equal