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Math Help - Total Present Value of Income Flows

  1. #1
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    Total Present Value of Income Flows

    A company is considering a major investment programme, with a choice of two projects A and B.

    Project A : Initial outlay $7800 but then promises income streams of

    Year 1: Income Stream $11500
    Year 2: Income Stream $8200
    Year 3: Income Stream $5000
    Year 4: Income Stream $2400

    Project B : Initial Outlay $6600 with income streams of

    Year 1: Income Stream $10300
    Year 2: Income Stream $8000
    Year 3: Income Stream $4000
    Year 4: Income Stream $850

    The current rate of interest is 4.5 %

    I have to calculate the total present value of the income flow from project A and project B, over the four year period.

    Could anyone just point me in the right direction as to what Iím supposed to do here? In other examples Iíve looked at there always seems to be a discount rate (is that the same as the interest rate?)
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  2. #2
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    Yes, if the "current rate of interest" (4.5%, as stated) is the only rate you're given, then the textbook is contemplating that you'll use that as your discounting rate.

    And it's not unreasonable...the phrase "current rate of interest" usually means, in this context, "the market yield / rate of return on projects of similar risk and time horizon is currently 4.5%".

    So use that given rate to discount the cash flows, just as you've seen in those examples you consulted. Best of luck, and check back in if you need further guidance.
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  3. #3
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    You'll also have to be more specific. What does "Year 1" mean? It amy be reasonable to assume that it does NOT mean the beginning of the year. There are many other things it could mean.
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  4. #4
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    Quote Originally Posted by Apache View Post
    Project A : Initial outlay $7800 but then promises income streams of
    Year 1: Income Stream $11500
    Year 2: Income Stream $8200
    Year 3: Income Stream $5000
    Year 4: Income Stream $2400
    Hint: 7800 + 11500 / 1.045^1 + ..... + 2400 / 1.045^4
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  5. #5
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    Thanks all,

    Quote Originally Posted by TKHunny View Post
    You'll also have to be more specific. What does "Year 1" mean? It amy be reasonable to assume that it does NOT mean the beginning of the year. There are many other things it could mean.


    All Iím told is what Iíve provided, I guess it just means the income stream will be $11500 after the first year, $8200 after the second year... and so on....

    This is what Iíve got

    Project A

    Year 0 = -7800/(1.045)^0 = -7800
    Year 1 = 11500/(1.045)^1 = 11004.78
    Year 2 = 8200/(1.045)^2 = 7508.99
    Year 3 = 5000/(1.045)^3 = 4381.48
    Year 4 = 2400(1.045)^4 = 2012.55

    Project B


    Year 0 = -6600(1.045)^0 = -6600
    Year 1 = 10300/(1.045)^1= 9856.46
    Year 2 = 8000/(1.045)^2 = 7325.84
    Year 3 = 4000/(1.045)^3 = 3505.19
    Year 4 = 850/(1.045)^4 = 712.78

    Summing up the total of the income flows for their respective years gives me,

    Project A = 24907.80

    Project B =21400.27

    Should I then subtract the initial outlay from this to give me the total present value of the income flow?

    Now the second part of my question asks to calculate the net return of investment on each project, to do this do I just simply subtract the initial outlay of each project away from the total income flows Iíve calculated above, or is there a separate formula I should use?
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  6. #6
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    Quote Originally Posted by Apache View Post
    Should I then subtract the initial outlay from this to give me the total present value of the income flow?
    Yes; 1st one = 24,908 - 7,800 = 17,108

    You can look at it as borrowing $7,800 at start (year 0), then this being
    repaid from the 1st flow, interest cost being $351 (7800 * .045):

    0 00000 0000 -7800
    1 11500 -351 03349
    2 08200 0151 11700
    3 05000 0527 17227
    4 02400 0775 20402 ; 20402 / 1.045^4 = 17,108

    The "net return" calculation is nothing more than assuming a deposit of
    $7,800 in a savings account, and being able to close the account 4 years
    later by withdrawing $20,402; hence:
    7800(1 + r)^4 = 20402
    r = (20402/7800)^(1/4) - 1
    r = .271728883... ; so ~27.17%
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  7. #7
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    For the first part, you're right on the money (pun intended). You've calculated the Present Value of the cash inflows...now just as you say, deduct from that the PV of the immediate cash outflow (which is the same as its actual amount, since it occurs immediately) to get the overall Net Present Value of each project.

    When a problem like this doesn't specify otherwise, it almost always assumes the project's cost is a single cash outflow occuring immediately, and the subsequent inflows all occur at the end of each year or period. That's the assumption you're using in your calcs, so you should be good to go.

    For the second part of your question you're computing the project's yield (aka IRR, or return on investment). You'll use the same process you used in the first part, but now you need to come up with the discounting rate--something other than 4.5%--that makes the project's NPV equal to zero.

    Hint: Since both projects have a positive NPV when the cash flows are discounted at 4.5%, the yield on both projects is > 4.5%.

    Unfortunately, finding the correct discount rate that makes the NPV = 0 is a trial-and-error thing in most cases, because it amounts to finding the roots of a poly, where the poly's degree is the number of cash flows. So you'll need to just keep fiddling with the discount rate until you hit on the one that makes the NPV zero. (Of course, Excel can do that fiddling for you--check out the Goal Seek utility, or the IRR function.)
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  8. #8
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    Thanks,
    So the Total present values of income flow are calculated correctly

    Project A = 17108
    Project B = 14800

    I am a little bit confused regarding the net return on investment calculation

    I understand that 17108 * (1.045)^4 = 20402

    But, I donít grasp why that is the amount I would be withdrawing $20402 after the four years, wouldnít I be withdrawing $17108 which is the present value of in the income flow calculated in part 1?
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  9. #9
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    Quote Originally Posted by LochWulf View Post
    Unfortunately, finding the correct discount rate that makes the NPV = 0 is a trial-and-error thing in most cases,....
    But not in this case, since you have both PV and FV.
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  10. #10
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    Quote Originally Posted by Apache View Post
    But, I don’t grasp why that is the amount I would be withdrawing $20402 after the four years, wouldn’t I be withdrawing $17108 which is the present value of in the income flow calculated in part 1?
    No.
    You are theoretically "worth" $17108 NOW; like, you can theoretically
    "sell" this flow for $17108, then invest it @ 4.5% for 4 years.

    RELAX! Keep it simple: you are parting with $7800 NOW, ...nothing happens... ,
    and in 4 years you pick up $20402. Roger and out.
    Last edited by Wilmer; July 31st 2009 at 06:02 AM. Reason: none
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  11. #11
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    Apache, check very carefully your text's interpretation of the phrase "net return on investment". Return on Investment, or ROI, has multiple uses and definitions, and the correct computation for your question will be the one consistent with how your text uses the term.

    Very commonly, a "project's ROI" is synonymous with Internal Rate of Return ("IRR") or yield; and refers to that discount rate which causes the NPV of all the project's cash flows to sum to zero. In the context of your first project, the IRR would be the rate of return you'd be earning if you invested $7,800 today, and then received those four named cash inflows, one at the end of each of the next four years.

    Without even glancing at your calculator you can tell that this yield for Project A must be huge: The entire initial investment is completely returned one year later, along with an extra $3,700 to boot.

    To see this, assume for a moment that Project A only has two cash flows: The initial outlay of 7,800; and then a single cash inflow of 11,500 one year later (in other words, disregard for a moment the three subsequent inflows, and consider only the first). This project would thus be providing a return of

     \frac{11,500}{7,800}\ -\ 1 \approx \ 47.4%. And that's from the first year's cash inflow alone. Adding in the subsequent three cash inflows can't do anything but drive the project's IRR northward from there.

    Wilmer, on the other hand, is interpreting "net rate of return" differently and his calculation is consistent with his interpretation. So verify with your text exactly what it has in mind with that "net ROR" phrase.
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  12. #12
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    My question clearly states that i should calculate the net return on the investment and asks which product has the greatest net return on investment.

    If i follow your method LochWulf, would this be correct,

    Project A

    \frac{11500+8200+5000+2400}{7800} - 1 = 2.47

    Project A net return on investment = 247%
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  13. #13
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    No, Project A's yield, or IRR (Internal Rate of Return) would be the r which satisfies...

     -7,800\ +\ \frac{11,500}{1+r}\ +\ \frac{8,200}{(1+r)^2}\ +\ \frac{5,000}{(1+r)^3}\ +\ \frac{2,400}{(1+r)^4} = 0.

    Note that my one-period model I used in the previous example is derived from this same computation; viz

     -7,800\ \ +\ \frac{11,500}{1+r}\ =\ 0 which becomes

     \frac{11,500}{7,800}\ -\ 1\ =\ r\ \approx 47.4%. But don't dwell on the 'one-period' example...I just used it as a quick back-of-the-envelope to support the immediately intuitive notion that whatever the project's actual ROR turns out to be, you can see right away it'll be large, given the outsized relationship between the initial outlay and the subsequent cash inflows.

    The focus of the question, though, is to figure out that discount rate r which produces a -0- NPV for the project as a whole, as in the first equation above.

    By the way, you can see that the equation is precisely the one you used to determine the two projects' NPV, when you used 4.5% for r. In other words, with r = 4.5%, the equation evaluates to 17,108 for Project A, which is thus A's NPV when discounted at 4.5%. Now you just use the same computation, but figure out what value of r makes the equation evaluation to zero. Such r will the the project's IRR.
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  14. #14
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    Sorry about this, but if r = 0.474 satisfies the first period, wouldnt that be the rate used throughout the equation ?
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  15. #15
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    If you use 47.4% across the equation, you're just performing a different task--you're determining the project's NPV when discounted at 47.4% (which is 5,846, btw).

    That's the same task you performed when you calculated that Project A has a NPV of 17,108 when discounted at 4.5%, only now you're using a different discount rate (47.4%) to come up with a new NPV (5,846).

    But what your original question is requiring is for you to determine the appropriate discount rate such that the equation evaluates to zero. The r that makes this happen is the project's IRR. As you can see, 47.4% can't be the project's IRR, because the project still has a positive NPV when discounted at such rate.

    Notice that by increasing the discount rate from 4.5% to 47.4% we've seen that the project's NPV drops from 17,108 to 5,846. You just need to keep hitting that equation with still higher values for r until you get that NPV down to zero. Then you'll have the r that represents the project's NPV.
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