My Math Forum Net Present Worth Calculation (Economic Equivalence)

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November 5th, 2012, 02:50 PM   #1
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Net Present Worth Calculation (Economic Equivalence)

Hi guys,

I'm currently doing some work involving net present worth analyses, and I'm really struggling with calculations that involve interest and inflation, such as the question below. I feel that if anyone can set me on the right track, and once I've worked through the full method for doing one of these calculations, I should be able to do them all. Is there any chance that anyone may be able to guide me through the process of doing the question below, or give me any pointers?

Quote:
 You win the lottery. The prize can either be awarded as USD1,000,000 paid out in full today, or yearly instalments paid out at the end of each of the next 10 years. The yearly instalments are $100,000 at the end of the first year, increasing each subsequent year by$5,000; in other words you get $100 000-00 at the end of the first year,$105,000 at the end of the second year, \$110 000-00 at the end of the third year, and so on. After some economic research you determine that inflation is expected to be 5% for the next 5 years and 4% for the subsequent 5 years. You also discover that real interest rates are expected to be constant at 2.5% for the next 10 years. Using net present worth analysis, which prize do you choose? Further, given the inflation figures above, what will the real value of the prize of USD1,000,000 be at the end of 10 years?

 November 8th, 2012, 06:59 AM #2 Newbie   Joined: Jul 2010 Posts: 10 Thanks: 0 Re: Net Present Worth Calculation (Economic Equivalence) A basic concept--and one which applies in answering this question--is that you get the same present value whether you discount a set of nominal cash flows by the appropriate nominal rate, or discount their real equivalents by the real discount rate. So to compare the structured payout alternative against the 1M-immediately option, you could either • discount the nominal cash flows (as given by the problem: 100K, 105K, 110K, ...) by the nominal discount rate; or • first convert those nominal CFs to their constant-dollar ("real") equivalents, then discount 'em by the real rate. Same PV either way. In actually doing what I mentioned above, there are two fundamental relationships you'll need to use: • For any nominal rate k, inflation rate i, and real rate r, the relationship (1 + k) = (1 + i)(1 + r) holds. So given any two of the three you can easily solve for the third. • The real (today's dollars) equivalent of some nominal amount of cash C, occurring n periods from now, is C / (1 + i)^n. Incidentally, I'd urge you to explore the derivations of those two relationships. They're related (both stemming from the idea of "purchasing power" of a dollar, with respect to some basket of goods), pretty intuitive, and fundamental to moving back and forth between nominal and real flows and returns. So a simple example: What's the PV of a two-CF project, which throws off 1,000 one year from today and then 1,500 two years from today? (These are the nominal flows.) The real rate is 2.5% and inflation is expected to be 5% p.a. over the next two years. First, from the given real and inflation rates, we have that the nominal rate is 7.625%. We can then discount the nominal flows by the nominal rate... $\frac{1,000}{1.07625} + \frac{1,500}{(1.07625)^2}$ ...or first convert the nominal CFs to their constant-dollar equivs (1,000 / 1.05; 1,500 / 1.05^2) and then discount at the real rate... $\frac{952.381}{1.025} + \frac{1,360.544}{(1.025)^2}$. I'll let you bring both of those across the finish line to see their equivalence, as far as giving the PV of the CFs. The problem you posted is just an expanded version, and the fact that inflation isn't constant across the entire 10 years doesn't change the concept.
 November 8th, 2012, 08:13 AM #3 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 Re: Net Present Worth Calculation (Economic Equivalence) Expanding on LW's "accurate!" stuff... this can be done by formula; easier when flows are numerous: PV = {i[f(r^n - 1) - c(n - 1)] + cr[r^(n - 1) - 1]} / [(i^2)(r^n)] where: f = 1st payment (1000 in LW's example) c = constant change to payment (500 in LW's example) n = number of payments (2 in LW's example) i = interest (.07625 in LW's example) r = 1 + i (1.07625 in LW's example) : not really necessary, but makes formula less wieldy PV = {.07625[1000(1.07625^2 - 1) - 500(2 - 1)] + 500(1.07625)[1.07625^(2 - 1) - 1]} / [(.07625^2)(1.07625^2)] PV = 2224.1377.... With yours, the 1st 5 flows: f = 100000, c = 5000, n = 5, i = .07625, r = 1.07625 That should give you 440617.16 For the last 5 years: don't forget that f = 125000 and change i to reflect the reduced 4% inflation then discount result for a further 5 years to bring to present Did I goof, LW? Perhaps you can shorten that loooooong formula of mine?
 November 8th, 2012, 01:31 PM #4 Newbie   Joined: Jul 2010 Posts: 10 Thanks: 0 Re: Net Present Worth Calculation (Economic Equivalence) Spot on, D, nice closed-form workup! Search for an algebraically more condensed rendering? You overestimate my motivation level at the end of this h'yar workday, comrade After 5 pm, if it all fits onto a single line of code, it's efficient enuf for me.

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