# Net Present Worth Calculation (Economic Equivalence)

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?

Thanks very much in advance!

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 USD100,000 at the end of the first year, increasing each subsequent year by $5,000; in other words you get USD100 000-00 at the end of the first year, USD105,000 at the end of the second year, USD110 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? ## 1 Answer Using the discount rate (interest rate) calculate the present value of each payment. Let A = 100,000, first payment a = 5,000 (annual increase), and r = 2.5%, interest rate to simply the formulae. $$PV_1 = A/(1+r) \\ PV_2 = (A+a)/(1+r)^2 \\ ... \\ PV_{10} = (A+9a)/(1+r)^{10}$$ The total present value (PV) is just the sum of individual payment present values. To compute the expected real value (V) apply the expected inflation rates ($i_1 \dots i_{10}$) on this present value, i.e. in year one$V_1 = PV/(1+i_1), V_2=V_1/(1+i_2),$etc. You can do the simplifications since$i_1 = \dots = i_5\$ and write one equation. However, my suggestion is use a spreadsheet and do the computations individually to better comprehend the subject.

• Thanks very much karakfa. As I understand it, the inflation rate is only used for the last part of the question, to calculate the real value of the prize at the end of 10 years. For the decision on which prize to choose, the inflation rate doesn't come into it at all. Is that correct? I'll give the question a go with your method and I'll let you know if I have any issues. – JimRollins Nov 5 '12 at 20:51
• that's correct. discount (interest) rate is for time traversing the amounts. Inflation is what the amount will be worth (as buying power). – karakfa Nov 5 '12 at 21:28