I’ve recently been thinking about a certain problem in finance that requires some thought on net present value. This post is meant to both gather my thoughts and serve as a reference for the next post, which will hopefully be more interesting to those who don’t study financial theory. So, if this doesn’t float your boat, hang on for a week!
In finance there is an idea of the time value of money. One dollar today is not worth one dollar tomorrow. This is because there are, hopefully, better things I could do with a dollar than just put it in a shoebox and wait until tomorrow. Possibly the easiest way to think about this is with inflation. If prices are rising, then one dollar today can buy more stuff today than it can tomorrow. So the value of a dollar tomorrow will be reduced by the amount of inflation:
Value of a Dollar Tommorow,Today=(One Dollar Tomorrow)/(1+Inflation)
For example, if inflation is 10% per day (yikes!), $1.00 tomorrow is worth $0.91 today. If the price of chewing gum rises from $1.00 to $1.10, my dollar today can buy more chewing gum today than tomorrow.
If we want to use this idea to analyze investments, we need to introduce the idea of opportunity cost, that is, the thing you give up today to consume tomorrow. (I’ll do my best, but for a more thorough treatment of this, you might check out John Geanokoplos’ lecture series, he is a great speaker and gives the subject of financial theory a really thorough treatment. In particular this lecture: http://oyc.yale.edu/economics/econ-251/lecture-7. Jump to “Coupon bonds…” if you’d like, but you’ll miss the Merchant of Venice!)
If I am faced with a choice of two investments, one paying a return of and the other paying a return of , the amount of money I’m willing to pay for those investments should be proportional to their returns. For example if I am thinking about investing at a return of , I should think to myself “My other choice is to invest at a return of ”. So, my opportunity cost of investing at return is . If I am thinking about investing at the thought process is reversed. In the same way that I discounted my dollars today and tomorrow, I can discount my returns on these stocks. So, the present value of my investment at should be discounted by the opportunity cost.
Value of Investing at y = (1+y)/(1+x)
It’s only logical to say that if is greater than then I would prefer investing in , and vice versa. Now this may seem silly because this comparison seems simple. However, what if we are faced with a more complicated situation? For example, what if we have a bond that pays a fixed rate of interest for one period? How much is that bond worth? Lets say someone says, “Give me one dollar today and I will give you one dollar and ten cents tomorrow”, offering to sell me a bond for a dollar at a rate of 10%. Well, if I know that I can earn some rate of return, , on my investments if I don’t buy this bond, then I should discount the bond at the rate :
Value of the bond=1.1/(1+x)
Depending on how much I can earn if I don’t buy the bond (my discount rate, ), the value of the bond to me will be greater than, equal to, or less than one. If I can only earn 9% on my own investments, the value of the bond to me is greater than one and I should buy it. If I can earn 11% on my own investments, I would lose money by buying the bond and thus it is worth less than one dollar to me. If I can earn 10%, I’m indifferent and the bond is worth exactly $1.
Now that we’ve arrived at a state of complete confusion, let’s think about more complex situations. In particular, what if the bond has two periods?! If we have a bond that costs $1.00 and earns a compound rate of interest of 10%, what is its value? Well, if the bond will pay out at the end, that is after two periods, and I could invest my money at for two years, then the value of the bond to me today is:
Value of the bond=〖(1.1)〗^2/〖(1+x)〗^2
Ok, not so bad. What if the bond is going to pay me an annuity, or some fixed amount of return, every year? This is what one would call a “cash flow” in finance, and to think about how much this is worth we’ve got to look how many years it will pay out and what are my other opportunities, that is, my discount rate. To make things simple (and to apply to the idea I want to talk about next week!) let’s pretend the annuity will go on forever. Well, let’s look at a formula. If I’m going to receive C dollars every year forever and I have a discount rate of :
Net Present Value of Annuity to Perpetuity (NPV) = C/((1+x)) + C/〖(1+x)〗^2 + ⋯
Each year, the amount by which I discount the annuity C increases because if I had that money today I could have invested it at and earned the compounded interest, or , where n is the number of years. If this sum goes to infinity, we can rewrite it as:
NPV = ∑▒C/〖(1+x)〗^n = C [∑▒1/〖(1+x)〗^n ]
Avoiding the mathematical magic of summing an infinite geometric series, trust me when I say that the sum comes out to be:
NPV = C/x
So, for example, if you are offering to sell me a bond for $100 that will pay a 10% coupon, or $10, every year forever, the value of that to me today is wholly dependent on my discount rate. If I am discounting at 10%, the bond is worth exactly $100, because I could either invest my money in something else earning 10% or I could invest in the bond and earn 10%. Essentially, I am indifferent.
Finally, let’s think about an example where the coupon of the annuity grows at a rate of indefinitely. The formula for that would be:
NPV of Growing Annuity to Perpetuity = C/((1+x)) + (C(1+y))/〖(1+x)〗^2 + (C〖(1+y)〗^2)/〖(1+x)〗^3 + …
NPV = C ∑▒〖(1/〖(1+x)〗^n ) ((1+y)/(1+x))^(n-1) 〗
Again, sparing you the mathematical gibber gabber, trust me when I say that this geometric series sums to:
NPV = C/(x-y)
So, if you are offering to give me $10 a year every year forever, increasing at 1% per year, and I have a discount rate of 10%, today that is worth to me:
NPV = $10/(0.10-0.01) = $111.11
Phew! We made it! Although this may not have been the most interesting thing, it will help next week when I grapple a much more pertinent, real world subject. Thanks for reading, or at least for humoring my need to preface posts. Check back next week to see this applied to a real world problem.
Spoiler: Why will the University of Texas endowment be worth more thatn $400 billion in 50 years and whose money is that?!