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    {\bf Notes on Present Discounted Value      }\\
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                    October 12, 1998. September 8, 2003. Slight update, 11 Sept. 2006.   \\
                    \bigskip
                    Eric Rasmusen, erasmuse@indiana.edu, Indiana
University \\
                   
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   Time is money.  That statement has many meanings, but one of them is
that money now is worth more than money later. If you have a choice
between 800 dollars in 1998 and 800 dollars in 2098, you will prefer the
money you get 100 years in advance. This is so for many reasons. Three
of them are:
 \begin{enumerate}
 \item
  Impatience, or time preference.  You would prefer to be able to spend
the money and consume things now simply because that is part of your
utility function, part of your psychological  preferences.  \\
\item
   Option value. If you have the money earlier, you have the choice of
spending it earlier or waiting and spending it later.\\
\item
    Investment value.  If you  get the money in 1998, you can invest it
and have more than the original amount to spend in 2098.
\end{enumerate}
 
  We'll concentrate on Reason 3.  Suppose that money can be invested
risklessly (say, in U.S. government bonds)  for an annual interest rate
of $r = .10$, ten percent. It is easy to see that with a choice between
\$800 now and \$800 in 100 years, you would prefer your money now.
Also, there must exist some value $V$ such that you would be indifferent
between \$800 now and \$V in 100 years. What is the value of $V$?

     Let's start with an easier problem.  You have a choice between 1
dollar today and  $D$  dollars delivered a year from now. What is the
value of $D$?

  Well, if you had the  1 dollar today and invested it at interest rate
$r$, you would end up with $1(1+r)$ dollars tomorrow. So the value of
$D$ is
 \begin{equation}\label{e1}
  D = 1 (1+r) = 1 (1+.05) = 1.05. 
 \end{equation}
   What about if you had a choice between  1 dollar today and D dollars
to be delivered in two years?  With one more year of interest, the value
would be
  \begin{equation}\label{e2}
  D = 1 (1+r) (1+r) = 1 (1+.05) (1+.05)  = 1.1025. 
 \end{equation}
   What about 1 dollar today versus D dollars in 100 years?  The value
would be
 \begin{equation}\label{e3}
  D = 1 (1+r)^{100}  = 1 (1+.05)^{100}  \approx  131.50. 
 \end{equation}
  With this information, we can solve the original problem. 800 dollars
today is equivalent to   $(800) (1+r)^{100}$ dollars in 100 years, which
is \$105,200.

      More often, we want to do a different form of the calculation.
What value of D dollars today is equivalent to 1 dollar to be delivered
in one year? If I had D dollars today, I could invest it and have D(1+r)
dollar in one year, so D must solve the equation
\begin{equation}\label{e4}
  D  (1+r)  = 1,   
 \end{equation}
yielding 
 \begin{equation}\label{e5}
  D    = \frac{1}{1+r}.    
 \end{equation}
 That is the basic equation for {\bf present discounted value}: the
value, in terms of present dollars, of a future stream of money.

  What would D be if the 1 dollar were to be delivered in 2 years, not
1?  Then  it must be that
 \begin{equation}\label{e6}
  D  (1+r) (1+r)  = 1,   
 \end{equation}
so 
 \begin{equation}\label{e7}
  D    = \frac{1}{(1+r)^2}.    
 \end{equation}
  If the interest rate is 5 percent, this means 1 dollar to be delivered
in 2 years is equivalent to about 91 cents delivered immediately.

 It is easy to see that this means that 1 dollar to be  delivered in 100
years is worth only
  \begin{equation}\label{e8}
  D    = \frac{1}{(1+r)^{100}},    
 \end{equation}
 so 800 dollars to be delivered in 100 years is worth only \$6.08   in
present dollars.

 None of this, by the way necessarily involves {\bf inflation}.  You
should do the discounting using the nominal, not the real interest rate,
if the future dollars are not adjusted for inflation. If the inflation
rate is zero, however, the interest rate will still be positive, and a
dollar in the  present is still  better than a dollar in the future.

 A {\bf zero coupon bond} is a bond  that pays no interest. People buy
such bonds only if they sell at a {\bf discount}. For example, a company
might issue a  bond  with a face value of  1,000 dollars  maturing in 30
years.  If the bond is riskless and the market interest rate is 5
percent, a person would pay
$ P = \frac{1000}{(1+r)^{30}}   =\frac{1000}{(1+.05)^{30}}  \approx $
231 dollars  to the company for that bond.  That is fine with the
company-- it has obtained 231 dollars to use as capital for 30 years.
Over time, the value of the bond rises as maturity gets closer.  On the
day of maturity, the market value of the bond is 1,000 dollars.

  You will find yourself using an equation like (\ref{e8}) for most of
your discounting calculations, figuring out the present value of money
rather than the future value. That is because the idea is  helpful  in
calculating the value of  an entire stream of future payments, positive
and negative, arriving in different years.  Suppose  that  your company
makes an investment that you think will cost 200 thousand dollars now
and 30 thousand dollars a year from now, but can be sold for  500
thousand dollars nine years from now, and that the discount rate is  8
percent.  The present value is
   \begin{equation}\label{e9}
  P   =  -200 -  \frac{30}{(1+r) } + \frac{500}{(1+r)^{9}} \approx  -200
-  27.78 + 250.12 = 22.34.
 \end{equation}
  This means that you would only pay  \$22,340  for the opportunity to
make this investment, even though the undiscounted profit is \$270,000.
To put the same idea differently: if  the company  had a sudden windfall
of  \$22,340  in cash and bought a bond yielding  8 percent annually, at
the end of 9 years the company would end up just as well off as if it
made this  investment.
 Companies do this kind of calculation all the time to figure out
whether to  undertake a project  or use their funds in some other way.

    One final point: it is often convenient to  know the value of a {\bf
perpetuity} or {\bf consol}, a  stream of payments of 1 dollar each year
starting in one year and going on forever.  The present value of this
stream is {\it not} infinite, even though the total number of dollars
is.  Using the formula above, the value is
  \begin{equation}\label{e10}
  P   =   \frac{1}{(1+r)^{1}} + \frac{1}{(1+r)^{2}} +\frac{1}{(1+r)^{3}}
+...
 \end{equation}
 It turns out that this value equals the very simple expression, 
  \begin{equation}\label{e10}
  P   =   \frac{1}{r}.      
 \end{equation}
 Thus, if the interest rate is $r = .05$, you would pay $1/.05 $ for the
perpetuity, which is 20 dollars. That's a lot less than infinity.

       This makes sense, though.  Suppose you had a choice between 20
dollars or the perpetuity. The perpetuity yields one dollar per year
forever. If you invest the 20 dollars at the interest rate of $4=.05$,
what do you get? -- 1 dollar per year  forever. The values are the same.

     If interest rates go up, though,  your perpetuity falls in value.
If the interest rate rises to  $r=.10$, then the value of the perpetuity
falls to $p = 1/.10 = 10$.  That is because now the stream of 1 dollar
per year could be achieved even if you had only 10 dollars, not 20, to
invest.

        This is a  way to explain why  bond prices fall when interest
rates rise.

    

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