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finance-1.Rnw
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finance-1.Rnw
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\documentclass{beamer}
\usepackage{amsmath}
\usepackage{wasysym}
\setkeys{Gin}{width=0.9\textwidth,height=0.7\textheight}
\title{Finance}
\subtitle{Part 1: Basic Theory of Value}
\author{Isaac Carruthers}
\begin{document}
\frame{\titlepage}
\frame{\frametitle{What is Value?}
\begin{itemize}
\item Value is how much something can improve your life.
\item In a world where things can be traded and exchanged, value is
also related to how much something can improve {\em anyone's} life.
\item We can then approximate the value of something as equivalent to
the most valuable thing we can get in exchange for it.
\item To simplify things, we're generally going to talk about valuing things in terms of
dollars... but it's still not as simple as it sounds.
\end{itemize}
}
\frame{\frametitle{Things We're Going to Ignore}
\begin{itemize}
\item We're going to pretend that transactions are free and instant.
\item Alternatively, We're going to assume that we will make any trade that benefits us
financially, no matter how small the benefit. If something is for sale for any
amount less than its value to us, we will buy it. If someone offers to buy it at any
amount greater than its value, we will sell it.
\item We're going to assume that we are rational.
\item We're going to assume that we're not going to suddenly need money.
\end{itemize}
}
\frame{\frametitle{What is the Value of a Dollar?}
Are any two dollars today worth the same?
\vspace{12pt}\pause
Yes.
\vspace{12pt}\pause
Is a dollar today worth the same as a dollar next year?
\pause
\begin{itemize}
\item Inflation
\item Uncertainty
\item Opportunity cost
\end{itemize}
}
\frame{\frametitle{How do we Account for Changes in the Value of a Dollar?}
{\bf Inflation:} \pause Easy. If inflation is 2\%, then a dollar in a year is worth 2\cent
\; less today.
\pause \vspace{12pt}
{\bf Opportunity Cost:} \pause Easy. If we can invest 96\cent\; and get a dollar
back in a year (risk free), then a dollar in a year is worth 4\cent\; less today.
\pause \vspace{12pt}
{\bf Uncertainty:} \pause Hard. Usually we just make something up. Maybe I think that the
promise of a dollar from Sadie is worth 4\cent\; less than a dollar in my hand, but a
promise from Sean is only worth 2\cent\; less.
}
\frame{\frametitle{The Discount Rate}
For any future cash flows, we can combine our three concerns to arrive at a {\em discount
rate}, describing how the value of the money changes with when we're supposed to get it or
give it up.
\vspace{12pt}
We can use the discount rate to define the {\em present value} (PV) of any future transfer
of money. That is, if we could buy or sell the promise of some future transfer, we define
the value at which we would be willing to buy or sell it as the PV of that transfer.
}
\frame{\frametitle{Present Value}
There is a magic gosling which in one year will lay exactly one golden egg worth
\$1000, and then disappear. How much should you pay for the gosling? What is its {\em present
value}?
\vspace{12pt}\pause
If your discount rate is 8\%, then you should pay no more than $(1 - 0.08) * 1000 = 920$
dollars.
\vspace{12pt}\pause
We buy the gosling, but then when we go and double-check our grimoire, we find out that magic
geese don't lay their egg until they are {\bf 2} years old, not 1! What is the goose worth to
us now? How much would someone need to pay us to take it off our hands? (Hint: how much
will the goose be worth to us in a year?)
\vspace{12pt}\pause
If you wait a year, we'll be back in the situation we imagined when we bought the goose,
so in a year the goose will be worth \$920. How much would you pay now to get \$920 in
a year? Clearly it's $(1-0.08)*920 = (1-0.08)^2*1000\approx 846$ dollars.
}
\frame{\frametitle{Mathematical Digression}
You may have realized that not all contracts will last exactly a year. How do we deal with
that?
\vspace{12pt}\pause
What we're really doing with our $(1 - r)^n$ discount factor is approximating the formula
for {\em continuously compounded interest}, $e^{-rt}$.
\vspace{12pt}
This approximation is good when $r$ is small, and in general it is fine anyway, because we
cannot actually re-invest our money continuously.
}
\frame{\frametitle{Example: Valuing a Coupon Bond}
A {\em bond} is just a form of debt. If you own a bond, it means someone owes you money.
\vspace{12pt}
Bonds come in many forms. One common form is a {\em coupon bond}, which is a bond that
makes small interest payments (called the coupon) throughout the life of the bond
(often bianually), and then pays back the principal at the end.
\vspace{12pt}
For instance, the US treasury may offer a \$1Mil, 10-year bond at a 2\% yield. If we bought
this bond, the cash-flows would look like this:
\begin{itemize}
\item Spend \$1Mil to buy the bond at time 0.
\item Receive \$10k coupon at time 6 months.
\item Receive \$10k coupon at time 1 year.
\item \ldots
\item Receive \$1.01Mil at time 10 years.
\end{itemize}
}
\frame{\frametitle{Example: Valuing a Coupon Bond}
How do we value a complicated thing like a bond?\pause
\vspace{12pt}
Just come up with a discount rate $r$, and sum the values of the individual transfers!
\begin{align*}
PV(B) &= -1e6 + (1 - r)^{\frac12}1e4 + (1-r)1e4 + \cdots\\
&\hphantom{=} + (1-r)^{9.5}1e4 + (1-r)^{10}1.01e6
\end{align*}
}
\frame{\frametitle{Example: Valuing a Coupon Bond}
<<bond1,fig=TRUE,echo=FALSE>>=
r <- (0:500) / 10000
val <- function(r) {
s <- -1e6
for (i in 1:19) s <- s + 1e4 * (1 - r)^(i/2)
s <- s + 1.01e6 * (1 - r)^10 }
v <- val(r)
plot(r, v, ty='l', bty='n', las=1, col='red')
abline(v=.02, lty=2)
abline(h=0, col='gray'); abline(v=0, col='gray')
@
}
\frame{\frametitle{Internal Rate of Return (IRR)}
Looking at the PV of a deal is one way to look at what you can do with your resources, but
it's not a very good way (a deal where you put up \$1Mil to get back \$1.1Mil is very
different from one where you put up \$100k to get back \$200k, but both have the same PV).
\vspace{12pt}
A more standard way of evaluating options is the {\em Internal Rate of Return} (IRR).
\vspace{12pt}
The IRR is the discount rate at which a deal's PV becomes zero. In our bond example above,
the IRR is 2\%.
}
\end{document}