## Sunday, July 7, 2013

### Financial Mathematics: Time Value of Money

One of the critical ideas that drives much in financial mathematics is known as the Time Value of Money (TVM). So I think it's important to start here. In fact, if you do not understand TVM, you will not understand the remainder of this series as TVM is an implicit assumption in most of these models.

The key idea behind TVM is that a dollar today is not worth the same as a dollar tomorrow. Part of what motivates this idea is postulate about human behavior.1 That idea is that human beings prefer to have something (money) now rather than later. As a result, money in the future is worth less today than it is in the future.

A simple way to think of it is in terms of savings. When you save money (say in a savings account), and you put $\$1 $in there, you won't have$ \$1$ but rather it will be more due to interest. The idea is that this present value is equivalent to the future value when you factor in the interest.

In order to make future money equivalent to present money, we need a conversion factor. Just as you need a conversion factor to convert dollars to euros or to convert inches to centimeters, so you also need a conversion factor to convert future money to present money. One of the devices we'll use is a discount factor and it works like this:
$$PV = FV \times D$$
where $PV$ is the present value, $FV$ is the future value and $D$ is the discount factor.

Here's an example. Suppose you prefer $\$ 80 $now equally the same as$ \$100$ at some time $t$ in the future. Then discount factor is the number $D$ is given by:
$$D = \frac{PV}{FV}= \frac{\ 80}{\ 100} = 0.80$$
So where does this discount factor of $D=0.80$ come from? Apart from being a number I made up in an example, this idea can be related to two things that most people are familiar with: savings and loans.

When you save money you forgo present money to ultimately receive more money in the future. With loans you do the exact opposite; you accept money in the present but agree to forgo money in the future. The difference between the two is related to interest.2 In the case of savings, you receive interest; in the case of borrowing, you pay interest. Interest is directly tied to the discount factor.

You'll come across several ways of calculating interest. One is simple interest which doesn't get used too often but I'll mention it here. If the simple interest rate is $i$ and the time when the interest is applied is $t$, then simple interest is given by:
$$\text{Simple Interest} = (1+it)$$
The discount factor for simple interest would then be:
$$D = \frac{1}{(1+it)} = (1+it)^{-1}$$
What we'll typically see is compound interest. The formula for compound interest is given by:
$$\text{Compound Interest} = (1+i)^t$$
The discount factor for compound interest would then be:
$$D = \frac{1}{(1+i)^t} = (1+i)^{-t}$$
Unless otherwise stated, we will always be using compound interest.

Sometimes you'll see the following substitution: $v=(1+i)^{-1}$. This would enable us to write the discount factor as:
$$D = v^t$$
This can come in handy when we start manipulating the equations.

Lastly, we'll define the discount rate.  The term is sometimes used interchangeably with the interest rate but we're going to use an alternative definition. The discount rate, $d$ is given by:
$$d=(1-v)=\frac{i}{1+i}=iv$$
We can also express $i$ as a function of $d$:
\begin{align*} \frac{i}{1+i} &= d \\ i &= d(1+i) \\ i &= d+di \\ i(1-d) &= d \\ i &= \frac{d}{1-d} \end{align*}
Before ending this segment we'll look at four examples. The basic relationship has four variables so if you know 3 of the variables you can calculate the fourth.

Relationship 1
$$PV = \frac{FV}{(1+i)^t}$$
Example 1. Suppose you know $FV$, $i$ and $t$. An example might be you want to save up enough money for a vacation at some time in the future and you know what rate of return you can get on that money so you need to figure how much money you need to set aside now to get that return. Suppose that $FV=\$1000$,$i=5\%$and$t=3 \text{ years}$. Then you would need to save: $$PV = \frac{\1000}{(1+5\%)^3} = \ 863.84$$ Relationship 2 Next we may know$PV$,$i$and$t$and we want to find how much savings we'll have at a future date. $$FV = PV (1+i)^t$$ Example 2. Suppose I have$ \$1000$ and I want to save it for 5 years at a rate of $4\%$.. The amount I'll have saved will be:
$$FV = \1000 (1+4\%)^5 = \1216.65$$
Relationship 3

The third situation we might encounter is that we know $PV$, $FV$ and $t$ and need to find $r$:
$$i = \left( \frac{FV}{PV} \right)^{1/t}-1$$
Example 3. Suppose we have $\$ 1000 $and we want to turn that into$ \$2000$ within 6 years. Then the required rate of return is given by:
$$i = \left( \frac{\2000}{\1000} \right)^{1/6}-1=12.2\%$$
Relationship 4

The last situation is we might know $PV$, $i$ and $FV$ and want to know how long it will take to get there.
$$t = \frac{\ln(FV/PV)}{\ln(1+i)}$$
So let's suppose I want to turn $\$ 1000 $into$ \$2000$ and I can earn $3\%$ interest. How long will it take me to save?
$$t = \frac{\ln(\ 2000 / \ 1000)}{\ln(1+3 \%)} = 23.45 \text{ years}$$

1 With the exception of behavioral economics and behavioral finance, academics in the economic and financial world frequently make assumptions about how humans behave. These assumptions may not always hold up to empirical research on the matter. One should be aware of what assumptions are being made. I will make every attempt to be explicit about all assumptions surrounding these models.

2 Interest has both an interesting etymology and history. Various religious institutions have considered it immoral at different times. David Graeber offers a good discussion of it in his book Debt: The First 5,000 Years.