# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] The Present Value Rule

In our discussion today we considered the following three cases:

• No Bank, No Investment Opportunity: In absence of a bank, the sum of consumption in both time periods must be equal to the total available wealth. What one consumes is an individual decision, but sum of what is consumed ‘today’ and ‘tomorrow’ must equal total wealth (in our example, Rs. 5000). That is, (Consumption Today) + (Consumption Tomorrow) = 5000
• Only Bank, No Investment Opportunity: In this case we saw that an impatient person could consume all of Rs. 5000 today, and a patient one one could get Rs. 5500 tomorrow. Meaning that, when there is a bank, there is a ‘reward for waiting’, in that the impatient person can consume an extra Rs. 500 tomorrow. We also argued that in this case it doesn’t make sense to simply add money today and tomorrow, because there is an exchange happening between today and tomorrow and only after adjusting for exchange rate can we compare monies at two different time points. So in this case we would have (1 + Interest Rate) * (Consumption Today) + (Consumption Tomorrow) = 5500. This is one of the most important lessons in finance – that one can’t simply add absolute value of money today and tomorrow, one needs to adjust for interest rate.
• Both Bank and an Investment Opportunity: Here the lesson was that as long as Net Present Value of the investment opportunity is positive both the patient and the impatient ones should invest and use bank to plan consumption based on their preferences. This is what is called Fisher Separation.

Fisher Separation Theorem

Let’s first state the theorem in words:

Given perfect and complete capital markets, the production decision is governed solely by an objective market criterion (represented by maximizing wealth) without regard to individual subjective preferences that enter into their consumption decision.

That is, the consumer’s problem of determining the optimal level of investment and optimal consumption stream can be separated in two parts:

• First we choose the investment level that maximizes our wealth. This choice is independent of one’s preferences.
• And then select the consumption stream based on the maximized wealth

This is what allows managers to invest in projects on their own merit irrespective of the individual shareholders’ preferences. And that happens because existence of capital markets allow shareholders to plan their consumption according to their preferences.

Let’s recap the example we did in the class.

We are given the following:

• Wealth: $5000$
• Investment opportunity: $2000$
• Interest rate: $r = 10\% = 0.1$

Graphically, we ended up with a figure like this:

(Click on the graph to zoom.)

For a patient person, the investment opportunities allow him to transform $I = 2000$ into $4000$, and by putting the remaining $3000$ into the bank he ends up $7300$ at the end of the period.

For an impatient person there is an extra step of going to the bank to borrow against the promised $4000$ after the investment, but she also ends up richer. How? Out of the $5000$ that she has, she gives $2000$ to her friend for the business, and on her friend’s credibility goes to the bank and borrows against the $4000$ that she knows she will get from the business (which she can then return to the bank). So, in total she ends up with: $3000 + \displaystyle \frac{4000}{1.1} = 6636$.

So both our richer by $1800$ and $1636$ respectively. But then, as the graph above shows, they are equivalent. Being richer tomorrow by $1800$ is equivalent to being richer by $1636$ today.

Of course, we could have also found this by completely ignoring the preferences, and simply considering investment in its own right:

\begin{aligned} NPV &= -\mbox{Investment} + \frac{\mbox{Cash flow from investment}}{1.1} &= -2000 + \frac{4000}{1.1} &= 1636 \end{aligned}

So, if we apply the NPV rule, we should accept this opportunity, and then plan our consumption anywhere along line $P$ in the graph above by borrowing from (or lending to) the bank.

What follows is more than what we did in the class, and can be safely ignored. For those interested, carry on.

In general, if a consumer earns in both periods (let’s say a salary of $Y_0$ and $Y_1$) and also consumes in both periods (let’s say amount $C_0$ and $C_1$) then we have the following identity:

$\displaystyle C_0 + \frac{C_1}{1 + r} = Y_0 + \frac{Y_1}{1 + r}$

which is simply another way of saying that:

$\displaystyle \mbox{Present Value of Total Consumption} = \mbox{Present Value of Total Income}$

As you can imagine, this generally holds for all kinds of income/consumption patterns, and not just over two years. Loosely speaking, ala physics you can think of it as a kind of a ‘conservation equation’.

The Consumption Decision

We may also use the above equation to find our consumption pattern given our income in both periods.

As an example for the numbers we did in the class, we can think of it as, $Y_0 = 3000$ today and $Y_1 = 4000$ tomorrow (one of the cases that we considered in the class, as those awake would recall -:) ). Then if we want to consume $P_0 = 4500$ today, then we can find out the consumption tomorrow as:

\displaystyle \begin{aligned} P_1 &=Y_1 + (Y_0 - P_0) (1 + r) \\&= 4000 + (3000 - 4500) \times 1.1 \\&= 2350 \end{aligned}

And you can check that $4500 \times 1.1 + 2350$ indeed equals $7300$.

Next we’ll spend time learning the mechanics of time value of money – the Chapter 2 of the book essentially. Most of it relies on knowing your high school simple and compound interest, and summing up simple geometric series. Try and also recall how the Euler’s number, $e$, comes about as the limit $\lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n$.

We’ll just put it all together in the context of valuing certain future cash flows.

Written by Vineet

August 12, 2016 at 3:06 pm