# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Foundations of the NPV Rule (Wonkish)

If you came looking for the summary of session 2 and found this post instead, stop right now and click here.

(So yes, any blog post which you see here with ‘Wonkish’ attached to the title – just like this one – can be safely skipped without any loss of continuity. While such posts would add value in the sense that you’ll hopefully get a better understanding of  the ideas covered, the material would typically be slightly advanced compared to what we are doing in the class. So yes, in that sense, all such posts would be, what to say, well, completely useless as far as your exams etc. are concerned. But, then, people don’t come to WIMWI just because they get to read textbooks, right?)

This post describes a graphical proof of the idea that in presence of a bank, investment and consumption decisions can be separated. Consider it as supplement, and slightly advanced, material related to the Fisher Separation Theorem.  Your take away from this? Well, it shows that what we did using some nice round numbers in the class holds more generally. If you like microeconomics, and have time, go for it. That said, it can be safely skipped without any loss of understanding of what comes later.

‘Proving’ the Fisher Separation Theorem

Let’s consider the following decision problem:

The world just has two periods – today and tomorrow. Assuming that you have an income today of amount $X$, the question that is asked is, how much should you consume today?

Case 1: No Capital Markets, No Production Opportunities

In this case the best you can do is consume whatever you feel like today (your subjective preference for today), and maybe just put the remaining money in a trunk and save for the next day. Whatever you save today is available to you tomorrow for consumption.

So, in this world, for every one unit you save, you have that available to consume tomorrow. If you started out with $X$, the max you can consume today is $X$ (if you decide nothing to consume tomorrow), and the max you can consume tomorrow is also $X$ (if you decide nothing to consume today).

So, how does my choice problem look like graphically? The budget constraint equation is $Y_0 + Y_1 = X$, and the slope is -1.

(Click on the graph to zoom)

So, if your indifference curve is of the shape $U(C_0, C_1)$ as above, you just choose any point like A on the budget constraint. Of course you can’t do any better than that, as there is just no access to any other technology – neither in terms of investment/production opportunities, nor in form of a bank/capital markets.

Case 2: Capital Markets but No Production Opportunities

Let’s complicate things a little bit and introduce a bank in the system. That is, now you have the option to not just put your money in the trunk, but also to put it in bank that will earn you an interest at the rate of $r$\%. That is, for every unit of Rupee you put in the bank today, you get $(1 + r)$ tomorrow.

So, in our previous example, if you still decide to consume $Y_0$ today, now instead you can put that remaining $Y_1$ in the bank and get $Y^{'}_1 = Y_1 (1 + r)$ tomorrow. That is, you can consume more tomorrow.

So, what changes? Well, nothing but the budget constraint, because the tradeoff that was earlier $1$-on-$1$ (you put $1$ unit in the trunk today, and that allows you to consume $1$ unit extra tomorrow) has now become $1$-on-$(1 + r)$ – that is, for every unit of your income kept aside today you get a little more because the bank pays you interest.

And that extra amount is $(1 + r) - 1 = r$. So, the budget constraint now rotates around the x-axis, to reach a higher point on the y-axis. That is, the slope which was earlier now $-1$ is now $- (1 + r)$.

The budget constraint equation now becomes $\displaystyle Y^{'}_0 + \frac{Y^{'}_1}{1 + r} = X$, and we are clearly better of:

(Click on the graph to zoom)

That is, while earlier we could only reach the utility curve $U(C_0, C_1)$, now, because of the existence of the capital markets, we are able to move to the higher utility curve $V(C_0, C_1)$.

Case 3: Production Opportunities but no Capital Markets

Now, this is a slightly less familiar case of having no bank, but having production opportunities. That is, we also have the option to invest our money now in some venture (instead of a bank).

The problem, however, is that while the production function as you know from your microeconomics course was in the production plane, our choice problem is that of consumption, so we have to first move from the production plane to the consumption plane.

We can do this because we are investing all what we are not consuming today. So, if we consume $Y_0$ today, we invest $X - Y_0$ in the production opportunity. And this is the connection that allows us to bring what is there in the production plane onto the consumption plane.

We can do so by taking a mirror image of our production function with respect to the y-axis (i.e. the output $f(X)$) and we end up with a production function in the consumption plane. (You wanna try this now on your own?)

Because the marginal rate of return from production opportunities is initially high, as long as it is more than our bank rate of return in the beginning stages of investment, we can reach even a higher consumption tomorrow as compared to what was possible in a world where there was only a bank. That is, now, given the technology of production, we can reach a maximum point $f(X)$ instead of just $X(1 + r)$ on the y-axis (tomorrow). And this is how it looks like:

(Click on the graph to zoom)

That is, now we are able to move from a new (higher) tangency point $W(C_0, C_1)$.

(Although one can’t make direct comparisons with the capital markets case, as there is no capital market here, given that a production opportunity potentially offers a higher rate of return than a bank, we may think of it as being better off compared to putting money in a bank ‘had there been one’ where our utility was $U(C_0, C_1)$.)

Case 4: Production Opportunities in Presence of Capital Markets

Ok. Let’s put it all together now. We bring our Case 2 and Case 3 together and get:

(Click on the graph to zoom)

So, while we could reach the utility curve $W(C_0, C_1)$ in Case 3, here we area able to do a little bit better. We are able to move to a higher utility curve at $W^*(C_0, C_1)$. And how does this come about?

Again, as always we break the problem in parts. Tackle it step by step, right. We have the following two decisions to make to solve our consumption problem:

• How much to put in production opportunities? (Case 3)
• Of the amount left over, how much to put in the bank (Case 2), and how much to consume today (our subjective preferences)?

The Production Decision

We have two places to park our money in. We can either put it in the bank or in production opportunities.

That is, starting at $X$, our current wealth, we can either do what we did in Case 2 and go along the $X \leftrightarrow X (1 + r)$ schedule, or we can go along $X \leftrightarrow P \leftrightarrow f(X)$ schedule? Which one should we choose?

You know enough microeconomics by now to understand that we should keep investing in one or the other until at the margin we are getting the same return from both. (If you don’t understand this you are on very shaky grounds – time to pick up your micro text.)

We can see that to the right of point $P \equiv (P_0, P_1)$, the marginal rate of return from production opportunities is more than the bank return: slope of the production opportunity curve $f(X)$ is greater than $(1 + r)$, i.e. $f'(X) > (1 + r)$. And vice-versa for the left of point $P$.

So, we should go along the path of $X \leftrightarrow P$ and invest in production opportunities till we reach point $P$. After that because the slope of the production function is less than the bank rate of return, we should not invest in production opportunities. Because at that point it makes more sense to put the money in the bank rather than put in the production opportunities. So, the long and short of the argument is that whatever one’s subjective preferences the maximum we should invest in production opportunities is till we reach point $P$, i.e. the amount $X - P_0$.

But what about consumption? Isn’t that our real decision problem?

Borrowing Against Future Income: Present Value

Before we address this question, we need a concept which I am sure none of need to be taught about. Formally speaking, this is what is called the idea of Present Value (PV) in finance.

If the bank gives you a rate of return $r$, i.e. for every one unit of money you put in the bank you get $(1 + r)$. The bank not only allows you to deposit money, but also to borrow from it. So if you borrow $1$ from the bank today, you will have to return $(1 + r)$ tomorrow. Let’s now pose the question: how much do you have to borrow today to return $1$ tomorrow?

Well, it’s a simple high school problem. You just scale today’s borrowing by a factor of $\frac{1}{1 + r}$, i.e. you just borrow $\frac{1}{1 + r}$, and when the time comes (tomorrow) return $\frac{1}{1 + r} \times (1 + r) = 1$.

We say, then, that the PV, of $1$ is $\frac{1}{1 + r}$. That is, each unit $1$ tomorrow is worth a little less today, and that amount is $\frac{1}{1 + r}$. An entrepreneur understands this all too well. PV is just an economist’s way of telling a nit’s proverbial refrain:

A bird in the hand is worth two in the bush

To summarize, if we are expecting $1$ unit tomorrow, existence of a bank allows us to borrow it’s PV today, which is $\frac{1}{1 + r}$. When the time comes, we can use our expected income to return the $1$ unit we owe to the bank. This is called borrowing against future income.

The Consumption Decision

Ok, to our consumption decision problem. Choosing the production level, or the investment amount, at $X - P_0$, gives us $P_1$ tomorrow (as in the graph above).

Let’s use what we just learnt. Assuming we are living in a world without any uncertainties, we know that we are going to get $P_1$ if we invest $X - P_0$ today. So, we can now go to the bank and borrow against this future income of $P_1$. And how much will the bank give us for that?

Well, that’s simple. Because the PV of $1$ unit today is $\frac{1}{1 + r}$, $P_1$ is worth $\frac{P_1}{1 +r}$ today, and that’s exactly what the bank would be willing to offer us. That is, a bank allows us to increase our today’s consumption by borrowing against tomorrow’s earnings – the PV of $P_1$, i.e.  $\frac{P_1}{1 +r}$.

Remember that $P$ is the point where the marginal rate of transformation (MRT) from production just matches the ‘opportunity cost’ of putting that money in the bank. And as we argued, that is the maximum amount of money we should be investing in production opportunities. Any more than that and we are choosing an inferior solution to our production decision problem.

So, the total consumption possible today at the point of optimum production  is:

$\displaystyle Z_0 = P_0 + \frac{P_1}{1 +r}$

You are welcome to check, but $Z_0$ is the farthest point we can reach on the x-axis. That is, any other point on the locus of the production function results in a solution that will be to the left of (or inferior to) $Z_0$.

That is by starting out with the wealth $X$ we have invested $X - P_0$ which gives $P_1$ tomorrow, whose PV is just $\frac{P_1}{1 + r} = Z_0 - P_0$. That is, the maximum possible consumption today is $P_0 + (Z_0 - P_0) = Z_0$.

Net Present Value

This is now time to introduce another important concept in finance.

How much did we invest? We invested $X - P_0$ in our production opportunity. And what did we get out of it ? $X - P_0$ invested today gave us $P_1$ tomorrow. The PV of the income from investment tomorrow is $\frac{P_1}{1 +r}$.

The difference between our investment outlay ($X - P_0$) and the PV of our income from the investment $\frac{P_1}{1 +r}$ is called the Net Present Value, or by its abbreviation, as NPV. And our NPV with investment of $X - P_0$ is:

\begin{aligned} \displaystyle NPV &= \frac{P_1}{1 +r} - (X - P_0) \\&= \big(P_0 + \frac{P_1}{1 + r}\big) - X \end{aligned}

Again, you can show that any other level of production / investment results in a lower NPV. That is, investment at a point where MRT = $- (1 + r)$ maximises the NPV.

The Last Step: Back to the Consumption Problem

But what if one wants to consume the same amount of money today that one was consuming when there was no bank. That is the amount $Y_0$ as in Case 3. Well, in presence of a bank one can easily do that. How? Simple – by borrowing against future income. In fact, why just $Y_0$, with a bank one can actually attain any level of consumption today up to $Z_0$. And how is that?

Well, one doesn’t have to borrow against all of the future income $P_1$, right? One can borrow any fraction of the future income. So, if one wants to consume $Y_1$ tomorrow, one can borrow the PV of $P_1 - Y_1$, which as you are welcome to check is exactly $Y_0 - P_0$. This means that the consumption today is $P_0 + (Y_0 - P_0) = Y_0$ – again, exactly as wanted.

Since choice of $Y_0$ is arbitrary, to generalize, by borrowing part/full against one’s future income one can reach any point on the $P_0 \leftrightarrow Z_0$ line.

Of course, one can also do the other way around. If we don’t want to consume $P_0$ today, we can postpone our consumption by lending – by depositing the money in the bank. Again, the same logic applies.

Remember, we still invest till the point $P$. And if we want to consume less than $P_0$ today, we put the remaining money in the bank. (To the left of $P$, the marginal return from the production opportunity is less than the rate of return from the bank, so it doesn’t make sense to invest if we are to the left of $P$.)

This gives us the following Fisher Separation Theorem:

In presence of productive opportunities and capital markets, all consumers should choose the investment opportunities that maximises their net present value (the farthest point on abscissa: $Z_0$) irrespective of their individual subjective preferences. Having selected the level of investment that maximises their net present value (wealth), they should then borrow from / lend to the bank depending on how they want to plan (smooth) their inter-temporal (in between times) consumption.

But what happens at the margin: What if we only have a single investment opportunity – as for example, we did in the class?

So far we have considered a continuous production function – that is, we implicitly assume there is a continuous set of (or infinite) investment opportunities. We then argued that the NPV is maximised at the level of investments where MRT = $- (1 + r)$.

But what happens when we have only a single investment opportunity? What is the optimal investment decision criterion in that case? So what we are essentially saying is that instead of a production function schedule/curve, we have just a point.

Let’s consider two different cases. First is the case where the available investment opportunity, $A$, lies to the left  of the “bank line”, i.e. as below:

(Click on the graph to zoom)

The second case we consider when it’s the other way around – that is the investment opportunity $B$ lies to the right of the “bank line”, i.e. as below:

(Click on the graph to zoom)

It should be clear to you that the opportunity $A$ is inferior to opportunity $B$. How? As earlier, just draw the ‘tangency line’ on the available investment opportunity ‘set’. Of course, in this case it would be trivial and would just translate into drawing a line parallel to the “bank line” passing through the points $A$ and $B$. What do we get? Let’s see:

(Click on the graph to zoom)

So, if we choose investment $A$, we consume $A_0$ today, invest $X - A_0$ and get $A_1$ tomorrow, and the total PV is $X_A$. On the other hand if we choose investment $B$, we consume $B_0$ today, invest $X - B_0$ and get $B_1$ tomorrow, and the total PV is $X_B$. That is, the NPV from investment in $A$ and $B$ respectively are:

\displaystyle \begin{aligned} NPV_A &=A_0 + \frac{A_1}{1 + r} - X \\&= X_A - X \\& < 0 \end{aligned}

\displaystyle \begin{aligned} NPV_B &=B_0 + \frac{B_1}{1 + r} - X \\&= X_B - X \\& > 0 \end{aligned}

That is, if we choose investment $A$ we end up with a PV less than our original wealth $X$ and that PV is $X_A$. On the other hand, if we choose investment $B$ we end up with a PV which is more than $X$ and that PV is $X_B$. Investments of the kind $A$ with NPV  < 0 are clearly not desirable then – we end up with wealth less than what we start out with.

So if have only finite number of investment opportunities to choose from we should only choose the ones that have an NPV > 0.

This is indeed a more realistic situation. Typically a manager would only have select investment opportunities available. In that case, the NPV maximisation rule boils down to a very simple criterion:

Select all investment opportunities that have NPV > 0.

Assumptions in ‘proving’ the Separation Theorem

1. No uncertainty
2. All consumers have the same information
3. There are no transaction costs
4. One can borrow and lend at the same rate of interest
5. Capital markets are complete (we’ll see what that exactly means when we study the notion of market efficiency)

To lay the cards on the table, the theorem doesn’t work exactly (or as neatly) when there are transaction costs. Again, assumptions do not always test a theory, its predictions do. And clearly, when managers and entrepreneurs plan their investments and financing decisions their intuition is not too far from the Separation Theorem. And that’s why you should understand what it means and where is it coming from.

Even though we ‘proved’ the theorem in a certain world, it turns out it holds even in the uncertain world if instead of a certain payoff you are willing to substitute its certainty equivalent in the problem. But that’s more than you should worry about at this stage – unless you are one of the two Finance area FPMs in the class 🙂