# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] How markets really work

Enjoy!

Written by Vineet

September 9, 2016 at 9:21 pm

Posted in Stories, Teaching: FM

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## [PGP-I FM]: CML vs. SML

We earlier wrote the Capital Market Line (CML) as:

$\displaystyle E[R_p] = \underbrace{R_f }_{\mbox{Reward for Waiting}}+ \underbrace{\frac{E[R_m] - R_f}{\sigma_M}}_{\mbox{Reward per unit of Risk}} \sigma_p$

which describes expected return from efficient portfolios.

Later in the class we extended this idea of expected return as comprising reward for waiting $R_f$, and reward for bearing risk $(\frac{E[R_M] - R_f}{\sigma_M})\sigma_p$ to write the Security Market Line (SML) from the Capital Asset Pricing Model (CAPM) as:

$\boxed{E[R_i] = R_f + (E[R_M] - R_f) \beta_i}$

where,

\begin{aligned} \beta_i &= \frac{\mbox{Cov}(R_i, R_M)}{\sigma_M^2} \\&= \frac{\sigma_{iM}}{\sigma_M^2} \end{aligned}

where the only, but the key, difference was that instead of using standard deviation as a measure of risk we used $\beta_i$ – sensitivity of change in return from a stock to change in the market – as the measure of risk.

Another important distinction between CML and SML is that while the former can be used only for efficient/optimal portfolios, the latter can be used all assets, inefficient portfolios as well as efficient portfolios.

That is CML is like a subset of SML, or alternatively SML subsumes CML.

Before we can prove this result, however, we need another result – that market beta is a weighted average of beta of individual securities. In this part of the post we establish the relationship for $\beta$, and in the next part we prove that SML implies CML.

Market beta is a weighted average of beta of individual securities

Recall from your basic probability theory that for any three random variables, $I, J$ and $M$:

\begin{aligned} Cov[\omega_1 I + \omega_2 J, M] &= \omega_1 Cov[I, M] + \omega_2 Cov[J, M] \\ &= \omega_1 \sigma_{IM} + \omega_2 \sigma_{JM} \end{aligned}.

If we let $M = \omega_1 I + \omega_2 J$, and use the fact that $Cov[M, M] = \sigma^2_M$, it immediately follows that

\begin{aligned}\sigma^2_M &= \omega_1 \sigma_{IM} + \omega_2 \sigma_{JM} \\ \Rightarrow 1 &= \omega_1 \frac{\sigma_{IM}}{\sigma_M^2} + \omega_2 \frac{\sigma_{JM}}{\sigma_M^2} \\ \mbox{or } 1 &=\omega_1 \beta_1 + \omega_2 \beta_2 \end{aligned}

Our proof below is just a generalization of this result.

Taking as starting point the result that market variance is a weighted average of covariance of individual assets with the market (previous post), i.e.:

\begin{aligned} \sigma^2_M &= \sum_{i=1}^N \omega_i \sigma_{iM} \\ & = \omega_1\sigma_{1M} + \omega_2\sigma_{2M} + \omega_3\sigma_{3M} + ... + \omega_N\sigma_{NM} \end{aligned}

Dividing both sides by $\sigma^2_M$ gives:

\begin{aligned} 1 &= \omega_1\frac{\sigma_{1M}}{\sigma^2_M} + \omega_2\frac{\sigma_{2M}}{\sigma^2_M}+ \omega_3\frac{\sigma_{3M}}{\sigma^2_M}+ ... + \omega_N\frac{\sigma_{NM}}{\sigma^2_M} \end{aligned}

Using the intuition that $\beta$ is the sensitivity of change in a stock’s return to change in market return, that is, it can be interpreted as the regression coefficient, we have the result that:

$\displaystyle \beta_i = \frac{\sigma_{iM}}{\sigma^2_M}$

We can now substitute this formula for $\beta$ in the previous equation and write:

\begin{aligned} 1 &= \omega_1\frac{\sigma_{1M}}{\sigma^2_M} + \omega_2\frac{\sigma_{2M}}{\sigma^2_M}+ \omega_3\frac{\sigma_{3M}}{\sigma^2_M}+ ... + \omega_N\frac{\sigma_{NM}}{\sigma^2_M} \\ & = \omega_1\beta_1 + \omega_2\beta_2 + \omega_3\beta_3 + ... \omega_N\beta_N \end{aligned}

By definition market $\beta_M$ is $1$ (this is trivial, beta of the market represent change in the market when market changes), we have our required result that:

\begin{aligned} \beta_M & = \omega_1\beta_1 + \omega_2\beta_2 + \omega_3\beta_3 + ... \omega_N\beta_N = 1 \end{aligned}

Or, more succinctly, using the summation symbol:

$\boxed{\displaystyle \beta_M = \sum_{i=1}^N \omega_i \beta_i = 1}$

This result can be used to show that SML implies CML – which brings us to our last lesson for this module:

Moral of the Story 5: In the CAPM world, while CML describes expected return from efficient portfolios SML describes expected return from individual stocks, inefficient portfolios as well as efficient portfolios. SML implies CML.

Written by Vineet

September 9, 2016 at 9:07 pm

Posted in Teaching: FM

## [PGP-I FM] Diversifiable vs. Non-diversifiable Risk: The Math (Wonkish)

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The result that variance of the market portfolio is a weighted average of covariance of the underlying stocks with the market is a general result. This post fills in the mathematical blanks.

Diversifiable vs. Non-diversifiable Risk: The Math

Consider the variance of a Markowitz portfolio $M$ containing $N$ assets:

$\displaystyle \sigma^2_M = \sum_{i=1}^N \omega^2_i \sigma^2_i + \sum_{i=1}^N \sum_{\substack{j=1 \\ j \ne i}}^N \omega_i \omega_j \sigma_{ij}$

Now if we let the weight of each asset in the portfolio to be the same, i.e. $\omega_i = \omega = 1/N$ $\forall i$, and consider the “average variance” as:

$\displaystyle \overline{\sigma^2} =\frac{1}{N} \sum_{i=1}^N \sigma^2_i$

and “average covariance” as:

$\displaystyle \overline{\sigma_{ij}} = \frac{1}{N (N - 1)} \displaystyle \sum_{i=1}^N \sum_{\substack{j=1 \\ j \ne i}}^N \sigma_{ij}$

then the above portfolio variance simplifies to:

\begin{aligned} \displaystyle \sigma^2_M &= \sum_{i=1}^N \frac{1}{N^2} \sigma^2_i + \sum_{i=1}^N \sum_{\substack{j=1 \\ j \ne i}}^N \frac{1}{N^2} \sigma_{ij} \\ &= \frac{1}{N^2} \sum_{i=1}^N \sigma^2_i + \frac{1}{N^2} \sum_{i=1}^N \sum_{\substack{j=1 \\ j \ne i}}^N \sigma_{ij} \\ &= \frac{N \overline{\sigma^2}}{N^2} + \frac{N(N - 1)}{N^2} \overline{\sigma_{ij}} \\ &= \frac{ \overline{\sigma^2}}{N} + (1 - \frac{1)}{N} \overline{\sigma_{ij}} \end{aligned}

Then as $\displaystyle N \rightarrow \infty$, the portfolio variance $\sigma^2_M$ converges to:

$\displaystyle \sigma^2_M = \overline{\sigma_{ij}}$

That is, as the number of assets in the portfolio go up, the variance of individual assets become unimportant, and its the covariance terms that dominate. This is just our diversification. Graphically this can be represented as:

Diversification (Click on the figure to zoom; Source: Brealey-Myers, 9th Ed.)

Unique Risk (or alternatively, Diversifable Riskor Unsystematic Riskor Idiosyncratic Risk) is the “average variance” of the individual assets.

As number of assets in the portfolio increase, this “average variance” tends to zero. The only risk, then, that matters is the one that remains after diversification has done its work. And this is just the average covariance between all assets in the portfolio. This is called Market Risk (or alternatively, Systematic Risk, or Undiversifiable Risk). And accordingly, the covariance of an asset with the market portfolio is called its market risk.

The fact that portfolio variance after diversification is just the weighted average of covariance between assets can be seen by first noting that:

\displaystyle \begin{aligned} \sigma_{iM} &= E[r_i - \overline{r_i}][r_M - \overline{r_M}] \\ &= E[r_i - \overline{r_i}][\sum_{j=1}^N \omega_j r_j - \sum_{j=1}^N \omega_j \overline{r_j}] \end{aligned}

Since the expectations add up, we can take out the summation sign outside the expectation, and it follows that:

\displaystyle \begin{aligned} \sigma_{iM} &= \sum_{j=1}^N \omega_j E[r_i - \overline{r_i}][r_j - \overline{r_j}] \\ & \Rightarrow \boxed{\sigma_{iM} = \sum_{j=1}^N \omega_j \sigma_{ij}} \end{aligned}

That is the covariance of any asset with the market portfolio is nothing but the weighted average of its covariance with all other assets in the portfolio.

Next, note that we can write:

\displaystyle \begin{aligned} \sigma^2_M &= \sum_{i=1}^N \omega^2_i \sigma^2_i + \sum_{i=1}^N \sum_{\substack{j=1 \\ j \ne i}}^N \omega_i \omega_j \sigma_{ij} \\&= \sum_{i=1}^N \sum_{j=1}^N \omega_i \omega_j \sigma_{ij} \end{aligned}

(If you are looking for the variance terms, note the change in the limits in the summation operator, and recall that $\sigma_{ii} = \sigma^2_i$)

Then if we substitute our result that $\sigma_{iM} = \sum_{j=1}^N \omega_j \sigma_{ij}$, we see that:

\displaystyle \begin{aligned} \sigma^2_M &= \sum_{i=1}^N \sum_{j=1}^N \omega_i \omega_j \sigma_{ij} \\ &= \sum_{i=1}^N \omega_i \sigma_{iM} \end{aligned}

That is the variance of the market portfolio is just the weighted average of the covariance of all assets in the portfolio with itself. Again, this result is important enough to warrant a separate ‘box’:

$\boxed{\sigma^2_M = \sum_{i=1}^N \omega_i \sigma_{iM}}$

Combine this with our observation that in the limit individual variances (unique risks) disappear and we have our economic result that:

Moral of the Story 4: The risk of an individual asset is determined not by its individual variance, but by its covariance with the market portfoliobecause the diversifiable/unique/idiosyncratic risk can be diversified away.

Written by Vineet

September 9, 2016 at 8:57 pm

Posted in Teaching: FM

## [PGP-I FM] Capital Market Line

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At this stage, having introduced the new straight line efficient set, we are all but there to our final destination. So, let’s step back a bit and try and understand the larger picture.

In the beginning was the efficient frontier. Markowitz gave us that. Efficient frontier describes the maximum possible expected return for any given amount of risk from the portfolio of available assets. Or alternatively, the minimum amount of risk that one must live with for any given amount of expected return.

As a first step, we moved from individual assets to portfolios that lay on the efficient frontier. When we did that implicitly the x-axis (labeled as risk/standard deviation) then became the risk of the portfolio (and not the risk of the individual assets). There should be no cause for this confusion, but no harm emphasizing it nonetheless – because we would always lie on the frontier the right risk/standard deviation to consider is the risk/standard deviation of the portfolio and not the individual asset.

Moral of the Story 1: When we consider the efficient frontier the relevant quantities to consider are portfolio risk and portfolio expected return.

Then, of course, Tobin came along and introduced a risk free asset in the Markowitz world, and he said we could ignore all other points on the frontier except the tangency one – because everybody would hold some proportion of only the tangency portfolio $\mbox{M}$ (as all other points even on the envelope are now inferior), and the line connecting the return from the risk free asset $R_f$ and the tangency portfolio offers the best possible combinations of portfolio risk and expected return. Remember, the operative word here is portfolio.

This gave us our revised efficient set as:

(Click on the figure to zoom.)

The equation of the new efficient set immediately follows (it’s a linear line with intercept at $R_f$ and slope $\displaystyle \frac{E[R_M] - r_f}{\sigma_M}$)  as:

$\boxed{\displaystyle E[R_p] = R_f + \frac{(E[R_M] - R_f)}{\sigma_M}\sigma_P}$

What is the tangency portfolio $M$?

Having said that all investors should hold the tangency portfolio $M$, the next thing to understand is the meaning of this tangency portfolio. By saying that all investors should hold $M$, what we are essentially saying is that investors would demand only combinations of portfolio $M$ and the risk-free asset. (Holding any other risky portfolio other than $M$ is inefficient.) This is the demand side of the problem. What is the supply side? The supply side is just all the assets that exist in the market.

And by now you would know enough of microeconomics to understand that equilibrium requires that demand be same as supply. That is, assets demanded in the portfolio $M$ must exactly equal the supply of each asset in the market. And the supply of each asset in the market is given by its market capitalization. So, in equilibrium all assets must be held in $M$ in exactly the same proportion as their market capitalization. That is, in percentage terms weight of assets in the total market capitalization and in the portfolio $M$ must be the same.

Consider the case where you run your Markowitz optimizer and find that that weight of a particular asset, say $\omega_i$ is $0$. Is that possible? Mathematically, of course, yes. But what about economically? Let’s try and understand this.

Saying that the weight of an asset $\omega_i$ in the Markowitz portfolio $M$ is $0$ is saying that no investor wants to hold the $i^{th}$ asset. If no investor wants to hold that asset, but the asset exists in the market then we have a state of disequilibrium. And what happens in a state of disequilibrium? Prices adjust. So, if no wants to hold an asset, its price will drop. Once the price starts to drop its expected return:

$E[r_i] = \displaystyle \frac{E[P_i] - P_0}{P_0}$

will rise. As the price starts to fall, and expected return starts to rise, investors would start to find this asset more attractive. As its expected return $E[r_i]$ rises even more, then when you re-run your Markowitz optimizer again, you’ll find that this asset has a non-zero weight in the tangency portfolio $M$. That is, all assets that exist in the market must be held. This brings us to another important lesson:

Moral of the Story 2: The tangency portfolio $M$ is nothing but the market itself!

As another example consider a situation where the Markowitz optimizer prescribes a weight of $2.1\%$ for an asset whose market capitalization is $2\%$. What happens in that case? Well, now you know how to think about such disequilibrium situations. This is the case where the asset has more demand than supply. When demand is more than supply, prices rise. As price rises, the expected return will fall. As expected return falls, the Markowitz optimizer will prescribe a lower weight to this asset and in equilibrium the price and the market capitalization of the asset would adjust to make the demand exactly equal supply. That is:

When one imposes equilibrium, the line passing through the tangency portfolio has a specific name and it is called the Capital Market Line.

Note that at this stage, when we impose economic equilibrium, we have to necessarily assume that everybody has the same information – things don’t quite work the same way otherwise. And this brings us to the last moral of the story for today:

Moral of the Story 3: All efficient portfolios lie on the Capital Market Line.

Again, as in the case of the efficient frontier, the relevant quantities in the Capital Market Line are the expected return and risk of efficient portfolios. All individual stocks and other inefficient portfolios, however, would be anywhere below the efficient frontier, as say in the shaded portion of the graph below (from your book; think of $S$ in the plot below as the equilibrium market portfolio):

CML with the Efficient Frontier (Click to zoom; Source: Brealey-Myers, 9th Ed.)

Written by Vineet

September 3, 2016 at 10:39 pm

Posted in Teaching: FM

## [PGP-I FM] Portfolio Selection with a Risk-free Asset

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For all his insights on the portfolio choice problem, somehow Markowitz didn’t explicitly consider a bank in the system. In principle, of course, one could have solve the problem by just adding one more security in his set up. However, it turns out that having a bank in the system is not just a matter of adding one more security to the world – there is a bit more to it in terms of intuition.

Let’s first consider how a Markowitz-ian would handle this problem. A fan of Markowitz would just rerun the following optimization problem, but instead would consider $N + 1$ assets instead of $N$, i.e. nothing much really changes:

$\displaystyle \max_{\omega} \sum_{i = 1}^{N + 1} \omega_i \mu_i$

$\mbox{ s.t. } \displaystyle \sigma^2 = \sum_{i=1}^{N + 1} \sum_{j = 1}^{N + 1} \omega_i \omega_j \sigma_i \sigma_j = c$

So, we would need to rerun our optimization software and this will give us a new allocation of weights to all the securities. Today, of course, the problem is hardly difficult (you can even do it in Excel). But is it the best way to introduce a risk-free asset in the Markowitz world?

James Tobin, a colleague of Markowitz’s at the Cowles Foundation in the ’50s (and another Nobel Laureate) argued that it’s not. And brilliant as his device was, we can easily see its impact in a two stock world.

In our familiar two stock world, let one of the assets be risk-free, such that it’s rate of return is known ‘today’ as $R_f$ with variance, of course, zero. Then, since of the assets is no more a random variable, even the correlation between the two $\rho$ would also be 0. So, if in our set of equations:

\begin{aligned} E[R] &= \omega_X (\mu_X - \mu_Y) + \mu_Y \\ \sigma^2 &= \omega^2_X \sigma^2_X + (1 - \omega_X)^2 \sigma^2_Y + 2 \omega_X (1 - \omega_X) \rho \sigma_X \sigma_Y \end{aligned}

we let $\mu_Y = R_f \mbox{ , } \sigma_Y = 0 \mbox{ and } \rho = 0$ we are left with:

\begin{aligned} E[R] &= R_f + \omega_X (\mu_X - R_f) \\ \sigma &= \omega_X \sigma_X \end{aligned}

That is, the efficient frontier in this case is simply a straight line connecting the rate of return from the risk-free asset $R_f$ and the expected return from the asset $X$, with slope $= \frac{\mu_X - R_f}{\sigma_X}.$ If one could assume that people could both borrow and lend at the same risk-free rate, $R_f$, then we could even consider negative weights on the risk-free asset, and extend the Efficient Frontier to the right (the “blue dots” in the graph below). So, if an investor would extremely risk-loving he/she could borrow money from the bank and invest it in the second risky asset.

(Click on the graph to zoom.)

With this insight Tobin said that with the risk-free asset in the world in the $N$ asset Markowtiz-ian world, we can just consider such straight lines emanating from the intercept on the ordinate (return from the risk-free asset $R_f$) and connecting with all the points on Efficient Frontier. That is, he said, rather than re-running the Markowitz optimizer, let’s only consider following straight lines connecting the Efficient Frontier:

(Click on the graph to zoom.)

That is, instead of considering just single assets, Tobin argued we could consider connecting stratight lines to efficient portfolios. And lines of the kind $R_f - B$, $R_f - A$ and $R_f - M$ all such possibilities. By now it should be clear that we have a new Efficient Frontier which is the line $R_f - M$. So while points lying to the left of the “blue dots” mean that some of the wealth is invested in the risk-free asset and some in the portfolio $M$ (called the tangency portfolio), and points lying on the “blue dots” represent the points where an investor has put all of one’s wealth in the tangency portfolio $M$ and then some.

That is, as we see having a risk-free asset in the Markowitz world changes everything. Instead of a concave envelope, one ends up with a much-simpler frontier, which is a straight line.  For rational investors, then, only two assets should matter – the risk-free asset and the tangency portfolio.

Another way of stating the same thing is to say that the line from $R_f$ to $M$ is the steepest, or alternatively offers the maximum reward per unit of risk compared to any other point on the frontier. That is, the slope of the line $R_f - M$ is more than slope of both lines $R_f - A$ and $R_f - B$.

Financial market professionals have kinda made this idea their own and turned it into a measure of performance to gauge the excess return per unit of risk from investment choices made by fund managers. They call the slope of the lines emanating from $R_f$ and joining points on the efficient frontier, like $R_f - A$, $R_f - B$ and $R_f - M$, as the Sharpe ratio.

Since the slope of the line $R_f - M$ is the highest, so is the Sharpe ratio of investment in the market portfolio. Note that since Sharpe ratio is defined in terms of expected returns, ex-ante (or before-the-fact) Sharpe ratio of investment in the market portfolio is the highest. So, for a given point, say, $M$, on the efficient frontier the Sharpe ratio is given as:

$\displaystyle \mbox{Sharpe Ratio} = \frac{E[R_M] - R_f}{\sigma_M}$

This brings us to the second separation theorem in finance, and it goes by multiple names of Tobin/Two-fund/Mutual-fund Separation Theorem. It’s important enough to warrant a formal statement:

Mutual Fund Separation Theorem: Each investor will have a utility maximising portfolio that is a combination of the risk-free asset and a tangency portfolio $\mbox{M}$. All risky portfolios other than the portfolio $\mbox{M}$ are inefficient.

Note that all points to the ‘top’ of $\mbox{M}$ are unattainable. Our original Efficient Frontier presented all possibilities giving maximum return for any given level of risk. Having a risk-free asset implies that the line connecting the return from the risk-free asset and the tangency portfolio dominates all other possibilities. This is the new efficient frontier.

And now we can get rid of the original concave envelope, and we are left with just the $R_f - M$ line. And a quick Google Image search gives us this nice little picture presenting different possibilities combining the risk-free asset and the tangency portfolio:

[Click on the figure to zoom; Source: Wikipedia]

Post-script

Needless to say, by definition, Sharpe ratio coincides with the slope of the $R_f - M$ line when the investment manager chooses $M$ as the point on the frontier, i.e.:

\displaystyle \begin{aligned} \mbox{Sharpe Ratio for Tangency Portfolio} &= \frac{E[R_M] - R_f}{\sigma_M} \end{aligned}

Here is Nobel Laureate William Sharpe on the ratio that bears his name.

Written by Vineet

September 3, 2016 at 10:32 pm

## [PGP-I FM] Markowitz’s Portfolio Theory: II

With Markowitz having shown us that only expected return and variance of the gambles matter, we need not restrict ourselves to  considering 120-80 kind of gambles with only two possible states of the world. The only thing we need is estimates of expected return and variance of the gambles – and we can study their combinations more generally. Let’s do that now.

Consider two stocks $X$ and $Y$ with expected returns $\mu_X \mbox{ and } \mu_Y$, variances $\sigma^2_X \mbox{ and } \sigma^2_Y$ and correlation $\rho_{XY}$ between them. If we consider an investor with unit wealth,  with amount $\omega_X$ invested in stock $X$, and $\omega_Y = 1 - \omega_X$ invested in stock $Y$  then the portfolio expected return $latex\mathbb{E}[R_p]$ and portfolio variance $\sigma^2$ are easily obtained using some basic results from probability theory as:

\begin{aligned}\mathbb{E}[R_p] &= \omega_X \mu_X + \omega_Y \mu_Y \\ &= \omega_X \mu_X + (1 - \omega_X) \mu_y \\& = \omega_X (\mu_X - \mu_Y) + \mu_Y \\ \\ \sigma^2 &= \omega^2_X \sigma^2_X + \omega^2_Y \sigma^2_Y + 2 \omega_X \omega_Y \rho \sigma_X \sigma_Y \\& = \omega^2_X \sigma^2_X + (1 - \omega_X)^2 \sigma^2_Y + 2 \omega_X (1 - \omega_X) \rho \sigma_X \sigma_Y \\& < (\omega_X \sigma_X + (1 - \omega_X) \sigma_Y)^2 \end{aligned}

It should be clear that the relationship between the portfolio weight in any asset and portfolio expected return is linear, and that between portfolio weight and variance is quadratic. Our purpose, however, is to look at the trade-off between expected return and variance of the portfolio.

We are lucky that the relationship between expected return and weight in any asset is linear so we can eliminate the weights and express expected return as a quadratic function of variance. The algebraic expression is messy and lacks intuition, but for any given level of $\rho$, it can shown (and as we did in the class using Excel) that shape of the trade-off is something like this:

Efficient Frontier: Two Assets

That is, the opportunities available to an investor is a concave envelope. And this envelope captures the trade-off between expected return and risk available from the portfolio (geometrically speaking, it is a conic section – you can do the math and check which one!).

It should be clear that to a rational investor all the points below the minimum variance point should be inferior – as all those points represent a lower expected return for any given level of risk. That is, no rational investor would prefer to choose a portfolio that lie below the minimum variance point.

The envelope traced by the upper arm of the curve above the minimum variance point is called the Opportunity Set (or Efficient Set or Efficient Frontier). This is the set of opportunities available to a rational investor given the securities available in the market.

So, according to Markowitz all investors should choose one of the portfolios lying on the Efficient Frontier depending on their degree of risk aversion. So, if an investor is risk loving he/she should choose one of the points on the top right end (high risk, high expected return), or if he/she is risk-averse choose one of the points on the bottom left part of the frontier, but never below  the Frontier.

In a two-asset world it was easy to visually identify the Efficient Frontier – but for an $N$ asset world, the Markowitz portfolio selection boils to solving the following Quadratic Programming problem:

$\displaystyle \max_{\omega} \sum_{i = 1}^{N} \omega_i \mu_i$

$\mbox{ s.t. } \displaystyle \sigma^2 = \sum_{i=1}^{N} \sum_{j = 1}^{N} \omega_i \omega_j \sigma_i \sigma_j = c$

or alternatively,

$\displaystyle \min_{\omega} \sum_{i=1}^{N} \sum_{j = 1}^{N} \omega_i \omega_j \sigma_i \sigma_j$

$\displaystyle \mbox{ s.t. } \sum_{i = 1}^{N} \omega_i \mu_i = c$

And how does the Efficient Frontier looks like for the case of $N$ assets? As expected, all the set of opportunities available increase. But, importantly, luckily for us, Robert Merton showed that the Efficient Frontier retains the same concave shape whatever be the number of securities in the market. In general, the shape looks something like the following (from your book):

Efficient Frontier: Multiple Assets (Click on the graph to zoom; Source: Brealey-Myers, 9th Ed.)

Again, according to Markowitz, no investor should be on any point below the “pink line” (Efficient Frontier, traced by ABCD), i.e. in the shaded region, as all points on the curve ABCD offer a higher expected return for any given level of risk / variance.

Given that expected utility is also a function of expected return and variance, by clubbing the two together Markowitz had solved the Portfolio Selection problem for a rational investor. So, if an investor were risk-averse he/she would choose a portfolio like C or D, and if one were risk-loving then he/she would choose a portfolio like B or A, but never anything below the curve ABCD.

Special Cases

In the two-stock world two special cases are interesting.

Consider two stocks $X$ and $Y$ with expected returns $\mu_X \mbox{ and } \mu_Y$, variances $\sigma^2_X \mbox{ and } \sigma^2_Y$ and correlation $\rho_{XY}$ between them. We had the following relationships for the portfolio variance from the two stocks:

\begin{aligned} \mathbb{E}[R_p] &= \omega_X (\mu_X - \mu_Y) + \mu_Y \\ \\ \sigma^2 &= \omega^2_X \sigma^2_X + (1 - \omega_X)^2 \sigma^2_Y + 2 \omega_X (1 - \omega_X) \rho \sigma_X \sigma_Y \end{aligned}

1. Perfect positive correlation: $\rho = 1$

Setting $\rho = 1$ doesn’t change the expected return $latex\mathbb{E}[R_p]$, but simplifies the portfolio variance to:

\begin{aligned} \sigma^2 &= (\omega_X \sigma_X + (1 - \omega_X) \sigma_Y)^2 \\ \Rightarrow \sigma &= \omega_X \sigma_X + (1 - \omega_X) \sigma_Y \end{aligned}

i.e. the portfolio standard deviation is just a weighted average of the standard deviation of the two assets. If the two stocks are perfectly positively correlated, that is they move in lock-step in the same direction all the time, it’s as if they are the two same stocks.

2. Perfect negative correlation:$\rho = -1$

Again, setting $\rho = -1$ doesn’t change the expected return $latex\mathbb{E}[R_p]$, but simplifies the portfolio variance to:

\begin{aligned} \sigma^2 &= (\omega_X \sigma_X - (1 - \omega_X) \sigma_Y)^2\\ \Rightarrow \sigma &= \pm (\omega_X \sigma_X - (1 - \omega_X) \sigma_Y) \end{aligned}

While even in this case the portfolio standard deviation is just a weighted average of the standard deviation of the two assets, there are two possibilities (two roots) given the magnitude of $\sigma_X \mbox{ and } \sigma_Y$. While mathematically there are two possibilities, as the graph below shows us, economically there is only one possibility.

What’s more interesting, however, is that when $\rho = -1$we can reduce the portfolio variance to zero. How is that? We have:

$\sigma = \pm (\omega_X \sigma_X - (1 - \omega_X) \sigma_Y)$

And setting,

\begin{aligned} \omega_X &= \frac{\sigma_Y}{\sigma_X + \sigma_Y} \\ \\ \Rightarrow \sigma &= 0 \end{aligned}

That is, when there is perfect negative correlation (recall our earlier 120-80 example), by appropriately allocating our wealth in the two stocks we can reduce our portfolio variance to 0, i.e. remove all risk.

In general, depending on whether the value of the correlation is such that $\rho = 1$ or $\rho = -1$, or $|\rho| < 1$, the efficient frontier changes as below:

Efficient Frontier: Two Assets (Special Cases)

And here is Markowitz on Markowitz.

Written by Vineet

September 1, 2016 at 7:24 pm

## [PGP-I FM] Markowitz’s Portfolio Theory: I

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In our journey so far – from giving a justification for NPV for valuing investments to valuation of common stocks – we have been talking about risk all along, but we haven’t really done full justice to it. Fair enough, it’s been at the back of our mind in all our discussions all along, but we still don’t quite have a way to measure it.

Like for most fundamental ideas in finance, the first systematic treatment of risk can also be traced back to early/mid-twentieth century, which brings us to the third protagonist in our story.

In the early ’50s, a precocious graduate student at the University of Chicago named Harry Markowitz was looking for a suitable topic for his dissertation. After an encounter with a trader sparked an interest in financial markets, his would-be adviser suggested he read John Burr Williams’ Theory of Investment Value, whose Dividend Discount Model we learnt while talking about common stock valuation.

What struck Markowitz was that if John Burr Williams’ theory is correct, then people should buy just one stock – the one that offered the maximum possible expected return and nothing else. But, he noticed, obviously it’s not what people did (or do). To quote Peter Bernstein in his entertaining biography of finance, Markowtiz

…was stuck with the notion that people should be interested in risk as well as return.

This, of course, is nothing new to us now. We learnt that while talking about the St. Petersburg Paradox: that while valuing risky gambles we should not be looking at expected return from the gamble, but expected utility of returns from the gamble. One of the things that we noticed then was that for any risky gamble $X$, expected utility $E[U(X)]$ is always less than the utility of the sure thing $U(E[X])$ (mathematically known as the Jensen’s inequality for concave functions), i.e.:

$\mathbb{E}[U(X)] < U(\mathbb{E}[X])$

Markowitz had this figured this out too, and he exploited this idea to come up with a way to quantify the trade-off between risk and return.

Given a certain starting wealth $W$, Markowitz studied the change in expected utility to marginal investments in risky gambles.

That is, he considered the quantity $\mathbb{E}[U(W(1 + h))]$ for any small risky gamble $Wh$ relative to the starting wealth $W$.

As you may have done in your statistics courses, a useful way to think about a risky gamble is as a random variable (something that takes a different value depending on the ‘state of the world’). Since we can talk about $h$ as a random variable, we can talk about its expected value, say $\mu$, and variance, say $\sigma^2$.

With $h$ small we can evaluate $U(W(1 + h))$ as a Taylor series, and then the expected utility from wealth including the gamble can be written as:

\begin{aligned}\mathbb{E}[U(W(1 + h))] &=\mathbb{E}[U(W(1 + h))] \\& =\mathbb{E}[U(W) + WhU'(W) + \frac{1}{2}W^2h^2U''(W) + ...] \end{aligned}

With $h$ small we can ignore the the exponents of $h$ greater than $2$, and this gives us:

$\mathbb{E}[U(W(1 + h))] \approx\mathbb{E}[U(W) + WU'(W)h + \frac{1}{2}W^2U''(W){h}^2]$

Given a certain (sure) starting $W$, we can write the above as:

$\mathbb{E}[U(W(1 + h))] -\mathbb{E}[U(W)] \approx WU'(W)\mathbb{E}[\tilde{h}] + \frac{1}{2} W^2U''(W)\mathbb{E}[h^2]$

that is, the change in expected utility:

$\Delta\mathbb{E}[U(W(1 + h))] \approx WU'(W) \mu + \frac{1}{2}W^2 U''(W) (\mu^2 + \sigma^2)$

Given that $h$ is small, its expected value $\mu$ would be smaller still and we can ignore the higher powers of $\mu$ to give:

$\Delta\mathbb{E}[U(W(1 + h))] \approx WU'(W)\mu + \frac{1}{2}W^2U''(W) \sigma^2$

Since $U(W)$ is known, so are $W \mbox {and } U'(W)$, and we have:

\begin{aligned} \frac{\Delta\mathbb{E}[U(W(1 + h))]}{WU'(W)} & \approx \mu + \frac{WU''(W)}{2U'(W)} \\& \approx \mu - A \sigma^2 \end{aligned}

where $A = - \displaystyle \frac{WU''(W)}{2U'(W)} > 0$ because concavity of $U(W)$ implies $U'(W) > 0$ and $U''(W) < 0$. The coefficient $A$ defines a measure of relative risk aversion. That is, higher the value of $A$, more risk-averse the person, and lower its value more risk-loving the person. (What would be the value of $A$ for a risk-neutral person?)

That is, the change in expected utility from a marginal gamble depends only on the expected return and variance of the gamble. And expected utility goes up as the expected return from the gamble increases and goes down as variance increases.

For small gambles, then according to Markowitz people should only consider a single number to talk about risk, i.e. its variance $\sigma^2$irrespective of the number of states of the world. This turned out to be a revolutionary idea in the history of finance, and is a cornerstone in the theory of portfolio choice and asset pricing. For his efforts Markowitz was awarded the Nobel Prize in Economics in 1990.

This result, that however complex the world maybe, for small gambles people need only consider the expected return $\mu$ and variance of the gamble $\sigma^2$ will form the basis for our further discussions.

Not only did Markowitz notice that people care both about risk and return, he also observed that people held not one but a portfolio of stocks. Just on its own, the fact that people should care about expected return and variance of gambles doesn’t necessarily imply that people would hold multiple stocks. If they knew their degree of risk-aversion, they would just want to pick one that offered the ‘right’ trade-off for them.

The fact that people could and did hold a portfolio of stocks made ample economic sense. Consider the following two risky gambles:

It should be clear that by holding half of each $X_1$ and $X_2$, an investor could make his end-of-period payoff the same (= 100), irrespective of the end-of-period state of the world, i.e. the portfolio of $X_1$ and $X_2$ with equal percentage invested in each is completely risk-less.

This, of course, is an extreme example and in general such gambles would be rare that offered perfectly negatively correlated payoffs. However, Markowitz’s point had been made. As long as end-of-period payoffs are not perfectly positively correlated investors could reduce the variance or risk associated with the end-of-period payoffs by holding multiple stocks. We’ll generalize this idea in the next post.

Written by Vineet

September 1, 2016 at 7:12 pm