1.4 Modern portfolio theory and basic risk management strategies

1.4 Modern portfolio theory and basic risk management strategies $\diamondsuit$

The material in this section is intended for students at a more advanced level than your profile

With his conjecture that investment risk can be quantified as volatility from the standard deviation of the expected return, H. Markowitz started in 1952 at the University of Chicago what has become the Modern Portfolio Theory (MPT) and awarded him in 1990, the counterpart of the Nobel Prize in Economics [16].
Rather than looking at the risk from a single investment, he examined the stochastic evolution from different asset classes and found that seemingly random prices are sometimes correlated, for example when two industries compete or complement each other in the same market. This led him to distinguish specific risk associated with small groups of inter-dependent assets (e.g. bad weather affects the harvest of coffee and the valuation of all the shares in the food industry) from the non-specific risk that affects the market as a whole (e.g. a stock market crash). Using a judicious choice of anti- and uncorrelated securities, Markowitz showed that the total volatility of a portfolio can be minimized without reducing the expected return, by diversifying out the specific risks.
indent For a simple example, imagine a portfolio composed of two assets that are perfectly anti-correlated, but have the same expected return: by canceling each other's price fluctuations, the total volatility can be reduced to zero without changing the total expected return.
A second example illustrated in (1.4#fig.1) suggests how the combined risk from two assets can be minimized $(\sigma_b<\sigma_a)$ at a constant expected return

provided that the prices are partly de-correlated. By varying the proportions invested in each asset, an efficient frontier can be calculated where the highest return is expected for a given volatility (continuous line in blue).

**Figure 1.4#fig.1:** Expected return from a mix of two volatile assets $(\sigma _1,E_1)$ and $(\sigma _2,E_2)$ that are partly correlated. The plot shows how, for the same expected return , the combined risk from two assets is lower than the average that is calculated when the risks of the assets are evaluated separately $\sigma _b<\sigma _a$ . The tangent (broken line in red) drawn from the risk-free rate intersects the efficient frontier (line in blue) at $(\sigma _c,E_c)$ where the best portfolio is located.
$\includegraphics[width=7cm]{figs/MPTFrontier2.eps}$

The analysis is only slightly more complicated when more than two assets are involved: the applet below calculates the efficient frontier based on monthly variations of the price of raw materials, bonds and stock as they have been observed in the markets during the years 1986-1996.

**FRONTIER applet:** select the historical data (1986-1996) from **at least two** securities to display the efficient frontier where the maximum return E[r_e] was achieved as a function of the volatility sigma. Courtesy of Prof. C.Harvey, Duke University, USA.

Market history experiments: efficient frontier

Select "US LT Govt Bnd" and "US S&P500" to plot the joint expected return from a variable amount of money invested in bonds (10 years government bonds) and shares (stock market index) in the US.
Add the "US 30Day TBill" and explain the similarity with (1.4#fig.1).
Explain what happens with an investment in "Gold". You may check the glossary on-line for a complete list of other symbols appearing in the applet.

To find out which combination is most risk efficient under the present market conditions, another Nobel laureate, W. Sharpe, looked at the return from a variable amount invested partly for a risk-free rate

and partly in assets located on the efficient frontier (1.4#fig.1). He found that the ``best'' portfolios, located on the broken line in red, intersect the efficient frontier where the highest return

is achieved if the money has to be borrowed on the market. This led him to the definition of what is now known as the Sharpe ratio, where the reward-to-risk from a portfolio

is measured as the excess return over the risk free rate divided by the total volatility

$\displaystyle S_x=\frac{E_x-r}{\sigma_x}$

(1.4#eq.1)

To compare the relative performance from a portfolio $(\sigma_x,E_x)$ with the average performance from a market index $(\bar{\sigma},\bar{E})$ over different periods of time, Sharpe later developed the capital asset pricing model (CAPM)

$\displaystyle \alpha + \beta(\bar{E}-r) = E_{x}-r\$

(1.4#eq.2)

where a linear least-square fit is calculated to obtain the slope beta ( $\beta$ relative performance from taking risks) and the offset alpha ( $\alpha$ relative performance from arbitrage and costs). To ``beat the market'' alpha should always be positive and beta larger than unity. Unfortunately for most of the investors, commercial funds generally have a negative alpha because of the 1-2% management costs that are payed in the form of commissions. Some say that expertise justifies the costs because of additional earnings made from arbitrage: this is in general not true, because arbitrage is a zero-sum game, so that whatever makes one manager look better only make another look worse! The value added by the fund manager therefore resides mainly in beta, i.e. in the management of risk.
All together, the modern portfolio theory largely justifies investments in funds, provided that a large number of weakly correlated assets are managed at a very low cost. It also explains how different funds can be classified according to their growth (drift), standard deviation (volatility), reward-to-risk (or Sharpe ratio 1.4#eq.1) and every fund can always be compared with a market index using the CAPM parameters (alpha and beta). The correlation between individual assets and the portfolio as a whole provides a more detailed description (execise 1.03). For the layman, the theory leads to the simplest and best known risk management strategy: diversify your portfolio by investing in a variety of weakly or anti-correlated securities. To maximize the reward, it is better to blend different types of investments, for example by selecting assets according to the time that it will take to average out fluctuations. Some investors subtract their age from 100 to determine the percentage to invest in stocks and put the rest in bonds: the younger the an investor is, the more risk he can afford to take.

Quants: statistics.
Remember that the correlation coefficient between two random variables X,Z is given by

$\displaystyle \mathrm{Corr}[X,Z]= \frac{\mathrm{Cov}[X,Z]}{\sqrt{\mathrm{Var}[X]\mathrm{Var}[Z]}} \hspace{1cm} \in[-1; 1]$

(1.4#eq.3)

the covariance, the variance and the expectancy operators are defined by

$\displaystyle \mathrm{Cov}[X,Z]$	$\displaystyle = E[XZ]-E[X]E[Z] = E[(X-\mu_x)(Z-\mu_z)]$
$\displaystyle \mathrm{Var}[X]$	$\displaystyle = E[X^2]-(E[X])^2 = E[(X-\mu)^2]\equiv \sigma^2$
$\displaystyle E[X]$	$\displaystyle = \int_\Omega x f(x) \equiv \mu$	(1.4#eq.4)

where $\mu$ is the mean, $\sigma^2$ the variance and $\sigma$ the standard deviation. Higher order central moments are defined from $\mu_k=E[(X-\mu)^k]$ , such as the skewness $\gamma_1=\mu_3/\sigma^3$ and the curtosis $\gamma_2=(\mu_4/\sigma^4)-3$ . Under general conditions, the sum of a large number of random variables is approximatively normally distributed $f\approx \mathcal{N}[\mu,\sigma^2]$

$\displaystyle \mathcal{N}[\mu,\sigma^2](x) = \frac{1}{\sigma}\varphi\left(\frac{x-\mu}{\sigma}\right), \hspace{1cm} \varphi(x)=\frac{1}{\sqrt{2\pi}}\exp(-x^2/2)$

(1.4#eq.5)

and the normalized probability $P(-a\leq (X-\mu)/\sigma\leq a)= \int_{-a}^{+a} \varphi(x) dx = \mathrm{erf}(a/\sqrt{2})$ . Unbiased estimates for the mean and variance of

data points $\{x_1,x_2,\hdots, x_n\}$ generated by a normally distributed process can be calculated from

$\displaystyle \bar{\mu}=\frac{1}{n}\sum_{i=1}^{n} x_i ,\hspace{1cm} \bar{\sigma}^2=\frac{1}{n-1}\sum_{i=1}^{n} (x_i-\bar{\mu})^2$

(1.4#eq.6)

Finally, a least-square fit to a linear model $\alpha+\beta x=y$ is obtained by solving the system of normal equations

$\displaystyle \left(\begin{array}{rcr} 1 & x_1\ 1 & x_2\ . &\hdots \ 1 & x_n... ...y} \qquad\Rightarrow\qquad \mathbf{X^{T}}\mathbf{X}\mathbf{c} = \mathbf{X^{T}y}$

(1.4#eq.7)

In the sixties, E. Fama [8] proposed another important conjecture following market observations, called the Efficient Market Hypothesis: at any time, the price of a security fully reflect all the information that is available about this security. The reason is that the market is inhabited by arbitrageurs, whose highly paid job is to seek out and exploit possible mis-pricings. Under the efficient market hypothesis, no-arbitrage arguments state that it is not possible to find a self-financing trading strategy leading to an immediate risk-less profit. This means that there is no way for investors to buy securities at a bargain price: even if the prices just fell, there are equal chances for them to move back up or fall down even further: there is no way to make a statement such as ``the market is too high''.
Of course, not all the markets are efficient and human psychology is such that investors tend to buy more in rising than falling markets: buying stocks in a falling stock market sounds easy, but very few people have the stomach to do it! To avoid arbitrageurs taking advantage of the psychology, portfolio managers sometimes perform a cost averaging by regularly buying a fraction of the security they want to buy or sell - independently of the short time fluctuations of the market (exercise 1.02). This strategy has, however, also its limits since investors should imperatively minimize the transaction costs.
In conclusion, simple management strategies can be used to reduce the investment risk in a portfolio: ignoring the advice to diversify and regularly pay large commissions and transaction costs have the worst long term effects. For a more quantitative and a flexible approach of managing investment risk, the next chapter will examine a new class of securities: so-called options, which can be combined with other assets to hedge a portfolio to any level and type of risk chosen by the investor.

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