# In the pursuit of the Perfect Portfolio: Modern Portfolio Theory

### Mikel Unibaso Berrueta

#### 09/03/2022

Historically, portfolio managers have carried out asset allocation combining intuition with deep knowledge about specific market or asset classes in order to beat the market. However, is it possible to design a quantitative investment strategy that provides the optimal balance in the asset allocation?

Let’s find out!

## Modern Portfolio Theory (MPT)

This question was addressed by Henry Markowitz (1927), Nobel Prize winner in 1990 for his pioneering theoretical contributions to the Modern Portfolio Theory (MPT). He formulated the choice of an optimal portfolio as an optimization process, in terms of two parameters, mean and variance, i.e., return and risk. Thereby, it is also known as the Mean-Variance Theory.

### Key ideas: diversification and risk types

The risk of a security can be analyzed in two ways, stand-alone basis (an asset is considered in isolation) and portfolio basis. In the context of a portfolio, the total risk of a security can be divided into two basic components:

• systematic or market risk. Macro-level form of risk affected by inflation, interest rates, unemployment levels, exchange rates,etc, inherent to the entire market, unpredictable and impossible to completely avoid.
• unsystematic or idiosyncratic risk. Micro-level form of risk such as business risk, financial risk or operational risk.

Markowitz showed that the individual performance of a particular stock wasn’t as important as the performance and composition of an investor’s entire portfolio.

By means of a non-perfect correlation between assets, it is possible to build a portfolio such that its volatility is smaller than stand-alone volatility of the components. Reducing portfolio’s risk by selecting uncorrelated assets to portfolio is known as diversification. The idea of “not putting all the eggs in the same bag” was soon widely accepted by the financial industry. 1. Figure: Standard deviation of the portfolio splitted into systematic and unsystematic risk in terms of the number of stocks in the portfolio.

### Mathematical framework, building the Efficient Frontier

Let’s make a fast review on the essential mathematics behind the MPT model. As said before, we have two important parameters, risk and return. We compute the expected return $$E[R_i]$$ and the volatility $$\sigma_i$$ of each asset based on their historical time series returns. For a portfolio containing N different assets the expected return and variance:

$$E[R_p] = wE[R_i], \ \ \sigma_p^2 = w\Sigma w^T,$$

being $$w = (w_1 w_2 .. w_N)$$ , the vector of weight allocated to each asset , $$E[R_i] = (E[R_1] …E[R_N])^T$$ the expected return of each asset and $$\Sigma$$ the covariance matrix.

Markowitz’s model rests on the concept of efficient portfolio, that is, minimum risk (measured by the standard deviation) for a given level of expected return. In other words, the problem resides in the selection of weights such that for a certain level of return we achieve the minimum possible risk:

$$min (w\Sigma w^T) \ \ s.t. \ E[R_p]= \phi, \ \ \Sigma w_i=1,$$

• The concatenation of these efficient portfolios leads to the so called efficient frontier.
• Any individual asset and non-efficient portfolio will lay under the efficient frontier and they are known as dominated portfolios.
• MVP is the Minimum Variance Portfolio.

### Incorporating risk-free space to the picture

Up to this point we just spoke about assets that are risky, but once we have the tools to build the efficient frontier, the concept of risk-free asset must be introduced.

Roughly speaking, a risk-free asset is one that has a certain future return, it is the theoretical rate of return of an investment with zero risk, so it tends to have low rates of return. Due to their safety investors are not compensated for taking a chance. Risk-free is associated to fixed-income and a typical example of a risk-free asset could be a 3-month government Treasury bill.

Translated to equation:

$$E[R_f]=R_f, \ \ \sigma_f = 0.$$

It is now when the Capital Asset Line (CAL) comes into play as the bridge that links both worlds, risky and risk-free. Let’s say we want to compute the expected return of a combination of an optimal portfolio and a risk-free asset:

$$E[R_{prf}] = wE[R_{p}] + (1-w)E[R_f], \ \ \sigma^2_{prf} = w^2\sigma^2_{p},$$

combining both equations we have that,

$$E[R_{prf}] = R_f + \underbrace{\frac{E[R_{p}]-R_f}{\sigma_{p}}}_{Sharpe} \sigma_{prf},$$

The Sharpe ratio represents the premium per unit of risk and it is the slope of the CAL. Therefore it is natural that the optimal portfolio or tangent portfolio is the one that maximizes such ratio.

### Utility function and indifference curves

Once knowing that the optimal portfolio maximizes the risk/return tradeoff we might think that we are done, but how can we decide the exact allocation between the tangent portfolio and the risk-free one?

According to Markowitz, the degree of risk aversion will determine those final weights. Risk averse investors will not accept fair gambles (expected payoff is zero), they would instead require a premium to buy risky assets.

Rational choice theory is associated with maximizing self-interest to provide the greatest benefit and satisfaction, and that satisfaction can be measured by the so-called utility function. A risk averse investor will pick the investment with the highest utility. By assuming a quadratic utility:

$$U(\mu_p,\sigma_p)= \mu_p – \frac{1}{2}A\sigma_p^2,$$

where $$\mu_p$$ stands for expected return, $$\sigma_p$$ the volatility and $$A$$ risk-aversion coefficient. A risk-neutral investor would base the ‘happiness’ on the expected return, the higher the better. However, by this second order correction it is possible to add a penalty to the taken risk.

Indifference curves define the collection of points with the same level of utility. All portfolios that lie on the same indifference curve are equally desirable to the investor. In MPT it is assumed that investors are risk averse $$A\gt0$$ , thus, indifference curves are positively sloped and convex.

## Wrapping up

Interestingly enough, it turns out that the election of the tangent portfolio does not depend on the investors’ view, neither personal preference, risk aversion do not affect. Once the shape of the utility function is defined it is time to set up the allocation in both risk and risk free assets.

Finally, it is important to be aware of the theoretical limitations based on the assumptions under which the MPT is constructed. In practice, perfect information, investors’ irrationality, efficient markets, absence of taxes or transaction costs, etc, are not fulfilled. However, MPT continues to be a cornerstone for portfolio managers by applying statistical techniques to portray to investors the benefits of diversification.

Thanks for reading and if you are interested in the implementation, stay tuned! See you around!

## References

 Markowitz, Harry. “Portfolio Selection.” The Journal of Finance, vol. 7, no. 1, [American Finance Association, Wiley], 1952, pp. 77–91, https://doi.org/10.2307/2975974.

 Mangram, Myles E., A Simplified Perspective of the Markowitz Portfolio Theory (2013). Global Journal of Business Research, v. 7 (1) pp. 59-70, 2013.

 Frank J. Fabozzi, Francis Gupta, Harry M. Markowitz. The Legacy of Modern Portfolio Theory. The Journal of Investing Aug 2002, 11 (3) 7-22.

 Prof. Karaivanov, Department of Economics . The utility function and indifference curves.