Markowitz came up with a research paper "Portfolio Concepts" in 1952 which is considered to be pioneer work in the field of financial economics. Markowitz explained risk of a portfolio in term of volatility in returns. His risk return analysis is also known as mean-variance analysis.
Underlying principle of his analysis was that any investment value should be equal to the present value of its cash flows. If the investment is in stocks, then its value should be determined by the present value of dividends. Dividends should be discounted at the expected rate of return to get its intrinsic value.
To get the maximum value of a portfolio, investment should be made in the security that yields maximum return. Since portfolio is a set of securities, a single security in a portfolio with maximum value can yield that highest return. So a single security portfolio could be used. But this is not appropriate because it does not minimize risk. Investors want to diversify their investment to minimize risk and maximize return. Diversification means that rather than putting all eggs in one basket, investment should be made on securities of multiple companies, working in different sectors.
Markowitz mean-variance framework works on a strict set of assumptions which simplify the analysis. Underlying assumptions are
All investors are risk averse: Risk aversion refers to the investor's preference for lower risk or higher risk at a given level of expected returns. Risk averse investors will take higher risk if they are compensated with higher return. It is not assumed that investor is willing to take higher risk at the cost of lower expected return.
Utility Maximization: Risk taken by an investor is represented in the form of utility curve where each curve represents increase in return with the level of increase in risk. Investors maximize their expected utility over a given time period. Moreover their indifference curve is convex, representing diminishing marginal utility of wealth.
The utility curve that gives maximum return at a given level of risk is the most preferred. In the figure above, the curve I3 is the most preferred that given higher return at lower risk and I1 is the least preferred because that gives lower return at given level of risk. From practical point of view, there are infinite numbers of indifferent curves and indifference curves for any given investor do not overlap or cross each other.
Returns Distribution: Investors look at each investment as a probability distribution of expected returns. If the probability is higher, returns are also high and investors will invest in those stock. On the other side, the investment may have higher return but the probability may not be very high. So in that case higher return at a low probability will yield lower total expected return.
Risk variability: Risk is measured as the volatility in returns. In other words, risk is measured as the variance or standard deviation in return. If the variance or standard deviation in returns is high, then risk is greater. But if volatility in returns is less, then it means risk is also low.
Expected return, variances and covariances are known for all assets: Investors create optimal portfolio by relying solely on expected return, variances and covariances. Mean variance analysis is based on the returns that are normally distributed. Any irregularity in distribution in the form of kurtosis or skewness renders the analysis invalid. Investors know the future values of all the parameter required to value the investment. They know the value of risk and return associated with their investment. Investors should also know the covariance of one asset with the other.
Return Calculation
As mentioned earlier in the assumptions that investors know the return of their investments, so they diversity their portfolio in such a way that maximizes return. Since diversification is also important, so the maximum weightage is given to those securities give highest return. Investment in large number of securities insures that actual yield of the portfolio will be almost the same as the expected yield.
Mathematically, expected return is given as
Return (R) on a portfolio as a whole is a weighted sum of random variables, where the investor chooses the weights.
Risk Calculation
Initially it was considered that risk of a portfolio, like expected return calculation, is simply a weighted average of the individual risks of the assets. Markowitz disagrees with the previously given methodologies.
To determine risk, Markowitz argued that diversification is used to minimize risk, but diversification does not eliminate risk, it only reduces it. Moreover returns from securities are also correlated. So other than probability of return, there is another factor called covariance. Covariance is the co-movement or volatility in asset returns with respect to another asset. If one asset goes up, the other asset with higher covariance is also expected to go up and if same asset goes down, same path is followed.
So the introduction of covariance factor means that while creating a portfolio, each security should not be considered isolated from other securities, but it also depends upon the risk-return characteristics of other securities. Diversification benefits are maximum when each asset added to portfolio has minimum covariance with every other asset in a portfolio. Covariance between two asset returns is calculated as
So for n-asset portfolio, the total risk of a portfolio in terms of variance and covariance can be given as
Where xj are the portfolio proportions and is the covariance of the returns between the pair of assets.
It can be implied from the above equation that higher the correlation between assets, greater is the risk; and lower the covariance, lower would be the total risk. So for minimum risk, the correlation between the assets should be perfectly negative i.e. -1.
If a risk-return combination is graphically drawn by take taking total risk on x-axis and expected return on y-axis, just by changing covariance value between the pair of assets, the following diagram can drawn
The amount of bulge in a risk-return trade-off depends upon the correlation between assets. Lower the correlation, higher the bulge and vice versa.
Considering above risk and return calculation, Markowitz went on to present efficient frontier which is the primary contribution. Efficient frontier is a graphical representation of set of portfolio that gives the maximum return at a given level of risk. Any portfolio above or below violates the risk-return trade-off, resulting in higher risk at a given level of risk or lower return at a given level of risk. Any optimum portfolio, that gives maximum return at minimum risk, must lie on the efficient frontier.
In terms of investor's utility curve, optimal portfolio for an investor is at point where the investor's highest indifference curve is tangent to the efficient frontier.
Just like two asset portfolio, efficient frontier of n-number of assets can be created by using mean-variance analysis. We need to have each assets expected return, variance and correlation with every other asset. For n-number of assets, we need to have n-number of expected return & variances, and n(n-1)/2 number of correlations.
Portfolio optimization technique of Sharpe
Sharpe expanded the legendary work of Markowitz and addressed in detail the components of risk of a portfolio which in turn separately influence the total return.
Like Markowitz, Sharpe's framework also applicable only under strict set of assumptions. Assumptions are
Markowitz Investors: Markowitz mean-variance analysis is used to select appropriate portfolio in securities. Only those portfolios are selected that maximize utility and are tangent to the efficient frontier.
Unlimited Risk-free Lending & Borrowing: Investors can lend and borrow any amount at risk free rate.
Homogenous Expectations: All investors use same risk/return distribution.
One Period Horizon: All investors have same one period time horizon.
Divisible Assets: All investments are infinitely divisible.
No taxes and Transaction Costs: Investors are in tax free environment, facing no transaction costs.
No inflation and Constant interest: There is no inflation and interest rates do not change.
Equilibrium: There are no inefficiencies in market and markets are in equilibrium.
Markowitz frontier only consists of risky assets. Sharpe introduced a risk-free asset in Markowitz efficient frontier which ended up in a straight line which is also known as Capital Market Line.
Because of the introduction of risk-free asset, the expected return is divided into two components: risk free return and market return. Risk free return is given as return on long term government bonds because government bonds are considered virtually default free.
Using the Markowitz expected return equation; expected return with the introduction of risk free asset is given as
Similarly, the addition of risk free assets in risk calculation formula minimizes it to the risk attributed to weightage given to market portfolio, because risk free asset's correlation with any other risky asset is zero. Mathematically
Equation for capital market line CML can be derived from expected return equations, which is given as
Where RFR which is a risk free asset is a slope intercept while the slope itself is also known as Sharpe Ratio i.e.
Sharp ratio is the excess return per unit of market risk. When comparing two assets each with the expected return against the same benchmark with risk free return, the asset with the higher Sharpe ratio gives more return for the same risk.
For individual securities, security market line (SML) is derived which helps in calculating reward-to-risk ratio compared to market. SML is different from CML as CML compares return to that of total risk while SML utilizes systematic risk in the form of Beta.
Systematic risk is the risk that cannot be diversified away. It applies to individual securities as well as to portfolios. Systematic risk is represented as Beta which is sensitivity to the market changes. Assets that are highly sensitive to market changes have high betas. Similarly assets that are independent or less influenced by market changes have low betas.
For example, luxury products, like luxury cars, usually have high betas. If the economy goes up, then wages increases and when wages increases people would prefer to buy their own cars. If car industry goes up, so does their returns. On the other hand, tobacco industry has very low betas. People always use tobacco no matter in what direction market changes. Demand for tobacco always remains there.
So mathematically, total risk and beta are given as
Total Risk = Systematic Risk + Non-Systematic Risk
OR Total Risk = Beta + Non-Systematic Risk
As beta is sensitivity to market changes, or in other words, it is a covariance of an asset returns with market.
Mathematically,
OR
The above equation which captures systematic risk in the form of beta is known as Capital Asset Pricing Model (CAPM), where beta is the covariance or sensitivity of the returns on the asset with the returns on the market.
The CAPM equation shows that optimal portfolio based on investors risk tolerance can be constructed either by fully investing in risk-free asset or by using some combination of risky or risk free asset where risky asset is a beta adjusted market risk premium.
For portfolio optimization in Sharpe framework, investor must select those assets that have minimum betas or in other words; minimum systematic risk. Betas can be minimized by creating market neutral portfolios by taking long and short positions.
