This page contained a detailed discussion of various concepts used in managing your investment portfolios:
* Harry Markowitz and the Efficient Frontier
* Modern Portfolio Theory (MPT)
* William Sharpe and the Capital Asset Pricing Model (CAPM)
* Beta, Standard deviation
* Tools to measure risk: The Treynor, Sharpe and Jensen ratios
* Arbitrage Pricing Theory (APT)
* Capital Market Line (CML)
* covariance

In 1952, Harry Markowitz published in the Journal of Finance, his theories on modern portfolio theory (MPT) under the title “Portfolio Selection”. It was there that he introduced the terms “risk, returns, and diversification.”

Efficient Frontier

The Efficient Frontier and the level of risk. Under MPT, there is just one portfolio that offers the lowest possible risk, and for every level of risk, there is a portfolio that offers the highest return, calculated as the risk/return ratio. In the graph below, each dot represents just one possible portfolio. The connection of the topmost points creates a curve known as the “efficient frontier”. This curve represents the most optimal portfolios. A portfolio above the curve is not possible. Portfolios below the curve results in inferior returns for a given level of risk and therefore are not efficient.

Chart A

Capital Market Line (aka Capital Allocation Line)

In 1958, James Tobin expanded on Markowitz’s work by introducing the concept of risk-free asset. This is typically represented by a 10-year government bond yield which resides on the y-axis and a -0- risk factor on the x-axis. The straight line intersecting this point and the efficient frontier is the Capital Market Line (CML). The CML shows the most efficient portfolio that lies on the efficient frontier line.

Chart B

During the 1964-66 period, William Sharpe, John Lintner, and Jan Mossin simultaneously and independently introduced the Capital Asset Pricing Model (CAPM). CAPM helps us to explain investment risk and expected return on investments.

Sharpe identified two types of risk: unsystematic risk and systematic risk.

Unsystematic Risk – Also known as “specific risk”, this risk is specific to individual stocks. Under modern portfolio theory (MPT) specific risk can be diversified away as you increase the number of stocks in your portfolio. It represents the component of a stock’s return that is not correlated with general market moves.

Systematic Risk – These are market risks that cannot be diversified away. Interest rates, recessions and wars are examples of systematic risks. CAPM evolved as a way to measure this systematic risk.

The Formula. Sharpe found that the return on an individual stock, or a portfolio of stocks, should equal its cost of capital. The standard formula remains the CAPM, which describes the relationship between risk and expected return.

The CAPM formula:

Sample CAPM Computation:
Risk-free rate (typically a 10-year government bond yield) = 3%
Beta (risk measure) of the security (or portfolio) = 2
Expected market return = 10%

Then the stock (or portfolio) is expected return = (3%+2(10%-3%)) = 17%

Measurement of risk – – Beta vs. Standard Deviation

Beta. According to CAPM, beta is the measurement of a stock’s (or portfolio) risk. It measures a stock’s relative volatility. If a share price moves exactly in line with the market, then the stock’s beta is 1. A stock with a beta of 1.5 would rise by 15% if the market rose by 10%, and fall by 15% if the market fell by 10%.

The Treynor and Jensen ratios uses beta as a measure of risk.
The Sharpe ratio uses standard deviation as a measure of risk.

Portfolio Variance and Standard Deviation

The variance/standard deviation of a portfolio reflects not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together. Two measures of how the returns on a pair of stocks vary together are the covariance and the correlation coefficient.

The Covariance between the returns on two stocks can be calculated using the following equation:  

• 12 = the covariance between the returns on stocks 1 and 2,
• N = the number of states,
• pi = the probability of state i,
• R1i = the return on stock 1 in state i,
• E[R1] = the expected return on stock 1,
• R2i = the return on stock 2 in state i, and
• E[R2] = the expected return on stock 2.

The Correlation Coefficient between the returns on two stocks can be calculated using the following equation:

• 12 = the correlation coefficient between the returns on stocks 1 and 2,
• 12 = the covariance between the returns on stocks 1 and 2,
• 1 = the standard deviation on stock 1, and
• 2 = the standard deviation on stock 2.

The Treynor, Sharpe and Jensen Ratios

The success of a portfolio should be based on both performance and risk. The investors have available three different tools to measure portfolio performance: The Treynor, Sharpe and Jensen ratios.

Treynor Ratio (Portfolios A, B, C)

Jack L. Treynor was the first to provide investors with a composite measure of portfolio performance that also included risk. Treynor suggested that there were really two components of risk: the risk produced by fluctuations in the market and the risk arising from the fluctuations of individual securities.

Treynor introduced the concept of the security market line, which defines the relationship between portfolio returns and market rates of returns, whereby the slope of the line measures the relative volatility between the portfolio and the market (as represented by beta). The beta coefficient is simply the volatility measure of a stock, portfolio or the market itself. The greater the line’s slope, the better the risk-return tradeoff.

The formula: Treynor Ratio = (Portfolio Return – Risk-Free Rate) / Beta

The numerator identifies the risk premium and the denominator corresponds with the risk of the portfolio. The resulting Treynor ratio represents the portfolio’s return per unit risk.

The higher the Treynor measure, the better the portfolio. Between Portfolios A, B, or C, Portfolio C (15%) has the superior performance. However, when considering the risks Portfolio B demonstrated the better outcome (0.087). Note that all three portfolios have outperformed the aggregate market. See computation in the chart below.

Because this measure only uses beta to measure systematic risk, it assumes that the investor already has an adequately diversified portfolio. Unsystematic risk (also known as diversifiable risk) is not considered.

Sharpe Ratio (Portfolios D, E, F)

The Sharpe ratio is similar to the Treynor ratio, except that standard deviation is used as a measure of the risk. Standard deviation measures total risk which includes both systematic risk and unsystematic risk. The Sharpe ratio evaluates the portfolio manager on the basis of both rate of return and diversification (as it considers total portfolio risk as measured by standard deviation in its denominator).

The formula: Sharpe Ratio = (Portfolio Return – Risk-Free Rate) / Standard Deviation

The higher the Sharpe ratio, the better the portfolio. Between Portfolios D, E, or F, Portfolio F (19%) has the superior performance. However, when considering the risks Portfolio D demonstrated the better outcome (0.818). Note that all three portfolios have outperformed the aggregate market. See computation in the chart below.

Jensen Ratio (aka Jensen’s Alpha) (Portfolios G, H, C)

The Jensen ratio measure calculates the excess return that a portfolio generates over its expected return. The Jensen ratio measures how much of the portfolio’s rate of return is attributable to the manager’s ability to deliver above-average returns, adjusted for market risk. The higher the ratio, the better the risk-adjusted returns. A portfolio with a consistently positive excess return will have a positive alpha, while a portfolio with a consistently negative excess return will have a negative alpha.

The computation:
formula: Jensen’s Alpha = Portfolio Return – Benchmark Portfolio Return
Where Benchmark Return or portfolio’s expected return:
CAPM = Risk Free Rate of Return + Beta (Return of Market – Risk-Free Rate of Return)

The higher the Jensen ratio, the better the portfolio. Between Portfolios G, H, or C. Both portfolios H and C have a 15%. Portfolio H has a higher Jensen ratio and is the superior portfolio. See computation in the chart below.


Monte Carlo Simulation

Risk analysis is part of every decision we make. We are constantly faced with uncertainty, ambiguity, and variability. And even though we have unprecedented access to information, we can’t accurately predict the future. Monte Carlo simulation lets you see all the possible outcomes of your decisions and assess the impact of risk, allowing for better decision making under uncertainty.

Monte Carlo simulation is named after the city in Monaco, where the primary attractions are casinos that have games of chance. Gambling games, like roulette, dice, and slot machines, exhibit random behavior.

What is Monte Carlo simulation?

Monte Carlo simulation is a computerized mathematical technique that allows people to account for risk in quantitative analysis and decision making. The technique is used by professionals in such widely disparate fields as finance, project management, energy, manufacturing and other fields.

Monte Carlo simulation furnishes the decision-maker with a range of possible outcomes and the probabilities they will occur for any choice of action.. It shows the extreme possibilities—the outcomes of going for broke and for the most conservative decision—along with all possible consequences for middle-of-the-road decisions.

How Monte Carlo simulation works
Monte Carlo simulation performs risk analysis by building models of possible results by substituting a range of values—a probability distribution—for any factor that has inherent uncertainty. It then calculates results over and over (iteration), each time using a different set of random values from the probability functions. Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands or tens of thousands of recalculations before it is complete. Monte Carlo simulation produces distributions of possible outcome values. In this way, Monte Carlo simulation provides a much more comprehensive view of what may happen. It tells you not only what could happen, but how likely it is to happen.
References (Efficient Frontier, MPT, CAPM, Treynor, Sharpe, Jensen, CML, etc):
Harry Markowitz. “Portfolio Selection”, 1952, Vol. 7, Issue 1 of the Journal of Finance.
William Sharpe, “Portfolio Theory and Capital Markets.

References (Monte Carlo Simulation):

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