Risk of portfolios

A treasury bill is one of the safest investments you can make because they bear no risk of default and because of their short maturity that keeps their prices almost stable. But there is still uncertainty about how inflation will evolve in the future. When investors decide to buy other kind of assets which fluctuate in price as the interest rates vary (like bonds) they are exposed to more risk and, if they buy assets like common stocks, their investments will be influenced by all the ups and downs of those stocks.

ARITHMETIC AVERAGES VERSUS COMPUNDED ANNUAL RETURNS

Suppose that you are considering a stock priced a $100 and that, at the end of the year, the stock price could grow to $110 or $130 or could decrease at $90. So the possible returns an investor can experience are: -10%, +10% or 30%. If you compute the expected return assuming that all these outcomes are equally likely to occur, you will get: 1/3*(-10+10+30)=10%. So we can use 10% as the discount for the expected cashflows of the company. This expected return also represents the opportunity cost of capital in investing in assets that bear the same risk as the stock we are considering. If the stock returns behave as predicted, then in the long term 1/3 of the times the stock will generate -10% revenues, for another 1/3 of the times the stock will generate +10% and, for the remaining 1/3, the stock will generate +30%. The arithmetic average measures the average return by simply computing the average of the returns for every single year. The compound annual interest (also called geometric average return) does in principle the same job but it takes into account compounding: the possibility to reinvest the gains of each year in the future. In our case this average can be computed as: (0.9+1.1+1.3)^(1/3)=0.88. But more in general it can be computed as:

(sum of the returns in each year elevated to the power of 1 divided by the number of years)

PAST DATA AND THE FUTURE COST OF CAPITAL

You can think of estimating the market return (denoted as rm) by assuming that the future will follow the same trend as in the past and that investors today can expect the average returns from periods in the past. But rm generally is not stable over time and therefore, if you use that assumption, you will obtain misleading results. A better way to estimate the return expected by investors is to add to the interest rate on treasury bills 7.7% which represents the average risk premium. By doing so, you are assuming that there is a stable risk premium offered by markets. It is important to consider that the 7.7% is an average value. This means that the actual risk premiums during the various years could vary around that value. An increase or decrease in the expected market risk premium will modify also the realized rates of return. To understand why, imagine that you own the market portfolio and that it pays an aggregate dividend of $100. If the portfolio yields 5% and the dividends are expected to grow forever at a rate of 4% each year, the expected return of this portfolio will simply be: r = 5+4 = 9%. The present value of the portfolio can be computed in the usual way as a growing perpetuity: PV=DIV/(r-g)= 100/(0.09 – 0.04) = 2000. Our 9% return includes the risk premium of the market: if we the return on risk free assets is 2%, the risk premium will be 7%. Imagine now that people suddenly start trusting the market more and therefore demand a lower risk premium of 6% making the return fall from 9% to 8%. The present value will become: DIV/(r-g) = 100/(0.08-0.04) = 2500. So, as you can see from our example, a small change in the required risk premium has modified the present value by a large margin.

HOW DIVIDEND YELDS AND RISK PREMIUM ARE RELATED

Dividend yields are subjected to sharp fluctuation but they historically followed a downward trend. When the dividend yields fall, the price to dividend ratio increase and the returns realized increase. This ratio however can not increase forever because otherwise the risk premiums and therefore the average returns will be lower in the future than they were in the past. Furthermore it is also impossible to predict how the short sun fluctuations of the dividend yields will affect the price dividends ratios.

HOW TO MEASURE RISK

In order to measure how much an investment is risky, we have to measure how much the actual returns are far from the expected ones. Thanks to statistics we can measure the variance (σ^2) of an asset. The variance is computed as the expected value of the squared difference between expected (rm) and actual returns (ȓm):

Another way to measure the variability of an asset is by using the standard deviation (σ)which is simply the square root of the variance:

In order to calculate the expected returns however you need to know the various returns that the market could offer and the probabilities associated with each outcome. Obviously you can not know in advance which will be the probability of realising a certain return when investing on the market.

HOW YOU CAN REDUCE RISK WHEN INVESTING

Generally individual stocks are more variable than the market indexes. But why if the market is composed by the single individual stocks it is less variable than the stocks composing it? Diversification reduces variability because the prices of the stocks do not always change exactly in the same way. Stocks are in fact less than perfectly correlated to one another. Often a decline in the price of a stock was offset by the rise in the price of another stock. Therefore, thanks to diversification, you can eliminate some risk called the specific risk. Specific risk represents the risks related to single firms and that therefore can influence that single firm or its close competitors. There is however some risk that can not be eliminated even with diversification. This risk is called market risk and it influences all the firms in the market.

HOW TO CALCULATE THE RETURN AND THE RISK OF A PORTFOLIO

If you have a portfolio made up of stocks each with its own expected return you can calculate the expected return on the portfolio by computing a simple weighted average of the returns of the single stocks. In order to calculate the variance of the portfolio instead you have to cum all the values in the cells of the matrix as in the figure (where we consider a portfolio made up of only 2 stocks).

The upward-left cell and the downward-right cell you only need the percentage of the portfolio made up of that stock (x) and the variance of the single stock. When computing the other cells you need also the covariance of the two stocks which is computed as the product of the correlation coefficient ρ and the standard deviations of both the stocks:

If the prices of the stocks moved in the direction, the correlation coefficient and the covariance would be positive. If instead they moved in opposite directions, both the correlation coefficient and the covariance would be negative. When computing the portfolio variance you have to weight the covariance using the proportions of the stocks in the portfolio. So the variance of a portfolio composed by 2 stocks is computed as follows:

YES BUT…

This method of summing up the values in the cells is easy to compute when you have 2, 3 or 4 stocks but when you consider a large number of stock, for example 500, you would need to fill and add up 500x500=2500 cells. This is obviously impossible and this is the reason why in general investors do not compute the variance of stock portfolios. And in general you do not need that data to make smart investment decisions, you just need to know that, the more you diversify, the more you reduce risk. This is why many investors buy index funds that contain all the stocks in a specific market like the S&P500 that includes the stocks of the 500 most important companies in the USA.

HOW A SINGLE STOCK CAN INFLUENCE THE RISK OF THE PORTFOLIO

As we saw, the risk of a portfolio depends on the risk of the stocks included in it. When you are asking yourself which is the risk of a single stock it is not sufficient to measure the risk of that isolated company: you need to measure how that single stock reacts to market fluctuations. This relationship is indicated by beta β. Stocks having a β>1 amplify the movements of the market, stock with 0>β>1 follow the market but not perfectly. The portfolio made up with all the stocks in the market has β exactly equal to 1. As we said, with diversification you can eliminate the specific risk but not the market risk. The market risk depends on the beta of the stocks in the portfolio. Therefore, if we consider a portfolio made up of all the stocks in the market, its beta would be 1 and its standard deviation will be exactly equal to the standard deviation in the market. If instead we diversify using 500 stocks that have and average beta of 1.2, the standard deviation of this portfolio would be 20% higher than the one of the market and therefore will amplify also the movements of the market by 20% and therefore would bear 120% of the market risk. The risk of a portfolio is therefore proportional to the portfolio beta which is computed as the average beta of the securities in it. Another way to define beta is as follows:

Where the numerator is the covariance of the stock and market returns and the denominator is the variance of market returns. This ratio can be used as a measure of how much the stock influences the portfolio risk.

VALUE ADDITIVITY

For investors diversifying is essential but for firms? While investors can simply diversify their portfolios by buying stocks of different companies. In order to diversify, firms have to expand their businesses. The difference between diversification for investors and for firms is that, if investors recognise that they made a bad investment they can easily recover their money (or part of it) in a fast way, firms instead can not demolish the new factory they are build or fire the new hired workers that easily. Because of their possibility to diversify on their own, investor have no incentive into paying a premium for the stocks of firms that diversify. Therefore the present value of a firm having multiple assets or businesses namely A and B is simply the sum of the present values of the single businesses:

PV(AB) = PV(A) + PV(B)