Ƴ12 is the covariance between two returns and is related to the correlation coefficientƱ by the formula:
2 1 12 ƳƳ Ƴ
Ʊ / (I.A.3.1)
1Keith Cuthbertson is Professor of Finance and Dirk Nitzsche is Senior Lecturer at the Cass Business School, City University, London.
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The PRM Handbook – I.A.3 Capital Allocation
The covariance and correlation coefficient will both have the same sign, but the covariance has the annoying property that it is dependent on the units used to measure returns (e.g. proportions or percentages), whereas the correlation coefficient is ‘dimensionless’ and must always lie between +1 and –1. In the extreme case where Ʊ = +1, the two assets’ returns are perfectly positively (linearly) related and therefore the asset returns always move in the same direction (but not necessarily by the same percentage amount). ForƱ = –1 the converse applies and for Ʊ = 0 the asset returns are not (linearly) related. As we see below, the ‘riskiness’ of the portfolio consisting of both asset 1 and asset 2 depends crucially on the sign and size ofƱ. If Ʊ = –1, risk may be completely eliminated by holding a specific proportion of your wealth in both assets.
Even ifƱ is positive (but less than +1) the riskiness of the overall portfolio is reduced (although not to zero).
Consider the reason for holding a diversified portfolio consisting of a set of risky assets. Assume for the moment that you use all your ‘own funds’ to invest in stocks. Putting all your wealth in asset 1, your forecast or ‘expected’ return is ER1with risk of Ƴ . Similarly, holding just asset 2 you expect to earn ER
2 1
2 and incur risk Ƴ . Let us assume a two-asset world where there is a negative covariance of returns . (This also implies a negative correlation coefficient .) When the return on asset 1 rises, that on asset 2 tends to fall, so if you hold both assets the risks are partially offsetting. Hence, if you diversify and hold both assets, this would seem to reduce the variance of the overall portfolio return (i.e. of asset 1 plus asset 2). To simplify even further, suppose that ER
2 when the return on asset 1 increases by 1%, that on asset 2 falls by 1% (i.e. returns are perfectly negatively correlated,Ʊ = –1). Under these conditions when you hold half your initial wealth in each of the risky assets, the expected return on the overall portfolio is ER
2 2 2
1 Ƴ
p = 0.5 ER1 + 0.5 ER2
= 10%. However, diversification has reduced the risk on this portfolio to zero; an above average return on asset 1 is always matched by an equal below average return on asset 2 (sinceƱ = –1).
Our example is a special case. But, in general, even if the correlation between returns is zero or positive (but not perfectly positively correlated), it still pays to diversify and hold a combination of both assets.
So, the benefits of diversification in reducing risk depend on returns having less than perfect (positive) correlation. In fact, even a little diversification quickly reduces risk. Suppose we randomly choose a one-stock portfolio, a two-stock portfolio, … , n-stock portfolio from stocks in the S&P 500 index, by throwing the required number of darts at the stocks page of the Wall Street Journal. Each time we throw the darts, we calculate the portfolio standard deviation Ƴp (with weights = 1/n). We find that ‘risk’ drops to a level C with only about 20–30 stocks (Figure I.A.3.1).
wi
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This is because the risk that is specific to each firm/industry is random around zero (e.g. risk due to strikes, unfilled orders, managerial incompetence, the weather, computer failures, production line malfunctions, etc.). When averaged across many firms (stocks), these specific risks cancel each other out (i.e. good and bad luck) and hence do not contribute to overall portfolio risk. Hence, we can eliminate specific (or diversifiable or idiosyncratic) risk, simply by adding more stocks to our portfolio. Hence, in general, portfolio diversification arises from Ʊ < +1, and the law of large numbers, n o f.
Figure I.A.3.1: Increasing size of portfolio
Standard deviation
No. of shares in portfolio Diversifiable / idiosyncratic risk
Market / non-diversifiable risk
20 40
0 1 2 ...
C
For example, consider n = 2 for illustrative purposes. If the proportion of our own wealth held in asset 1 is w1, then portfolio variance is:
Ƴ2p w12Ƴ12 w22Ƴ22 2w1w2ƱƳ1Ƴ2
w1 w2 1
(I.A.3.2)
x if Ʊ = +1, then Ƴp = w1Ƴ w1 2Ƴ2 andƳp is a (linear) weighted average of Ƴ1 and Ƴ2
and there are no diversification benefits;
x if Ʊ <+1, then Ƴp < w1Ƴ w1 2Ƴ2 and Ƴpmust be less than the weighted average ofƳ1
andƳ2.This is the portfolio diversification effect.
IfƱ = 0, equation (I.A.3.2) becomes:
(I.A.3.3)
2 2 2 2 2 2 2 1
2 Ƴ Ƴ
Ƴp w w
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For example, supposeƳ1 = Ƴ2 = Ƴ and w1 = ½, then Ƴp Ƴ / 2 . In general, as we see below, for Ʊ = 0 we have Ƴp Ƴ/ n. This is the ‘law of large numbers’ or ‘insurance effect’. A large number of uncorrelated (Ʊ = 0) events has a low variance (i.e. Ƴpo 0 as n o f). This is why your car insurance premium is low relative to the replacement value of the car. If the car insurer has a large number of customers who have accidents that are largely independent of each other, then the risk of the whole ‘portfolio’ of customers is relatively small. (The reason, in practice, why the risk does not actually reach zero as the number of customers increases is that there is some small positive correlation between accidents from claimants within the same company, e.g. on the relatively few days when road conditions are particularly bad throughout the entire country in which the insurer operates.)
The systematic risk of a portfolio is defined as the risk that cannot be diversified away by adding extra securities to the portfolio. (It is also referred to as ‘non-diversifiable’, ‘irreducible’,
‘portfolio’ or ‘market’ risk.) There is always some non-zero risk even in a well-diversified portfolio, and this is because different firms are affected by economy-wide factors (e.g. changes in interest rates, exchange rates, tax laws). This is what gives rise to the correlation between different stock returns – but the correlation is not perfect, because the economy-wide variables affect different firms (profits) by differing amounts. To see the influence of these correlated
‘events’ on portfolio variance consider a portfolio of n assets held in proportions wi (0 < wi < 1):
(I.A.3.4)
With n assets there are n variance terms and n(n – 1)/2 covariance terms that contribute to the variance of the portfolio. The number of covariance terms rises much faster than the number of assets in the portfolio. To illustrate the dependence of on the covariance terms, consider a simplified portfolio where all assets are held in the same proportion (w
It follows that as n o f the influence of the variance term V approaches zero and Ƴ equals the (constant) covariance, C. Thus the variance of individual securities that represents (idiosyncratic or specific) risk particular to that firm or industry can be diversified away – you might surmise at
2 p
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this point that if you hold a diversified portfolio then you should not earn any return from holding the specific risk of these firms. When we discuss the CAPM in Chapter I.A.4, we do indeed find that this holds true. However, covariance risk cannot be diversified away, and it is the covariance terms that (in a loose sense) give rise to market/non-diversifiable or systematic risk (Figure I.A.3.1). As we shall see in Chapter I.A.4, this ‘covariance’ or market risk can be represented by the beta of the security.
Above we have shown that, in general, the investor can reduce portfolio risk Ƴp by including additional stocks in her portfolio. In fact the portfolio variance Ƴ falls very quickly as one increases the number of stocks held from 1 to around 25 and thereafter the reduction in portfolio variance is quite small (Figure I.A.3.1). This, coupled with the brokerage fees and information costs of monitoring a large number of stocks, may explain why individuals tend to invest in only a relatively small number of stocks. Individuals may also obtain the benefits of diversification by investing in mutual funds (i.e. unit trusts), ‘closed-end’ mutual funds (i.e. investment trusts) and pension funds, since these take funds from a large number of individuals to invest in a wide range of financial assets and each individual then owns a proportion of this ‘large portfolio’.
2 p