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In finance, the variance of the returns of a security are used as a proxy for the associated risk of the security. I've seen some books include sentences like "if you take variance as a measure of risk...". In what way might variance be an incomplete measure of risk? What alternatives do we have?

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  • $\begingroup$ It would be helpful if you included a definition of "risk of a security" so that one can contemplate measures of risk that will be consistent with the definition of it. $\endgroup$ – Alecos Papadopoulos Jan 9 '15 at 23:11
  • $\begingroup$ @Alecos That's the point of the question. What is this thing we call "risk"? How do I quantify it? Colloquially, we assume stocks (for example) are riskier than bonds. The standard way to quantify risk is the variance of returns. Under this definition, yes, stocks on average have a higher variance of returns. But does the variance really capture the notion of "risk" that we have in mind? Is there more to it? Why or why not? $\endgroup$ – jmbejara Jan 9 '15 at 23:20
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    $\begingroup$ This is a broad, but good, question - with hidden depths. Why? Because "risk" means different things to different people at different times, when applied to different situations. During 2007/2008../20XX one could read things like "the risk of Greece leaving the Euro". Other examples are "what is the risk of oil dropping below $50", "what is the (gap)risk of my interest rate hedge on pension liabilities", and "what is the risk of my investment portfolio". In a sense there are as many alternative risk measures as there are "alternatives". For interest, look at: Knightian uncertainty & Taleb. $\endgroup$ – Rusan Kax Jan 13 '15 at 23:38
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A variance is an incomplete measure of risk in a sense, that it measures uncertainty in security payoffs, rather than uncertainty in holder's welfare. In the simplest way we can demonstrate this point as follows.

Suppose that agents want to marginally increase her holding of an asset by $\xi$ and a unit of asset provides a payoff of $x$, which is a random variable. Now we model agent as caring about volatility in her consumption $c$, which is more reasonable, than assumption that she cares about volatility in the payoff directly. Then $$Var(c+\xi x) = Var(c) + 2\xi\cdot cov(c,x) + \xi^2Var(x)$$ and the last term is negligible as $\xi$ is marginal change.

So, as marginal changes are considered in equilibrium, it is more reasonable to take covariance between consumption and payoff as measure of riskiness of the security, rather than just variance of the payoff. In practice the aggregate consumption is often used as a proxy for individual consumption.

Yet another more accurate measure of riskiness can be introduced if we were to presume not direct care for volatility in consumption, but consider an agent with a utility dependent upon his consumption stream. It is a rather lengthy technical exercise and those who are interested should look it up in an excellent textbook "Asset Pricing" by John Cochrane. I will limit myself with presenting the result for multiperiod discrete time model here.

If we were to defined a random variable $m_t = \beta\frac{u'(c_{t+1})}{u'(c_t)}$ called stochastic discount factor for agent with von Neumann-Morgenstern utility and discount factor $\beta$ then by the similar argument as above we may derive that $cov(m_t,x_t)$ should be considered to be even a better measure of riskiness of the security.

Obviously, any nonrandom transformation of the above may be considered to be measure of riskiness, namely, standard deviation and correlation instead of variance and covariance are quite popular. Also different nonrandom normalizations may be applied to $m_t$. For example $u'(c)$ may be used, as $\beta$ is nonrandom and $c_t$ is known at time $t$

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For alternative measurements of risk, consider: 1. Maximum Adverse Excursion [MAE]- the largest historical loss suffered by a system, trade or investment whether real or back-test. 2. Average True Range [ATR] a measure of price change capturing high/low/close and gaps: http://stockcharts.com/school/doku.phpd=chart_school:technical_indicators:average_true_range_atr

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  • $\begingroup$ Welcome to Econ.SE Michael.H., and thanks for this answer—I didn't know about either of these measures. I am having difficulty accessing the link that you posted. Do you think you should check that out and perhaps provide an alternative for the benefit of future visitors to the site? $\endgroup$ – Ubiquitous Feb 10 '15 at 15:30
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For an alternative approach:

Assume we have wealth $W_0$ which is certain. Assume away inflation and things to that effect. If we invest an amount $A$ somewhere (security or whatever), whose future is uncertain, our wealth becomes a random variable

$$W_r = W_o -A + A(1+r) = W_0 + Ar, \;\;\; r \geq -1$$

where $r$ is the proportional return, and it can be as low as $-1$, i.e. we can even completely lose the amount we have invested. This also reflects the cases of "limited liability" of the investor, which is what happens when ones contemplates buying bonds, stocks etc (but in other forms of investing, e.g. a Personal Business, the whole wealth of the investor may be risked, irrespective of the amount invested in the business). The source of randomness is $r$.

Now, a "conservative" point of view would ignore opportunity cost, and think as follows: "I understand that the "risk" I am undertaking, is the possible reduction in my wealth". From this definition, it follows naturally that a measure of risk should be based on the change of wealth. The change in wealth is (by a "before and after" approach)

$$\Delta W = W_r - W_o = Ar$$

This is a random variable. Accepting the expected value as a reasonable stochastic analogue of the level of a deterministic variable (it is not the only one of course, but that is another discussion), we can reasonably say that "a quantification in wealth-units of my risk, is the expected value of negative change in wealth ($r<0$) given that such a thing happens, times the probability that it will indeed happen". In symbols

$$\text {Risk} = E(\Delta W \mid r< 0) \cdot P(r< 0) = A\cdot E(r \mid r< 0) \cdot P(r< 0) = A\cdot E(r \,; r< 0)$$

$$\Rightarrow \text {Risk} = A\cdot \int_{-1}^0rf_r(r)dr$$

where $f_r(r)$ is the probability density function of $r$.

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  • $\begingroup$ +1 Your formula gives a measure of risk in the important sense that the ordinary person thinks of it, that is, risk of loss or downside risk, as opposed to statistical risk which includes both upside and downside. One question: why do you assume $r \geq -1$? This assumption is valid for say an investment in shares in a limited liability company, but (although it may depend on a country's legal and institutional framework) there are situations in which losses can go beyond the initial investment, eg underwriting of insurance, spread betting? $\endgroup$ – Adam Bailey Jan 11 '15 at 0:34
  • $\begingroup$ @AdamBailey Yes you are right, I implicitly assumed liability limited to the amount of investment, since essentially this is the "flavor" I got from the question ("impersonal" investing in bonds, stocks, etc). I will update to stress that. $\endgroup$ – Alecos Papadopoulos Jan 11 '15 at 0:48
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Forward variance is a valid measure of risk for fixed coupon and zero coupon bonds, but it is not at all a measure of risk for stocks, or antiques for that matter.

Variance is a property of a distribution, like noses are a property of most vertebrates. You would not expect to see a nose on a tree. Not all distributions have a variance, just as not all living things have noses. Returns for equity securities cannot have a variance. For an extended discussion, see https://ssrn.com/abstract=2828744

In order to understand why, it is first important to note that returns on stocks are not data. The prices involved are data, but returns are not data, but rather the transformation of data, in particular and buy and a sell price. To understand the distribution of returns, we must first understand the distribution of prices, which returns are a transformation of.

The distribution of prices would be determined by the rules that govern how those prices are created. See the above paper for a discussion of different rules. Taking a Markowitzian view of the world for a moment, with infinite liquidity and hence no bid or ask price, no bankruptcy and no forced mergers, with many buyers and many sellers in equilibrium, then at any static moment, prices will be normally distributed.

The reason is that stocks are sold in a double auction, with buyers competing against other potential buyers and the same with potential sellers. Since we are assuming the market is in equilibrium, not a mandatory assumption actually, there is no winners' curse and hence the only rational behavior is to bid your expectation. The book of orders will be a book of expectations and as we are assuming "many" the distribution will converge to normality. This implies that the "shocks" are appraisal errors, in Markowitz's world, by the counterparty because we are price takers.

The most basic equation of the CAPM, from which Black-Scholes and other option models are explicitly or implicitly derived, is $$\tilde{w}=R\bar{w}+\epsilon.$$ As Gauss pointed out, this is not a solvable problem, instead we must solve this at the limit, so assuming this process is repeated, we can discuss $$w_{t+1}=Rw_t+\epsilon_{t+1}.$$ If we assume $wealth=price\times{quantity}$ and without loss of generality assume $quantity=1$ at all points in time, then we can reduce this to a problem of prices such that $$p_{t+1}=Rp_t+\varepsilon_{t+1},$$ where $\epsilon$ is centered on zero with a finite variance greater than zero.

$R$ is the reward for investing with returns being the reward minus one. We are going to discuss the reward because the return is just a shifted variable and to discuss one is to discuss the other.

If we think of a realized reward at time $t$ as being $$r_t=\frac{p_{t+1}}{p_t}$$ then we are discussing a function that is a ratio for its distribution. The question is what is that distribution. If we add the assumption that the errors are independent, again see the above paper if you do not want to assume that or if you want to include bankruptcy, mergers or liquidity, and we assume they are in equilibrium so they are centered on (0,0) for the error component, then that distribution is well known and solved. It is $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(r_t-R)^2}.$$

Under the assumptions of a Markowitzian word, the distribution of rewards or returns has only a zeroth moment, that is it can have no expectation. Because of this, and because $R>1$, you can show that there does not exist an admissible non-Bayesian solution, so any Frequentist method, including the derivation of the CAPM or Black-Scholes, is an inadmissible solution to the problem UNLESS all parameters are truly known with perfect certainty.

This is not quite to the level where one would write quod erat demonstrandum as it lacks the rigor in this post, but look up the paper. Variance is not a property of stock returns. While you can invoke a process such as $$s^2=\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}$$ as an algorithm, since $\bar{x}\perp\mu$ this algorithm only generates a random number.

The closest Frequentist solution is the semi-interquartile range, but due to truncation by not allowing negative prices, underestimates variability by four percent over the population of annual returns in the CRSP universe for one year returns from 1925-2013.

For other assets, such as antiques sold at Sotheby's, there is still no variance, but the distribution is instead the ratio of two Gumbel distributions. You can see this, in the paper, by doing the trigonometric transformation instead of the ratio transformation. All solutions must have $\tan^{-1}(x)$ for some form of $x$ in it, which is the cdf of the Cauchy distribution, guaranteeing that, except for certain fixed payout instruments, no variance exists at all in finance or macroeconomics or any other model with capital in it. Other assets have other distributions. Learn the rules in the environment and you can derive the distribution(s).

Your alternative measure of risk is the spread on the Bayesian predictive distribution. It also has the nice property of coherence, so fair bets can be placed on it.

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Any meaningful measure of risk should be subjective and forward looking, not some measure of historical returns. Objective risk removes the people from the equation, whilst in essence risk is about how people behave (act and react) in financial markets. There is no constancy in the social sciences, unlike the physical sciences where a frequentist approach may be valid.

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I think what you are referring to a regression betas. Those are commonly used to look at the risk, but due to the high std error in many regression or a high r^2 traditional regression betas are somewhat obsolete.

However, if you take an industry average of all regression betas due to "the law of large numbers"(about 9+ betas, but the more the better) your std error will be much smaller. If you then unlever this industry average beta by: AvgBeta/(1+(1-NominalTaxRate)*(AvgD/E_Ratio)) you will have a pure industry beta.

Moreover, you could adjust this beta for company specific operating leverage by: PureBeta*(1+(1-NominalTaxRate)*(Company_Specific_D/E_Ratio)

If you calculate these betas for every industry your company operates in and then take a weighted average, you will have a better beta alternative to traditional regression betas.

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