The concept of high risk, high return is generally well understood by investors, but calculating risk is actually an almost impossible task

20 October 2022 - 05:00

byGRAHAM BARR and BRIAN KANTOR

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Risk is an elusive concept to pin down and for investors to grapple with in practical, measurable terms.

Investors who take a position on the stock market understand clearly what it means when they’re told their investment has produced a particular return over a particular period.

Most will also tell you they understand the notion of investment risk as an uncertainty of outcome; in particular, the higher the risk one is exposed to, the higher the chance that one loses one’s money. However, it is also accepted that in order to obtain good returns, one needs to take on extra risk. Then, in hindsight, risk is often used to explain why the realised return on an investment is high, on the one hand, or sometimes disastrously low on the other.

Underlying these perceptions of risk is the fundamental market tenet that one must expect to get rewarded for taking a position on an uncertain future. Therefore, markets must “price” risk into a share price, so that the higher the perceived risk of that investment, the higher the required future return on the investment. The problem is that neither this market-determined required return nor the associated risk is objectively measurable.

Still, plenty of people have tried. Quantitative financial analysts and the pioneering work of Nobel prize-winning economist Harry Markowitz use a statistical measure known as standard deviation (of return) as a proxy for risk. Typically, researchers in financial analysis will calculate an estimate of this standard deviation by using past values of share price returns.

In other words, to calculate the risk of a quoted company they would first compute the daily (or weekly) return of the share price over a certain period, and then compute the standard deviation of those returns. This measure, often termed volatility, is then taken as a measure of company risk. We will discuss below the flaws in this approach to measuring risk, but first consider some examples of situations where risk is much more precisely measurable.

In the game of roulette, played in casinos and assuming, of course, an unbiased wheel, we have constant probabilities of the ball landing on any of 36 numbers and zero at each spin of the wheel. It will be easily accepted that the risk involved in a bet on, say, red is much less than the risk of betting on the number 8-black, and one is rewarded accordingly.

If a red number comes up and you’ve bet R1 on red, you get R2 back. If you bet R1 on 8-black and it comes up, you get R36 back.

Because the probabilities are fixed at each spin of the wheel, you could precisely calculate the expected return of your bet and the associated standard deviation of that return, which proxies for risk.

When one plays a game with fixed probabilities, one always knows precisely what one’s expected return is. But in financial markets, event probabilities are not known precisely and change over time

The key point is that when one plays a game with fixed and known probabilities, and places a particular bet in that game, one always knows precisely what one’s expected return is, and also the risk one is exposed to.

But in financial markets, event probabilities are not known precisely and, in fact, change continuously over time. What’s more, there is no possible repetition within an economic system as there is in roulette; the clock cannot be put back, and no process can ever be repeated in exactly the same way.

However, in the case of measuring risk and return in financial markets, we can make some headway in certain circumstances.

For example, assuming the SA government does not default on its contractual payment obligation, one can calculate the exact realised return on an RSA government bond held to maturity.

This required return must reflect the chance of a country default (if there is a default, the return is zero). Apart from its local rand borrowing, the SA government also borrows money on foreign markets denominated in dollars. These SA “Yankee bonds” are traded in New York, along with similar dollar-denominated bonds from other countries.

The required premium of the return (the spread) over and above the return an investor obtains on US government bonds of similar tenure is termed the sovereign risk. It is a measure of the probability of the bonds being paid out, according to contract, in dollars.

One can then calculate the spreads for the different countries which have issued dollar bonds. So, in this case, one can quite precisely compare the market’s perceived risk of default for different countries in paying these dollar-denominated bonds, and compare sovereign risk across different countries.

In the share market, there is no similarly definitive way to obtain the expected return or the risk of any company share on the basis of share market prices. Market analysts often use proxies for comparative value (and hence comparative risk) such as p:es and various measures of yield, such as dividend yield or earnings yield. The underlying principle is that a high-risk company should be reflected in a comparatively low market price, given the current earnings or the dividend payout.

Quantitative portfolio analysts are, however, given the even more challenging task of combining different shares and instruments into a portfolio of assets expected to yield some overall return for some, often pre-mandated, risk.

They are thus faced with the problem of estimating portfolio (or share) risk in order to construct portfolios that fall within their given risk mandate. Given this problem, analysts almost always opt for using the estimated standard deviation of historical share price returns as a measure of volatility, which are then used as a proxy for risk.

There are plenty of problems associated with using past market data to measure risk or return

But there are plenty of problems associated with using past market data to measure risk or return; the underlying issue is that markets are assumed to be efficient. This means the share price at any time can be assumed to reflect known information about the underlying company, but that price will continuously change as new information flows into the market.

Given that this new information is, by definition, unexpected and hence not predictable in any way, the resulting movement in share prices is, in turn, unpredictable. Therefore, past returns can give no indication of what future returns might be.

Risk, in contrast, may have some momentum in that a dramatic event, such as 9/11, will generally give rise to an extended period of return volatility, as markets grapple to understand and price in the impact of the event on share values.

However, though we may be able to anticipate volatility in the short term, the ability to do so over time is confounded by the statistical requirement of parameter stationarity.

In other words, if one wants to estimate a parameter using observations of that parameter over time, the parameter one is measuring cannot itself change over that period.

It’s a bit like locating a target when it’s moving, but your locating method must assume that the target is stationary. In the case of share price (or portfolio) volatility, this is an untenable assumption.

The conclusion is that any attempt to measure risk is problematic, especially in the context of listed companies.

However, there is little acknowledgment of this fact by analysts. Analysts require estimates of risk as a key input into almost any comparative share valuation or portfolio recommendation, but carefully avoid any interrogation of the validity of their estimates of risk. Individual investors may believe they understand risk, but their perceptions are often governed by whatever return they receive.

So: risk is an elusive concept to pin down and for investors to grapple with in practical, measurable terms. Fortunately, investors can usually take comfort in the one clear truth offered up by financial analysis. This is that the only sensible investment strategy is to carefully diversify one’s portfolio across as many asset classes as possible.

Then, assuming the world continues to advance technologically in the same innovative and productive ways it has in the past, irrespective of what unexpected challenges may arise, one’s investment will yield attractive returns over a long period.

* Barr is emeritus professor of statistical sciences at the University of Cape Town (UCT); Kantor chairs the Investec Wealth & Investment Research Institute and is emeritus professor of economics at UCT

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## The elusive notion of risk

## The concept of high risk, high return is generally well understood by investors, but calculating risk is actually an almost impossible task

(digital only)isR30for the first month and thereafter you payR129 p/mnow ad-free for all subscribers.Risk is an elusive concept to pin down and for investors to grapple with in practical, measurable terms.

Investors who take a position on the stock market understand clearly what it means when they’re told their investment has produced a particular return over a particular period.

Most will also tell you they understand the notion of investment

riskas an uncertainty of outcome; in particular, the higher the risk one is exposed to, the higher the chance that one loses one’s money. However, it is also accepted that in order to obtain good returns, one needs to take on extra risk. Then, in hindsight, risk is often used to explain why the realised return on an investment is high, on the one hand, or sometimes disastrously low on the other.Underlying these perceptions of risk is the fundamental market tenet that one must expect to get rewarded for taking a position on an uncertain future. Therefore, markets must “price” risk into a share price, so that the higher the perceived risk of that investment, the higher the required future return on the investment. The problem is that neither this market-determined required return nor the associated risk is objectively measurable.

Still, plenty of people have tried. Quantitative financial analysts and the pioneering work of Nobel prize-winning economist Harry Markowitz use a statistical measure known as standard deviation (of return) as a proxy for risk. Typically, researchers in financial analysis will calculate an estimate of this standard deviation by using past values of share price returns.

In other words, to calculate the risk of a quoted company they would first compute the daily (or weekly) return of the share price over a certain period, and then compute the standard deviation of those returns. This measure, often termed volatility, is then taken as a measure of company risk. We will discuss below the flaws in this approach to measuring risk, but first consider some examples of situations where risk is much more precisely measurable.

In the game of roulette, played in casinos and assuming, of course, an unbiased wheel, we have constant probabilities of the ball landing on any of 36 numbers and zero at each spin of the wheel. It will be easily accepted that the risk involved in a bet on, say, red is much less than the risk of betting on the number 8-black, and one is rewarded accordingly.

If a red number comes up and you’ve bet R1 on red, you get R2 back. If you bet R1 on 8-black and it comes up, you get R36 back.

Because the probabilities are fixed at each spin of the wheel, you could precisely calculate the expected return of your bet and the associated standard deviation of that return, which proxies for risk.

The key point is that when one plays a game with fixed and known probabilities, and places a particular bet in that game, one always knows precisely what one’s expected return is, and also the risk one is exposed to.

But in financial markets, event probabilities are not known precisely and, in fact, change continuously over time. What’s more, there is no possible repetition within an economic system as there is in roulette; the clock cannot be put back, and no process can ever be repeated in exactly the same way.

However, in the case of measuring risk and return in financial markets, we can make some headway in certain circumstances.

For example, assuming the SA government does not default on its contractual payment obligation, one can calculate the exact realised return on an RSA government bond held to maturity.

This required return must reflect the chance of a country default (if there is a default, the return is zero). Apart from its local rand borrowing, the SA government also borrows money on foreign markets denominated in dollars. These SA “Yankee bonds” are traded in New York, along with similar dollar-denominated bonds from other countries.

The required premium of the return (the spread) over and above the return an investor obtains on US government bonds of similar tenure is termed the

sovereign risk. It is a measure of the probability of the bonds being paid out, according to contract, in dollars.One can then calculate the spreads for the different countries which have issued dollar bonds. So, in this case, one can quite precisely compare the market’s perceived risk of default for different countries in paying these dollar-denominated bonds, and compare sovereign risk across different countries.

In the share market, there is no similarly definitive way to obtain the expected return or the risk of any company share on the basis of share market prices. Market analysts often use proxies for comparative value (and hence comparative risk) such as p:es and various measures of yield, such as dividend yield or earnings yield. The underlying principle is that a high-risk company should be reflected in a comparatively low market price, given the current earnings or the dividend payout.

Quantitative portfolio analysts are, however, given the even more challenging task of combining different shares and instruments into a portfolio of assets expected to yield some overall return for some, often pre-mandated, risk.

They are thus faced with the problem of estimating portfolio (or share) risk in order to construct portfolios that fall within their given risk mandate. Given this problem, analysts almost always opt for using the estimated standard deviation of

historicalshare price returns as a measure of volatility, which are then used as a proxy for risk.But there are plenty of problems associated with using past market data to measure risk or return; the underlying issue is that markets are assumed to be efficient. This means the share price at any time can be assumed to reflect known information about the underlying company, but that price will continuously change as new information flows into the market.

Given that this new information is, by definition, unexpected and hence not predictable in any way, the resulting movement in share prices is, in turn, unpredictable. Therefore, past returns can give no indication of what future returns might be.

Risk, in contrast,

mayhave some momentum in that a dramatic event, such as 9/11, will generally give rise to an extended period of return volatility, as markets grapple to understand and price in the impact of the event on share values.However, though we may be able to anticipate volatility in the short term, the ability to do so over time is confounded by the statistical requirement of parameter stationarity.

In other words, if one wants to estimate a parameter using observations of that parameter over time, the parameter one is measuring cannot itself change over that period.

It’s a bit like locating a target when it’s moving, but your locating method must assume that the target is stationary. In the case of share price (or portfolio) volatility, this is an untenable assumption.

The conclusion is that any attempt to measure risk is problematic, especially in the context of listed companies.

However, there is little acknowledgment of this fact by analysts. Analysts require estimates of risk as a key input into almost any comparative share valuation or portfolio recommendation, but carefully avoid any interrogation of the validity of their estimates of risk. Individual investors may believe they understand risk, but their perceptions are often governed by whatever return they receive.

So: risk is an elusive concept to pin down and for investors to grapple with in practical, measurable terms. Fortunately, investors can usually take comfort in the one clear truth offered up by financial analysis. This is that the only sensible investment strategy is to carefully diversify one’s portfolio across as many asset classes as possible.

Then, assuming the world continues to advance technologically in the same innovative and productive ways it has in the past, irrespective of what unexpected challenges may arise, one’s investment will yield attractive returns over a long period.

* Barr is emeritus professor of statistical sciences at the University of Cape Town (UCT); Kantor chairs the Investec Wealth & Investment Research Institute and is emeritus professor of economics at UCT## ALSO READ:

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