Ekkehard Kopp

Portfolio Theory and Risk Management: Risk and return

Financial investors base their activity on the expectation that their investment will increase over time, leading to an increase in wealth. Over a fixed time period, the investor seeks to maximise the return on the investment, that is, the increase in asset value as a proportion of the initial investment. The final values of most assets (other than loans at a fixed rate of interest) are uncertain, so that the returns on these investments need to be expressed in terms of random variables. To estimate the return on such an asset by a single number it is natural to use the expected value of the return, which averages the returns over all possible outcomes. Our uncertainty about future market behaviour finds expression in the second key concept in finance: risk. Assets such as stocks, forward contracts and options are risky because we cannot predict their future values with certainty. Assets whose possible final values are more ‘widely spread’ are naturally seen as entailing greater risk. Thus our initial attempt to measure the riskiness of a random variable will measure the spread of the return, which rational investors will seek to minimise while maximising their return. In brief, return reflects the efficiency of an investment, risk is concerned with uncertainty. The balance between these two is at the heart of portfolio theory, which seeks to find optimal allocations of the investor’s initial wealth among the available assets: maximising return at a given level of risk and minimising risk at a given level of expected return.

Value at Risk

Until now we have focused our attention on variance, or equivalently, standard deviation of the return, as a tool for measuring risk. The standard deviation measures the spread of the random future return from its mean. In portfolio selection we seek to minimise the variance while maximising the return. However, an investor, seeking to measure the risk inherent in an asset he holds, is naturally more concerned to place a bound on his potential losses, while remaining relaxed about possible high levels of profit. Thus one looks for risk measures which focus on the downside risk, that is, measures concerned with the lower tail of the distribution of the return. Variance and standard deviation are symmetric, so they are not good candidates in this search. In looking for quantitative measures of the overall risk in a portfolio, we seek a statistic which can be applied universally, enabling us to compare the risks of different types of risky portfolio. Ideally, we look for a number (or set of numbers) that expresses the potential loss with a given level of confidence, enabling the risk manager to adjudge the risk as acceptable or not. In the wake of spectacular financial collapses in the early 1990s at Barings Bank and Orange County, Value at Risk (henceforth abbreviated as VaR) became a standard benchmark for measuring financial risk.