Risk measures

In this chapter we elaborate the idea that the risks of investments may partly cancel out when they are held together in a portfolio. The strategy of diversification is, of course, common knowledge and many languages have a proverb for not putting all one’s eggs in one basket or not staking everything on one card. Markowitz (1952) was the first to formalize this idea in a pioneering paper that became the basis of modern portfolio theory.

In 1990 Markowitz was awarded the Nobel prize in economics. We look at his portfolio selection model and the pricing models that are based on it, the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT).

3.1 Risk and return

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3.1.1 Risk measures

Recall that finance studies how people choose between risky future values. Risk means that the future outcomes of choices can deviate from their expected values. Hence, risky future outcomes have to be expressed, or modelled, in a probabilistic manner. This can be done with discrete variables, that simply enumerate all possible future values and their associated discrete probabilities. It can also be done with continuous variables, that spec-ify a probability distribution of possible values. Time can also be modelled in a discrete and continuous manner. Discretely modelled, time is a series of moments (points in time) at which the uncertainty over the previous period is resolved. In the periods between the moments ‘nothing happens’. For example, a calendar year can be modelled as one period of a year, with two moments: beginning and end. But it can also be modelled as two peri-ods of six months, with three moments, or as four quarters, twelve months or 365 days.

The limiting case is continuous time, in which each period is infinitesimal and the number of periods is infinite. Table 3.1 places some well-known models and modelling techniques in the double dichotomy of discrete and continuous time and variables. We will meet most of them later on. Portfolio theory usually analyzes the continuous returns of investments over discrete holding periods. However, to keep the calculations easy to follow we will derive the main results using a small number of discrete possible outcomes.

The risk of investments is often depicted as in Figure 3.1, that shows the closing prices of the Nasdaq-100 stock index (ticker: NDX) over an arbitrary period of 253 trading days. As we have seen in Chapter 1, this index includes 100 of the largest US and inter-national non-financial securities listed on the Nasdaq stock exchange. At the start of the period the index had a value of 1996.6 and at the end it was 2139.8, a gain of more than 7 per cent. However, in the beginning of the period the index rallied to almost 2400, a gain of 20 per cent, and later it plunged more than 10 per cent. Those were, with hindsight, the 51

Table 3.1 Models and modelling techniques

Discrete time Continuous time Discrete variables State preference theory Bankruptcy

Binomial Option Pricing processes

Continuous var’s Portfolio theory, CAPM Black and Scholes Capital structure models Option pricing

risks faced by somebody who wanted to invest in this index for a couple of months during that period.

0 100 200

2000 2200 2400

days NDX

Figure 3.1 Nasdaq-100 index, daily closing prices, adjusted for dividends, for 253 trading days from 1 October 2010 to 30 September 2011

To analyze investments we usually transform the prices (p) into returns (r) by taking percentage first differences:¹

r_t = 100 × p_t+1− pt

p_t

Recall from the previous chapter that such discretely compounded returns have the advantage over logarithmic returns that they are easily aggregated over investments in a portfolio. These daily returns are depicted in Figure 3.2 for the same period (minus one day). We see that most returns are between −2 per cent and +2 per cent, with (quite) a few spikes in both directions.

Although illustrative, Figure 3.2 is difficult to compare with similar graphs for other indices or periods. Comparability is provided by the statistical properties of the returns, particularly their distributional properties. Figure 3.3 plots the relative frequency of days with returns in each one-percentage interval. For instance, there were 101 days with a

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1 Return also includes dividends; we use closing prices that are adjusted for dividends.

53 3.1 Risk and return

0 100 200

–6 –4 –2 0 2 4

days r

Figure 3.2 Daily returns Nasdaq-100 index for 252 days from 4 October 2010 to 30 September 2011

return between 0.5 per cent and 1.5 per cent; this is depicted as the peak diamond (✸⁾ with coordinates 1, 0.4 (101/252 = 0.4). The solid line connects these frequencies. The dashed line is a normal density function with the same mean and standard deviation as the returns (∽N(0.036, 1.327)). When return data are used in this way it becomes natural to use a statistical measure of dispersion as risk measure. Indeed, the variance (or its square root, the standard deviation) is the most frequently used quantitative measure of risk.

–6 –4 –2 0 2 4 6

0.1 0.2 0.3 0.4

r frequency

Figure 3.3 Frequency of daily returns Nasdaq-100 index over 252 days from 4 October 2010 to 30 September 2011

Variance can be used in a backward looking way (measuring the dispersion around an observed, historical mean) as well as in a forward looking way (measuring the dispersion around an expectation). It is easily calculated in both cases and has well-known statistical

properties. However, using variance as risk measure in financial analyses also has clear disadvantages. It gives equal weights to the upward and downward deviations, but receiv-ing a larger than expected return would hardly be considered a risk by most investors.

Also, using the variance ignores the higher moments of the distribution (skewness and kurtosis). Most investors would prefer a distribution skewed towards higher returns.

In response to these disadvantages, some other risk measures have been suggested.

One of these is semi-variance, which is calculated in the same way as normal variance, but using only the returns that are lower than the mean or expectation. The concept of semi-variance can be adapted to measure deviations not from the mean or expectation but from some target return set by the investor. This is called lower partial variance or, when the higher moments are included, lower partial moments. An alternative measure is Value at Risk (VaR) which is defined as the maximum loss if the x per cent left tail of the distribution is ignored. However, variance/standard deviation remain widely used measures of risk; they are also the risk measures in Markowitz’s portfolio model, that we look at next.