A Calculating mean returns
A.2 A forward looking example
A more elaborate numerical example is presented in Lattice 3A1. It gives a binomial tree of the possible values, V , over three periods of a e100 investment with a return rate of either +50 per cent or −25 per cent per period. Both returns are equally likely. The probabilities, p, geometric averages, rg, and arithmetic averages, ra, corresponding to the values are also included. The bottom line gives the expectations of the values and returns, calculated as the probability weighted sum of the possible values. For example, the expectation of the geometric average after two periods is (0.25 × 50) + (0.5 × 6.1) + (0.25 × −25) = 9.3. The lattice illustrates a number of general characteristics of returns over time. First, when the return rates are constant, the geometric and arithmetic averages are the same. This is the case in all upper and lower nodes, including the nodes after one period.
Second, when the return rate fluctuates over time, the geometric average is always lower than the arithmetic.19 This is the case in the three middle nodes (with V = 112.5, 168.75 and 84.38). Third, the expected geometric average decreases with time, even if the expected return is the same in each period and each node, as it is in Lattice 3A1. Since the expected arithmetic average is constant, the difference between the two expectations increases with time, as the bottom line of Lattice 3A1 shows. The difference also increases with the volatility of returns, but that is not shown.
Lattice 3A1 also shows what the proper discount rate is. That rate is usually referred to as the opportunity cost of capital, and it is the return that investors demand, and get, on investments with the same risk characteristics as the investment we are looking at.
This rate includes a risk premium as well as the time value of money (and inflation, if any). Investors express their return demands in the price they are willing to pay for the investment. So if e100 is the proper market price now given the expected values in future periods, then the proper discount rate is the rate that connects present and expected
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19 We can write the returns of +50 per cent or −25 per cent as an expected return of +12.5 per cent ± a deviation of 37.5 per cent. The first fluctuating compound return then becomes 1.1252− .3752 = 1.125. Repeating the procedure for the fluctuating compound returns in the other two middle nodes is possible, but there the cross terms do not cancel out and the process becomes messy.
337.5 0.125 50 50 ր
225 0.25 50 50
ր ց
150 0.5 50 50 168.75 0.375 19.1 25
ր ց ր
100 112.5 0.5 6.1 12.5
ց ր ց
75 0.5 −25 −25 84.38 0.375 −5.5 0
ց ր
56.25 0.25 −25 −25 ց
42.19 0.125 −25 −25
E[.] 112.5 12.5 12.5 126.6 9.3 12.5 142.39 8.23 12.5
Lattice 3A1 Expected values, geometric and arithmetic return rates (in %)
93 Exercises
future values: P V = E[F VT]/(1 + r)T. Clearly, the rate that satisfies this equation for all periods is the arithmetic average rate of 12.5 per cent. The geometric average rate would understate the required return and overstate the value of the investment for all periods
>1. So this answers the question which average should be used when return rates are calculated from time series of stock or index returns: the arithmetic average.
Conceptually, the opportunity costs of capital is a forward looking rate that reflects the required reward for risk. It is therefore independent of the ‘history’ of the investment. In the lattice example, the risk is expressed as a return of either +50 per cent or −25 per cent.
That risk is the same in all nodes and periods, independent of the return that was realized in the previous period. So the opportunity cost of capital should also be the same in all nodes and periods. The rate that equates the expected value next period to the value now in all nodes is 12.5 per cent, the arithmetic average.
A more formal basis for using the arithmetic average is provided by probability theory.
Suppose the continuously compounded, logarithmic returns per period on an asset are independently, identically and normally distributed with mean μ and variance σ2.These are strong but not unusual assumptions. They mean that the expected return in each period E[ln (Vt+1/Vt)] = μ where V is the asset’s value. The iid assumption (indepen-dently and identically distributed) implies that cumulative return over T periods is also normally distributed with mean μT and variance σ2T. If the logreturns are normally dis-tributed, then, by definition, the ‘raw’ returns (Vt/Vt−1)are lognormally distributed. By the properties of the lognormal distribution (see Aitchison and Brown, 1969) the expected return in each period E[Vt+1/Vt] = eμ+12σ2and the cumulative return over T periods is E[Vt+T/Vt] = e(μ+12σ2)T. Since Vt is the known value today, we can write these as E[Vt+1] = Vteμ+12σ2and E[Vt+T] = Vte(μ+12σ2)T. The return rate μ +12σ2is the arith-metic average rate. So it can be concluded that an unbiased estimate of the asset’s future value is found by compounding the present value forward at the arithmetic average rate of return.
However, the correctness of this conclusion depends on the assumptions that the returns are independently, identically and normally distributed (iid) and that the parameters of the distribution (expectation, variance) are known.20In practice, the iid assumption may not obtain. Moreover, the distribution’s parameters usually are not known and have to be esti-mated, with error, from e.g. historical time series. If that is the case then the historical arithmetic average yields an upwardly biased estimate of the true return and more com-plex estimators have to be used. Such estimators, and a more thorough discussion of the matter, can be found in Blume (1974), Jacquier et al. (2003) and Cooper (1996).
Exercises
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1. Mrs Pennymoney has invested 50 per cent of her money in a risk-free bank account and 50 per cent in an index fund that invests in all shares on the stock exchange. The risk-free interest rate is 5 per cent and the expected return on the market index is 15 per cent with a standard deviation of 20 per cent.
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20 For a large number of periods, normality may also follow from the central limit theorem.
(a) Is Mrs Pennymoney’s portfolio mean-variance efficient?
(b) What is the expected return on the portfolio?
(c) What is the portfolio standard deviation?
(d) What is the portfolio β (relative to the market as a whole)?
2. Mr Poundmoney has invested 150 per cent of his money in an index fund that invests in all shares on the stock exchange. The risk-free interest rate is 5 per cent and all investors can borrow and lend unlimited amounts at this rate. The expected return on the market index is 15 per cent with a standard deviation of 20 per cent.
(a) How can Mr Poundmoney invest more than 100 per cent of his money in the index fund?
(b) Is Mr Poundmoney’s portfolio mean-variance efficient?
(c) What is the expected return on the portfolio?
(d) What is the portfolio standard deviation?
(e) What is the portfolio β (relative to the market as a whole)?
3. Ms Stonemoney has invested 50 per cent of her money in ADC (Aker Dissolution Company) and 50 per cent in NAC (Northern Aluminium Company). ADC has an expected return of 20 per cent and a standard deviation of 40 per cent and NAC has an expected return of 12.5 per cent and a standard deviation of 25 per cent. A statistical analysis of the returns in the dissolution and the aluminium industries over the period 2004–2007 (which was a period of stable growth) has shown that the two industries are statistically independent (i.e. uncorrelated – correlation coefficient is 0).
(a) Is Ms Stonemoney’s portfolio mean-variance efficient?
(b) The correlation coefficient between the returns in the dissolution and the alu-minium industries was measured over a period of stable growth. How does this correlation coefficient change in times of financial crises, such as the 2007–2008 credit crunch?
(c) What is the expected return on the portfolio?
(d) What is the portfolio standard deviation?
4. Suppose your financial newspaper of last Monday shows that the company with the highest β coefficient on your stock exchange was ZX Co. The company has a β of 2.1 and a market value of e4.25 billion (109). Assume that the company is financed only with equity, no debt, that the risk-free interest rate is 5 per cent and that the required rate of return of an appropriate index of all shares on the stock exchange is 15 per cent.
(a) What is the required rate of return on ZX Co stocks?
ZX Co’s management proposes to raise e4.25 billion in additional equity capital and to invest this amount in risk-free government bonds. Ignoring taxes and transaction costs:
(b) Calculate the required rate of return of ZX Co stocks after the proposed investment.
What are the company’s market value and its β coefficient?
(c) What general phenomenon does this example illustrate?
95 Exercises
5. Take another look at your uncle Bob’s portfolio problem in Table 3.7.
(a) Using the same setting as in the main text, where short sales are not allowed, what is the maximum return portfolio?
(b) Do you see a practical way to find the efficient frontier?
(c) The portfolio that maximizes return under the restriction that portfolio standard deviation is ≤ 0.25 consists of 0.02 Cisco, 0.36 Amazon and 0.62 Apple. Calcu-late, for each of these three stocks, its contribution to portfolio variance and its β relative to this three-stock portfolio. Check your results.
6. Download weekly closing prices over the last year of Microsoft (ticker: MSFT), Yahoo (YHOO) and the Nasdaq-100 index (NDX). Assuming an annual risk-free interest rate of 5.2 per cent and using the index as a proxy for the marked portfolio, calculate for each stock:
(a) the characteristic line (b) the Sharpe ratio (c) the Treynor ratio (d) Jensen’s alpha.
7. In the arbitrage example in the text (section 3.4.3) we had an arbitrage portfolio of .2P1 + .3P2 + .5P3, but we did not say how the portfolio weights were obtained.
The equilibrium (i.e. arbitrage-free) factor model (pricing relation) is E(ri) = .075 + .06bi1+ .03bi2 and the arbitrage portfolio’s sensitivities to the two factors are .75 and .7.
(a) Show how the portfolio weights are calculated.
(b) A variation on the same theme: show the composition of the two pure factor portfolios.
(c) Finally, just to make sure that you get the multi-factor concept, what are the risk and return of an equally weighted portfolio of the three portfolios P1–3 in Table 3.11? Check your return answer with an alternative calculation.
Table 3.11 Portfolios r and b
Portf.1 Portf.2 Portf.3
rp .18 .15 .12
b1 1.5 0.5 0.6
b2 0.5 1.5 0.3