# Risk, expected return, and the cost of equity capital.

1. Introduction

An important determinant of investment expenditure is the cost of the capital employed in undertaking any investment project. For firms considering a specific project, the cost of capital is the discount rate applied to all future cashflows that allows estimation of the project's profitability. For regulators such as the New Zealand Commerce Commission, the cost of capital effectively determinines the price path a regulated firm is required to follow and thus has significant implications for the firm's incentives to invest.

Unfortunately, the cost of the equity component of capital is not directly observable and hence must be estimated from a theoretical model, the most commonly used of which is the celebrated Capital Asset Pricing Model (CAPM). (1) The standard version of the CAPM has the familiar form:

E[[R.sub.i]] = [R.sub.f] + [[beta].sub.i] {E[[R.sub.m]] - [R.sub.f]} (1)

where [R.sub.i] is the random return on asset i, [R.sub.f] is the riskless rate of interest, [R.sub.m] is the random rate of return on the market portfolio of risky assets, [[beta].sub.i] is the asset i 'beta', equal to the covariance of [R.sub.i] and [R.sub.m] divided by the variance of [R.sub.m], and E[x] is the expectations operator. Equation (1) states that the asset i risk premium is proportional to the market risk premium where the factor of proportionality is equal to [[beta.sub.i]. In this formulation, [[beta].sub.i] is the quantity of asset i risk and the market risk premium {E[[R.sub.m]] - [R.sub.f]} is the price of that risk.

Equation (1) is a relative pricing model. Specifically, it relates one market price (the risk premium on asset i) to another market price (the risk premium on the market portfolio). But because the market portfolio includes all assets (including asset i), the second price, if correctly measured, incorporates the first price, so there is an element of circularity in this process. Practical applications of the CAPM, such as estimating the cost of capital, typically ignore this problem--on the grounds that any individual asset is an infinitesimally small portion of the market portfolio--and treat the market risk premium as a free parameter to be estimated from data. (2) But as Cochrane (2001) points out, this procedure ignores the CAPM predictions for the market portfolio itself; taking the market premium as given neglects its underlying dependence on more fundamental CAPM parameters.

The implications of this argument are explored in the remainder of the paper, but an intuitive overview may be helpful. Underlying the CAPM is the separation result from modern portfolio theory that all investors optimally allocate their funds between two portfolios, one risky and one riskless. Investor-specific risk attitudes determine the split between the two portfolios, but have no effect on the composition of the risky asset portfolio which depends only on the distribution of future asset returns. If all investors perceive the same returns distribution, they must then wish to hold the same portfolio of risky assets. Equilibrium in the market for these assets requires that this common portfolio be the so-called market portfolio, an observation that leads directly to equation (1). However, this process makes no explicit reference to equilibrium in the market for riskless assets. Incorporating this additional condition in the model places an exact restriction on the allowable value of the market risk premium in terms of the underlying variance of market returns. As a result, applications of the CAPM need only estimate the value of the latter parameter and not the market risk premium itself.

The link between the market risk premium and the returns variance is not in itself new. Assuming quadratic utility, or exponential utility with normal returns, (e.g., Friend and Blume, 1975; Huang and Litzenberger, 1988), or continuous trading opportunities (Merton, 1980), other authors have also shown that the market risk premium is proportional to the variance of market returns. However, I demonstrate that this result is a simple consequence of riskless asset equilibrium, for all preferences defined over the mean and variance of single-period wealth, and that additional assumptions about utility or the returns structure are unnecessary. Perhaps more importantly, I examine the implications of the result for practical applications such as estimating the cost of equity capital, an issue that none of the authors above, with the partial exception of Merton, considers.

Expressing the market risk premium as a function of the variance of market returns is potentially valuable for calculating the cost of capital. (3) Estimating the variance of returns is much easier than estimating the mean, particularly when these parameters vary through time, so the alternative approach holds out the promise of identifying risk-based shifts in the cost of capital. Unfortunately, I find that the estimated variance of returns in New Zealand data is extremely volatile; at times the implied cost of capital is implausibly high while at other times it is implausibly low. In my view, this is not good news for CAPM-based approaches to estimating the cost of capital: seemingly-reasonable estimates based on (1) are obtained only by ignoring an important part of CAPM content.

The next section explicitly derives the link between the market risk premium and the variance of market returns. In Section 3, I discuss how this might help obtain more accurate cost of capital estimates and apply this to data. The final section contains some concluding remarks.

2. CAPM and equilibrium in the market for riskless assets

If [[lambda].sub.ik] is expenditure on asset i by investor k = 1, ..., m, and [[lambda].sub.fk] is expenditure on the riskless asset, then end-of-period wealth [W.sub.k] satisfies:

[W.sub.k] = [n.summation over (i=1)] [[lambda].sub.ik] (1 + [R.sub.i]) + [[lambda].sub.fk] (1 + [R.sub.f]) (2)

The asset expenditures must sum to initial wealth [W.sub.0k]. Setting the latter to unity for convenience, (2) can be written as:

[W.sub.k] = 1 [R.sub.f] + [n.summation over (i=1)] [[lambda].sub.ik] ([R.sub.i] - [R.sub.f]) (3)

so that:

E[[W.sub.k]] [equivalent to] [[bar.W].sub.k] = 1 + [R.sub.f] + [n.summation over (i=1)] [[lambda].sub.ik] (E[[R.sub.i]] - [R.sub.f]) (4)

var([W.sub.k]) [equivalent to] [sigma] = [n.summation over (i=1)] [n.summation over (j=1)] [[lambda].sub.ik][[lambda].sub.jk] cov([R.sub.i], [R.sub.j]) (5)

Each investor has a utility function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that depends only on the mean and variance of end-of-period wealth, and chooses the portfolio that maximises the value of this function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first-order conditions for this problem are:

[[gamma].sub.k] (E[[R.sub.j]] - [R.sub.f]) = [n.summation over (j=1)] [[lambda].sub.jk] cov([R.sub.i], [R.sub.j]) i =1, ..., n (6)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is investor k's marginal rate of substituting risk for return. That is, 1/[[gamma].sub.k] [equivalent to] is the additional mean return that investor k would need to be no worse off following a marginal increase in variance. If this is high (low), then investor k is relatively intolerant (tolerant) of risk. We can therefore interpret 1/[gamma] as a measure of investor k's risk aversion.

Writing (6) in matrix form and rearranging yields the two-fund separation result of Tobin (1958), i.e., the composition of investor k's portfolio of risky assets is unaffected by risk attitudes ([[gamma].sub.k]) and depends only on perceived means, variances and covariances of returns. If these parameters are the same for all investors, then they all hold the same portfolio of risky assets. Market clearing then requires that this portfolio must include each asset in an amount equal to its weight in the portfolio of total invested wealth in risky assets, i.e., the market portfolio of risky assets. Then [[gamma].sub.jk] is equal to asset j's weight in this portfolio multiplied by investor k's total investment in risky assets (1-[[gamma].sub.fk]) and equation (6) can be rewritten as:

[[gamma].sub.k] (E[[R.sub.i]] - [R.sub.f]) = (1 - [[lambda].sub.fk])cov([R.sub.i], [R.sub.m]) i=1, ..., n (7)

where [R.sub.m] is the return on the market portfolio of risky assets.

As (7) holds for all risky assets, it must also apply to the market portfolio:

[[gamma].sub.m] (E[[R.sub.m]] - [R.sub.f]) = (1 - [[lambda].sub.fk])[[sigma].sup.2.sub.m] (8)

where [[sigma].sup.2.sub.m] is the variance of [R.sub.m]. Combining (7) and (8) then yields equation (1):

E[[R.sub.i]] = [R.sub.f] + [[beta].sub.i](E[[R.sub.m]] - [R.sub.f])

Equation (1) is, of course, the standard formulation of the CAPM, but it overlooks an important part of the underlying pricing process. Going from (6) to (7) requires that supply equal demand for each risky asset, but no corresponding requirement is imposed on the riskless asset.

To determine the implications of imposing riskless asset equilibrium, return to equation (8) and note that this can be written as

[[lambda].sub.fk] = 1 - (E[[R.sub.m]] - [R.sub.f])/[[sigma].sup.2.sub.m] / (1/[[gamma].sub.k]) (9)

which expresses investor k's demand for the riskless asset as a function of risk aversion

and market portfolio characteristics. Intuitively, (E[[R.sub.m]] - [R.sub.f]) / [[sigma].sup.2.sub.m] is the rate at which the market portfolio trades off risk and return while (1 / [[gamma].sub.k]) is the rate at which investor k is willing to make this tradeoff. If, for example, the former exceeds the latter, then investor k desires more risk than is offered by the market portfolio alone and thus borrows at rate [R.sub.f] to finance a larger holding of that portfolio, i.e., [[gamma].sub.fk] < 0.

Equilibrium in the riskless asset market requires that total borrowing equal total lending, i.e, [m.summation over (k=1)] [[lambda].sub.fk] = 0. (4) Applying this to (9) yields:

E[[R.sub.m]] - [R.sub.f] = [([m.summation over (k=1)] [[lambda].sub.k]/m).sup.-1] [[sigma].sup.2.sub.m] = (1/[gamma]) [[sigma].sup.2.sub.m] (10)

where [gamma] [alpha] [m.summation over (k=1)] [[gamma].sub.k] [alpha]/m is the average value of [[gamma].sub.k] i.e., 1/[gamma] is the average risk aversion of all investors. Thus, equation (10) states that riskless asset equilibrium constrains the market risk premium to a value equal to the product of market risk and market risk aversion.

To understand (10) intuitively, suppose that it is not satisfied, e.g., (E[[R.sub.m]] - [R.sub.f]) < (1/[gamma]) [[sigma].sup.2.sub.m]. Then the rate at which the market portfolio offers to trade off risk and return is below the rate required by investors to make this tradeoff. Investors will thus wish to substitute from the market portfolio to the riskless asset, i.e., there is excess demand for the riskless asset. With supplies fixed, equilibrium is re-established by a rise in (E[[R.sub.m]] - [R.sub.f]) until the excess demand is eliminated. This occurs only when the available risk-return tradeoff is equal to the required tradeoff, i.e., when (10) holds. More succinctly, any violation of (10) implies excess demand or supply in the riskless asset market, so equilibrium in that market requires that (10) be satisfied. In such an equilibrium, the market risk premium (E[[R.sub.m]] - [R.sub.f]) must equal the "market price of risk". ([[sigma].sup.2.sub.m]/ [gamma]). (5)

Substituting equation (10) into (1) yields:

E[[R.sub.i]] = [R.sub.f] + [[beta].sub.i] ([[sigma].sup.2.sub.m] / [gamma]) (11)

which relates the expected excess return on asset i to beta and the market price of risk.

The link between (1) and (11) is worth emphasizing. Equation (1) requires only that risky asset markets clear, and is thus a partial equilibrium statement that relates one endogenous price variable (E[[R.sub.i]] - [R.sub.f]) to another (E[[R.sub.m]] - [R.sub.f]). Equation (11) takes this insight a step further by requiring that supply also equal demand for the riskless asset, thereby allowing E[[R.sub.m]]-[R.sub.f] to be expressed in terms of underlying exogenous parameters and transforming (1) into a general equilibrium statement.

Friend and Blume (1975) and Merton (1980) also report an equation similar to (11), and Huang and Litzenberger (1988) explicitly derive it in the context of either quadratic utility or exponential utility with normal returns. However, none explicitly makes the link to riskless asset equilibrium, and only Merton recognises its implications for practical applications such as estimating the cost of capital. The latter issue is the subject of the remainder of this paper.

3. Estimating the cost of capital

Suppose one wishes to estimate a firm's cost of capital. The usual approach estimates the market risk premium E[[R.sub.m]]-[R.sub.f] as a free parameter and uses this in equation (1), along with estimates of [[beta].sub.i] and [R.sub.f]. However, as section 2 demonstrates, the market risk premium is not a free parameter; rather it is an endogenous function of underlying CAPM parameters, as described by (10). Thus, applications of the CAPM can, in principle, use either (1) or (11), estimating either the market risk premium (E[[R.sub.m]] - [R.sub.f]) or the market price of risk ([[sigma].sup.2.sub.m]/[gamma]). Although the latter approach is theoretically superior, the only relevant consideration in practical situations is the reliability of estimates. More precisely, is it better to estimate the market risk premium or the market price of risk?

There are two reasons to favour the latter approach. First, as discussed at length by Merton (1980), Black (1993) and Campbell et al (1997), it is much easier to estimate the variance of returns than it is to estimate expected returns. The essence of their argument is that the precision of the variance estimate increases with the number of observations while the precision of the expected return estimate increases only with the length of the data series. In other words, a good estimate of variance can be obtained even with a short time series so long as the data are sufficiently high-frequency, but the only way to get a similarly good estimate of the mean is to have a long time series. (6) Consequently, given the usual constraints on available data, variance estimates will be considerably more accurate than expected return estimates.

The second reason for favouring (11) follows from the first. In calculating the cost of capital, the relevant distribution of returns is the conditional distribution, since it is this that describes the current risk outlook: if risk is high at a particular date, then the market risk premium, and hence the cost of capital, should also be high at that date. More precisely, if one wishes to use equation (1) to estimate the cost of capital, then an estimate of the current market risk premium is required (i.e., the conditional mean of excess market returns); equation (11) requires, instead, an estimate of current risk (i.e., the conditional variance of market returns). In the case of (1) however, because of the long time period needed to estimate expected returns, the best one can feasibly do is obtain a single estimate of the unconditional market risk premium. Consequently, applications of (1) are unable to incorporate variation over time in the market risk premium and thus do not reflect the current risk environment. (7) By contrast, equation (11) is potentially able to identify this variation because of the shorter time series required for estimating variance.

Of course, there is also a significant disadvantage to using (11): the parameter [gamma] is unobservable. However, it may be possible to estimate this fairly accurately, so long as one is prepared to assume that it is constant (i.e., independent of wealth or consumption). This seems reasonable, at least as a first approximation; as Campbell and Viceria (2002) point out, there have been large increases in per capita consumption and wealth in the last 100 or so years, but no corresponding trends in risk premia or interest rates consistent with investors having changed their attitudes towards risk.

This assumption can be exploited by rearranging (10) to obtain 1/[gamma] as the ratio of the market risk premium and the variance of market returns and then following a two-step procedure. First, I use the unconditional version of (10) to estimate the constant 1/[gamma] applicable to a given market, i.e., as the ratio of the unconditional market risk premium and the unconditional market variance. Second, I multiply this parameter by the conditional variance and substitute into (11) to estimate the current cost of capital. (8) For example, suppose one obtains, from 100 years of data, unconditional estimates of {E[[R.sub.m]] - [R.sub.f]} and [[sigma].sup.2.sub.m], equal to 0.06 and 0.03 respectively. Then the implied value of 1/[gamma] is two and, from (11), the current (date t) expected return on asset i is [R.sub.ft] + 2([sigma])[[beta].sub.it], where the t subscripts denote date t values.

To provide a concrete illustration and assessment of this process, I use equation (10) to estimate an annual series for the New Zealand market price of risk over the last 30 years. This requires, first, estimation of 1/[gamma] as described above, and, second, annual estimates of [[sigma].sup.2.sub.mt], To calculate 1/[gamma], I use the recent study of Lally and Marsden (2004) on NZ returns during the 1931-2002 period. Their estimates of the unconditional values of {E[[R.sub.m]] - [R.sub.f]} and [[sigma].sup.2.sub.m] mply a value of 1/[gamma] equal to 1.4. (9)

Turning to the conditional value of [sigma], I use monthly real stock returns on the NZ stockmarket since 1967 to calculate a moving average variance of returns for the 34 years from January 1970 to December 2003. (10) Specifically, for each month during this period, I calculate the sample variance of returns over the previous 36 months. These monthly variances are first multiplied by twelve and then averaged across the 12 months of each calendar year to obtain a single estimate of the conditional variance for each year. (11)

Combining these estimates of [[sigma].sup.2.sub.mt] with 1/[gamma] = 1.4 gives a time series of annual estimates of the market price of risk, and the results from this procedure are outlined in Figure 1 and Table 1. The former depicts the year-by-year variation in the market price of risk estimate; the latter summarises these data for both the full period and three subperiods.

[FIGURE 1 OMITTED]

The primary impression from these results is one of considerable volatility in the variance of market returns. For the period as a whole, the average market price of risk is a reasonable-sounding 6.4%, but this hides significant intra-period variation. (12) Throughout the 1970s, the maximum value was 3.7% and the average only 2.1%. In the 1980s, the price of risk ranged from less than 1% to more than 33%; the period since 1990 is similarly volatile. Although the most extreme values occur in the 1980s, the remaining periods are also characterized by both high and low values. For the most recent year in the sample (2003), the estimated market price of risk is 2.3%, implying a very low cost of capital for most projects.

If one wishes to use the CAPM to capture time variation in the market price of risk, these results are thought-provoking. For instance, is it really plausible that the price of risk went from less than 1% in the early part of the 1980s to more than 30% by the end? Answering in the affirmative implies an acceptance of very large swings in the cost of capital. Similarly, does it seem reasonable that the average-risk firm (i.e., [[beta].sub.i] = 1) in 2003 had a cost of equity only 2.3 percentage points above the riskless rate of interest?

Measurement error in the two crucial variables - [gamma] and [[sigma].sup.2.sub.m] - seems unlikely to resolve these problems. Any error in the estimate of y changes the market price of risk at each date by a scalar, and thus has no effect on the volatility depicted in Figure 1. Moreover, the estimated volatility in [[sigma].sup.2.sub.m] is unaffected by the use of alternative estimation periods, higher-frequency data, or more sophisticated estimation methods. (13) Certainly, estimates of variance contain estimation error and are thus likely to be more volatile than true values, but while this potentially rescues the theoretical validity of the CAPM pricing process, it simply re-states the practical problems encountered when trying to use the CAPM to capture time variation in the cost of capital.

A more interesting possibility is that, contrary to the assumption maintained above, 1/[gamma] is not constant, but instead covaries negatively with market risk. In this case, high realisations of [[sigma].sup.2.sub.m] would be offset by low values of 1/[gamma] and the market price of risk series would be considerably more stable than it appears in Figure 1. One way this might occur is if the presence of noise traders (see, for example, Black, 1986; De Long et al, 1990) in the market waxes and wanes with the extent of their over-optimism, e.g., as bullishness rises, more noise traders enter the market and increase market volatility. However, such an effect would have to dominate the possibly countervailing influence of intertemporal considerations for other investors. As Boyle and Young (1992) point out, rational risk-averse investors respond to consumption shocks by altering their demand for, and hence the price of, equities; higher risk aversion elicits a greater demand response and more dispersed equity prices across possible states of the world. For such investors, shifts in risk aversion are positively correlated with market volatility, thereby exacerbating the original problem.

If the above factors are not driving the results, then only two possibilities remain. One is that the true cost of capital is subject to much higher volatility, and takes on more extreme values, than has previously been thought. While this is by no means impossible (and the swings depicted in Figure 1 appear qualitatively consistent with practitioners' ex-post assessment of time variation in the stock market's risk environment), movements of the kind described above would require a seismic change in mindset among managers and regulators used to dealing with a more-or-less constant cost of capital. The other possibility is that the simple product of [gamma] and [[sigma].sup.2.sub.m] fails to adequately capture the market price of risk. Or, to put it another way, the relationship between market risk and expected return is considerably more complex than envisaged by the CAPM. According to this interpretation, my results are simply another manifestation of the lack of evidence for the CAPM tradeoff between risk and expected return (e.g., French et al, 1987; Campbell and Hentschel, 1992).

4. Concluding Remarks

I interpret the results of this paper as bad news for practical applications of the CAPM, particularly those that require estimates of the current cost of capital rather than its long-run mean. The usual approach takes the market risk premium as given and uses this, along with other exogenous parameters, to estimate the risk premium on some other asset or project. However, this suffers from two drawbacks. First, the market risk premium is an endogenous variable in CAPM equilibrium, so applications that treat it as exogenous are effectively ignoring part of the CAPM. Second, because the market risk premium can only be accurately estimated using a long time series of data, the estimate used in applications is likely to reflect little of the current risk environment. The second problem can, in principle, be resolved by explicitly dealing with the first, a procedure that links the market risk premium to the variance of market returns. Unfortunately, although this approach seems to yield plausible results when measured over a long time period, annual estimates of the variance are extremely volatile, resulting in significant swings in the market risk premium. At the same time, the fact that market risk seems to vary through time at all is at odds with an approach that ignores the effect of risk on the market risk premium.

In short, the observed volatility in market risk casts doubt on the usual approach that implicitly assumes the market risk premium is a constant. But relaxing this assumption leads to the opposite problem: high variation in the market risk premium resulting in alternately high and low values of the cost of capital. Of course, some of this excessive variation is due to the extreme conditions of the 1980s, a period often associated with various forms of investor irrationality. This suggests the possibility that noise trader or other behavioural characteristics may vary over time in such a way so as to induce the pattern observed in Figure 1. However, even if this is true (a subject well beyond the scope of this paper), it need not resurrect the case for using the CAPM to estimate the cost of capital. First, in order to be able to apply the CAPM in these circumstances, one must deal with the practical difficulty of 'stripping out' the behavioural component from the estimate of variance. Second, as Stein (1996) points out, the CAPM may not be the appropriate model for estimating the cost of capital in the presence of behavioural investors, even if it describes 'fundamental' values perfectly.

To summarise, the conventional usage of the CAPM in applications seems to largely reflect a willingness to ignore both empirical reality (time-variation in market risk) and theoretical consistency (the implications of the CAPM for pricing market risk). At best, it seems able to generate realistic estimates of the unconditional risk premium, but this may bear little relationship to the current risk environment. Overall, the inability of the CAPM to capture the short-term relationship between risk and expected return in an empirically plausible manner must cast doubt on its suitability for cost of capital calculations. Its enduring popularity for this purpose may largely reflect the lack of an easily-understood, and viable, alternative.

References

Black, F., (1986). "Noise", Journal of Finance, 41: 529-543.

Black, F., (1993). "Estimating expected return", Financial Analysts Journal, 49: 36-38.

Bollerslev, T., (1986). "Generalized autoregressive conditional heteroskedasticity". Journal of Econometrics, 31: 307-327.

Boyle, G. and Young, L., (1992). "Risk aversion and stock market volatility". Journal of Macroeconomics, 14: 593-606.

Campbell, J. and Hentschel, L., (1992). "No news is good news: An asymmetric model of changing volatility in stock returns". Journal of Financial Economics, 31: 281-318.

Campbell, J, A. Lo, and Mackinlay, A., (1997). The Econometrics of Financial Markets. New Jersey: Princeton University Press.

Campbell, J. and Viceira, L., (2002). Strategic Asset Allocation. Oxford: Oxford University Press.

Chay, J., Marsden, A., and Stubbs, R., (1993). "Historical rates of return to equities, bonds, and the equity risk premium: New Zealand evidence". Pacific Accounting Review, 5: 27-46.

Claus, J. and Thomas, J., (2001). "Equity premia as low as three percent? Evidence from analysts' earnings forecasts for domestic and international stock markets". Journal of Finance, 56: 1629-1666.

Cochrane, J., (2001). Asset Pricing. Princeton: Princeton University Press.

De Long, J., Shleifer, A., Summers L., and Waldmann, R., (1990). "Noise trader risk in financial markets". Journal of Political Economy, 98: 703-738.

Dimson, E., Marsh, P., and Staunton, M., (2002). Triumph of the Optimists. Princeton: Princeton University Press.

Fama, E. and French, K, (2002). "The equity premium". Journal of Finance, 57: 637-659.

French, K., Schwert, G., and Stambaugh, R., (1987). "Expected stock returns and volatility". Journal of Financial Economics, 19: 3-29.

Friend, I. and Blume, M., (1975). "The demand for risky assets". American Economic Review, 65: 900-922.

Graham, J. and Harvey, C., (2001). "The theory and practice of corporate finance: evidence from the field". Journal of Financial Economics, 61: 187-243.

Huang, C. and Litzenberger, R., (1988). Foundations for Financial Economics. New York: Elsevier.

Lally, M., (2004). The weighted average cost of capital for gas pipeline businesses. NZ Commerce Commission.

Lally, M. and Marsden, A., (2004). "Tax-adjusted market risk premiums in New Zealand: 1931-2002". Pacific-Basin Finance Journal, 12: 291-310.

Lally, M., Roush, M., and van Zijl, T., (2004). "The market risk premium in New Zealand". INFINZ Journal, 1: 5-12.

Merton, R., (1980). "On estimating the expected return on the market". Journal of Financial Economics, 8: 323-361.

Officer, R., (1973). "The variability of the market factor of the New York stock exchange". Journal of Business, 46: 434-453.

Patterson, C., (1989). "Investment decision criteria used by listed New Zealand companies". Accounting and Finance, 29: 73-89.

Stein, J., (1996). "Rational capital budgeting in an irrational world". Journal of Business, 69: 429-455.

Tobin, J., (1958). "Liquidity preference as behavior towards risk". Review of Economic Studies, 65-86.

Glenn Boyle * ([dagger])

* NZ Institute for the Study of Competition and Regulation, Victoria University of Wellington, PO Box 600, Wellington. glenn.boyle@vuw.ac.nz fax 04 463 5560

([dagger]) I am grateful to Russell Investment Group Ltd for providing me with data used in this study. For helpful comments, I am indebted to two NZEP anonymous referees, Paul Hocking, Martin Lally, Leo Krippner, Abdullah Mamum, Alireza Tourani-Rad, and participants at NZ Finance Colloquium and ISCR seminars. Hanqing Wang provided invaluable research assistance. Despite all this assistance, the responsibility for any remaining errors or ambiguities is mine alone.

(1) For example, Graham and Harvey (2001) report that 73.5% of United States CFOs "always or almost always" use the CAPM for estimating the cost of equity capital. An older survey by Patterson (1989) based on a much smaller sample finds that 38% of NZ firms do the same. Examples of the NZ Commerce Commission's dependence on the CAPM can be found on its website (www.comcom.govt.nz).

(2) See. for example, Lally (2004).

(3) Somewhat loosely, I henceforth use 'cost of capital' as a shorthand convenience for 'cost of equity capital'.

(4) This assumes the riskless asset is in zero net supply. Nothing substantive in what follows is lost by allowing for a positive net supply.

(5) Of course, this result is dependent on the assumption of rational, utility-maximising investors. The presence of behavioural investors may drive a wedge between the true, fundamentals-based, risk premium and observed market volatility. I discuss this point further in the next section.

(6) This suggests the use of daily, rather than monthly, data in the analysis that follows. However, two considerations act against this. First, daily data are available for a much shorter period of time, thereby constraining the procedure outlined below that I use to estimate [gamma]. Second, the high degree of noise in daily data offsets the advantages of greater frequency. In unreported work, I use daily data from 1986 and obtain similar results to those appearing in this paper.

(7) This problem also applies to so-called forward-looking methods of estimating the market risk premium that require historical averages of dividend yields and growth; see, for example, Claus and Thomas (2001) and Fama and French (2002). As Fama and French point out, this approach is not well suited to estimating the conditional risk premium. Other forward-looking methods that utilise analyst forecasts do allow for time variation in the premium, but do not explicitly relate this variation to risk shifts.

(8) This procedure assumes, consistent with existing data, that both the risk premium and the variance change through time, but that their ratio is constant.

(9) Lally and Marsden (2004) report post-tax estimates of the unconditional risk premium and variance equal to 0.074 and 0.059 respectively. However, their calculation of the former is with respect to bonds rather than bills. For the countries examined in Dimson et al (2002), this understates the market premium by approximately one percentage point. Hence, I add back this difference and use 1/[gamma] = 0.084/0.059 = 1.4.

(10) The nominal returns data are obtained from the NZ Gross Index compiled by Russell Investment Group Ltd. I am grateful to Craig Ansley and Fiona Lintott for providing me with access to these data. Details of its construction can be found in Chay et al (1993). Nominal returns are deflated using movements in the CPI.

(11) This simple procedure is similar to that used by Officer (1973), Merton (1980), and others. Averaging the squared returns (rather than calculating the sample variance), as those authors do, has essentially no effect on the results. An alternative approach is to estimate a GARCH process, e.g., Bollerslev (1986). As the latter method yields virtually identical conclusions in this case, I report only the simple approach described above.

(12) Interestingly, this figure is very similar to the mean market risk premium of 6.5% reported in the survey evidence of Lally et al (2004).

(13) Although he does not discuss implications for cost of capital estimates, Merton's (1980) tables 4.7 and 4.8 indicate that similar volatility is present in US data. Thus, my results do not appear to be an artifact of NZ data. See also French et al (1987, Fig. 1a).

An important determinant of investment expenditure is the cost of the capital employed in undertaking any investment project. For firms considering a specific project, the cost of capital is the discount rate applied to all future cashflows that allows estimation of the project's profitability. For regulators such as the New Zealand Commerce Commission, the cost of capital effectively determinines the price path a regulated firm is required to follow and thus has significant implications for the firm's incentives to invest.

Unfortunately, the cost of the equity component of capital is not directly observable and hence must be estimated from a theoretical model, the most commonly used of which is the celebrated Capital Asset Pricing Model (CAPM). (1) The standard version of the CAPM has the familiar form:

E[[R.sub.i]] = [R.sub.f] + [[beta].sub.i] {E[[R.sub.m]] - [R.sub.f]} (1)

where [R.sub.i] is the random return on asset i, [R.sub.f] is the riskless rate of interest, [R.sub.m] is the random rate of return on the market portfolio of risky assets, [[beta].sub.i] is the asset i 'beta', equal to the covariance of [R.sub.i] and [R.sub.m] divided by the variance of [R.sub.m], and E[x] is the expectations operator. Equation (1) states that the asset i risk premium is proportional to the market risk premium where the factor of proportionality is equal to [[beta.sub.i]. In this formulation, [[beta].sub.i] is the quantity of asset i risk and the market risk premium {E[[R.sub.m]] - [R.sub.f]} is the price of that risk.

Equation (1) is a relative pricing model. Specifically, it relates one market price (the risk premium on asset i) to another market price (the risk premium on the market portfolio). But because the market portfolio includes all assets (including asset i), the second price, if correctly measured, incorporates the first price, so there is an element of circularity in this process. Practical applications of the CAPM, such as estimating the cost of capital, typically ignore this problem--on the grounds that any individual asset is an infinitesimally small portion of the market portfolio--and treat the market risk premium as a free parameter to be estimated from data. (2) But as Cochrane (2001) points out, this procedure ignores the CAPM predictions for the market portfolio itself; taking the market premium as given neglects its underlying dependence on more fundamental CAPM parameters.

The implications of this argument are explored in the remainder of the paper, but an intuitive overview may be helpful. Underlying the CAPM is the separation result from modern portfolio theory that all investors optimally allocate their funds between two portfolios, one risky and one riskless. Investor-specific risk attitudes determine the split between the two portfolios, but have no effect on the composition of the risky asset portfolio which depends only on the distribution of future asset returns. If all investors perceive the same returns distribution, they must then wish to hold the same portfolio of risky assets. Equilibrium in the market for these assets requires that this common portfolio be the so-called market portfolio, an observation that leads directly to equation (1). However, this process makes no explicit reference to equilibrium in the market for riskless assets. Incorporating this additional condition in the model places an exact restriction on the allowable value of the market risk premium in terms of the underlying variance of market returns. As a result, applications of the CAPM need only estimate the value of the latter parameter and not the market risk premium itself.

The link between the market risk premium and the returns variance is not in itself new. Assuming quadratic utility, or exponential utility with normal returns, (e.g., Friend and Blume, 1975; Huang and Litzenberger, 1988), or continuous trading opportunities (Merton, 1980), other authors have also shown that the market risk premium is proportional to the variance of market returns. However, I demonstrate that this result is a simple consequence of riskless asset equilibrium, for all preferences defined over the mean and variance of single-period wealth, and that additional assumptions about utility or the returns structure are unnecessary. Perhaps more importantly, I examine the implications of the result for practical applications such as estimating the cost of equity capital, an issue that none of the authors above, with the partial exception of Merton, considers.

Expressing the market risk premium as a function of the variance of market returns is potentially valuable for calculating the cost of capital. (3) Estimating the variance of returns is much easier than estimating the mean, particularly when these parameters vary through time, so the alternative approach holds out the promise of identifying risk-based shifts in the cost of capital. Unfortunately, I find that the estimated variance of returns in New Zealand data is extremely volatile; at times the implied cost of capital is implausibly high while at other times it is implausibly low. In my view, this is not good news for CAPM-based approaches to estimating the cost of capital: seemingly-reasonable estimates based on (1) are obtained only by ignoring an important part of CAPM content.

The next section explicitly derives the link between the market risk premium and the variance of market returns. In Section 3, I discuss how this might help obtain more accurate cost of capital estimates and apply this to data. The final section contains some concluding remarks.

2. CAPM and equilibrium in the market for riskless assets

If [[lambda].sub.ik] is expenditure on asset i by investor k = 1, ..., m, and [[lambda].sub.fk] is expenditure on the riskless asset, then end-of-period wealth [W.sub.k] satisfies:

[W.sub.k] = [n.summation over (i=1)] [[lambda].sub.ik] (1 + [R.sub.i]) + [[lambda].sub.fk] (1 + [R.sub.f]) (2)

The asset expenditures must sum to initial wealth [W.sub.0k]. Setting the latter to unity for convenience, (2) can be written as:

[W.sub.k] = 1 [R.sub.f] + [n.summation over (i=1)] [[lambda].sub.ik] ([R.sub.i] - [R.sub.f]) (3)

so that:

E[[W.sub.k]] [equivalent to] [[bar.W].sub.k] = 1 + [R.sub.f] + [n.summation over (i=1)] [[lambda].sub.ik] (E[[R.sub.i]] - [R.sub.f]) (4)

var([W.sub.k]) [equivalent to] [sigma] = [n.summation over (i=1)] [n.summation over (j=1)] [[lambda].sub.ik][[lambda].sub.jk] cov([R.sub.i], [R.sub.j]) (5)

Each investor has a utility function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that depends only on the mean and variance of end-of-period wealth, and chooses the portfolio that maximises the value of this function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first-order conditions for this problem are:

[[gamma].sub.k] (E[[R.sub.j]] - [R.sub.f]) = [n.summation over (j=1)] [[lambda].sub.jk] cov([R.sub.i], [R.sub.j]) i =1, ..., n (6)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is investor k's marginal rate of substituting risk for return. That is, 1/[[gamma].sub.k] [equivalent to] is the additional mean return that investor k would need to be no worse off following a marginal increase in variance. If this is high (low), then investor k is relatively intolerant (tolerant) of risk. We can therefore interpret 1/[gamma] as a measure of investor k's risk aversion.

Writing (6) in matrix form and rearranging yields the two-fund separation result of Tobin (1958), i.e., the composition of investor k's portfolio of risky assets is unaffected by risk attitudes ([[gamma].sub.k]) and depends only on perceived means, variances and covariances of returns. If these parameters are the same for all investors, then they all hold the same portfolio of risky assets. Market clearing then requires that this portfolio must include each asset in an amount equal to its weight in the portfolio of total invested wealth in risky assets, i.e., the market portfolio of risky assets. Then [[gamma].sub.jk] is equal to asset j's weight in this portfolio multiplied by investor k's total investment in risky assets (1-[[gamma].sub.fk]) and equation (6) can be rewritten as:

[[gamma].sub.k] (E[[R.sub.i]] - [R.sub.f]) = (1 - [[lambda].sub.fk])cov([R.sub.i], [R.sub.m]) i=1, ..., n (7)

where [R.sub.m] is the return on the market portfolio of risky assets.

As (7) holds for all risky assets, it must also apply to the market portfolio:

[[gamma].sub.m] (E[[R.sub.m]] - [R.sub.f]) = (1 - [[lambda].sub.fk])[[sigma].sup.2.sub.m] (8)

where [[sigma].sup.2.sub.m] is the variance of [R.sub.m]. Combining (7) and (8) then yields equation (1):

E[[R.sub.i]] = [R.sub.f] + [[beta].sub.i](E[[R.sub.m]] - [R.sub.f])

Equation (1) is, of course, the standard formulation of the CAPM, but it overlooks an important part of the underlying pricing process. Going from (6) to (7) requires that supply equal demand for each risky asset, but no corresponding requirement is imposed on the riskless asset.

To determine the implications of imposing riskless asset equilibrium, return to equation (8) and note that this can be written as

[[lambda].sub.fk] = 1 - (E[[R.sub.m]] - [R.sub.f])/[[sigma].sup.2.sub.m] / (1/[[gamma].sub.k]) (9)

which expresses investor k's demand for the riskless asset as a function of risk aversion

and market portfolio characteristics. Intuitively, (E[[R.sub.m]] - [R.sub.f]) / [[sigma].sup.2.sub.m] is the rate at which the market portfolio trades off risk and return while (1 / [[gamma].sub.k]) is the rate at which investor k is willing to make this tradeoff. If, for example, the former exceeds the latter, then investor k desires more risk than is offered by the market portfolio alone and thus borrows at rate [R.sub.f] to finance a larger holding of that portfolio, i.e., [[gamma].sub.fk] < 0.

Equilibrium in the riskless asset market requires that total borrowing equal total lending, i.e, [m.summation over (k=1)] [[lambda].sub.fk] = 0. (4) Applying this to (9) yields:

E[[R.sub.m]] - [R.sub.f] = [([m.summation over (k=1)] [[lambda].sub.k]/m).sup.-1] [[sigma].sup.2.sub.m] = (1/[gamma]) [[sigma].sup.2.sub.m] (10)

where [gamma] [alpha] [m.summation over (k=1)] [[gamma].sub.k] [alpha]/m is the average value of [[gamma].sub.k] i.e., 1/[gamma] is the average risk aversion of all investors. Thus, equation (10) states that riskless asset equilibrium constrains the market risk premium to a value equal to the product of market risk and market risk aversion.

To understand (10) intuitively, suppose that it is not satisfied, e.g., (E[[R.sub.m]] - [R.sub.f]) < (1/[gamma]) [[sigma].sup.2.sub.m]. Then the rate at which the market portfolio offers to trade off risk and return is below the rate required by investors to make this tradeoff. Investors will thus wish to substitute from the market portfolio to the riskless asset, i.e., there is excess demand for the riskless asset. With supplies fixed, equilibrium is re-established by a rise in (E[[R.sub.m]] - [R.sub.f]) until the excess demand is eliminated. This occurs only when the available risk-return tradeoff is equal to the required tradeoff, i.e., when (10) holds. More succinctly, any violation of (10) implies excess demand or supply in the riskless asset market, so equilibrium in that market requires that (10) be satisfied. In such an equilibrium, the market risk premium (E[[R.sub.m]] - [R.sub.f]) must equal the "market price of risk". ([[sigma].sup.2.sub.m]/ [gamma]). (5)

Substituting equation (10) into (1) yields:

E[[R.sub.i]] = [R.sub.f] + [[beta].sub.i] ([[sigma].sup.2.sub.m] / [gamma]) (11)

which relates the expected excess return on asset i to beta and the market price of risk.

The link between (1) and (11) is worth emphasizing. Equation (1) requires only that risky asset markets clear, and is thus a partial equilibrium statement that relates one endogenous price variable (E[[R.sub.i]] - [R.sub.f]) to another (E[[R.sub.m]] - [R.sub.f]). Equation (11) takes this insight a step further by requiring that supply also equal demand for the riskless asset, thereby allowing E[[R.sub.m]]-[R.sub.f] to be expressed in terms of underlying exogenous parameters and transforming (1) into a general equilibrium statement.

Friend and Blume (1975) and Merton (1980) also report an equation similar to (11), and Huang and Litzenberger (1988) explicitly derive it in the context of either quadratic utility or exponential utility with normal returns. However, none explicitly makes the link to riskless asset equilibrium, and only Merton recognises its implications for practical applications such as estimating the cost of capital. The latter issue is the subject of the remainder of this paper.

3. Estimating the cost of capital

Suppose one wishes to estimate a firm's cost of capital. The usual approach estimates the market risk premium E[[R.sub.m]]-[R.sub.f] as a free parameter and uses this in equation (1), along with estimates of [[beta].sub.i] and [R.sub.f]. However, as section 2 demonstrates, the market risk premium is not a free parameter; rather it is an endogenous function of underlying CAPM parameters, as described by (10). Thus, applications of the CAPM can, in principle, use either (1) or (11), estimating either the market risk premium (E[[R.sub.m]] - [R.sub.f]) or the market price of risk ([[sigma].sup.2.sub.m]/[gamma]). Although the latter approach is theoretically superior, the only relevant consideration in practical situations is the reliability of estimates. More precisely, is it better to estimate the market risk premium or the market price of risk?

There are two reasons to favour the latter approach. First, as discussed at length by Merton (1980), Black (1993) and Campbell et al (1997), it is much easier to estimate the variance of returns than it is to estimate expected returns. The essence of their argument is that the precision of the variance estimate increases with the number of observations while the precision of the expected return estimate increases only with the length of the data series. In other words, a good estimate of variance can be obtained even with a short time series so long as the data are sufficiently high-frequency, but the only way to get a similarly good estimate of the mean is to have a long time series. (6) Consequently, given the usual constraints on available data, variance estimates will be considerably more accurate than expected return estimates.

The second reason for favouring (11) follows from the first. In calculating the cost of capital, the relevant distribution of returns is the conditional distribution, since it is this that describes the current risk outlook: if risk is high at a particular date, then the market risk premium, and hence the cost of capital, should also be high at that date. More precisely, if one wishes to use equation (1) to estimate the cost of capital, then an estimate of the current market risk premium is required (i.e., the conditional mean of excess market returns); equation (11) requires, instead, an estimate of current risk (i.e., the conditional variance of market returns). In the case of (1) however, because of the long time period needed to estimate expected returns, the best one can feasibly do is obtain a single estimate of the unconditional market risk premium. Consequently, applications of (1) are unable to incorporate variation over time in the market risk premium and thus do not reflect the current risk environment. (7) By contrast, equation (11) is potentially able to identify this variation because of the shorter time series required for estimating variance.

Of course, there is also a significant disadvantage to using (11): the parameter [gamma] is unobservable. However, it may be possible to estimate this fairly accurately, so long as one is prepared to assume that it is constant (i.e., independent of wealth or consumption). This seems reasonable, at least as a first approximation; as Campbell and Viceria (2002) point out, there have been large increases in per capita consumption and wealth in the last 100 or so years, but no corresponding trends in risk premia or interest rates consistent with investors having changed their attitudes towards risk.

This assumption can be exploited by rearranging (10) to obtain 1/[gamma] as the ratio of the market risk premium and the variance of market returns and then following a two-step procedure. First, I use the unconditional version of (10) to estimate the constant 1/[gamma] applicable to a given market, i.e., as the ratio of the unconditional market risk premium and the unconditional market variance. Second, I multiply this parameter by the conditional variance and substitute into (11) to estimate the current cost of capital. (8) For example, suppose one obtains, from 100 years of data, unconditional estimates of {E[[R.sub.m]] - [R.sub.f]} and [[sigma].sup.2.sub.m], equal to 0.06 and 0.03 respectively. Then the implied value of 1/[gamma] is two and, from (11), the current (date t) expected return on asset i is [R.sub.ft] + 2([sigma])[[beta].sub.it], where the t subscripts denote date t values.

To provide a concrete illustration and assessment of this process, I use equation (10) to estimate an annual series for the New Zealand market price of risk over the last 30 years. This requires, first, estimation of 1/[gamma] as described above, and, second, annual estimates of [[sigma].sup.2.sub.mt], To calculate 1/[gamma], I use the recent study of Lally and Marsden (2004) on NZ returns during the 1931-2002 period. Their estimates of the unconditional values of {E[[R.sub.m]] - [R.sub.f]} and [[sigma].sup.2.sub.m] mply a value of 1/[gamma] equal to 1.4. (9)

Turning to the conditional value of [sigma], I use monthly real stock returns on the NZ stockmarket since 1967 to calculate a moving average variance of returns for the 34 years from January 1970 to December 2003. (10) Specifically, for each month during this period, I calculate the sample variance of returns over the previous 36 months. These monthly variances are first multiplied by twelve and then averaged across the 12 months of each calendar year to obtain a single estimate of the conditional variance for each year. (11)

Combining these estimates of [[sigma].sup.2.sub.mt] with 1/[gamma] = 1.4 gives a time series of annual estimates of the market price of risk, and the results from this procedure are outlined in Figure 1 and Table 1. The former depicts the year-by-year variation in the market price of risk estimate; the latter summarises these data for both the full period and three subperiods.

[FIGURE 1 OMITTED]

The primary impression from these results is one of considerable volatility in the variance of market returns. For the period as a whole, the average market price of risk is a reasonable-sounding 6.4%, but this hides significant intra-period variation. (12) Throughout the 1970s, the maximum value was 3.7% and the average only 2.1%. In the 1980s, the price of risk ranged from less than 1% to more than 33%; the period since 1990 is similarly volatile. Although the most extreme values occur in the 1980s, the remaining periods are also characterized by both high and low values. For the most recent year in the sample (2003), the estimated market price of risk is 2.3%, implying a very low cost of capital for most projects.

If one wishes to use the CAPM to capture time variation in the market price of risk, these results are thought-provoking. For instance, is it really plausible that the price of risk went from less than 1% in the early part of the 1980s to more than 30% by the end? Answering in the affirmative implies an acceptance of very large swings in the cost of capital. Similarly, does it seem reasonable that the average-risk firm (i.e., [[beta].sub.i] = 1) in 2003 had a cost of equity only 2.3 percentage points above the riskless rate of interest?

Measurement error in the two crucial variables - [gamma] and [[sigma].sup.2.sub.m] - seems unlikely to resolve these problems. Any error in the estimate of y changes the market price of risk at each date by a scalar, and thus has no effect on the volatility depicted in Figure 1. Moreover, the estimated volatility in [[sigma].sup.2.sub.m] is unaffected by the use of alternative estimation periods, higher-frequency data, or more sophisticated estimation methods. (13) Certainly, estimates of variance contain estimation error and are thus likely to be more volatile than true values, but while this potentially rescues the theoretical validity of the CAPM pricing process, it simply re-states the practical problems encountered when trying to use the CAPM to capture time variation in the cost of capital.

A more interesting possibility is that, contrary to the assumption maintained above, 1/[gamma] is not constant, but instead covaries negatively with market risk. In this case, high realisations of [[sigma].sup.2.sub.m] would be offset by low values of 1/[gamma] and the market price of risk series would be considerably more stable than it appears in Figure 1. One way this might occur is if the presence of noise traders (see, for example, Black, 1986; De Long et al, 1990) in the market waxes and wanes with the extent of their over-optimism, e.g., as bullishness rises, more noise traders enter the market and increase market volatility. However, such an effect would have to dominate the possibly countervailing influence of intertemporal considerations for other investors. As Boyle and Young (1992) point out, rational risk-averse investors respond to consumption shocks by altering their demand for, and hence the price of, equities; higher risk aversion elicits a greater demand response and more dispersed equity prices across possible states of the world. For such investors, shifts in risk aversion are positively correlated with market volatility, thereby exacerbating the original problem.

If the above factors are not driving the results, then only two possibilities remain. One is that the true cost of capital is subject to much higher volatility, and takes on more extreme values, than has previously been thought. While this is by no means impossible (and the swings depicted in Figure 1 appear qualitatively consistent with practitioners' ex-post assessment of time variation in the stock market's risk environment), movements of the kind described above would require a seismic change in mindset among managers and regulators used to dealing with a more-or-less constant cost of capital. The other possibility is that the simple product of [gamma] and [[sigma].sup.2.sub.m] fails to adequately capture the market price of risk. Or, to put it another way, the relationship between market risk and expected return is considerably more complex than envisaged by the CAPM. According to this interpretation, my results are simply another manifestation of the lack of evidence for the CAPM tradeoff between risk and expected return (e.g., French et al, 1987; Campbell and Hentschel, 1992).

4. Concluding Remarks

I interpret the results of this paper as bad news for practical applications of the CAPM, particularly those that require estimates of the current cost of capital rather than its long-run mean. The usual approach takes the market risk premium as given and uses this, along with other exogenous parameters, to estimate the risk premium on some other asset or project. However, this suffers from two drawbacks. First, the market risk premium is an endogenous variable in CAPM equilibrium, so applications that treat it as exogenous are effectively ignoring part of the CAPM. Second, because the market risk premium can only be accurately estimated using a long time series of data, the estimate used in applications is likely to reflect little of the current risk environment. The second problem can, in principle, be resolved by explicitly dealing with the first, a procedure that links the market risk premium to the variance of market returns. Unfortunately, although this approach seems to yield plausible results when measured over a long time period, annual estimates of the variance are extremely volatile, resulting in significant swings in the market risk premium. At the same time, the fact that market risk seems to vary through time at all is at odds with an approach that ignores the effect of risk on the market risk premium.

In short, the observed volatility in market risk casts doubt on the usual approach that implicitly assumes the market risk premium is a constant. But relaxing this assumption leads to the opposite problem: high variation in the market risk premium resulting in alternately high and low values of the cost of capital. Of course, some of this excessive variation is due to the extreme conditions of the 1980s, a period often associated with various forms of investor irrationality. This suggests the possibility that noise trader or other behavioural characteristics may vary over time in such a way so as to induce the pattern observed in Figure 1. However, even if this is true (a subject well beyond the scope of this paper), it need not resurrect the case for using the CAPM to estimate the cost of capital. First, in order to be able to apply the CAPM in these circumstances, one must deal with the practical difficulty of 'stripping out' the behavioural component from the estimate of variance. Second, as Stein (1996) points out, the CAPM may not be the appropriate model for estimating the cost of capital in the presence of behavioural investors, even if it describes 'fundamental' values perfectly.

To summarise, the conventional usage of the CAPM in applications seems to largely reflect a willingness to ignore both empirical reality (time-variation in market risk) and theoretical consistency (the implications of the CAPM for pricing market risk). At best, it seems able to generate realistic estimates of the unconditional risk premium, but this may bear little relationship to the current risk environment. Overall, the inability of the CAPM to capture the short-term relationship between risk and expected return in an empirically plausible manner must cast doubt on its suitability for cost of capital calculations. Its enduring popularity for this purpose may largely reflect the lack of an easily-understood, and viable, alternative.

References

Black, F., (1986). "Noise", Journal of Finance, 41: 529-543.

Black, F., (1993). "Estimating expected return", Financial Analysts Journal, 49: 36-38.

Bollerslev, T., (1986). "Generalized autoregressive conditional heteroskedasticity". Journal of Econometrics, 31: 307-327.

Boyle, G. and Young, L., (1992). "Risk aversion and stock market volatility". Journal of Macroeconomics, 14: 593-606.

Campbell, J. and Hentschel, L., (1992). "No news is good news: An asymmetric model of changing volatility in stock returns". Journal of Financial Economics, 31: 281-318.

Campbell, J, A. Lo, and Mackinlay, A., (1997). The Econometrics of Financial Markets. New Jersey: Princeton University Press.

Campbell, J. and Viceira, L., (2002). Strategic Asset Allocation. Oxford: Oxford University Press.

Chay, J., Marsden, A., and Stubbs, R., (1993). "Historical rates of return to equities, bonds, and the equity risk premium: New Zealand evidence". Pacific Accounting Review, 5: 27-46.

Claus, J. and Thomas, J., (2001). "Equity premia as low as three percent? Evidence from analysts' earnings forecasts for domestic and international stock markets". Journal of Finance, 56: 1629-1666.

Cochrane, J., (2001). Asset Pricing. Princeton: Princeton University Press.

De Long, J., Shleifer, A., Summers L., and Waldmann, R., (1990). "Noise trader risk in financial markets". Journal of Political Economy, 98: 703-738.

Dimson, E., Marsh, P., and Staunton, M., (2002). Triumph of the Optimists. Princeton: Princeton University Press.

Fama, E. and French, K, (2002). "The equity premium". Journal of Finance, 57: 637-659.

French, K., Schwert, G., and Stambaugh, R., (1987). "Expected stock returns and volatility". Journal of Financial Economics, 19: 3-29.

Friend, I. and Blume, M., (1975). "The demand for risky assets". American Economic Review, 65: 900-922.

Graham, J. and Harvey, C., (2001). "The theory and practice of corporate finance: evidence from the field". Journal of Financial Economics, 61: 187-243.

Huang, C. and Litzenberger, R., (1988). Foundations for Financial Economics. New York: Elsevier.

Lally, M., (2004). The weighted average cost of capital for gas pipeline businesses. NZ Commerce Commission.

Lally, M. and Marsden, A., (2004). "Tax-adjusted market risk premiums in New Zealand: 1931-2002". Pacific-Basin Finance Journal, 12: 291-310.

Lally, M., Roush, M., and van Zijl, T., (2004). "The market risk premium in New Zealand". INFINZ Journal, 1: 5-12.

Merton, R., (1980). "On estimating the expected return on the market". Journal of Financial Economics, 8: 323-361.

Officer, R., (1973). "The variability of the market factor of the New York stock exchange". Journal of Business, 46: 434-453.

Patterson, C., (1989). "Investment decision criteria used by listed New Zealand companies". Accounting and Finance, 29: 73-89.

Stein, J., (1996). "Rational capital budgeting in an irrational world". Journal of Business, 69: 429-455.

Tobin, J., (1958). "Liquidity preference as behavior towards risk". Review of Economic Studies, 65-86.

Glenn Boyle * ([dagger])

* NZ Institute for the Study of Competition and Regulation, Victoria University of Wellington, PO Box 600, Wellington. glenn.boyle@vuw.ac.nz fax 04 463 5560

([dagger]) I am grateful to Russell Investment Group Ltd for providing me with data used in this study. For helpful comments, I am indebted to two NZEP anonymous referees, Paul Hocking, Martin Lally, Leo Krippner, Abdullah Mamum, Alireza Tourani-Rad, and participants at NZ Finance Colloquium and ISCR seminars. Hanqing Wang provided invaluable research assistance. Despite all this assistance, the responsibility for any remaining errors or ambiguities is mine alone.

(1) For example, Graham and Harvey (2001) report that 73.5% of United States CFOs "always or almost always" use the CAPM for estimating the cost of equity capital. An older survey by Patterson (1989) based on a much smaller sample finds that 38% of NZ firms do the same. Examples of the NZ Commerce Commission's dependence on the CAPM can be found on its website (www.comcom.govt.nz).

(2) See. for example, Lally (2004).

(3) Somewhat loosely, I henceforth use 'cost of capital' as a shorthand convenience for 'cost of equity capital'.

(4) This assumes the riskless asset is in zero net supply. Nothing substantive in what follows is lost by allowing for a positive net supply.

(5) Of course, this result is dependent on the assumption of rational, utility-maximising investors. The presence of behavioural investors may drive a wedge between the true, fundamentals-based, risk premium and observed market volatility. I discuss this point further in the next section.

(6) This suggests the use of daily, rather than monthly, data in the analysis that follows. However, two considerations act against this. First, daily data are available for a much shorter period of time, thereby constraining the procedure outlined below that I use to estimate [gamma]. Second, the high degree of noise in daily data offsets the advantages of greater frequency. In unreported work, I use daily data from 1986 and obtain similar results to those appearing in this paper.

(7) This problem also applies to so-called forward-looking methods of estimating the market risk premium that require historical averages of dividend yields and growth; see, for example, Claus and Thomas (2001) and Fama and French (2002). As Fama and French point out, this approach is not well suited to estimating the conditional risk premium. Other forward-looking methods that utilise analyst forecasts do allow for time variation in the premium, but do not explicitly relate this variation to risk shifts.

(8) This procedure assumes, consistent with existing data, that both the risk premium and the variance change through time, but that their ratio is constant.

(9) Lally and Marsden (2004) report post-tax estimates of the unconditional risk premium and variance equal to 0.074 and 0.059 respectively. However, their calculation of the former is with respect to bonds rather than bills. For the countries examined in Dimson et al (2002), this understates the market premium by approximately one percentage point. Hence, I add back this difference and use 1/[gamma] = 0.084/0.059 = 1.4.

(10) The nominal returns data are obtained from the NZ Gross Index compiled by Russell Investment Group Ltd. I am grateful to Craig Ansley and Fiona Lintott for providing me with access to these data. Details of its construction can be found in Chay et al (1993). Nominal returns are deflated using movements in the CPI.

(11) This simple procedure is similar to that used by Officer (1973), Merton (1980), and others. Averaging the squared returns (rather than calculating the sample variance), as those authors do, has essentially no effect on the results. An alternative approach is to estimate a GARCH process, e.g., Bollerslev (1986). As the latter method yields virtually identical conclusions in this case, I report only the simple approach described above.

(12) Interestingly, this figure is very similar to the mean market risk premium of 6.5% reported in the survey evidence of Lally et al (2004).

(13) Although he does not discuss implications for cost of capital estimates, Merton's (1980) tables 4.7 and 4.8 indicate that similar volatility is present in US data. Thus, my results do not appear to be an artifact of NZ data. See also French et al (1987, Fig. 1a).

Table 1 Time-Variation in the Market Price of Risk: Summary Statistics. This table calculates the average, maximum and minimum values of the market price of risk series appearing in Figure 1. Average Maximum Minimum Full Sample 0.064 0.336 0.009 1970-2003 Sub-Samples 0.021 0.037 0.011 1970-79 1980-89 0.096 0.336 0.009 1990-2003 0.071 0.327 0.020

Printer friendly Cite/link Email Feedback | |

Author: | Boyle, Glenn |
---|---|

Publication: | New Zealand Economic Papers |

Geographic Code: | 8NEWZ |

Date: | Dec 1, 2005 |

Words: | 5663 |

Previous Article: | Efficiency of New Zealand Universities and the impact of the Performance Based Research Fund. |

Next Article: | Agency theory and trust ownership of shares. |

Topics: |