4  Models for calculating the social opportunity cost

While CAPM currently dominates the other models, the other models are continually challenging this dominance. Lally (2000 p. 47) concluded that:

“All versions of the CAPM along with APT suffer from considerable ambiguity in empirical testing. However, parameter estimation problems appear to be considerably less for the CAPM than for APT, multi-factor models (such as Fama-French) and the dividend growth model. These considerations do not favour any alternative to the CAPM, and this is consistent with the CAPM’s dominance in practice.”

In addition, it is useful to understand why the CAPM is the preferred choice. Therefore this section briefly covers the alternative models.[3]

4.1  Capital Asset Pricing Model

CAPM is concerned with the way different investments move in relation to the market. The expected return derived using CAPM assumes all risks that can be removed by diversification are done so. This means the resulting expected return include only an allowance for the risk that cannot be removed by diversification.

The CAPM approach gives an expected return equal to the risk-free return (tax adjusted) plus a market related risk premium. This risk premium is based on how the security or investment moves in relation to the market. The way the security or investment moves in relation to the market is the β_e. The difference between the expected return on the market (R_m) and the after tax risk free rate (R_f(1-T_c)) is the after tax market risk premium (R_m – (1-T_c)R_f). The Equity beta (β_e) and the market risk premium are multiplied together to get the additional return for systematic risk. T_c is the corporate tax rate. The variables used in the capital asset pricing model are explained in more detail in section 6. As a formula this is:

(6)    k_e = R_f (1 - T_c) + [R_m – R_f(1-T_c)] β_e

When taking into account that the government does not pay corporate income tax, the formula becomes the following:

(7)     K_e = [R_f (1 - T_c) + (R_m – R_f(1-T_c)) β_e] / (1 - T_c)

The Capital Asset Pricing Model is widely used in the private and public sector.

This model is based on a choice of securities from the efficient set. This means that if there are two securities with the same expected return but different standard deviations, only the one with the smaller standard deviation is considered because the variance is taken into account in the market beta used.

The Roll (1977) critique implies that caution should be used in interpreting the results from testing CAPM rather than that the theory is invalid. Copeland and Weston (1992 p. 219) comment that:

“In fact, the only way to test the CAPM directly is to see whether or not the true market portfolio is ex post efficient. Unfortunately, because the market portfolio contains all assets (marketable and nonmarketable, e.g., human capital, coins, houses, bonds, stocks, options, land, etc.), it is impossible to observe.”

There is going to be some bias or error in the estimation but the direction is unclear. The estimation of the various variables in the calculations adds to any problems there might be in the underlying model. For this reason, it is important to use sensitivity analysis when working through any analysis.

4.2  Arbitrage Pricing Theory

Arbitrage Pricing Theory, developed by Ross (1976), is an alternative model to CAPM. This is an equilibrium-pricing model. The APT assumes that returns on securities are generated by a number of industry-wide and market-wide factors. The APT is a factor model enabling the use of multiple factors to explain or determine the expected return.

The APT theory does not specify exactly which factors determine the expected return or how many should be used. The application of the theory could include the use of the GDP or GNP, inflation or interest rates.

CAPM is a special case of the APT where the only factor is the market risk premium and how the security or investment moves in relation to the market.

The Arbitrage Pricing Theory could be of help, but it is unclear what factors would be appropriate to use in this case. This means that any results from using this approach are open to question more than by using CAPM.

4.3  Fama and French’s multi-factor models

Fama and French (1993) have developed several multifactor models designed to predict the expected return of particular market investments. They are like the APT in that they are factor models. However, they are more specific about which factors to use. The Fama and French (1993) multi-factor model uses five factors to explain average market performance of particular stocks. There are three stock market related factors (overall market performance, firm size, and book-to-market equity) and two-bond market related factors (default risk and affect of unexpected changes in interest rates).

However, the present situation is not concerned with a true market investment, so the model is not appropriate. For example it is impossible to derive a book value of equity relative to the share value of equity for the government, as government equities (share values) are not listed in the market (except Air New Zealand).

The benefit of a tax system with few investment distortions in particular industries, if any, is the ability to assume that the return on this project is the same as the return of the project being displaced.

Overall using the capital asset pricing model and the weighted average cost of capital is a robust method for determining a social opportunity cost. This means that even though the social rate of time preference is the preferable discount rate in certain circumstances, the lack of a robust estimate means that at a minimum the social opportunity cost should be used in all cases discussed.

In addition, the use of the risk-free interest rate in the capital asset pricing model brings in an element of the social rate of time preference to the estimation of the social opportunity cost. The social rate of time preference and social opportunity cost approaches could yield similar results if a 100% debt-funding scenario were considered.