# ON THE RISK ADJUSTED DISCOUNT RATE FOR DETERMINING LIFE OFFICE APPRAISAL VALUES BY M. SHERRIS B.A., M.B.A., F.I.A., F.I.A.A. 1.

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2 582 On the Risk Adjusted Discount Rate where its present value is V and the risk adjusted discount rate is E(Rx), given by, and Rf is the risk free rate, E(RM) is the expected rate of return on all assets in the economy and ß is equal to covariance (X, RM) = covariance (Rx, RM) V x variance (RM) variance (RM) The risk free asset is an asset whose return is not affected by economic conditions so that its beta is zero. 2.4 In words this result says that we discount the expected value of the cash flow at a risk adjusted discount rate which is determined as the risk free rate plus a margin for risk. The margin for risk is related to the covariance of the cash flow with the returns on all assets in the economy. In fact, the adjustment is the product of the difference between the expected return on all assets of the economy and the risk free rate, often referred to as the market risk premium, and a measure of covariability with market returns, the beta of the cash flow. 2.5 It should be noted that in the one period case the relationship between rate of return and cash flow is 1 + Rx = X/V where the cash flow X is a random variable. This result allows the CAPM rate of return result to be expressed in cash flow form as above. 2.6 Some of the assumptions of the model are restrictive when it comes to practical application of the results, amongst these is the assumption that we live in a one period world. However, the model can be extended to a multi-period setting. 2.7 The paper by Fama (1977) examines the use of the CAPM in the valuation of multi-period cash flows. His results were: (a) the current market value of any future net cash flow is the current expected value of the flow discounted at risk adjusted discount rates for each of the periods until the flow is realized; (b) the discount rates for the periods preceding the realization of a cash flow need not be the same in each period and the rates for any given period can differ across cash flows; (c) the risk adjustments in the discount rates arise from uncertainties about reassessments of the expected value of a cash flow and the relationship between these reassessments and the corresponding reassessments of the expected cash flows of all assets in the economy. 2.8 Fama s results come from applying the one period CAPM result recursively to get the current value, at time 0, of a single cash flow in t periods time. In symbols:

3 for Determining Life Office Appraisal Values 583 where Ek [Xt] is the expected value of the cash flow at time k which is reassessed over time and is therefore not necessarily constant over time, and E(Rk) is the risk adjusted discount rate for time period k given by: and ßk = covariance variance (Rk, RMk) (RMk) It should be noted that the risk free rates and the beta values can vary in each period, and for different cash flows, but that the application of the CAPM to the multi-period case assumes that they are non-stochastic through time. 2.9 Another assumption of the CAPM that can be restrictive in practical applications is the allowance for taxation. The initial development of the CAPM ignored taxation. However, because the CAPM is based on the actual returns realized by investors it is effectively based on after tax returns when we allow for the existence of taxes in the model. The taxes to be allowed for are both company and personal taxes as investors will receive their returns net of both these taxes. Bowers and Ball (1987) discuss the adjustments to the CAPM to allow for taxes Under the Classical tax system equity returns are subject to taxation as profit at the company tax rate and as dividends in the hands of investors at the personal tax rate. Under an imputation system, as exists in the United Kingdom and as has recently been introduced in Australia, this double taxation of dividends is removed. Hence the adjustment to the CAPM for taxes will depend on the tax regime in the country where the asset is taxed If we assume that the company tax rate is Tc, the personal tax rate for interest income is Td and the personal tax rate for equity returns is Te, then Bowers and Ball give the risk adjusted interest rate as: where the returns Rx, RM and Rf are the actual returns observed in the market. Hence Rx and RM are net of company tax. The tax rates used are the effective rates applying to the investor and are adjusted to allow for the effect of financial intermediaries, such as life offices and superannuation funds, on these rates. These tax rates are those that apply to the marginal investor who is assumed to set the risk adjusted rate required. The beta value is defined as previously since the effect of tax is assumed to affect cash flows and total market returns in an identical way Under an imputation system the assumption could be made that Te is zero since this is the aim of such a system. The risk adjusted interest rate would then simplify further.

5 for Determining Life Office Appraisal Values 585 outgoings, death claim payments, surrender payments and changes in reserves. If we consider the cash flows at a particular point of time and let: NCF be the net cash flow, P be premium income, E be expense outgoings, D be death claim payments, S be surrender payments, and R be the change in reserves, we have NCF= P E D S R and V = E(NCF) 1 + E(Rx) where E(Rx) is the risk adjusted discount rate for the net cash flow given by with 3.5 We also have that where with and with similar definitions for Ve, Vd, Vs, Vr. 3.6 From the definition of ß we have that Hence ß for the net cash flow is a weighted average of the ß s of the component cash flows using the ratio of the component present values to the net cash flow present value as weights. It can be demonstrated that, if the weighted average ß is used to obtain the risk adjusted discount rate for the net cash flow, then the

9 for Determining Life Office Appraisal Values 589 (v) the tax regime of the country in which the cash flows occur will affect the rate and, hence risk adjusted rates will be country specific. 5.2 This paper will hopefully allow actuaries to adopt a scientific approach to the calculation of appraisal values, at least in the determining of risk adjusted discount rates. REFERENCES BOWERS, J. & BALL, R. (1987) The Cost of Equity Capital Under the New Imputation Tax System. Draft. Australian Graduate School of Management. BUNCH, T. S. (1986) The Valuation of With-Profits Business and the Estate. Paper presented to the Staple Inn Actuarial Society on 2nd December BURROWS, R. P. & WHITEHEAD, G. H. (1987) The Determination of Life Office Appraisal Values. J.I.A., 114, 411. COPELAND, T. E. & WESTON, F. J. (1983) Financial Theory and Corporate Policy, (2nd Edition). Addison-Wesley. FAMA, E. F. (1977) Risk-Adjusted Discount Rates and Capital Budgeting Under Uncertainty. Journal of Financial Economics. 5, LINTNER, J. (1965). The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Review of Economics and Statistics. Vol XLVII, No 1, ROBICHEK, A. A. & MYERS, S. C. (1966) Conceptual Problems in the Use of Risk Adjusted Discount Rates. Journal of Finance SHARPE, W. F. (1964) Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance. Vol XIX, No 3,

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