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Risk-Based Valuation
Statement of the problem 2 The CAPM-based capital budgeting theory does not work for capital- constrained firms Idiosyncratic risk may be costly Cash flows in different periods may be correlated (this matters if idiosyncratic risk matters) Benchmarks and comparables should be used when available Using forwards in a CAPM context can be challenging Options and other nonlinear relationships are difficult to include Some CAPM parameters are unknown (e.g. correlation between project and market return ) Project data normally occur in prices and levels, not returns Firms lack an integrated and consistent framework for valuing projects in capital-constrained environments. This presentation uses simulation as a unifying framework to achieve this objective
Simplest case: Obtaining the CAPM
4 One-year project Simulated levels of cash flow (C 1 ) and market index (M 1 ) Regress C 1 on M 1 C 1 = [C – LM] + LM 1 + (note L is in levels not returns; L=cov(C 1 ,M 1 )/var(M 1 )) Discount risk-free and market-correlated cash flows assuming residual risk is unpriced V 0 = [C – LM]/(1+rf) + LM 0 + 0 Replication pricing V 0 = [C – L{M – M 0 (1+rf)}]/(1 + rf) Risk-neutral pricing V 0 = C/(1+rf) – L{M/(1+rf) – M 0 } Time-neutral pricing Substitute L = V 0 /M 0 , C 1 = V 0 (1+rV), M 1 =M 0 (1+rM) Convert levels to returns E(rV) = rf + (E(rM) – rf) CAPM expected return eq. V 0 = C/(1+E(rV)) CAPM valuation
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Valuing a one-year oil project using forwards 5 W=WTI (West Texas Intermediate Crude Oil) Simulated levels of cash flow (C 1 ) and oil prices (W 1 ) Cash flow depends on revenues and costs, both of which are functions of oil prices Regress C 1 on W 1 C 1 = [C – LW] + LW 1 + Discount risk-free and oil-correlated cash flows assuming residual risk is unpriced and F W = forward price of oil V 0 = [C – LW]/(1+rf) + LFW/(1+rf) Replication V 0 = [C – L{W – FW}]/(1+rf) Risk-neutral Equivalent to time-neutral valuation if FWFW/(1+rf) Q: What if the relationship between C and W is nonlinear?
Adapting a general valuation equation 7 Every asset satisfies the general valuation equation GVE Expected return = Required return {Exp capital gain} + Exp cash flow = Cash opportunity cost + Risk compensation {E[Vt+1 ] – Vt}+ E[Ct+1] = r Vt + k Rt+1 at all times t In the CAPM example presented earlier, the risk compensation simplifies to k Rt+1 = (E(rM) – rf) V 0 = L[M – (1+rf)M 0 ] If we move the cost of risk (kR t+ ) to the left side of the GVE equation, we obtain risk-neutrality The time-neutral transformation of cash flows is achieved by discounting all the cash flows and risk measures at the riskless rate and then using an effective riskless rate of 0. Test: Discount cash flows and risk measures at the riskless rate in the GVE and apply a zero discount rate E[Vt+1 ]/(1+r) – Vt + E[Ct+1]/(1+r) = 0 + k Rt+1/(1+r) Multiplying by (1+r) and rearranging terms, this produces the original GVE
Pricing idiosyncratic risk 8 Normally distributed cash flow C 1 in one year ( C , C ) Apply the GVE: {C – V 0 } + 0 = rfV 0 + kC V 0 = [C – kC] /(1 + rf) Expected return equation E(rV) = rf + kC/V 0 Same cash flow, but now correlated with the market Apply the GVE: {C – V 0 } + 0 = rfV 0 + L[M – (1+rf)M 0 ] + k V 0 = [C – L[M – (1+rf)M 0 ] – k] /(1 + rf) The expected return equation E(rV) = rf + [E(rM) – rf] + k/V 0
Properties of these models 10 Idiosyncratic risk matters hedging adds value Correlations between cash flow periods matter Ordering of cash flows matters Values are non-additive a negative NPV incremental project can add value Easy to add market factors (multifactor risk) Easy to include option-like payoffs
Determining the private cost of risk (k) 11 k is a measure of the adverse impact caused by increased risk If an agent accepts a contract or purchases an asset, the incremental risk will generally Add to the risk of the agent’s cash flow Increase the risk of declines in future wealth Increase the likelihood of financial distress or bankruptcy The value of k is chosen on the margin so the agent is compensated for the cost to his income statement or balance sheet. Example Suppose each additional $100,000 of risk increases the likelihood of financial distress by 5%, and the cost of financial distress is $250,000. In this case k = expected loss per dollar of risk = (5% of $250,000)/100,000 = 12.5%. Most financial institutions have determined an explicit cost of risk which they use in their valuations of financial assets and contracts.
