A General Approach to Real Option Valuation with ... - ICMA Centre

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ICMA Centre Discussion Papers in Finance DP2012-04once per year and the last decision is taken at the fourth year. So that the four decision treesare nested the values considered for k are 12, 6, 3 and 1 representing decision opportunitiesonce per year and once every 6 months, 3 months and 1 month.The upper part of Table 2 reports the value of the option to invest at market price andthe lower part reports the value of the option to invest at a fixed cost K = $1m, for investorswith different levels of risk tolerance λ. Again the option value increases very rapidly with λ.As expected, it also increases when more decision nodes occur in the tree, i.e. as k decreases,and the percentage increase in option value is greatest for investors with low λ, whereas theabsolute increase in option value is greatest for high λ.4.4 The HARA Utility ClassIn all HARA utilities the local absolute risk tolerance coefficient is λw 0 , whereas it is justλ for the exponential utility. Thus, HARA utility values with w 0 = 1 are comparable withexponential utility values, since the initial absolute risk tolerance is identical. Setting w 0 =1,Figure 3 graphs the exponential, logarithmic, hyperbolic and power utility real option valuesas a function of λ, with01, hyperbolic utility values still lie between theCopyright c○ 2012 Alexander and Chen 22
ICMA Centre Discussion Papers in Finance DP2012-04exponential and logarithmic values, but the power utility values exceed even the logarithmicvalues, and as λ increases further the power values can become very large indeed, because therisk tolerance increases too rapidly with wealth.However, HARA utilities are not always well-behaved. For instance, when w1 the power utility (which has η = λ)has a risk tolerance that is much too sensitive to changes in wealth.Exponential utility values are very stable; indeed they can yield analytic solutions inmany cases. However, they have the CARA property which is widely regarded as beingunrealistic. Indeed, exponential utility values for real options are typically too low becausethey ignore the possibility that the decision maker’s risk tolerance could increase with wealth.If it does, the real option’s value will always be greater than it is for an investor with constant(CARA) risk aversion.We conclude that, of the four HARA utilities considered, power utilities produce themost reliable real option values for most reasonable values of risk tolerance, i.e. for 0 ≤λ ≤ 1, but for decision makers with very high risk tolerance logarithmic or hyperbolic utilityrepresentations would be more appropriate, the former giving real option values that aregreater than the latter.4.5 Relative Cost of the AssetHere we examine how the current price of the asset affects terminal wealth if the asset ispurchased now or in the future. When the price of the asset is high relative to initial wealththe terminal P&L becomes relatively more variable and this could increase or reduce theoption’s value, depending on the way that risk tolerance changes with wealth. The P&Larising from a low-cost asset would hardly change over the investment horizon, but it couldchange a lot for a high-cost asset with the same volatility. Thus, different optimal decisionsand option values may be obtained for assets with a current price that is low or high, relativeto initial wealth, even when the returns on the two assets have the same mean and variance.We set the asset price at time 0 to be p 0 =0.1, 1 and 10 and assume the investor’s initialwealth is w 0 = 1 so that the investments have different sizes relative to the initial wealth ofCopyright c○ 2012 Alexander and Chen 23
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