Page 196 is attached On page 176, we have an expression maximizes E[W] - bVar(W)/2. If you use the utilityfunction as U(x) = 1 - e-b², will this expression remain the same? If yes, indicate so; if no,give the analogous expression. 176 Valuing by Expected Utilitylimit theorem, a reasonable approximation. (It would also be exactlytrue if the X₁, i = 1, ..., n, have what is known as a multivariate normaldistribution.)Suppose now that the investor has an exponential utility functionU(x) = 1- e-bx, b>0,and so the utility function is concave. If Z is a normal random variable,then ez is lognormal and has expected valueE[e²] = exp{E[Z] + Var (Z)/2).Hence, as -bW is normal with mean -bE[W] and variance b² Var(W),it follows thatE[U (W)] = 1 E[e-bW] = 1 − exp{-bE[W] + b² Var(W)/2}.Therefore, the investor's expected utility will be maximized by choos-ing a portfolio thatmaximizes E[W] -b Var(W)/2.Observe how this implies that, if two portfolios give rise to randomend-of-period wealths W₁ and W₂ such that W₁ has a larger mean and asmaller variance than does W₂, then the first portfolio results in a largerexpected utility than does the second. That is,E[W₁] ≥ E[W₂] & Var(W₁) ≤ Var(W₂)E[U(W₁)] ≥ E[U(W₂)].(9.1)In fact, provided that all end-of-period fortunes are normal random vari-ables, (9.1) remains valid even when the utility function is not expo-

Answered: On page 176, we have an expression…

Advanced Engineering Mathematics

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Chapter2: Second-order Linear Odes

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Page 196 is attached

On page 176, we have an expression maximizes E[W] - bVar(W)/2. If you use the utility
function as U(x) = 1 - e-b², will this expression remain the same? If yes, indicate so; if no,
give the analogous expression.

176 Valuing by Expected Utility
limit theorem, a reasonable approximation. (It would also be exactly
true if the X₁, i = 1, ..., n, have what is known as a multivariate normal
distribution.)
Suppose now that the investor has an exponential utility function
U(x) = 1- e-bx, b>0,
and so the utility function is concave. If Z is a normal random variable,
then ez is lognormal and has expected value
E[e²] = exp{E[Z] + Var (Z)/2).
Hence, as -bW is normal with mean -bE[W] and variance b² Var(W),
it follows that
E[U (W)] = 1 E[e-bW] = 1 − exp{-bE[W] + b² Var(W)/2}.
Therefore, the investor's expected utility will be maximized by choos-
ing a portfolio that
maximizes E[W] -b Var(W)/2.
Observe how this implies that, if two portfolios give rise to random
end-of-period wealths W₁ and W₂ such that W₁ has a larger mean and a
smaller variance than does W₂, then the first portfolio results in a larger
expected utility than does the second. That is,
E[W₁] ≥ E[W₂] & Var(W₁) ≤ Var(W₂)
E[U(W₁)] ≥ E[U(W₂)].
(9.1)
In fact, provided that all end-of-period fortunes are normal random vari-
ables, (9.1) remains valid even when the utility function is not expo-

Expert Solution

Step 1: Explanation

Given

$U (x) = 1 - e^{- b^{2} x}$

Therefore

$E [U (W)] = 1 - E [e^{- b^{2} W}]$

Given that if $Z$ is a normal random variable then $e^{Z}$ is lognormal and has expected value

$E [e^{Z}] = e x p \{E (Z) + \frac{V a r (Z)}{2}\}$

Consider $Z = - b^{2} W$

$E (Z) = E (- b^{2} W) = - b^{2} E (W) V a r (Z) = V a r (- b^{2} W) = b^{4} V a r (W)$

Therefore;

$E (e^{- b^{2} W}) = e x p \{- b^{2} E (W) + \frac{b^{4} V a r (W)}{2}\}$

Therefore

$E [U (W)] = 1 - E [e^{- b^{2} W}]$

$= 1 - e x p \{- b^{2} E (W) + \frac{b^{4} V a r (W)}{2}\} = 1 - e x p \{- b^{2} (E (W) - \frac{b^{2} V a r (W)}{2})\}$

$∴$ Maximize $E (W) - \frac{b^{2} V a r (W)}{2}$

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On page 176, we have an expression maximizes E[W] - bVar(W)/2. If you use the utility function as U(z) = 1 - e-b², will this expression remain the same? If yes, indicate so; if no, give the analogous expression.