In a country its citizens can invest in pooled investment tax-efficient savings schemes, known as PITS. IIn one tax year, a citizen can choose only one PITS provider, and can invest a maximum of 10,000 euros. On 1 April there are two PITS providers: the Central Bank, offering a guaranteed return of $2\%$ per annum, and Aries Investment Management, offering a fund with a random return X, where $X \sim U(0, 6\frac{2}{3}\%)$.
a) A person with utility function $u(w) = 1 - e^{-0.0015w}$ seeks to maximize his utility. Show that his preference does not depend on the initial wealth $w_0$. b) If forced to invest his wealth on 1 April, which investor would he choose?
So how is a) even true? I guess it is meant to consider lotteries $w_0(1+T)$ where $T$ is a random variable. But clearly $\mathbb{E}[u((1+T)w_0)] = 1 - \mathbb{E}[e^{-0.0015(1+T)w_0}]$'s maximization depends on $w_0$ (we cannot take it out of the expectation or sth, as we would have been able to with $\log$ or $w^{const}$ functions.
And for b) when I write down the utilities in both cases (in the certain return one it is $1-e^{-0.0153w_0}$, in the second one - $1 + e^{-0.015w_0}\frac{e^{-0.001w_0}-1}{0.001w_0}$ and compare them, Wolfram Alpha gives that the first is bigger for some $w_0$-s and the other one for the others... What should I do?
Any help appreciated!
This type of utilility function implies constant absolute risk aversion. The absolute amount that the investor would pay to avoid risk is related to the so-called coefficient of risk aversion, which for initial wealth $W_0$, is given by
$$\gamma = - \frac{u''(W_0)}{u'(W_0)} = - \frac{(0.0015)^2e^{-0.0015W_0}}{-0.0015 e^{-0.0015 W_0}} = 0.0015$$
Notice that the coefficient is constant and independent of the level of wealth $W_0$.
To show that $\gamma$ is related to the quantity paid to avoid risk, consider a fair gamble paying $G = \pm \,\epsilon$ where $\displaystyle P(G=\epsilon ) = P(G = -\epsilon) = \frac{1}{2}.$
With initial wealth $W_0$ and utility function $u$, the amount $\delta$ the investor will pay to avoid the gamble is determined by
$$u(W_0 - \delta) = E[u(W_0+G)]=\frac{1}{2}u(W_0 +\epsilon) + \frac{1}{2}u(W_0- \epsilon).$$
Using the Taylor expansion for u around $W_0$ we have
$$u(W_0) -u'(W_0)\delta + \frac{1}{2} u''(W_0)\delta^2 + \ldots \\ = \frac{1}{2} [u(W_0) +u'(W_0)\epsilon + \frac{1}{2} u''(W_0)\epsilon^2 + \ldots ] \\ +\frac{1}{2} [u(W_0) -u'(W_0)\epsilon + \frac{1}{2} u''(W_0)\epsilon^2 + \ldots ] \\ = u(W_0) + \frac{1}{2}u''(W_0) \epsilon^2 + \ldots $$
Solving for $\delta$ for small $\epsilon$ we get
$$\delta \approx \frac{\epsilon^2}{2}\left[- \frac{u''(W_0)}{u'(W_0)} \right] = \gamma \frac{\epsilon^2}{2}$$
Selecting the investment with optimal expected utility
The utility function is $u(W) = 1 - e^{-\alpha W}$ where $\alpha = 0.0015$ and the initial wealth is $W_0 = 10000$.
The first investment offers a guaranteed return of $r = 2 \%$ with terminal wealth $W_0(1 + r)$ and utility
$$U_1 = u(W_0(1+r)) = 1 - e^{-\alpha W_0} e^{-\alpha W_0 r}$$
The second (risky) investment offers a random terminal wealth of $W_0(1+x)$ where $x$ is uniformly distributed in the interval $(0,b) = (0,6\frac{2}{3}\%)$ and expected utility
$$U_2 = E(u(W_0(1+x)) = 1 - \frac{1}{b}\int_0^b e^{-\alpha W_0} e^{-\alpha W_0(1+x)} \, dx \\ = 1 - e^{-\alpha W_0}\frac{1 - e^{-\alpha W_0 b}}{\alpha W_0 b} $$
We have $U_1 < U_2$ if and only if
$$1 - e^{-\alpha W_0} e^{-\alpha W_0 r} < 1 - e^{-\alpha W_0}\frac{1 - e^{-\alpha W_0 b}}{\alpha W_0 b}\\ \iff -e^{-\alpha W_0 r} < -\frac{1 - e^{-\alpha W_0 b}}{\alpha W_0 b}$$
Using Taylor expansions, it follows that $U_1 < U_2$ if and only if
$$-(1 - \alpha W_0r +\mathcal{O}(r^2)) < - \frac{1 - (1 - \alpha W_0 b + \frac{1}{2} \alpha^2 W_0^2 b^2 + \mathcal{O}(b^3))}{\alpha W_0 b} \\ \iff -1+\alpha W_0 r +\mathcal{O}(r^2) < -1 + \frac{1}{2}\alpha W_0 b +\mathcal{O}(b^2)\\ \iff \alpha W_0 r +\mathcal{O}(r^2) < \frac{1}{2}\alpha W_0 b + \mathcal{O}(b^2)$$
Neglecting the (small) second-order terms we have initial wealth canceling from both sides of the inequality to obtain
$$U_1 < U_2 \iff r < \frac{b}{2}$$
In this case, $r = 2\%$ is less than $b/2 = 3\frac{1}{3}\%$ and the risky investment is preferred.