1
$\begingroup$

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!

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ So why does CARA imply the statement in a)? It is still not clear to me. $\endgroup$ – DesmondMiles Apr 30 at 10:36
  • 1
    $\begingroup$ @DesmondMiles: I completed the problem showing that with this utility function the expected utility of the risky investment is greater than the utility of the risk-free investment at all levels of wealth when neglecting the terms of order of the rate-of-return squared. The notion that preferences under CARA are independent of the level of wealth holds for "small" gambles / investment returns. This concept originates from Pratt, J.W. (1964) "Risk-Aversion in the Small and in the Large'" Econometrica 55,143-54 $\endgroup$ – RRL Apr 30 at 18:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.