# Preference does not depend on wealth

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.

• So why does CARA imply the statement in a)? It is still not clear to me. – DesmondMiles Apr 30 at 10:36
• @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 – RRL Apr 30 at 18:43