Posts this month
A blog on financial markets and their regulation
Updated: corrected reference to absolute risk aversion instead of relative risk aversion and added a reference for the usage of the term return free risk.
The increased demand for US Treasuries after their credit rating was downgraded led some analysts to ask whether these assets are Giffen goods. The classic example of Giffen goods are staple foods like bread or potatoes where a rise in price depletes the spending power of the poor so much that they are no longer able to afford meat or other expensive food and are forced to consume more of the cheaper food. This means that the demand rises as the price rises – the income effect increases the demand of the inferior good so much that it outweighs the substitution effect of the higher price.
Can this happen with investment assets? For an investor trying to protect her capital, a rise in risk (without any change in the rate of return) of the safest asset is effectively an increase in the price of capital preservation. The idea is that a rise in risk of the safe asset consumes so much of the risk budget of the investor that she can no longer afford too much of the riskier asset. She therefore is forced to shift more of her portfolio into the safer asset. At a qualitative level, the story sounds plausible.
For a more rigorous analysis consider a portfolio choice model with two uncorrelated assets which we shall call the safer asset and the riskier asset. The following results can then be proved:
I have written up a complete mathematical demonstration of the mean variance result for those who are interested. The intuitive reason for this result is actually quite simple. In a mean variance framework, the optimal portfolio consists of two components (a) the minimum variance portfolio which minimizes risk without any regard for return, and (b) a zero investment purely speculative portfolio of long positions in high return assets financed by short positions in low return assets. The allocation to the speculative portfolio is proportional to the risk tolerance (reciprocal of the Arrow Pratt measure of relative risk aversion) of the investor. An investor with zero risk tolerance holds only the minimum variance portfolio. As the risk tolerance increases, the investor blends the minimum variance portfolio with more and more of the speculative portfolio.
Now if the risk of the safer asset rises, its weight in the minimum variance portfolio necessarily declines. The weights of the two uncorrelated assets in the minimum variance portfolio are proportional to the reciprocals of the variances of the two assets and so a rise in variances reduces the weight.
So an investor with zero risk tolerance will necessarily reduce his holding of the safer asset when its risk increases. What about other investors? What will happen to the short positions that they hold in the safer assets through the speculative portfolio? Increasing the risk of the safer asset makes this short position riskier and all risk averse investors will therefore reduce this position by buying the safer asset. The question is whether this can outweigh the sale of the safer asset via the minimum variance portfolio?
Clearly this can happen if and only if the risk tolerance is very high. We can show that at such high levels of risk tolerance, the initial total position in the asset would have been short. Such an investor is not increasing his long position; he is only reducing his short position. This is not a Giffen good situation at all. Moreover, with short sale restrictions, the initial position in the safer asset would have been zero and it would just remain zero.
So in a mean variance framework, the safe asset is never a Giffen good. As one thinks about it, this result is being driven by the fact that in this framework, the risk aversion is being held constant in the form of a fixed tradeoff between risk and return. This does not allow the income effect to play itself out fully. The principal mechanism for a Giffen phenomenon is likely to be a rapid rise in risk aversion as wealth declines.
So I shift to an explicit expected utility framework using a logarithmic utility function with a fixed subsistence level: U(x) = log(x – s). This functional form is characterized by rapidly increasing risk aversion as the subsistence level s is approached. I consider an up state and a down state for the terminal value of the safe asset u1 and d1 with probabilities p1 and q1=1 – p1 respectively. Independently of this, the riskier asset also has two states u2 and d2 with probabilities p2 and q2=1 – p2 respectively. The investor invests w1 in the safer asset and w2 = 1 – w1 in the riskier asset. Expected utility is therefore given by:
p1 p2 log(w1 u1+ w2 u2 – s) +p1 q2 log(w1 u1+ w2 d2 – s) +q1 p2 log(w1 d1 +w2 u1 – s) +q1 q2 log(w1 d1+ w2 d2 – s)
The optimal asset allocation is determined by maximizing this expression with respect to w1. I did this numerically using this R script for specific numerical values of the various parameters. Specifically, I set:
s = 0.8, u1 = 1.01, d1 = 0.99, u2 = 5.00, d2 = 0.70, p1 = p2 = 0.50.
In keeping with the spirit of the times, the expected return on the safer asset is zero – instead of a risk free return, it represents return free risk. For these parameters, the weight in the safer asset is 81%. If we now reduce d1 to 0.90 (increasing the risk and reducing the return of the safer asset), the weight in the safer asset rises to 82%. Alternatively, if we change d1 to 0.85 and u1 to 1.15 (increasing the risk and leaving the return unchanged), the weight in the safer asset rises to 85%. The absolute risk aversion in the low wealth scenario rises from 7.4 when d1 = 0.99 to 15.7 when d1 = 0.90 and even further to 37.3 when d1 = 0.85. This is what drives the higher allocation to the safe asset. The safer asset is truly a Giffen good.