# Safe assets as Giffen goods

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:

- In a pure mean-variance optimization framework, the safer asset can never be a Giffen good. An investor who had a positive allocation to the safer asset will reduce his allocation if its risk rises.
- In a more general expected utility setting, the safer asset can be a Giffen good. An investor who had a positive allocation to the safer asset could under certain conditions allocate even more to that asset when its risk rises.

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 *u*p state and a
*d*own state for the terminal value of the safe asset
*u*_{1} and *d*_{1} with probabilities
*p*_{1} and
*q*_{1}=1 – *p*_{1}
respectively. Independently of this, the riskier
asset also has two states *u*_{2} and
*d*_{2} with probabilities *p*_{2} and
*q*_{2}=1 – *p*_{2} respectively. The
investor invests *w*_{1} in the safer asset and
*w*_{2} = 1 – *w*_{1} in the riskier
asset. Expected utility is therefore given by:

*p*_{1} *p*_{2}
log(*w*_{1} *u*_{1}+
*w*_{2} *u*_{2} – *s*)
+*p*_{1} *q*_{2}
log(*w*_{1} *u*_{1}+
*w*_{2} *d*_{2} – *s*)
+*q*_{1} *p*_{2}
log(*w*_{1} *d*_{1}
+*w*_{2} *u*_{1} – *s*)
+*q*_{1} *q*_{2}
log(*w*_{1} *d*_{1}+
*w*_{2} *d*_{2} – *s*)

The optimal asset allocation is determined by maximizing this
expression with respect to w_{1}. I did this numerically
using this R script
for specific numerical values of the various parameters. Specifically,
I set:

*s* = 0.8, *u*_{1} = 1.01,
*d*_{1} = 0.99, *u*_{2} = 5.00,
*d*_{2} = 0.70,
*p*_{1} = *p*_{2} = 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 *d*_{1} 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
*d*_{1} to 0.85 and *u*_{1} 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 *d*_{1} = 0.99 to 15.7 when
*d*_{1} = 0.90 and even further to 37.3 when
*d*_{1} = 0.85. This is what drives the higher allocation
to the safe asset. The safer asset is truly a Giffen good.

Posted at 09:01 on Mon, 29 Aug 2011 View/Post Comments (1) permanent link