A Chat about Measures of Risk Aversion
David Ronayne
7 November 2011
Here I chat about absolute risk aversion, relative risk aversion and then a comparison in that order.
Give it a read, understand it, do Q4 and Q5 and ask yourself if you can explain what type of person
each utility function in the question represents. If you cannot do that, you must ask me in my office
hour.
1. Absolute Risk Aversion (ARA)
In class we discussed Absolute Risk Aversion (ARA) or
We defined it as an approximation to the
:
that we studied in Q3 PS1. Points you must remember:
1. It is an approximate measure in the sense that you cannot interpret the exact number you
compute when you compute
. What you can interpret is how it changes with , the
expected value of the lottery:
a.
If
depends positively on
i.e.
then this guy is getting more risk
averse as the expected outcome of the lottery increases. It is called increasing
absolute risk aversion (IARA)
b. If
depends negatively on
i.e.
then this guy is getting less risk
averse as the expected outcome of the lottery increases. (DARA) d for decreasing
c. If
does not depend on
i.e.
then this guy’s risk aversion is constant
regardless of the size of the expected outcome. (CARA) c for constant
d. Which one are you? Economics is supposed to be applied and useful, not some
abstract art. Think about what these 3 cases mean and see which suits you best!
2. Why do we bother with it? Why not just compute
explicitly and interpret the number
precisely instead of being approximate and using
? Because the utility function you are
given may be really hard to use to get the exact . Also, you may not care about the exact
numbers, but just want to know how the guy’s risk aversion changes as the expected
outcome of the lottery changes i.e. if his preferences are IARA, DARA or CARA. So,
computing
may be the easiest route to your goal.
2. Relative Risk Aversion (RRA)
We spoke about ARA in the context of lotteries that I was offering you in class, the flip of the coin
for example, or the continuously distributed lottery of Q3. Now RRA is not thought about in the
same setup. We need a new idea in mind to think about it. So what is it?
Imagine you have some wealth, call it
. Now, consider a world where you will be hit with random
shocks to your wealth, but these shocks are told to you in terms of proportions of your initial wealth.
For example, I am sitting happily with
wealth level after the shock is
is negative, any
, then I experience a shock (call it ) of 0.8, my
, so that was a negative shock. As you can see, any shock
is positive, and
does nothing to our wealth (we only discuss
because we are not talking about debt (negative wealth) here). The link to lotteries is that you
face a random variable shock
but
, so I will call this the multiplicative-shock lottery
world. So we are still talking about preferring lotteries or expected outcomes, its just that now we
are doing it with this new way of stating a lottery.
This setup is talked about in models where shocks are multiplicative (the shock multiplies your old
wealth to work out your new wealth). In this world, we would like a measure of aversion to these
shocks, just like in the lottery world where we found ARA. Here, in the multiplicative-shock lottery
world it is called relative risk aversion (RRA).
Now this RRA term is more or less the same measure as ARA but in this new world. I mean that if
you define your utility function to be a function of
i.e. plug in
wherever you see
rather than
, let call this function
in the original utility function just like Mas-Colell (your
text book) does. Now taking derivatives with respect to , compute (for any utility function you like)
the following term:
This is just the ARA term we used to approximate
before! Except this time everything was done
in terms of proportions of your initial wealth, rather than absolute amounts (can you see why this is
called RRA now?). So you could think of a facing shocks (the new kind of lottery) with expected
outcome
, and think about what proportion of your initial wealth you would use to insure
yourself against the shock (just like talking about
for ARA). Just as with ARA, this is an
approximation, so do not start interpreting the actual number, but do notice how this value changes
with
, i.e. how your risk-aversion changes with wealth. Now plug in
. What does this refer
to? Recall the meaning of , this means you are asking about the individual’s attitude toward risk
when he has wealth
. So, RRA is asking about risk aversion when initial wealth is
and the individual faces multiplicative shocks. So at the moment this seems really annoying because
to get ARA you need to differentiate
but for RRA you need to differentiate
. But the final
trick is to notice that:
How cool is that? So, even though we are in a different world when thinking about RRA rather than
ARA, the terms turn out to be really similar, and we can compute both ARA and RRA from each
other by multiplying or dividing by . Good stuff.
Example
Here’s an example of the equation above in case you do not believe me, you can try any other one
yourself to check. Let’s use the utility function
to get
first lets compute the lottery-world way
:
Now let’s find
in the proportional-shock world. First define
.
Now if you let initial wealth (in the proportional-shock world) be the expected outcome (in the
lottery world) so that
, then you can see that for whatever utility function you pick (choose
one and try it), you will get this equivalence. So to keep your life simple we just use the fact that
so you only need to compute
to get both ARA and RRA.
3. Comparison and Discussion
The calculation of RRA (as we saw) is very similar to ARA, indeed
. So when you
do Q4 and Q5 of PS1, you can find the answers to Q5 really fast using your answers to Q4!!!
talks about an individual’s attitude to risk as the expected value of a lottery changes, e.g. if he
exhibits IARA he likes lotteries less and less compared to just being given the expected outcome as
the expected outcome increases.
talks about an individual’s attitude to risk as his wealth level
changes when he is in a world where he faces multiplicative-shock lotteries to his wealth, e.g. if an
individual exhibits IRRA he likes facing shocks to his wealth less and less compared to just keeping
for sure as his wealth increases.
Why do we bother having these two terms? It is good to be able to discuss risk in different settings.
If we are in the lottery world, we will talk about ARA, if in the multiplicative shock world we will
talk about RRA. They are tools made to suit the environment. Essentially they are both trying to
answer the same question: “how do people’s feelings towards risk change” but they do so in
different settings. Which one you want to use depends on the application you are talking about.
Finally, by studying the statement
notice that:
But also notice that:
Summarising in a table, these are the combinations of properties one given utility function can have:
IARA
CARA
DARA
IRRA
Yes
Yes
Yes
CRRA
No
Yes
Yes
DRRA
No
No
Yes
4. What should you remember from all this for the exam/your life?
That’s all very interesting (or not!) to you. The more you know the better, as always. But what do
you need to know for the exam? You need to have a feel for the different worlds in which ARA and
RRA are used, how to interpret them, to know which one’s which, and know how to compute the
two terms
and
.