behavioural finance problem sets
Two behavioural finance problem sets related to Temporal Discounting and Bayesian Learning vs Reinforcement Learning in Financial Decision making
Temporal Discounting Consider a person who has the following inter-temporal
utility function,
U(Co, C1, C2) = ln Co + fl(6ZnC1 + 62 ln 02)
where Ct is the amount of consumption, measured in dollars, that they get
in period t, and 6 < 1 is the individual’s discount factor (that is, how much
they discount future consumption, relative to current time 0 consumption).
5 is an extra parameter included to incorporate the notion of “hyperbolic”
discounting. Assume that B = 0.6 and 6 = 0.5
1. Calculate the present value of 100 CHF in period 0, 1, and 2, according
to this function. Explain how the shape of your graph is different from
what results from standard exponential discounting (B = 1).
(5 marks)
2. Suppose this person has 244 CHF in period 0. How much will he consume
in each period?
Hint: Set the discounted marginal utility of consumption between period
0 and 1 equal, and also the discounted marginal utility of consumption
between period 1 and 2 equal. That gives you 2 equations and 3 un-
knowns. The third equation comes from the constraint. Recall that the
derivative of lna: is 1/30.
(5 marks)
3. Now consider this person at the beginning of time period 1, instead of 0.
So, he has already had period 0 consumption. Now he is deciding about
what’s left over. How much will they consume in period 1 and in period
2? Does that implement what he planned to do at time 0?
Fl NS 3655 Behavioural Fl na nce:
Optional Problem Set 2
Bayesian Learning vs Reinforcement Learning in Financial
Decision-Making: A Simple Example
1\/Ir Spout is a private investor. He envisions investing in two stocks, A and B.
He does not have enough capital to invest in both stocks today. So, he buys
one share of either A or B today, and he will buy another share tomorrow
-either of the same stock, or of the other one.
There are two possible states for today’s economy, and no one knows (even
Nouriel Roubinil) whether we are in deep trouble: maybe we are in a deep
structural crisis; but some argue that the situation might not be that catas-
trophic. What is certain however is that the current situation is not something
transient: everybody knows it will be the same state tomorrow.
At each period -today and tomorrow- investing one share in stock A returns
immediately 1$ with probability 2/3 and nothing (0 $) otherwise, unless the
economy is in deep trouble, in which case investing one share in stock A returns
immediately 1$ with probability 1/3 and nothing otherwise. Investing in stock
B returns immediately 1$ for sure, unless the economy is in deep trouble, in
which case it returns nothing for sure.
1. Imagine the following scenario: 1\/Ir Spout finally decides to invest in
stock A today, and he receives 1$. (He is not informed of what he would
have got, if he had invested in stock B.)
Assume that Mr Spout is a Bayesian learner and that he’s risk-neutral.
Is 1\/Ir Spout going to invest again in stock A tomorrow, or is he going to
switch to stock B?
Hint: Take the prior probability to be in a deep crisis to be 1/2. (This
assumption is a natural way to formalise the fact that investors are ag-
nostic about the state of the economy.)
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