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?
Bayesian Learning vs Reinforcement Learning in Financial Decision-Making: A Simple Example
Mr 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 Roubini!) whether we are in deep trouble: maybe we are in a deep structural crisis; but some argue that the situation might not be that catastrophic. 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: Mr Spout nally decides to invest in stock A today, and he receives 1$. (He is not informed of what he would have got, if he had invested in stock B.) Assume that Mr Spout is a Bayesian learner and that he’s risk-neutral. Is Mr Spout going to invest again in stock A tomorrow, or is he going to switch to stock B?
Hint : Take the prior probability to be in a deep crisis to be 1/2. (This assumption is a natural way to formalise the fact that investors are agnostic about the state of the economy.)
(6 marks)
1
2. Now imagine the following scenario: Mr Spout nally decides to invest in stock A today, and he receives 0$. (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 Mr 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 agnostic about the state of the economy.)
(6 marks) 3. Mr Spout is actually a Reinforcement Learner, not a Bayesian; consider again the two scenarios above: a) What will Mr Spout do after receiving 1$ from investing in stock A? b) What will he do after receiving 0$ from investing in stock A? (5 marks) 4. Comment on the dierences in behaviour, if any: do a Bayesian and a Reinforcement Learner behave similarly? Why? (3 sentences max ) (3 marks)
Instructions
1. Questions 1&2: your answer in itself is important, but how you arrived at it is as important. I want all the steps of your reasoning and all the details of your calculations. 2. Questions 1&2: it takes only 2 steps and 5 lines max to arrive at the correct answer. Your answer should thus be very short. 3. Question 3: Answering that question does not require any calculation; if you’ve understood the concept of Reinforcement Learning, the answer should be obvious.
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