Prospect Theory: The Reference Point and the Value Function

Prospect theory keeps the idea that outcomes are weighed and replaces what they are weighed against. Value is measured from a reference point rather than over final wealth, the curve bends more steeply below that point than above it, and stated probabilities are not used raw. The reference point, the bend in the curve and the reweighting of chances explain most of what the older account could not.

Prospect theory rests on one substitution, and the substitution is small enough to write in a line. The older account scores a choice by the total a person would be holding afterwards. Prospect theory scores it by the change from a reference pointThe level outcomes are measured against, which decides what counts as a gain.. Everything else in prospect theory is a consequence of that swap, and so is most of what it manages to explain.

Here is why so much rides on something so small. A total is a single number climbing in one direction, and nothing about it says where the person started. A curve drawn over such a total therefore has one shape. A curve drawn about a point has two sides, and the two sides are free to behave differently from each other. The freedom of the two sides to differ is exactly what the evidence had been demanding for twenty years. Kahneman and Tversky set the account out in 1979, in Prospect Theory: An Analysis of Decision under Risk, in Econometrica.

What does Expected Utility Theory actually say?

Expected Utility Theory is the account prospect theory replaces, and it is far better work than the way it is usually summarised. Stating it properly comes before contradicting it. von Neumann and Morgenstern set it out formally in 1944, in the book Theory of Games and Economic Behavior, and it does four things in a fixed order.

First it lists every outcome a choice can lead to. Second it attaches to each outcome the chance of that outcome happening, and uses that chance at face value, exactly as given. Third it scores each outcome, not by how many rupees it is but by a number standing for how good it would be to be holding that much afterwards. The account takes the name expected utilityThe older account, which weighs outcomes over final wealth. from that number. The input to that scoring is final wealthThe size of the whole holding once the dust settles, rather than what was won or lost along the way., the total held once the dust settles. Fourth it averages those scores using the chances and takes the choice with the highest average.

Take it out of finance for a moment. A vendor with a stall outside one office building is offered a fixed daily payment by a caterer, or he can keep whatever the stall takes that day. He runs all four steps in his head: what a day can bring, how often each kind of day happens, how much a very good day is worth to him against how bad a very bad one feels, and which arrangement comes out higher on average. Expected utility theory is that, written down carefully enough to be argued with.

The third step is where the theory does its cleverest work, and it is worth sitting with. Scoring a total rather than counting rupees is what lets the account explain caution at all. If each further rupee is worth a little less than the one before it, then a certain amount can beat a gamble with a higher average in money, and nothing irrational has happened anywhere. Preferring a sure Rs 5,000/- to a half chance of Rs 11,000/- is not a departure from expected utility theory. Expected utility theory predicts exactly that preference as soon as the curve is permitted to bend.

So the answer to the question everybody asks first is no. Expected Utility Theory is an excellent standard and a poor description of what people do, and prospect theory only replaces the second job. As a standard it is still what every effect in this subject is defined as a departure from. Relaxing one assumption inside it is therefore not the same as throwing it away.

What single substitution does prospect theory make?

Prospect theory leaves the first step and the fourth step of expected utility theory exactly where they were. Outcomes are still listed, and the choice with the highest weighted average is still the one taken. The change is in what happens between those two steps. Before any outcome can be scored, prospect theory inserts a step that the older account has no room for: the outcome has to be sorted into a gain or a loss first. Sorting an outcome that way is called codingDeciding whether an outcome counts as a gain or a loss before it is valued., and coding needs something to sort against.

Everybody does this constantly without noticing it. A pay packet is a rise or a cut against last month's, and nobody who hears the new number asks what the total in the bank is before reacting. A shop that took Rs 18,000/- yesterday and Rs 15,000/- today had a bad day, and the shopkeeper does not feel richer for having taken Rs 15,000/-. In every one of those the number is being read as a change, and the thing it is being read against was chosen before anybody thought about it.

The consequence is that the input to the scoring changes from a total to a change. The whole substitution is this: final wealth out, change from a reference point in. Say it that plainly and the rest of the theory reads as bookkeeping. If value is a function of the change, then the function has to be defined on both sides of zero, and once it has two sides there is no reason at all for them to be mirror images.

What are the three Prospect Theory Components?

Prospect theory is not one idea but three, bolted together, and they can be taken apart. The first is the reference point, the level everything gets measured against, and it decides what counts as a gain before anything is felt. The second is the value function, the curve relating a coded gain or loss to how much it is actually felt, and the famous bend lives in that curve. The third is probability weighting: what happens to a stated chance between being read and being used.

Keeping them separate matters more than it looks. Two effects that both get called loss aversion in ordinary conversation can come from different components, and the remedy for one is useless against the other. Most of the named effects in this subject trace to exactly one of the three components, and naming which one is what turns a label into an explanation.

What is a reference point, and who sets it?

A reference point is the level an outcome gets measured against. A reference point is not a property of the holding, and it is not printed on any statement. The usual candidates are short to list: what was paid for something, what it was worth at some earlier moment, what somebody expected it to be worth by now, and what a person nearby appears to have. Any of the four can be sitting underneath a decision, and only one of them can be active at a time for one outcome.

The Palash decision log, an invented record of 60 investors, eight quarters and 240 logged decisions, carries one in plain sight. On 12 October Meera Sundaram keeps Kesari Logistics Limited and says she will sell it when it gets back to Rs 3,00,000/-. The Rs 3,00,000/- she names is what she paid for it. A decision about keeping Kesari Logistics Limited can turn only on what happens to it next, and nothing in that sentence is about that. The purchase cost has quietly become the level everything is being measured from, and she did not choose it in any deliberate sense. The cost arrived with the purchase and stayed.

A reference point is a fact about the person and not a fact about the holding, so two people looking at the same price on the same afternoon can be looking at a gain and a loss at the same moment. The disagreement is not a paradox. Measuring change rather than totals produces it directly, and one curve can therefore predict two opposite behaviours.

One record makes this concrete. At 30 September the four positions in the Palash decision log stand at the Vindhya index scheme up Rs 36,000/-, Suvarna Chemicals Limited up Rs 60,000/-, the Nilgiri mid-cap scheme down Rs 45,000/-, and Kesari Logistics Limited down Rs 1,05,000/-, each one measured from what that position cost. Two sit on the gentle arm and two on the steep one, and which arm a position landed on was settled by its purchase cost rather than by anything about the holding itself.

What shape does the value function have, and why does the kink matter?

The value functionThe curve relating a gain or loss to how much it is felt. is the curve that turns a coded gain or loss into how much it is felt. The value curve has three properties and each one is doing separate work. The curve is defined over changes rather than totals, so the horizontal axis reads in rupees gained or lost and its zero is the reference point. The curve flattens with distance from that point in either direction, and the flattening is called diminishing sensitivityEach further rupee mattering less than the one before it.. The curve is also steeper below the point than above it, and that change of slope is the kinkThe sharp change of slope at the reference point, where losses begin..

Diminishing sensitivity is easy to feel outside finance. The difference between finding Rs 500/- and finding Rs 1,000/- in an old coat is noticeable. The difference between a bonus of Rs 50,000/- and one of Rs 50,500/- is the same Rs 500/- and is barely there at all. The flattening happens on the losing side too. The first Rs 500/- of an unexpected repair bill stings; the five hundredth rupee of the same bill has almost stopped registering. Notice that both of those sentences are about a distance from somewhere, never about a total.

The kink is the property with consequences. In the drawing above, one move out from the middle in each direction settles it. The two moves are the same number of rupees. The rise is small and the drop is more than twice as deep. The asymmetry is not a mood and not a weakness of character; it is a shape, and once the shape is known it supports prediction.

The kink and the flattening are two properties and not one, and separating them is worth the effort. Two straight arms meeting at an angle carry a kink and no flattening at all. A smooth bend with the same slope either side of the point flattens on both sides and carries no kink. The value function carries both at once.

How steep is the loss arm? The Palash decision log carries an answer for the 60 investors in it. The 60 were asked what gain would make a fifty-fifty gamble against a Rs 10,000/- loss worth taking. The median answer was Rs 22,000/-. Divide Rs 22,000/- by Rs 10,000/- and the measured coefficient is 2.2. The loss arm on that record is 2.2 times as steep as the gain arm. A gain of Rs 10,000/- reads 10,000 and a loss of Rs 10,000/- reads minus 22,000, so the two sides of the reference point are not mirror images of each other. The 2.2 is a property of 60 people on one afternoon in that log. The coefficient is not a constant of nature; it is simply the number this particular record carries.

What is probability weighting, and what does it do to a small chance?

The third component has nothing to do with the curve, and mixing the two up is the commonest mistake made about prospect theory. Probability weightingUsing a stated chance at more or less than its face value. is what happens to a chance between being read and being used. Expected utility theory uses a stated chance exactly as given: a one in a hundred chance enters the arithmetic as one in a hundred. Prospect theory says it does not. Small stated chances are used at more than their face value, and large ones at less. Tversky and Kahneman set the cumulative form of this out in the Journal of Risk and Uncertainty in 1992.

Ask why the same household buys a ticket in a draw on Monday and renews the fire cover on the shop on Tuesday. One is paying for a very small chance of a very large gain and the other is paying to remove a very small chance of a very large loss, and on the older account those two are opposite temperaments. Under probability weighting they are the same thing seen twice: a one in a hundred chance is being treated as though it were bigger than one in a hundred, whichever direction it points in. The value curve never sees a probability at all, so nothing about the curve produces that pairing.

The other end of the scale does work too, and it is quieter. Moving a chance from ninety five in a hundred to a certainty feels a much bigger step than moving it from sixty to sixty five, though both are five in a hundred. The large end of the same weighting curve has flattened. Probability weighting is a separate distortion from the curve, and an effect that traces to it cannot be explained by the kink however hard the kink is pushed.

Prospect Theory vs Expected Utility Theory: where exactly do they part?

The comparison is worth doing slowly, because both accounts are respectable and the difference between them is one row deep. Expected utility theory scores the total held afterwards. Prospect theory scores the change from a reference point. Everything below that row is a consequence, not a second disagreement.

The second way to see it is to draw the two curves next to each other. A curve over a total climbs from left to right and bends the same way at every point along it. Nothing in the input says where the person is standing, so no place on that curve changes character. A curve about a point has a middle, and the middle is where its character changes.

What can the older account not do, worked through the log?

On one afternoon the same 60 investors in the Palash decision log were put through two choices. In the first, a certain Rs 5,000/- sat against a fifty-fifty chance of Rs 11,000/-. Half of Rs 11,000/- is Rs 5,500/-, so counting rupees alone the gamble is worth Rs 500/- more than the certain amount. Forty two of the sixty took the certain amount anyway. Divide 42 by 60 and that is 70.0 per cent.

In the second choice the arithmetic was turned upside down. A certain loss of Rs 5,000/- sat against a fifty-fifty chance of losing Rs 11,000/-. The gamble now carries an average loss of Rs 5,500/-, and that is Rs 500/- worse than taking the certain loss. Thirty nine of the sixty took the gamble. Divide 39 by 60 and that is 65.0 per cent. The same people, the same afternoon, and the same Rs 500/- disadvantage attached to the risky option both times.

Now put the older account to work on those two numbers. Its input is a total. In the first choice the totals on offer are the starting amount plus Rs 5,000/- for certain, or the starting amount plus either Rs 11,000/- or nothing. In the second they are the starting amount minus Rs 5,000/- for certain, or minus Rs 11,000/- or nothing. All four are just points on one climbing curve, and the curve does not know which of them the person happens to be standing at right now. Nothing in a total tells a curve which side of anything the person is standing on, so one curve over a total cannot be cautious in the first choice and chance-taking in the second.

Code the same four outcomes as changes from a reference point and the contradiction simply stops existing. Above the point the curve flattens, so a certain Rs 5,000/- gain is worth more than half of a Rs 11,000/- gain that has already begun to flatten out. Below the point the curve is steeper and it is also still flattening, so a certain Rs 5,000/- loss is felt as worse than half of a Rs 11,000/- loss whose second half has flattened. One curve, two behaviours, all from drawing the curve about a point instead of about zero.

One caution before leaving the pair. A bending curve over totals already predicts caution in gains, so the first choice on its own refutes nothing. The second choice arriving beside the first is what does the work, and the two are therefore always quoted together and never singly.

What does prospect theory not claim?

A theory this useful attracts claims it never made, and they are worth refusing one at a time. A description carries no verdict, so prospect theory does not say anybody decided badly. The theory has no view on which reference point is correct, so it never says the point somebody is using is the right one. And every part of it is a statement about a person rather than about a market, so it says nothing about what any price will do next.

Prospect theory is a description of how a decision gets made, and a description is never an instruction. Prospect theory is unusually good at making somebody feel they have been handed a lever, so the distinction matters more in this account than in most. It has not. The theory has handed them a better question to ask about a decision already taken.

Where does the reference point come from, and why is that the loose part?

Here is the honest weakness, and it is not a small one. Prospect theory predicts sharply once the reference point is fixed, and it says very little about how the point gets fixed in the first place. Fix the point and the rest follows almost mechanically: which side of the curve applies, how steeply it is felt, how much the next rupee will add. Coding is the first step, and nothing downstream of it can run. Leave the point unfixed and the theory makes no prediction at all.

Take Kesari Logistics Limited at Rs 1,95,000/- on 30 September. Measured against the purchase cost of Rs 3,00,000/-, it is a loss of Rs 1,05,000/- or 35.0 per cent, and the steep arm of the curve applies to every rupee of it. Measured against a lower value from an earlier quarter, the very same Rs 1,95,000/- would be sitting on the gentle arm as a gain. Measured against what Meera Sundaram expected it to be worth by now, it is something else again, and the Palash decision log records no expectation for it anywhere. Three readings, one price, and the log settles only the first of the three.

Prospect theory predicts sharply once the reference point is fixed and says very little about how it gets fixed, and that is a real limitation rather than a detail to be tidied away. The loose part is also the part doing the most work, and the limitation is worth being blunt about. Every prediction the theory makes runs through the point, and the point is the one thing the theory does not supply.

How does an adviser or a person deciding alone use this?

Devika Rao advises at Palash Advisory Services Private Limited, and she does not reach for prospect theory to decide what anybody ought to hold. She uses it to work out what a conversation is actually about. When a client says they will sell something once it gets back to what they paid, she does not argue. She writes down the number. The number a client names is the reference point and the only part of the sentence doing any predictive work.

Somebody deciding alone, without an adviser and without a committee, has the same three lines and one advantage: nobody has to be persuaded. The level the holding is being measured against gets written down before anything is decided about it. If the honest answer is the purchase price, that is worth knowing. A purchase cost is a fact about a day in the past, and the decision at hand is entirely about days in the future. The point is allowed to be the purchase cost. The purchase cost just should not get there by default and then go unmentioned.

An unnamed reference point still does all of the work and cannot be argued with while it stays unnamed, so the technique is to say the point out loud before predicting anything. The technique is a smaller claim than it sounds and a more useful one. Naming the point does not tell anybody what to hold. It tells them what question their own sentence was already answering.

Kesari Logistics Limited, Suvarna Chemicals Limited, the Vindhya index scheme, the Nilgiri mid-cap scheme, Meera Sundaram, Devika Rao, Palash Advisory Services Private Limited and the Palash decision log are invented.
Educational material. Not advice on any investment, tax, budget or market position.