Dr. P.V. Viswanath



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A State Space Approach to Asset Pricing


We have already looked at the fact that a future dollar is not always the same, even if the likelihood of realization is the same. It also matters as to the circumstances in which we receive the money. This leads to the suggestion that perhaps we can price assets simply in terms of the states in which they receive their cashflows. How do we define states? Obviously, the range of states is, in general, quite broad. However, if we define states restrictively in terms of the performance of the entire economy, we could make this dependence more explicit.

Once we see that the difference between debt and equity is in terms of the different kinds of states in which bondholders receive their flows as opposed to stockholders, we can see that differing capital structures don't necessarily affect the total value of equity and debt combined. Lets look at this more carefully.

Think of the series of three quizzes that I gave in the webnote entitled "Perspectives on Diversification."

From the first quiz, one can infer that the cost of a given item like time is not always the same. It depends on what our opportunities are on that occasion, what else we want to do at that time, like rest or be with our family, etc. These desires can change from occasion to occasion. This modifies our valuation of what is otherwise an 'objective' item, i.e. time.

From the second quiz, we can see that the value of future monies can depend on the needs we will have for those monies at those future times, and in the scenarios in which we will get that money. Thus, in the problem, if we put down $3500 on the deal, a profit of $6500 ($10,000 - $3500) is of value to us, but not in an objective, state-independent manner. Neither can we evaluate the potential loss of $3500 in a state-independent manner. The loss of $3500 is more painful when it is going to be in a scenario where we are also going to lose our job.

We can apply this further to the circumstances laid out in the third quiz. The probability distribution of returns that we get from the investment cannot be investigated blindly. If the good returns are going to come at a time when the market is doing well (and by inference, our diversified portfolio is also doing well), each dollar is going to be worth less to us, because we will be 'relatively' rich at that time. Similarly, if we stand to make losses or low returns when the market is doing badly (and our other investments, i.e. the diversified portfolio, are also doing badly), those low returns will hurt us more than if they came in circumstances when we weren't doing so badly.

We have already made these points earlier in a different context. What we learnt there was that beta risk or market risk was more relevant than unsystematic risk or asset-specific risk. This makes much more sense now.  An asset that is highly correlated (high beta) with the market portfolio will have cashflows that are high when the market return is high (and we value a dollar less), and cashflows that are low when the market return is low (and we value a dollar more).  The reverse will be true for a low beta security.  Consequently, the expected cashflows from a high beta security will have to be discounted to a greater degree than the cashflows from a low beta security.

Let's take this idea further.  

Suppose we assume that most investors have well-diversified portfolios, then we can define times where wealth means a lot to us as times when the market return is low, and we are relatively poor; similarly, times when wealth means less to us are times when the market return is high. Many of you may know about options. For example, a call option on a stock is a security that has a positive cashflow when the price of the underlying stock is higher than a certain specified level called the exercise price. That is an example of a contingent security, i.e. a security whose payoffs in different scenarios depend on the performance of other securities in those scenarios. Consider, now, the following contingent security. Suppose we had a security whose payoff was conditional on how the market was doing. Instead of thinking of all the possible returns that we could get on the market, let us do the following. For convenience, let us think of the S&P 500 Index as representing the market portfolio. Suppose it is now trading at 800. (Actually, yesterday Mar. 25, it closed at 789.07.) Let us define scenarios/states in the following manner relative to the level of the S&P 500.

  Return Range Index Range Probability of Occurrence
State Low High Low High  
1 -∞ -0.4 -∞ 480 0.01
2 -0.4 -0.3 480 560 0.04
3 -0.3 -0.2 560 640 0.06
4 -0.2 -0.1 640 720 0.07
5 -0.1 0 720 800 0.08
6 0 0.1 800 880 0.09
7 0.1 0.2 880 960 0.1
8 0.2 0.3 960 1040 0.1
9 0.3 0.4 1040 1120 0.1
10 0.4 0.5 1120 1200 0.09
11 0.5 0.6 1200 1280 0.08
12 0.6 0.7 1280 1360 0.07
13 0.7 0.8 1360 1440 0.06
14 0.8 0.9 1440 1520 0.04
15 0.9 1520 0.01

Suppose we defined "state securities" 1 through 15 in the following manner. Security 1 pays $1 one year from now if the S&P is in state 1, one year from now; i.e. if the S&P is at a level less than 480. Similarly we define states 2 through 15. Now think of the maximum amount of money that we would be willing to offer for each of these securities. If we assume, as is reasonable, given what we know about diversification that all investors hold well-diversified portfolios, then, keeping probabilities constant, we would price these securities in decreasing order of state number. In other words, if we had two states with equal probabilities, e.g. states 8 and 9, we would price security 8 higher than security 9 because security 8 would give us a payoff when we needed it more, relative to security 9. We would also take into account the probability of occurrence of the state. Keeping these two factors in mind, we might come up with the following list of prices for state securities.

The numbers in column two as simply numbers that decrease as the state increase (keeping in mind the lower value of money when investors are wealthy). The third column is obtained by multiplying the second column numbers by the state probability and then normalizing, so that they add to 0.9433962 (which corresponds to a 6% risk-free rate, as explained below).


Price before taking probability into account

Price after taking probability into account

Alternative state prices

































































The prices in the second to last column are called state prices. Of course, it is possible to have a different set of state prices, depending on how much investors as a whole disliked being in low income states or desired being in high income states. The numbers in the last column provide another set of state prices.

Now suppose you bought one unit of every state security. Obviously then, you'd have $1 for sure in one year. What would you pay for all these state securities? Obviously, what we are asking is what is the riskfree rate of interest. With the state prices in the third column, we would assign a value of $0.9434 to the set of all state securities. This menas that by plunking down $0.9434, we get $1 for sure in one year. This translates to an interest rate of 6%.

Once we have the prices of all these state securities, we can pretty much price any security by crudely writing the cashflows to the security as a linear combination of the cashflows to the state security. Thus, suppose we have a security (call it security A) that pays one-eighth of the future index value. Column 4 gives us the future values of this security in the 15 states. The average (expected) payoff on this security is $125. Now suppose we construct another security (security B) with the same expected value of $125; however, in each state it's payoff exceeds the average payoff by 1.5 times the amount that the payoff on security A exceeds its mean payoff of $125. Then, security A will have a beta of 1 with respect to the S&P 500 Index, while security B would have a beta of 1.5.

State Average Return Average Index Value Value of security with payoff =index/8

Security A

Value of security with price change = 1.5 times relative to average value (Security B) Option on S&P 500 with exercise price =900 State price
1 -0.5 400 50 12.5 0 0.011009
2 -0.35 520 65 35 0 0.042759
3 -0.25 600 75 50 0 0.06272
4 -0.15 680 85 65 0 0.071756
5 -0.05 760 95 80 0 0.080515
6 0.05 840 105 95 0 0.088965
7 0.15 920 115 110 20 0.097065
8 0.25 1000 125 125 100 0.095236
9 0.35 1080 135 140 180 0.093311
10 0.45 1160 145 155 260 0.082099
11 0.55 1240 155 170 340 0.071095
12 0.65 1320 165 185 420 0.060259
13 0.75 1400 175 200 500 0.049512
14 0.85 1480 185 215 580 0.030875
15 1 1600 200 238 700 0.00622
      115.275 114 146.1054  

Using the state prices in the last column, we can get the current prices of the two securities as being $115.275 and $114 respectively. The expected returns, therefore are

125/115.275 - 1 or 8.44% and 9.65% respectively. Since the return on security A is also the market return, according to the CAPM, security B should have a return of 0.06 + 1.5(0.0844-0.6) or 0.0966 ot 9.66%, which is about what we get from our state price procedure.

The last column shows the generality of the procedure by pricing an option on the level of the S&P 500 Index with an exercise price of 900. Such an option, we see would be priced at $146.11. Since the expected return on the option can be computed (using the probability distribution) as equal to $170.2, we see that the expected return equals 170.2/146.1054 - 1 = 16.5%, or an implied beta of 4.3

This procedure can be generalized even further to take into account a broader definition of states. Thus, we can, if we wanted define a state not only in terms of the level of the market, but maybe the inflation level as well. We would then have to specify average cash flows for a given security for different combinations of market returns and inflation rates in order to price that security.