Comparison with Banz & Miller

In this section, we will compare estimated state price matrices using the framework of our pro-posed model to the original state price matrices in Banz and Miller (1978). Many years have passed since Banz & Miller printed their original results, and we find it interesting to compare our approach to theirs, using the same data and estimation window. More specifically, we will use real value weighted total returns for the S&P 500, with an estimation window running from 1926 to 1976. We compute real daily returns by subtracting daily inflation (disaggregated from monthly). All data is from the Center for Research in Security Prices (CRSP). We note that the state price matrices in Banz and Miller (1978) are based upon the predictive distributions of Ibbotson and Sinquefield. Since these predictive distributions originate from a different time, there may be some discrepancies between our data and the data used by Banz and Miller (1978).

As the risk-free rate is an important parameter in our proposed model, i.e. the value of our state prices are quite sensitive to it, we choose to let our real risk-free rate correspond to that of Banz and Miller (1978). To do so, we choose the risk-free rate as the implied risk-free rate from the perpetuity matrix in Banz & Miller’s original paper. Obviously, this is not a state-contingent risk-free rate (since the rate has converged) as in Banz and Miller (1978). However, as we are interested in comparing the results from our proposed model, which utilizes state-contingent volatility to model state-contingent preferences (instead of a state-contingent risk-free rate), with the result from Banz and Miller (1978), we believe that our choice of risk-free rate is a sensible one. The state-contingent risk-free rates from Banz and Miller (1978) can be seen in table 6.5.

Low Mid High

1-Year 0.7900% 0.1800% -0.4400%

Perpetuity 0.4000% 0.4000% 0.4000%

Table 6.5: State-contingent risk-free rates from Banz and Miller (1978)

Note that we set our state boundaries to be equal to the ones defined in Banz and Miller (1978). There is one exception, which is the lower bound for the Low state, where we increase the truncation from Banz and Miller (1978) slightly, as we need to make sure that we do not omit any observations of daily returns using our estimation method for state-contingent volatility from section 5.1.5. However, this is of little importance since the truncations are relatively arbitrary. We let this truncation be -121.8091% instead of -86.4700%.

CHAPTER 6. EXPLORATORY ANALYSIS OF THE MODEL

The exact state boundaries are thus

Low Mid High

[−121.8091%, 0.6000%] (0.6000%, 20.4200%] (20.4200%, 171.8300%]

Table 6.6: Return bounds

Next, we report the summary statistics from our model estimation, which can be seen in the table below.

Market State Low Mid High Unconditional

# observations 4796 4866 3758 13420

% of observations 35.7377% 36.2593% 28.0030% 100.0000%

Volatility 20.9886% 10.6626% 14.5112% 15.4306%

Mean return -2.5811% 6.2242% 17.5654% 6.2533%

Estimation window 02/01/1926 - 31/12/1976

Table 6.7: Summary statistic from the implementation of our proposed model with the S&P 500 as the underlying asset

As can be seen from table 6.7, the estimated volatility parameters are largely in line with our previous conclusions. However, a perhaps striking difference compared to our estimations in section 6.1.1, is that volatility seems to be markedly greater in the High state. We will not discuss this observation any further, but we note that there is evidence in the literature for the fact that volatility tends to increase whenever the market deviates from its ’normal’ regime. See for example Chu et al. (1996). Furthermore, we can see that we have roughly the same amount of observations in each state, although the High state contains slightly less observations, and that the estimated mean returns are in accordance with what one would expect in the defined state boundaries.

We will now compare our state price estimates to those of Banz & Miller. We limit our com-parison to the 1-year state price matrices and the perpetuity matrices. However, the remaining state prices matrices can be seen in appendix .4 and .5. A comparison between the 1-year matrices and perpetuity matrices, using both our proposed model and the original model from Banz and Miller (1978), can be seen in table 6.8.

CHAPTER 6. EXPLORATORY ANALYSIS OF THE MODEL

Our State Prices Banz & Miller’s State Prices

1 Year (V) Low Mid High 1 Year (V) Low Mid High

Low 0.5022 0.2350 0.2588 Low 0.5251 0.2935 0.1735

Mid 0.4454 0.4185 0.1321 Mid 0.5398 0.2912 0.1672

High 0.4725 0.3271 0.1963 High 0.5544 0.2888 0.1612

Perpetuity (V∞) Low Mid High Perpetuity (V∞) Low Mid High Low 119.8350 77.9353 51.7293 Low 132.5000 72.4100 42.0500 Mid 119.7710 78.1437 51.5852 Mid 133.3100 72.8500 42.3000 High 119.8020 78.0401 51.6580 High 134.1400 73.2900 42.5500

Table 6.8: Comparison of state price matrices using our proposed model, and the model proposed in Banz and Miller (1978)

From table 6.8, we can see that with our model, the Low state price is greatest when we are standing in Low. Oppositely, the Low state price is greatest when standing in High, in Banz

& Miller’s estimates. We can also see that when using our model, standing in Low gives a greater state price for High than Mid, which is the opposite of Banz and Miller’s results. The reader should note that these are all manifestations of the fact that we choose to model state-contingent preferences through volatility instead of the real risk-free rate. We will now try to give an intuitive explanation of the mathematical and economic differences arising from mod-eling state-contingent preferences using volatility instead of the real risk-free rate.

Consider the following. Everything else held equal, a lower risk-free rate implies a higher risk premium. Furthermore, since rational investors only require compensation for systematic risk, a higher risk premium must imply that investors are more concerned with having a certain payoff in the Low state one period from now. Consequently, if the risk-free rate is the lowest in the High state (as in Banz and Miller’s model), the price of a state-contingent claim in the Low state one period from now, must be more worth when we are standing in the High state today, compared to if we were standing in the Mid or Low state. The mechanics behind this effect can be seen from the following figure.

CHAPTER 6. EXPLORATORY ANALYSIS OF THE MODEL

Figure 6.1: The figure shows the mechanics behind changes in the risk-free rate, and what implications such changes have for state prices. Note that the area to the left of the low bound indicates theLow state, the area to the right of the high bound indicates theHighstate, and the area inbetween indicates theMid state

Figure 6.1 shows what happens when we increase the risk premium through a decrease in the risk-free rate. Mechanically, what we are doing is moving the risk-neutral distribution further to the left of the physical distribution. Thus, if the state boundaries are kept constant, we effectively increase the risk-neutral probability mass in the Low state relative to the two other states, which should make the prices of state-contingent claims in Low at a future point in time relatively more expensive. Note that there is also an effect from reducing the risk-neutral discounting. However, when the physical and risk-neutral distributions are not too far apart, this effect is less important than the the effect arising from a shift in the risk-neutral probability mass. The fact that the risk-neutral probability mass has increased for Low, relative to Mid and High, directly translates to a higher state price for Low.

To provide an explanation for the economic effect from changes in volatility, we can appeal to an argument which is often used for options. Since the intrinsic value of an option always has a lower bound, meaning options are limited liability securities when bought, an increase in the volatility of the underlying asset generally leads to an increase in the price of an option. This results from the fact that we have a higher probability of a higher payoff with limited downside risk. Hence, we know that for our state-contingent claims in Low and High, an increase in volatility tends to increase their price, since our High and Low claims are essentially binary options. Note however, that the above is not the case for the Mid claim. The reason is that the Mid state is bounded from both sides. Consequently, an increase in volatility reduces the probability that a unit claim in the Mid state one period from now will be in the money, everything else held equal. Thus, the price of such a claim tends to decrease with increased volatility.

CHAPTER 6. EXPLORATORY ANALYSIS OF THE MODEL

The mechanics behind the effects we have just described can be seen in the following figure.

Figure 6.2: State price under risk-neutral measure and changed volatility. Unconditional distribution on top

By increasing volatility, the distributions become wider, but are not shifted as in the previous case, i.e no change in the mean of the risk-neutral distribution. The increased volatility makes extreme outcomes more likely, and thus increases the risk-neutral and physical probability for bothLow and High. Consequently, the state prices for the extreme states are higher when the volatility is higher. For example, this is why when using our proposed model, the state price is higher forLow and High than for Mid, when standing in Low.