RISK NEUTRAL PROBABILITIES, THE MARKET PRICE OF RISK, AND EXCESS RETURNS

Transcription

1 ASAC 2004 Quebec (Quebec) Edwin H. Neave School of Business Queen s University Michael N. Ross Global Risk Management Bank of Nova Scotia, Toronto RISK NEUTRAL PROBABILITIES, THE MARKET PRICE OF RISK, AND EXCESS RETURNS This paper shows how risk-return measures can be extracted from relations between objective and risk-neutral probability distributions (RNPDs). We distinguish unit claims with non-positive or non-negative excess returns as respectively investment- or insurance-based, and disaggregate reference portfolio payoffs to show how market consensus pricing of upside potential and downside risk changes through time. Introduction Although it is becoming increasingly common to estimate market consensus attitudes from risk-neutral probability distributions, financial theory has not yet specified how information conveyed by risk neutral probability distributions (RNPDs) is related to traditional measures such as the market price of risk and the risk of individual securities. This paper extracts information from and assesses the impact of changing RNPDs on risk and return measures for: arbitrary securities, reference portfolios, specialized components of reference portfolios, and unit contingent claims. The paper begins by recalling the known linear relation between risk and return based on using an affine transformation of the state price density (L) as a reference portfolio. We refer to the relationship as a claims-based market line (CBML) and specialize the reference portfolio to L itself. The paper next shows that a second reference portfolio (I), whose components are inverses of the components of L, also defines a CBML, even though 1 is not an affine transformation of L. We show next that the variance of either L or I defines the market price of risk as measured by the slope of the CBML. The variance of L is calculated under the objective probability measure, the variance of I under the RNPD. Either is a sufficient statistic for determining the excess returns on its respective reference portfolio, and is also related to a covariance between the two reference portfolios. Finally, we show how changes in the RNPD can affect measurements of the market price of risk, either through rotations of the CBML or through movements along an unchanging CBML. Reference portfolio L is known to have a negative expected excess return, and we show that I has a positive expected excess return. We say that L is valued on an insurance basis, I on an investment basis. Both L and I are packages of unit claims, some claims having negative excess returns and being valued on an insurance basis, some claims having positive excess returns and being valued on an investment basis. To disentangle insurance and investment effects, we represent the payoffs to either L or I as payoffs to two options, one purely investment-based, one purely insurance-based. We can then describe how changes in the RNPD are manifest in risk and 1

2 return measures for the purely investment-based and the purely insurance-based components of either reference portfolio. Finally, we obtain risk-return relationships for unit contingent claims. The expected excess returns on unit claims can be read directly from a function of the state- price density, as can risk measures for the unit claims. Brief literature review In what are now classic papers Banz and Miller (1978) and Breeden and Litzenberger (1978) develop a claims-based theory of asset pricing. While subsequent theoretical and empirical progress has been limited, further research is desirable. Even though a claims-based theory provides risk-return information similar to that of the CAPM, the former uses less restrictive assumptions since it only postulates that securities prices present no arbitrage opportunities (Duffie, 1996; Pliska, 1997). At the same time, RNPDs are no more difficult to estimate empirically than are the data for the CAPM. Madan and Milne (1994) offer a factor-based approach to estimating RNPDs and the state price density. Longstaff (1995) examines relations between the prices of options and of their underlying assets while assuming an absence of arbitrage opportunities. Bahra (1997) examines techniques for estimating RNPDs from option prices. Jackwerth (1999) reviews methods for recovering implied RNPDs from option prices, and observes that it is difficult to distinguish among RNPDs. Ross (2000) estimates claim values using the Breeden-Litzenberger method. Neave and Ross (2001) suggest both that L can be estimated at successive points in time to help interpret changes in market sentiment, and that differences among risk-neutral probability distributions can be measured. Tarashev, Tsatsaronis, and Karampatos (2003) distinguish among RNPDs using an empirically based criterion. However, they provide little economic interpretation of the differences they observe, and their criterion is defmed over an arbitrarily specified range. This paper proposes risk preference measures that are both soundly based on theory and can distinguish sharply among apparently similar distributions. Preliminaries Let p and q be K-vectors of objective and risk-neutral probabilities respectively. Securities are valued at time 0; their payoff is realized at time 1. The payoff distribution to a generic security is described by a K-vector X; a payoff in a particular state k is denoted X k. Assuming the time zero prices admit no arbitrage opportunities, q satisfies v(x) = E q (X) / (l+r), (1) where v(x) is the time zero market value of X, E q (X) means the expectation of X under q, and 1 + r is the riskless discount factor between times 0 and 1. For later use, define L = q / p and I = p / q as reference portfolios. The properties of RNPDs are reflected in securities returns and their risk measures. To illustrate, we show below that the expected excess returns on unit contingent claims can be read from: E(R c ) - r = (1 +r)(i 1 ) (2) 2

3 where E(R c ) is the K-vector of expected returns on unit claims, r is the riskless rate, and 1 is a K- vector whose components are all unity. Clearly, the sign of the excess return for a given unit claim, say claim k, is determined by I k = p k / q k. By aggregating the information for individual claims we can obtain information about excess returns for arbitrary securities, for reference portfolios, and for options that separate out the investment and insurance components of reference portfolios. Henceforth we assume the partition defining the K states, the objective probabilities p, and the riskless rate r all remain unchanged. Under these conditions, we obtain additional theoretical results that help trace the impact of changes in RNPDs. Again to illustrate, we can use the following to examine changes in excess returns: R c (2) - R c (1) = (1+r)[I(2) - I(1)]; (3) where R c (t) is the K-vector of expected returns on state claims at time t. That is, differences in observed values of q at two points in time are reflected both as changes in I and as changes in claims' excess returns. Deriving CBMLs The following shows that the excess return on any security can be related to either L or I, and hence either can be regarded as a natural choice of reference portfolio. E(R X ) r = - COV(R X, L) = COV q (R X, I), (4) where E(. ) refers to the expectation under p, E q (. ) the expectation under q, COV(. ) is covariance calculated using p and COV q (. ) is covariance calculated using q. Moreover, there is also a natural relation between the two reference portfolios: σ 2 (L) = -COV q (L, I); σ 2 q(i) = - COV(L, I). Given the foregoing, it is easy to establish: [E(R X ) r ] = β XL [E(R L ) r], where β XL COV(R X, R L ) / σ 2 (L) and that (5) (6) [E(R X ) r ] = β XI [E(R I ) r], (7) where β XI COV q (R X, R I ) / σ 2 q (I) and the subscript q means the covariance is calculated using the RNPD q. The first of the two CBML s just stated is established in Pliska (1997); the second is new to this paper. 3

4 Estimating either β XL or β XI presents problems similar to those of estimating the traditional forms of β used in the CAPM. However and as has long been recognized (cf. e.g. Fama-Miller 1972), with different possible reference portfolios a scale dependency problem arises. For example: E(R X ) r = β XL [E(R L ) r]; E(R X ) r = β XI [E(R I ) r]. (8) The left-hand sides of the last two equations reflect the same market-determined data, but the products on the right-hand side combine market-determined data and a risk measure whose magnitude depends on the particular choice of reference portfolio. Similar problems can arise even if the analyst compares betas at different points in time for the same reference portfolio. To avoid such problems, this paper principally analyzes excess expected returns rather than the products involving betas. Moreover, for purposes of standardizing our references henceforth we focus on studying the expected excess returns for the reference portfolios themselves. Reference Portfolios and the Market Price of Risk Relevant financial data for L are E(L) = 1; v(l) = E q (L) / (l+r) = (1 + σ 2 (L)) / (l+r); E(R L ) -r = - σ 2 (L) (1 + r) / (1 + σ 2 (L) ) 0. (9) The term σ 2 (L) (1 + r) / (1 + σ 2 (L) ) defines the slope of the CBML, and is hence the market price of risk. An increase in σ 2 (L) increases the magnitude of the (negative) excess return on L. The effect is represented by rotating the security market line downward at β LL = -1 while leaving it unchanged at β LL = 0; i.e. by rotating the security market line counterclockwise about the riskless rate. Relevant financial data for I are E(I)= 1 + σ 2 q(i); +v(i) = E q (I)/(1 +r) = 1/(1 +r); E(R I ) -r = σ 2 q(i)(1 +r) 0, (10) The term σ 2 q(i)(1 +r) provides a second measure of the market price of risk. An increase in σ 2 q(i) has a quantitative impact on the market price of risks similar to the impact of an increase in β LL = -1 while leaving it unchanged at β LL = 0; i.e. by rotating the β LL = -1 while leaving it unchanged at β LL = 0. The product of one market price of risk with the appropriate measure of risk gives the same excess return as the product of the other market price of risk with its relevant risk measure. 4

5 Disaggregating Reference Portfolios use To disaggregate the mixed valuation effects incorporated in the expected excess on L, we L -1 == (L -1) + -(1 -L) + = C L -P L, (11) where (L - 1)+ and (1 - L)+ are non-negative K-vectors, and as before 1 is a K-vector with every component equal to unity. The call represents a pure insurance security, since it only pays off in states k for which L k -I > 0; i.e., states whose unit claims have a negative excess return. The put represents a pure investment security since it only pays off in states k for which L k -1 < 0. The risk and return measures for the component portfolios provide, among other information, a decomposition of the reference portfolio excess return into its insurance- and investment-based components: E(R L ) -r = w CL β CL [E(R L ) -r] -w PL β PL [E(R L ) -r] where w CL = v(c L ) / v(l) and w PL = v(p L ) / v(l). (12) The reference portfolio I can similarly be disaggregated I -1 == (I -1) + -(1 -I) + = C I -P I (13) Here the call payoffs are payoffs to a package of investment-valued claims, and the put payoffs are to a package of insurance-valued claims. As before: E(R I ) -r = w CI β CI [E(R I ) -r] -w PI β PI [E(R I ) -r]. Preliminary Empirical Tests (14) This section illustrates the kinds of market consensus information the tools in this paper can uncover. The preliminary tests estimate risk neutral probabilities from the prices of options written on the S&P500 Index between January 1, 1995 and December 31, Our price data is taken from the CME, and option prices are adjusted to reflect that we are estimating risk neutral probabilities using European options when the instruments written on the CME are American futures options (see Yang 2004 for details). After obtaining both risk neutral and objective probabilities for states represented as annualized rates of return, we form the reference portfolios L and I for each of the sixty dates in our sample, and then calculate excess returns for both the reference portfolios and their investment and insurance components. The data presented here are offered strictly for illustrative purposes; for detailed descriptions of the testing and estimation procedures see Yang (2004) and Neave-Yang (2004). Casual examination of the next two graphs, showing the performance of the S&P500 Index and its volatility over the sample period, suggest a distinction between market pricing in approximately the first half and the second half of the data. The S&P 500 Index data itself shows faster and more variable growth in the second thirty months than it does in the first, and the CBOE volatility data presented in the second graph support this interpretation. We are interested to learn if and how our new tools indicate the possible changes in market consensus. 5

6 6

7 7

8 2.5 Annualized Excess Returns (I as Reference Portfolio) Data Source: S&P Index 01/95-12/99 Prepared by Jun Yang 2 Market Price of Risk Investment Premium Insurance Cost The excess returns on the reference portfolios (the graph presents an example for I; the graph for L is not dissimilar) show a secular decline over the sixty months and are relatively higher in the first thirty months than in the second. The disaggregate information shows the differences much more dramatically. The upper line in the graph represents the excess returns on the investment portfolio; the lower line the excess returns on the insurance portfolio. To us 1995 and 1996 appear very different from 1997 through Recall that the S&P VIX considerably was lower in than in Our tools present trend differences between investment returns and insurance costs in , but an apparently much higher correlation between investment returns and insurance costs in Correlation tests bear this out. Note also that in market sentiment seemed to shift frequently between confidence and conservativeness, whereas this behaviour was not exhibited in the earlier period. To us the third graph suggests the following interpretations. Market participants apparently go for high returns when confident, safe plays when less confident. They go for either one or the other, not both at the same time. Disaggregate information further suggests that more uncertainty in the market is accompanied by greater absolute value of excess return on insurance. There appear to be differences between pricing upside potential and pricing downside risk (cf. Markowitz, 1959). It is gratifying to note that at month 45, September 1998, when the Long Term Capital Management Crisis was at its works, the disaggregated graphs show a distinct pattern: relatively high investment returns and the highest insurance returns posted over the entire sample period. The normal correlation between the two series does not hold at this point. References Bahra, Bhupinder (1997), Implied Risk-Neutral Probability Density Functions from Option Prices: Theory and Application, Bank of England Working Paper ISSN

9 Banz, R., and Merton Miller (1978), Prices for State-Unit claims: Some Estimates and Applications, Journal of Business 51, Black, Fischer (1976), The Price of Commodity Contracts, Journal of Financial Economics 3, Breeden, Douglas and Robert H. Litzenberger (1978), Prices of State-Unit claims Implicit in Option Prices, Journal of Business 51, Duffie, Darrell (1989), Futures Markets, Englewood Cliffs, NJ: Prentice-Hall, Duffie, Darrell, (1996), Dynamic Asset Pricing Theory (Second Edition), Princeton University Press. Fama, Eugene F., and Merton H. Miller (1972), The Theory of Finance, Hinsdale, Ill.: Dryden Press. Jackwerth, Jens Carsten and Mark Rubinstein (1996), Recovering Probability Distributions from Options Prices, Journal of Finance LI, Jackwerth, Jens Carsten (1999), Option-Implied Risk-Neutral Distributions and Implied Binomial Trees: A Literature Review, The Journal of Derivatives, Longstaff, Francis A. (1995), Option Pricing and the Martingale Restriction, Review of Financial Studies 8, Madan, Dilip and Frank Milne (1994), Unit claims Valued and Hedged by Pricing and Investing in a Basis, Mathematical Finance 4, Markowitz, Harry Max (1959), Portfolio Selection: Efficient Diversification of Investments. New York : Wiley. Neave, Edwin H. and Jun Yang (2004) Disaggregating Market Sentiment: Investment versus Insurance, Queens University School of Business Working Paper. Pliska, Stanley M., (1997) Introduction to Mathematical Finance, Oxford: Blackwell. Ross, Michael N., Security Market Lines: From CAPM to Arrow-Debreu, Queen s University Department of Economics MA Thesis, Ross, Stephen (1976), Options and Efficiency, Quarterly Journal of Economics 90, Tarashev, Nikola, Kostas Tsatsaronis, and Dimitrios Karampatos (2003), Investors Attitude Toward Risk: What Can We Learn from Options? BIS Quarterly Review, June. Yang, Jun (2004) Queens University School of Business Ph. D. Thesis. 9