Pricing Kernels

In June of 2011, I was invited to give the keynote speech at the annual meeting of the European Financial Management Association in Braga, Portugal. The talk was to be followed by an article for the EFMA journal. I of course chose to speak about my favorite topic. Here is the reference to the published version:

"Post-retirement Financial Strategies: Forecasts and Valuation", European Financial Management, Vol. 18, No. 3, 2012, pp. 324-351.

A pre-publication version is available here.

In the paper I tried to succinctly and somewhat formally summarize the approach to asset pricing that I and my coauthors Jason Scott and John Watson took in our earlier publications, that I have taken in my subsequent research and that I have used for the present value calculations in the RIS software. Here I'll use portions of the EFMA paper with added comments to provide a semi-formal description. 

Warning: some of this will be technical. Feel free to skim or ignore it, as needed.

Here goes. 

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Building on the approach of [Arrow 1952] and [Debreu 1959], I consider first a one- period setting in which there are n possible states of the world. Assume that securities are priced based on their payoffs in the states of the world, using a set of state prices, where p_s is the price today of $1 if and only if state s occurs at the end of the period. If the probability of state s is π_s, define the pricing kernel value for state s at time t as m_st≡ p_st/π_st,  a value that I called in [Sharpe, 2007] the price per chance (PPC). The set of n values of m_st is the pricing kernel for time t. To say that only the market portfolio is priced in a single period is to assert that all states with the same market return have the same price per chance. Moreover, societal risk-aversion implies that the higher a state’s market return, the lower should be its price per chance. More fundamentally, if markets are to clear, prices must adjust so that income in a state of scarcity (low market return) costs more than income in a state of plenty (high market return).

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The references are as follows:

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Arrow, Kenneth J., 1952, “Le Role de valeurs boursiers pour la repartition le meillure des risques,” Econometrie, Colloques Internationaux du Centre National de la Recherche Scientifique 11, pp. 41-47.

Debreu, Gerard, 1959, The Theory of Value, Wiley and Sons, New York.

Sharpe, William F., 2007, Investors and Markets: Portfolio Choices, Asset Prices and Investment Advice, Princeton University Press, 2007. 
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There are two major approaches to asset pricing in Financial Economics. One, starting with the Capital Asset Pricing Model (CAPM), builds on the mean/variance portfolio theory of Harry Markowitz. The other, using Pricing Kernels, builds on the state/preference theory developed by Kenneth Arrow and Gerard Debreu. It may surprise some that I favor the latter for applications such the analysis of retirement income strategies; after all, I was awarded a Nobel Prize in Economics for my work on the CAPM. But over the years I have found the state/preference approach better suited to the task of modeling the determination of asset prices in a one-period setting and even moreso in settings involving many periods, as with retirement income analysis.

I labored long and hard to justify this preference in a series of lectures given at Princeton and the subsequent 2007 book referenced above. However, the focus of the book was on a traditional one-period setting in which investment is made at the present time based on estimates of the probabilities of various returns over a single period such as a year. As I showed there, those who have been brought up on the CAPM need not fear obsolescence, since its major qualitative conclusions hold in the state/preference setting. The market portfolio is still an efficient investment for the average investor. Moreover, security and portfolio expected returns are linearly related to a measure of return sensitivity to the returns on the market. There is of course a difference. In the CAPM the appropriate measure (beta) is based on the covariance of a security or portfolio's return with that of the market portfolio. In the Pricing Kernel version, the measure is based on the covariance of a security or portfolio's return with a function of the return on the market portfolio, a result that I called its kernel beta. But the main qualitative message holds in both approaches: securities and portfolios are priced so that higher expected returns are associated with greater probabilities of doing badly in bad times. 

The CAPM can be considered a special case of the more general pricing kernel approach in which the kernel for a single investment period is a linear function of the return on the market portfolio over that period. In the book I argued that it is much  more plausible to assume that the relationship is non-linear, with a positive price for every possible future state of the world.

The case for such a relationship is far stronger when one attempts to model the determination of asset prices in a multi-period world -- a necessity for analysis of retirement income strategies. This requires some assumption about the efficiency of alternative strategies for investors with different horizons. Moreover, the assumption must be consistent with market clearing -- that is, the collective demands for available securities must equal the supplies. To return to the EFMA paper:

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Consistent with the focus on strategies that utilize only the market portfolio and a riskless asset, I assume that only market returns are priced, both in any single period and also for any multi-period horizon. This could be consistent with a model of multi-period equilibrium in the capital markets, although I have no aspirations to develop one here (or elsewhere). In any event, as I will show, the assumption greatly restricts the characteristics of the pricing function. 
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This may be far too simple a characterization of equilibrium in capital markets. But to value different combinations of income over future years one must make some assumptions and to be credible they should be consistent with market clearing. The approach taken in the EFMA paper seems reasonable. And some such model is better than none. Interestingly, in a world of this sort, it is quite simple to determine the manner in which assets should be priced. Here is the formal analysis from the paper:

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Now, consider a two-period case. Letting m_t and r_t represent vectors of pricing kernel values and market total returns (value-relatives) respectively for time t, the pricing kernels for periods 1 and 2 can be written as: 

        m₁ = f₁(r₁ )
        m₂ = f₂(r₂)

The pricing kernel for a horizon that includes both periods 1 and 2 will be the (dot) product of the two kernels. Thus: 

       m₁⋅ m₂ =  f₁(r₁ )⋅ f₂(r₂)
    

In order for (1) the market to be priced in the same manner for each period and (2) for only the market to be priced for any multi-year horizon, it must be the case that: 

       f(r₁ ) ⋅ f(r₂) =  g(r₁⋅ r₂)

A necessary and sufficient condition for this to be the case is that the one-period pricing function be isoelastic: 

      m_t  = Ar_t^-b 

More generally, if M_t represents the pricing kernel for payments t periods hence and V_t the cumulative market return over that horizon: 

     M_t  = A^t V_t ^-b
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While this may look formidable, it is quite sensible. Back to the paper:

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Taking the logarithms of both sides of the equation:

    log(M_t ) = log(A^t ) − b⋅log(V_t )

Clearly, the b coefficient indicates the elasticity of the pricing kernel with respect to cumulative market return – for every one percent increase in the latter, the pricing kernel decreases by approximately b percent. As is well known, this can be interpreted as indicating that a “representative investor” who holds the market portfolio has a utility function with a constant relative risk-aversion coefficient of b (for a further discussion, see Sharpe, 2007).
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The relationship is log-linear. Among other things, this means that every state price is positive, no matter how large or small the return on the market portfolio. This is not the case if the kernel is linear, which can be if one assumes that all investors have quadratic utility functions (and hence care only about the mean and variance of portfolio returns). 

This is not the place to pick a fight between proponents of traditional mean/variance portfolio theory with a strict interpretation of the of the CAPM and those (including me) who prefer the state/preference pricing kernel approach. The key point is that we now have a pricing kernel with desirable properties that could be consistent with a multi-period equilibrium. For any period in the future, we can use it to determine the price per chance (PPC) for $1 to be received at that future time if and only if the state occurs.  PPC is a function of the cumulative compounded return on the market portfolio from the present to that future time and the function is of the form shown above. And clearly, the price for $1 to be received at a future time and in a given state is equal to the PPC times the chance that the state will occur at the time in question.

To return to the paper:
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I will use the pricing kernel primarily to estimate the values of the distributions of spending (also called “payments” or “paychecks”) provided by a strategy. This requires a set of state prices – each of which represents the cost today of obtaining $1 at a given future time and state. Since the pricing kernel is simply a set of ratios of state prices to state probabilities: 

     P_t = M_t⋅∏_t

Where P_t  is a vector of state prices for payments at a future time t, and M_t and ∏_t are, respectively, vectors for the pricing kernel and probabilities of the states for that time. In the simulations ... market returns were drawn randomly from the underlying probability distributions for n multi-year scenarios. Considering each scenario as a state, the probabilities all equal 1/n so that:

P_t  = M_t /n
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But how to determine the key parameters in the equation (A and b)? In the simulations for the EFMA paper, I ran a million simulations for each year for every strategy analyzed. Moreover, my Matlab program made it possible to run all the results for each year at one time, then find the pricing kernel that best fit the resulting million simulated compound market returns. For each future year, I found values for the two variables (A and b) that would be consistent with valuations of the market portfolio and the riskless asset for that horizon. As I indicated in a footnote in the paper:

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In the simulations ... to reduce sampling errors of draws from the underlying return distribution, the values of A and b used for each horizon were found iteratively based on the requirement that the implied present values of both the cumulative market returns and the cumulative risk-free return were both within a very small distance from 1.0.
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John Watson is currently doing research on the efficacy of this and other possible approaches to valuation when using a simulated sample to approximate the universe of all possible scenarios. This procedure produces pricing kernels that may fit the simulated scenarios but may violate the underlying assumption about the multi-period equilibrium. I shall have more to say about this in subsequent posts. But the discussion is not particularly germane for those using the RIS software for two reasons. First, it is infeasible to run a million scenarios. And second, since the software generates results a scenario at a time, this fitting procedure is not feasible since it would require storing at least 40 or 50 million results (a matrix with a row for each scenario and a column for each year). 

Fortunately, there is another way, using a formula derived by John Watson in our earlier paper:

Scott, Jason S. with William F. Sharpe and John G. Watson, 2009, "The 4% Rule -- At What Price?", Journal of Investment Management, Vol. 7, No. 3, Third Quarter 2009, pp. 31-48 

 
It is available here.

As derived in the paper, for the special case in which the return on the market portfolio is assumed to be lognormally distributed in each period (year), the coefficients of the asset pricing equation can be calculated directly. Here is the EFMA paper's description:

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The values of the coefficients in the pricing equation can be computed directly from the parameters of the assumed distributions of annual returns as follows:

In the above equations, R_f is the total risk-free return, E_m = E[R_t] is the yearly expected total market return, and S_m = (Var[R_t])^1/2 is the annual market volatility.

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The RIS software uses these formulas to compute the parameters for the pricing kernel (computing b first, then A). While producing multiple scenarios, the formula is used to compute the state price for each income and fee payment, obtain the resulting present value and update the appropriate cumulative sum.

Unfortunately, with feasible numbers of scenarios the results will be imperfect since the sample scenarios will not be fully representative of the universe of possible future scenarios. As indicated in the previous post, this is reflected in the design of the RIS  software. First, present values are shown only when 5,000 or more scenarios have been analyzed. Second, the computed values are not shown, only the percentages of the total value associated with the prospects for three recipients (the household, the estate and those who receive fees). Third, the actual percentages are not shown, only a pie chart indicating their magnitudes. And finally, the user is encouraged to generate additional sets of multiple scenarios to determine the variation in the allocation of the present values from case to case. 

Enough caveats. Despite these limitations, the breakdown of present values in the RIS software should provide valuable information in most cases. And some of the analyses that I will describe in subsequent posts will be obtained using Matlab software,  employing large numbers of scenarios and, in some cases using different and hopefully more accurate approximations of present values. 

I will have much to say about the present values of future possible outcomes in the future analyses of alternative retirement income strategies. You should expect nothing less from an economist.