# LO 62.3: Explain how stochastic discount factors are created and apply them in the

LO 62.3: Explain how stochastic discount factors are created and apply them in the valuation of assets.
Multifactor models define bad times over multiple factors. They use the concept of a pricing kernel, also known as the stochastic discount factor (SDF), which represents a random variable used in pricing an asset. The SDF represents an index of bad times, where the bad times are indexed by a multitude of different factors and states. The SDF is denoted as m in the multifactor model, where m is a single variable that captures all bad times for any given a and b constants:
m = a + b x Rm
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2018 Kaplan, Inc.
Topic 62 Cross Reference to GARP Assigned Reading – Ang, Chapter 6
The CAPM is a special case of this model, where m moves linearly with the market return. However, modeling returns as linear is a shortcoming of the CAPM, which can be improved upon by using the pricing kernel which allows for the assumption of nonlinearity.
We can expand this model to include various factor exposures {fv f 2> etc.) where SDF depends on a vector of these factors, where all the k factors represent different bad times:
m = a + bjfj + b2f2 +
+ bkfk
With multifactor pricing kernels, bad times can be defined as periods when an additional \$ 1 income becomes very valuable. Looking at bad times this way interprets SDF as a marginal utility. Periods of high marginal utility could arise from the loss of a job (resulting in low income, where the value of an extra dollar is high), low GDP growth, low consumption (resulting in current consumption below past consumption), or generally low economic growth.
Pr ic in g Ke r n e l s v s . D is c o u n t Ra t e M o d e l s
In a traditional discount rate model, the price of an asset is determined by discounting its future cash flows at the appropriate discount rate:
P : = E
payoff^ l + E(Ri)
The discount rate is determined through the CAPM as:
E(Ri) = RF +(3i x[E(RM) – R F
The SDF model can also be used to predict an assets price, where we use the SDF as the relevant factor:
Pj = E mxpayoff^] This equation helps explain the name stochastic discount factor, since the payoffs are discounted using m as the relevant factor. The SDF is called a pricing kernel, using the term kernel from statistics where we estimate m using the kernel estimator. Because the kernel is used to derive asset pricing, it is called a pricing kernel.
2018 Kaplan, Inc.
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Topic 62 Cross Reference to GARP Assigned Reading – Ang, Chapter 6
If we divide both sides of the equation by the assets current price, constant payoff formula, which we can then use to derive the risk-free asset:
the equation gives us a
E mx payoff^
P ;
P ; = E m x (l + R|)] 1 = E m x (l + R|)] + R 1 1 + R
E[m X l], when payoffs are constant
We can also model an assets risk premium similar to the CAPM, where [fy = cov(Rp m) / var(m)]:
E ( R i) – R F =
^
l ^
var(m)
) x
var(m)^ , E(m) ,
Pi,m X Xm
Beta is multiplied by the price of the bad times risk, determined as:
var(m) E(m)
This equation represents the inverse of factor risk (denoted by the negative sign). In short, assets that have a positive payoff in bad times are valuable to hold, leading to high prices and low expected returns.
The equation for expected return can also be modeled as having exposure to the risk-free rate and multiple betas in the SDF model. Each beta represents a different macroeconomic factor, such as inflation, economic growth, the market portfolio, or investment strategy:
E(Ri) Rp + (3^ x E (fj) + (3i 2 x E(f2) + … +
xE(fjc)
Ef f ic ie n t M a r k e t Th e o r y