1. The Stochastic Discount Factor for the Capital Asset Pricing Model
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UNSW Actuarial Research Symposium
19 November 2004
The Stochastic Discount Factor for the Capital Asset Pricing Model
Mark Johnston
PricewaterhouseCoopers and UNSW

2. Introduction to Asset Pricing The CAPM (Traditional Formulation)
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Agenda
Introduction to Asset Pricing
The CAPM (Traditional Formulation)
Marginal SDF
Marginal SDF for the CAPM
Examples
Financial strength of a firm
Debt-financed corporate
Insurance firm
Conclusions

3. Basic terminology and notation
An asset is defined by the payoff (x) it provides to its owner
The payoff typically:
occurs in the future
is uncertain (a random variable)
The price or value (V) of an asset is the amount of cash we would pay today for the right to the asset’s risky future payoff
e.g. The call option, with payoff distribution shown here, has a value of $0.09
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4. “Risk-adjusted discount rate” “Stochastic discount factor”
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Asset Pricing 101
Old way:
“Risk-adjusted discount rate”
New way:
“Stochastic discount factor”
Cash flow info required
Expected payoff
E(x)
Payoff distribution x
Discount factor
Deterministic – but different for each asset: 1 / (1 + rj)
Stochastic – but prices all assets of interest: m
Pricing formula
V = E(x) / (1 + rj)
V = E(m x)
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5. …sounds good… but where do we find the discount factor?
Risk-adjusted discount rate approach
Typically a factor pricing model is used
Most common is CAPM
Others include Fama-French 3-factor model, etc.
Stochastic discount factor approach
???
In the accompanying paper we show how to derive a stochastic discount factor for “CAPM assets”
This lets us price a broader set of assets – namely portfolios of derivatives of portfolios of CAPM assets
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A useful characterisation of an asset’s price – the Sharpe Ratio or “market price of risk”
It’s not a weighted average as the “weights” don’t add up to one, as sigma_x can be less than sigma_y plus sigma_z.
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7. The CAPM: “the wealth portfolio is mean-variance efficient”
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8. Marginal Stochastic Discount Factor
The stochastic discount factor is defined over a very high-dimensional space
All “states of the world”
m = m(FTSE100, Nokia stock price, El Nino SOI, Sydney house prices, price of fish in China, …)
To price an asset, or derivatives of it, we only need the SDF over the states of that asset
Define “marginal stochastic discount factor” mx for an asset x as the conditional expectation of the stochastic discount factor with respect to that asset
Then we can price derivatives g(x) of x via: E(mx g(x))
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9. It’s easier to graph a marginal stochastic discount factor !
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It’s easier to graph a marginal stochastic discount factor !
Observe that the SDF varies by “state of the world” (here represented by S - stock price at time t).
Note that it will slope down iff mu > rf, and will slope up if mu < rf (i.e. if the underlying stock is “negative-beta”).
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10. 16/11/2018
We can derive a marginal stochastic discount factor for “normal” or “CAPM” assets
Model
Assumptions (nested)
Discount factor
Stochastic discount factor
Expected utility, smooth utility function u
m = A u’(c*)
c* = agent’s optimal consumption
Buhlmann’s model (1980)
Exponential utility, closed market
m = A exp(-α c),
c = total consumption
Wang’s specialisation of Buhlmann’s model (2003)
Total consumption normal, normal copula with assets
mx = Ax exp(- x h-1(x))
x = h(z), z unit normal,
x = z ,
= (E(Rc) – Rf) /  (Rc)
Capital Asset Pricing Model (1965-ish)
Asset payoffs normal
mx = Ax exp(- x (x-x)/x)
The stochastic discount factor (in Wang’s specialisation) has a log-normal distribution
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11. Derivation of marginal SDF for Wang assets
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12. Rather than nested models, consider nested sets of assets
Wang-Buhlmann economy: exponential utility, total consumption normal
“Normal assets” / “CAPM assets”: assets jointly normal with consumption
This set is closed under linear combinations, and so forms a vector subspace of the set of all assets
i.e. a portfolio of normal assets is a normal asset
“Wang assets”: assets normal-copula with consumption
These are derivatives of normal assets and can hence be priced
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13. Relationship with Wang Transform
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14. We can use the CAPM SDF to calculate equilibrium prices for derivatives of CAPM assets
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15. A simple limited-liability firm
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A simple limited-liability firm
Time T cash flow
Cash In
Cash Out
Assets AT
Liabilities
LT = min(CT, AT)
= CT – DT
Equity
ET = max(0, AT – CT)
= (AT – CT)+ DT
Creditors will claim an amount CT (the “claims”)
They will be paid LT = min(CT, AT) (the “liabilities”)
DT is the deficit between what creditors were promised and what they will be paid
Subscript zero will denote the value at time zero of each of these cash flows
D0 is the value if the deficit or the value of the “insolvency exchange option”
It’s the value of a call option on the claims less the assets
ς = D0 / C0 – call this the “credit discount ratio” ?
It’s the percentage by which creditors discount the value of their claim to work out the value of what they’ll actually get (the liabilities).
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16. Value-based measures of financial strength
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Value-based measures of financial strength
Comment on per-annum vs. total measure, and resolution of uncertainty over time
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17. Calculating credit discount for a firm with normal surplus
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Calculating credit discount for a firm with normal surplus
Assume the surplus is a normal asset with value S0
The value of the deficit is the value of a strike zero call option on the negative of the surplus
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18. Examples rf = 6%; MRP = 6%; Market volatility 10%
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Examples rf = 6%; MRP = 6%; Market volatility 10%
Firm financed with 2-year zero-coupon bond: credit spread 60bps (CD 1.2%)
Assets: μ: 100, σ: 20, ρ: 0.5
Value 81; β: 0.88; λ: 0.46
Solve for bond face value: 63
Def. prob: 3%; risk-neutral: 8%
Surplus: μ: 37, σ: 20, ρ: 0.5
Insurance firm. Same assets. Target same CD.
Claims: σ/μ: 35%, ρ: 0.1
Asset-claim correlation: 0.15
Solve for expected claims: 53
Def. prob: 3%; risk-neutral: 5.7%
Surplus: μ: 47, σ: 25, ρ: 0.32
60bps corresponds roughly with AA-rated debt.
n.b. the spread and CDR apply to all liabilities, not just the bond (i.e. they apply to the residual claim also – the negative part of the liability payoff).
Comment on high default probabilities – in practice, spreads are much larger than would be implied by observed expected losses. i.e., given a spread, the losses should be much higher than they are. In this example, we are calculating the default probabilities consistent with a specified spread, so we get numbers much higher than observed in practice. In other words – something else must be driving spreads – e.g. inability to diversify credit risk (so something other than normal copula may need to be considered).
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19. Conclusions What have we done ? Derived SDF for the CAPM
Shown how to value derivatives of CAPM assets
Introduced credit discount ratio
value-based characterisation of financial strength
showed how to calculate it for a firm with normal “net assets”
Next steps
See if it’s any good for pricing
Generalise: incomplete markets
Generalise: imperfect information
Capital allocation examples
So what ?
CAPM is the valuation model most commonly used in practice, so it’s nice to know how it fits into the SDF framework
Clarifies the theoretical context of the Wang Transform
Gives a stochastic discount factor – with some theoretical basis – that can be used in examples
Can calibrate financial strength for firm with risky claims
It might even be useful for calculating some asset prices !
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