The Stochastic Discount Factor for the Capital Asset Pricing Model 16/11/2018 UNSW Actuarial Research Symposium 19 November 2004 The Stochastic Discount Factor for the Capital Asset Pricing Model Mark Johnston PricewaterhouseCoopers and UNSW mark.e.johnston@au.pwc.com

Introduction to Asset Pricing The CAPM (Traditional Formulation) 16/11/2018 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

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 16 November 2018

“Risk-adjusted discount rate” “Stochastic discount factor” 16/11/2018 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) 16 November 2018

…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 16 November 2018

16/11/2018 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. 16 November 2018

The CAPM: “the wealth portfolio is mean-variance efficient” 16 November 2018

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)) 16 November 2018

It’s easier to graph a marginal stochastic discount factor ! 16/11/2018 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”). 16 November 2018

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 16 November 2018

Derivation of marginal SDF for Wang assets 16 November 2018

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 16 November 2018

Relationship with Wang Transform 16 November 2018

We can use the CAPM SDF to calculate equilibrium prices for derivatives of CAPM assets 16 November 2018

A simple limited-liability firm 16/11/2018 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). 16 November 2018

Value-based measures of financial strength 16/11/2018 Value-based measures of financial strength Comment on per-annum vs. total measure, and resolution of uncertainty over time 16 November 2018

Calculating credit discount for a firm with normal surplus 16/11/2018 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 16 November 2018

Examples rf = 6%; MRP = 6%; Market volatility 10% 16/11/2018 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). 16 November 2018

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 ! 16 November 2018