UntitledMIT OpenCourseWarehttp://ocw.mit.edu 15.997 Practice of Finance: Advanced Corporate Risk Management Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Pricing Risk MIT Sloan School of Management 15. 997 Advanced Corporate Risk Management John E. Parsons Overview  Two Alternative Methods for Discounting Cash Flows  Risk-Neutral Pricing – an Introduction ¾ State Prices ¾ The Risk-Neutral Probability Distribution ¾ The Forward Price as a Certainty-Equivalent  Implementing Risk-Neutral Valuation ¾ The Risk-Neutral Distribution in Binomial Trees 9 Random Walk Example 9 Mean Reversion Example ¾ Valuation Mechanics – when risk-adjusted works 9 Example #1: Single-Period, Symmetric Risk 9 Example #3: Two-Period, Compounded Risk ¾ Valuation Mechanics – when risk adjusted doesn’t work 9 Example #2: Single-Period, Skewed Risk 9 Example #4: Two-Period, Non-compounded Risk  Turbocharged Valuation  Equilibrium Risk Pricing, Arbitrage Pricing & Risk-Neutral Valuation 2 1Two Alternative Methods for Discounting Cash Flows Risk-adjusted discount rate method Discount for risk and time Future cash flow Present value Discount Discount for risk for time Certainty-equivalent method 3 Illustration of the Two Methods Table 7.1 Hejira Corporation Two Alternative Methods for Valuing Oil Production @ Forecasted Spot Prices Method #1: Risk Adjusted Discount Rate Method -- simultaneously adjust for risk and time Year 1 2 3 4 5 Forecasted Production (000 bbls) 10,000 9,000 8,000 7,000 6,000 Forecasted Spot Price ($/bbl) -- current price $38 35.00 33.50 32.75 32.38 32.19 Forecasted Spot Revenue ($ 000) 350,000 301,500 262,000 226,625 193,125 Risk-adjusted Discount Rate, ra 10.0% 10.0% 10.0% 10.0% 10.0% Risk-adjusted Discount Factor 0.9048 0.8187 0.7408 0.6703 0.6065 PV ($ 000) 316,693 246,847 194,094 151,911 117,136 Total PV Spot Sales ($ 000) 1,026,682 Method #2: Certainty Equivalent Method -- separately adjust for risk then for time Forecasted Spot Revenue ($ 000) 350,000 301,500 262,000 226,625 193,125 Certainty Equivalence Risk Premium, λ 6.0% 6.0% 6.0% 6.0% 6.0% Certainty Equivalence Factor 94.2% 88.7% 83.5% 78.7% 74.1% Certainty Equivalent Revenue 329,618 267,407 218,841 178,270 143,071 Riskless Discount Rate, rf 4.0% 4.0% 4.0% 4.0% 4.0% Riskless Discount Factor 0.9608 0.9231 0.8869 0.8521 0.8187 PV ($ 000) 316,693 246,847 194,094 151,911 117,136 Total PV Spot Sales ($ 000) 1,026,682 4 2 Figure 9.1 from Lecture Notes on Advanced Corporate Financial Risk Management. Used with permission.From Lecture Notes on Advanced Corporate Financial Risk Management. Used with permission.The Risk-Premium  The risk-adjusted discount rate equals the risk-free rate plus a risk-premium: ra = rf + λ    Where does this risk-premium come from? How general is this risk-premium? If I have a risk-premium for oil projects, can I apply it to all oil projects? The risk-neutral methodology is one incarnation of the certainty-equivalent method, where we drill down beyond the risk-premium, λ, and then implement the certainty-equivalent calculations in a specific way. 5 State Prices 3The General Idea       Start with a project that has cash flows tied to an underlying risk factor, S. Suppose we can successfully value that project by discounting the cash flows in the usual manner using a risk-premium, λ. Unfortunately, we cannot use the same risk-premium, λ, to value all other projects with cash flows tied to S! As the structure of the dependency on S changes, the risk-premium has to change. But how? We need to dig deeper. It turns out that we are ignoring some informationthat we already have. The risk structure of S – is it a random walk with a given drift and standarddeviation, or is it mean reverting, or something else – is the information that we are ignoring. We can combine information about this risk structure together with the knowledge of the risk-premium, λ, for any single project to recover the extra information that allows us to value all other projects with a dependence on S. What is the extra information? It is a complete vector of state-contingent prices for risk, ϕ. With this vector in hand, we can easily value all projects dependent on S. There is no contradiction between risk-adjusted discounting using a risk-premium, λ, and risk-neutral pricing using state prices. They are tied together and, used correctly, do the same thing. But risk-neutral pricing is the more convenient and robust route for handling many different types of projects with different contingencies and changing risk profiles. 7 The Starting Point The binomial branching shown is for an underlying risk factor S. The risk factor starts in t=0 at the value S0, and branches in t=1 to either SU or SD. This figure shows the branching of S, but not those values. Instead, we overlay onto the branching diagram the characteristics of a project with cash flows contingent on S. At t=1 we show those contingent cash flows. At t=0 we show the project value. 8 4 From Lecture Notes on Advanced Corporate Financial Risk Management. Used with permission.The Starting Point (cont.) Assuming rf = 4%, the fact that VP = $10 implies ra = 7% so that λ = 3%. 9 10 The Complication: How Do We Handle Other Projects Also Contingent on S? 5 Figures from Lecture Notes on Advanced Corporate Financial Risk Management. Used with permission.Applying the Same Risk-premium is Wrong  It is tempting to apply the same risk-adjusted discount rate or risk- premium to the expected cash flows for each of these derivative projects. But this would be incorrect: E[CF ]e−ra =(CFU ×πU ) e−ra =(CFU ×πU ) e−λPe −rf ≠ VU E[CF ]e−ra =(CFD ×π D ) e−ra =(CFD ×π D ) e−λPe −rf ≠ VD  How do we know this? 11 Revalue the Project Using “Forward State Prices”  Rewrite the valuation of the project. But instead of discounting the whole expected cash flow by the risk-premium, λ, discount the two separate contingent cash flows, CFU and CFD, by two different discount factors, ϕU and ϕD: VP =(CFU ×πU ×φU + CFD ×π D ×φD ) e −rf  If this were a single equation, there would be no sense in trying to separate the two cash flows and applying two different discount factors. But, in fact, we can combine that one equation together with the