Recognizing the Uncertainty of Future Interest Rates I. Stochastic Interest Rate Valuation Models “How to value assets

1. Recognizing the Uncertainty of Future Interest Rates • I. Stochastic Interest Rate Valuation Models “How to value assets when interest rates change randomly.” • A. Simulation Approaches to the Term Structure • Insert 7.1, 7.2, & 7.3 • Simulation must be based upon an interest rate distribution that is consistent and converges to the term structure. • The advantage is that we are then able to value instruments that have cash flows which are sensitive to interest rate paths. • Insert 7.4 • B. Discrete-Time Models of the Term Structure • Lattice Models / Binomial Trees • In a lattice, a node is positioned at the beginning of each discrete unit of time • At each node, interest rates are assumed to increase, decrease or stay the same • Information about a unique, possible term structure is implicit in every node

2. II. The Importance of Interest-Sensitive Cash Flows • Interest-sensitive cash flows have their amounts and/or timing dependent upon the level of interest rates and/or the interest rate path. • Examples of interest-sensitive instruments: • floating rate instruments • callable bonds • mortgages • III. Valuation with Interest Sensitive Cash Flows • Arrow & Debreu: “State Preference Theory” Cash flows are dependent on the state of nature that occurs. • Insert 7.5 • Present value can deviate from the expected value! • Insert 7.6, 7.7, & 7.8 • This valuation technology for financial instruments is more versatile than the standard approach. • In general, interest-sensitive investments should not be valued by fixed spot rates because rates are not fixed and payments may vary with rate movements. • Insert 7.9 & 7.10

3. Yield-to-maturity is sensitive to the interest rate path taken. • Insert 7.11 & 7.12 • Volatility can also affect yields. • Interest-sensitive cash flows should be valued by the interest rate paths that give rise to them. • The overall value is then calculated by weighting and summing the resultant present values by the probability associated with each path. If the marketplace does not display risk neutrality, these probabilities should be adjusted further to reflect the market price of risk. • A good interest-sensitive model should converge to the spot rates revealed by the term structure. Question: How bad is the old method? Contrast valuation using fixed rate cash flow and the current approach. When are the answers close? Will they ever be the same?

4. IV. Appendix • Continuous-Time Approaches to the Term Structure • Models assume that time is continuous. Sophisticated techniques are needed to incorporate interest rate randomness and this continuity. • Single factor models focus on a single source of uncertainty in the economy. • An example is in Cox, Ingersoll, and Ross’ mean reverting process: • Insert A.1 & A.2 • Multi-factor models introduce other random variables. These often don’t have closed-form solutions and must therefore be solved by numerical methods.

5. V. Summary • Advanced valuation techniques incorporate the randomness of interest rates. • If the cash flows are of fixed size, the instrument can be valued by either the interest rate path or the relevant spot rates. • If the cash flows are path dependent, then only the interest rate path method should be used. The probabilities of these paths can then be adjusted to reflect the market price of risk. Use of the spot term structure will result in an error that is influenced by variables such as volatility and flow size. • Additionally, incorporation of interest rate volatility changes the shape of the term structure.