PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

184 MODELING FINANCIAL SCENARIOSrate models and determine that based on monthly data from 1964through 1989 the value of ° is approximately 1.5.Models of the type shown in Formula (2.1) are called “equilibriummodels” since investors price bonds by responding tothe known expectations of future interest rates. Using the assumedprocess for short-term rates, one can determine the yieldon longer-term bonds by looking at the expected path of interestrates until the bond’s maturity. To determine the full-term structure,one can price bonds of any maturity based on the expectedevolution in short-term rates over the life of the bond: 2" Ã Z !#TP(t,T)=E exp ¡ r u du(2.2)where P(t,T) isthetimet price of a bond that pays $1 in (T ¡ t)years. One of the primary advantages of equilibrium models isthat bond prices and many other interest rate contingent claimshave closed-form analytic solutions. Vasicek and CIR evaluateFormula (2.2) to find bond prices:P(t,T)=A(t,T)e ¡r t B(t,T) , (2.3)where A(t,T) andB(t,T) are functions of the known process parameters·, , and¾. Therefore, given a realized value for r t ,rates of all maturities can be obtained.One immediate problem with equilibrium models of the termstructure is that the resulting term structure is inconsistent withobserved market prices, even if the parameters of the model arechosen carefully; while internally consistent, equilibrium modelsare at odds with the way the market is actually pricing bonds.Where equilibrium models generate the term structure as an output,“arbitrage-free models” take the term structure as an input.All future interest rate paths are projected from the existing yieldcurve.2 It should be noted that the expectations in Formula (2.2) are evaluated under the riskneutral measure. See chapter 9 of Tuckman [42] for an introduction to risk neutral valuationof bonds.t