Indexed sinking fund debentures: valuation and analysis - includes appendices - Security Design Special Issue

Financial Management (Financial Management Association), Summer, 1993 by John D. Finnerty

(A-5) The instantaneous market risk premium on any default-free security may be expressed as

(1/2 ||Sigma~.sup.2~|Lambda~(t)r/P)|Delta~P/|Delta~r

where |Lambda~(t) |is less than~ 0.

Equation (2) states that changes in r have a deterministic component and a random component. The deterministic component, -|Kappa~(r - |Theta~(t))dt, states that r tends to drift toward a steady-state mean value, |Theta~(t), which may itself vary over time. The rate r tends to revert to |Theta~(t) at a rate equal to |Kappa~; the magnitude of the expected adjustment equals the mean-reversion speed |Kappa~ multiplied by the magnitude of the differential, r - |Theta~(t). The random component of the change in r is, under the assumption of a standard Wiener process, distributed normally with mean zero and instantaneous variance ||Sigma~.sup.2~rdt.

Under the foregoing assumptions, the price P(r, t) of a default-free security, expressed as a function of rate and time, must satisfy the equation

1/2||Sigma~.sup.2~r||Delta~.sup.2~P/|Delta~|r.sup.2~ |Kappa~(|Theta~(t) -r)|Delta~P/|Delta~r |Delta~P/|Delta~t - 1/2||Sigma~.sup.2~|Lambda~(t)r|Delta~P/|Delta~r c(r, t) - rP = 0 (3)

where c(r, t) is the cash debt service payment at time t. The solution to Equation (3) must satisfy the boundary condition

P(r, T) = 1, all r |is greater than or equal to~ 0. (4)

In order to apply Equations (3) and (4) to value a default-free security, the parameters in Equation (3) must be calibrated to the prices of default-free (i.e., Treasury) bonds observed in the marketplace. The parameters are chosen so as to produce the Treasury term structure that is consistent with the prices at which the on-the-run Treasury securities are trading on the day of estimation.(4)

Denote the discount function that corresponds to the current Treasury term structure by d(T), 0 |is less than or equal to~ T |is less than or equal to~ 30 years. Note that for the special case of a lump sum payment, Equations (3) and (4) yield an analytic solution of the form

d(r, t; T) = exp(a(t ; T) b(t ; T)r), (5)

where d(r, t; T) denotes the value of the discount function at time t when the lump sum payment will be received at time T and the interest rate is r. The coefficients a and b solve the system of ordinary differential equations

da/dt = -|Kappa~|Theta~(t) b (6a)

db/dt = 1 (|Kappa~ 1/2||Sigma~.sup.2~|Lambda~(t))b -1/2||Sigma~.sup.2~|b.sup.2~ (6b)

with boundary condition a(T ; T) = b(T ; T) = 0. The Vasicek-Fong |20~ procedure is used (on a daily basis) to determine the Treasury discount function implied by the observed prices of the on-the-run Treasury securities. Then the parameters |Theta~(t) and |Lambda~(t) can be chosen so that the analytic solution (5) exactly reproduces the Treasury discount function:

d(|r.sub.0~, 0; T) = d(T), all T |is greater than or equal to~ 0,

where d(T) is today's discount function and |r.sub.0~ = -d|prime~(0) is today's short-term rate. In fact, once the mean-reversion speed |Kappa~ is specified, |Theta~(t) and |Lambda~(t) can be chosen to reproduce the discount function (and hence the term structure) for all levels of rate volatility |Sigma~, as explained in Appendix A.