LO 13.7: Calculate the Vasicek M odel rate change, standard deviation o f the rate

LO 13.7: Calculate the Vasicek M odel rate change, standard deviation o f the rate change, expected rate in T years, and half life.
The previous discussion encompassed the rate change in the Vasicek model and the computation of the standard deviation when solving for the parameters in the recombining tree. In this section, we turn our attention to the forecasted rate in T years.
To continue with the previous example, the current short-term rate is 6.2% with the mean-reversion parameter, k, of 0.03. The long-term mean-reverting level will eventually reach 18%, but it will take a long time since the value of k is quite small. Specifically, the current rate of 6.2% is 11.8% from its ultimate natural level and this difference will decay exponentially at the rate of mean reversion (11.8% is calculated as 18% – 6.2%). To forecast the rate in 10 years, we note that 11.8% x e(-0-3xio) _ 8.74%. Therefore, the expected rate in 10 years is 18% – 8.74% = 9.26%.
Page 172
2018 Kaplan, Inc.
Topic 13 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 9
In the Vasicek model, the expected rate in T years can be represented as the weighted average between the current short-term rate and its long-run horizon value. The weighting factor for the short-term rate decays exponentially by the speed of the mean-reverting parameter, 9:
rQe
+ 0(1 – e kT)
A more intuitive measure for computing the forecasted rate in T years uses a factors half- life, which measures the number of years to close half the distance between the starting rate and mean-reverting level. Numerically:
(18% – 6.2%)e–03T = V4(18% – 6.2%)
e 0,03t = Vi (cid:31) t = In(2) / 0.03 = 23.1 years
Professor’s Note: A larger mean reversion in a shorter h alf life.