Yield to Maturity of an Infinitely Lived Bond in the Vasicek Model: Practice Problems and Solutions

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3F / Exam MFE, authored by Mr. Stolyarov. This is Section 85 of the Study Guide. See an index of all sections by following the link in this paragraph.

An important formula to memorize for the Vasicek model is the formula for r^- (r-bar), the yield to maturity of an infinitely-lived bond. This formula was already given in Section 70, but more repeated exposure to it in actual applications is needed to facilitate adequate memorization. The following five problems use the exact same formula and procedure. They are intended to immerse you in the usage of the formula. You are advised to try to solve the problems without looking at the formula or at the solutions. By the time you solve all five of the problems, you will have the formula memorized - adding another tool to your arsenal for this exam.

The formula for the yield to maturity of an infinitely lived bond in the Vasicek model is

r^- = b + σφ/a - 0.5σ²/a².

Meaning of variables:
φ = Sharpe ratio (assumed to be constant).

r^- = yield to maturity on an infinitely lived bond.

σ = volatility factor.

a = drift factor.

b = the mean around which mean reversion occurs.

r = the short-term interest rate.

Sources: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 24, pp. 786-787.

Here are some ideas to help with memorizing the formula.

Note that b is the only term that stands alone. The only term that ever appears in a denominator is a. The numerators of the non-b terms contain at least one σ.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem YMILBVM1. A particular Vasicek model has the following parameters:
a = 0.56; b = 0.347; σ = 0.66; φ = 0.99. Find the yield to maturity on an infinitely lived bond in this model.

Solution YMILBVM1. We use the formula r^- = b + σφ/a - 0.5σ²/a².

Thus, r^- = 0.347 + 0.66*0.99/0.56 - 0.5*0.66²/0.56² = 0.8129704082