101 Concepts for the Level I Exam

Essential Concept 66: Confirming the Arbitrage-Free Value of a Bond

To value an option free bond, we can either use the sport rates or a binomial tree. Since both the methods are arbitrage free, the two values should be the same.

Consider an option-free bond with four years remaining to maturity, a coupon rate of 2%, and a par value of $100. Assume spot rates are as shown in Exhibit 3.

Maturity (Years)	One-Year Spot Rate
1	1.000%
2	1.201%
3	1.251%
4	1.404%
5	1.819%

Then the bond value can be calculated as:

$\frac{\$2}{{(1.01)}^1}+\ \frac{\$2}{{(1.01201)}^2}+\ \frac{\$2}{{(1.01251)}^3}+\ \frac{\$100+\ \$2}{{(1.01404)}^4}=\$102.33$

Next consider a binomial interest rate tree calibrated to the same spot curve (Exhibit 13).

We can then use the backward induction discounting process to obtain the bond value.

Time 0	Time 1	Time 2	Time 3	Time 4
102.3254	102.6769	101.7639	101.1892	102
	104.0204	102.8360	101.9027	102
		103.6417	102.4380	102
			102.8382	102
				102

The tree produces the same value for the bond as the spot rates and is therefore consistent with our standard valuation model.