# Yearly ytm calculation on stock using binomial model

So I have been given this problem in class, and although I have no issues doing the binomial model on options, I cannot seem to get my head around the problem when its calculating ytm on just a stock.

Problem:

Assume a binomial model for two periods (t= 0,1,2) in which the equity hodlers hold 20% of the company and debt holders hold 80%. The company value is \$1.000. The UP branch increases value by 20% and DOWN branch decreases value by 30%. Risk free rate for any period is 10%.

Question

Prove that YTM at t=0 equals 15,21% annually.

Thanks in advance guys!

## 1 Answer

At the terminal date, the value will be: 1.44, 0.84, 0.84, 0.49 in the four states: UU, UD, DU, and DD, respectively.

The probability of an up move is: (1.1-0.7)/(1.2-0.7)=0.8

So the probability of the four terminal states are: 0.64, 0.16, 0.16. 0.04.

Easy to verify that the value is 1 at time zero: $$\frac{1}{1.1^2}\sum_{s=1}^{4}{V_{s,2}Q_{s,2}}$$

At 15.21%, the principal + interest amount of debt at t=2 is 1.062 (0.8*1.1521*1.1521). Now the debt holders get paid full amount in the UU state but less, which is the value of the company in the other terminal states. Calculate its discounted expected value and you get 0.8 as desired.