Deriving Debt Yield to Maturity in a Binomial Firm-Value Model

Summary

The document works through a two-period binomial example in which a company’s value can rise or fall, shareholders own a stated fraction, and debt holders own the remainder. It first derives the risk-neutral probability of an up move from the up and down factors and the risk-free rate, then assigns probabilities to the terminal company-value states. Discounting the expected terminal value confirms the initial company value in the example.

To obtain the debt yield to maturity, the answer compares the promised terminal debt payoff at a candidate annual rate with the actual payoff across states. Debt is paid in full when company value is sufficient and receives only the available company value in lower-value states. Discounting the risk-neutral expected debt payoff at the risk-free rate and matching it to the initial debt value supports the stated yield. This is an illustration tied to its specific assumptions and state payoffs; it does not establish a general yield for equity or debt outside the model.

Key ideas

• Risk-neutral probabilities in the binomial model are derived from the risk-free rate and the up and down factors.
• Terminal firm values and their state probabilities determine the risk-neutral expected payoff.
• Debt repayment is capped by firm value in states where the company cannot meet its promised amount.
• The candidate debt yield is checked by discounting the expected state-contingent debt payoff to its initial value.
• The result depends on the example’s specific binomial assumptions and payoff structure.

Tags

# Yearly ytm calculation on stock using binomial model


# 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!

## Answer by Magic is in the chain (score 1)

https://quant.stackexchange.com/a/46823

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.