Limited Time Offer: Save 10% on all 2021 and 2022 Premium Study Packages with promo code: BLOG10    Select your Premium Package »

# Value of a Bond and its Credit Spread

The arbitrage-free framework is applied for credit analysis of a risky bond, assuming that interest rates are volatile. A binomial interest rate tree is constructed assuming no arbitrage. The tree is then verified if it has been correctly calibrated and used to value corporate bonds.

### Fixed coupon corporate bond

A fixed coupon corporate bond can be evaluated using the binomial interest rate tree in the following steps:

• Calculate the value of the bond assuming no default (VND)
• Determine the credit valuation adjustment (CVA)
• Work out the bond’s fair value, where fair value = VND – CVA
• Determine the yield to maturity (YTM) of the bond using fair value
• Calculate the credit spread of the bond using YTM
• Work out the credit spread, where credit spread = YTM of the risky bond – Benchmark YTM

Consider a four-year zero-coupon corporate bond with a par value of 1,000 and a flat government bond yield curve at 5%. The risk-neutral probability of default (hazard rate) for each date of the bond is 2%, and the recovery rate is 40%. 1. The fair value bond is closest to? 2. The YTM of the bond is closest to: 3. The credit spread of the bond is closest to: Solution 1. Fair value \begin{align*} \text{The value of the bond assuming no default (VND)} &= \frac{1000}{\left(1+0.05\right)^4} \\ & = 822.70 \end{align*} Recall the calculation of CVA from LOS (a) $$\begin{array}{c|c|c|c|c|c|c|c} \textbf{Year} & \textbf{EE} & \textbf{LGD} & \textbf{Hazard } & \textbf{PD} & \textbf{PS} & \textbf{EL} & \textbf{PV} \\ & & & \textbf{rate} & & & & \textbf{of} \\ & & & {} & & & & \textbf{EL} \\ \hline 1 & 863.84 & 518.30 & 2\% & 2.0000\% & 98.000\% & 10.37 & 9.8724 \\ \hline 2 & 907.03 & 544.22 & 2\% & 1.9600\% & 96.040\% & 10.67 & 9.6750 \\ \hline 3 & 952.38 & 571.43 & 2\% & 1.9208\% & 94.119\% & 10.98 & 9.4815 \\ \hline 4 & 1,000.00 & 600.00 & 2\% & 1.8824\% & 92.237\% & 11.29 & 9.2919 \\ \hline & & & & & & \textbf{CVA} & \bf{38.321} \end{array}$$ • EE = Expected exposure • LGD = Loss given default • PD = Probability of default • PS = Probability of survival • EL = Expected loss \begin{align*} \text{Fair value} & =\text{VND} -\text{CVA} \\ \text{Fair value} & = 822.70 – 38.32 =784.38 \end{align*} 2. YTM YTM of the bond can be determined as the rate that solves the equation: \begin{align*} 784.38 & =\frac{1000}{\left(1+YTM\right)^4} \\ YTM & = 6.26\% \end{align*} 3. Credit spread \begin{align*} \text{Credit spread} & = \text{YTM of the risky bond}\ – \text{Benchmark YTM} \\ \text{Credit spread} & = 6.26\% – 5\% = 1.26\% \end{align*} Thus, the compensation for credit risk received by the investor can be expressed in terms of: • The CVA, 38.32, which is the present value per 100 of par value on today • A credit spread of 126 basis points, which is an annual percentage rate for four years. ### Example 2: Volatility Assumption In the previous example, we have determined the fair value and the credit spread of a risky bond, assuming a flat benchmark yield curve. This example will use the binomial interest rate tree to calculate the VND and the expected exposure in a volatile interest rate environment. Given the spot rate curve for the annual payment benchmark Treasury, we can derive the discount Factors and forward rates as per the following table. $$\begin{array}{c|c|c|c} \textbf{Maturity} & \textbf{Spot rates} & \textbf{Discount Factors (DF)} & \textbf{Forward rates} \\ \hline 1 & 4.00\% & 0.96154 & 4.000\% \\ \hline 2 & 5.00\% & 0.90703 & 6.010\% \\ \hline 3 & 6.00\% & 0.83962 & 8.029\% \end{array}$$ Using the above forward rates, an iterative process has been used to generate the following binomial tree, assuming an interest rate volatility of 15%. $$\begin{array}{c|c|c} \textbf{Time 0} & \textbf{Time 1} & \textbf{Time 2} \\ \hline & & 10.69\% \\ \hline & 6.89\% & \\ \hline 4.000\% & & 7.86\% \\ \hline & 5.12\% & \\ \hline & & 5.90\% \end{array}$$ Consider a 5%, three-year corporate bond with a par value of1,000. The risk-neutral probability of default for this bond has been estimated as 2%, and the recovery rate as 40%

We can determine the fair value of the bond as follows:

The first step is to determine the value of the bond, assuming no default (VND). This is done through backward induction

N/B: The values in the above tree are subject to rounding errors

Year 3 cashflows are the principal (1,000) plus the coupon of 50

Year 2 values for the bond are obtained as follows:

\begin{align*} \frac{1050}{1.1069} &=948.59 \\ \frac{1050}{1.0786} &=973.48 \\ \frac{1050}{1.0590} & =991.50 \end{align*}

Year 1 Values:

\begin{align*} \frac{1}{2}\times\frac{\left(948.59+50\right)+\left(973.48+50\right)}{1.0689} & =945.86 \\ \frac{1}{2}\times\frac{\left(973.48+50\right)+\left(991.50+50\right)}{1.0512} &= 982.20 \end{align*}

Year 0 Value (VND)

$$\frac{1}{2}\times\frac{\left(945.86+50\right)+\left(982.20+50\right)}{1.04}=975.03$$

VND can also be determined by discounting the bond’s yearly cashflows at the relevant spot rates:

$$VND =\frac{50}{1.04}+\frac{50}{\left(1.05\right)^2}+\frac{1,050}{\left(1.06\right)^3}=975.03$$

However, the binomial tree is key for calculating the expected exposure at each node when computing the CVA.

\begin{align*} \text{The expected exposure at each year} & = \sum{\text{Value in node i at time t}} \\ & \times {\text{Probability}+\text{Coupon for year t}} \end{align*}

The next step is to calculate CVA.

$$\begin{array}{c|c|c|c|c|c|c|c} \textbf{Year} & \textbf{EE} & \textbf{LGD} & \textbf{PD} & \textbf{PS} & \textbf{EL} & \textbf{Discount} & \textbf{PV of} \\ & & & & & & \textbf{Factors} & \textbf{EL} \\ \hline 1 & 1,014.03 & 608.42 & 2.000\% & 98.00\% & 12.17 & 0.9615 & 11.70 \\ \hline 2 & 1,021.76 & 613.05 & 1.960\% & 96.04\% & 12.02 & 0.9070 & 10.90 \\ \hline 3 & 1,050.00 & 630.00 & 1.921\% & 94.12\% & 12.10 & 0.8396 & 10.16 \\ \hline & & & & & & \textbf{CVA} & \bf{32.76} \end{array}$$

\begin{align*} & {\text{The expected exposure for each year}} \\ & = \sum{\text{(Value at each node }}\times {\text{Probability}}) +{\text{Coupon}} \end{align*}

\begin{align*} \text{The expected exposure for year one} & =\left(945.88\times0.5\right)+\left( 982.18\times0.50\right) \\ & +50 \\ & =1,014.03 \\ \text{The expected exposure for year two} & = (948.57\times0.25)+\left( 973.48\times0.5\right)\\ & +\left(991.51\times0.25\right) +50 \\ & =1,021.76 \\ \text{The expected exposure for year three} & = 1,050 \end{align*}

\begin{align*} \text{Loss given default (LGD)} & = \left(1-\text{Recovery rate}\right)\times \text{Expected exposure} \\ LGD_1 & = 1,014.03\times\left(1-0.40\right)=608.42 \\ LGD_2 &= 1,021.76\times\left(1-0.40\right)=613.05 \\ LGD_3 &=1,050\times\left(1-0.40\right)=630.00 \end{align*}

The probability of default is calculated using the formula:

$$PD_t=PS_{t-1}\times \text{Hazard rate}$$

Where:

\begin{align*} \text{Probability of survival } (PS_t) &= 1- {\text{Cumulative conditional} \\ \text{probability of default}} \\ \text{Expected loss} & = LGD\times PD \\ \text{PV of expected loss} &=LGD\times\frac{PD}{\left(1+i\right)^t} \\ \end{align*}

CVA = sum of the present value of the expected loss for each period.

CVA = $32.76 VND =$975.03

\begin{align*} \text{The fair value of the bond} & = VND\ – CVA \\ \text{Fair value}&= 975.03-32.76=942.27 \end{align*}

Note that changes in the interest rate volatility have minimal effect on a corporate bond’s fair value. The volatility assumption has more weight on bonds with embedded options.

Similar to a fixed-coupon corporate bond, the arbitrage-free framework can also be used to analyze a floater as follows:

1. Determine the VND given the quoted margin
2. Calculate the CVA
3. Fair value = VND – CVA
4. Determine the discount margin by trial and error.

## Question

A $1,000 par, 6% annual coupon corporate bond matures five years from today. The bond is currently priced with a credit spread of 150 bps over the benchmark par rate of 3%. The bond’s CVA is closest to: 1.$43.63
2. $60.00 3.$1,084.86

#### Solution

$$CVA = VND- \text{Price of risky bond}$$

VND is the value of the cash flows arising from the bond, discounted at the benchmark par rate

In this case,

$$VND=\frac{60}{1.03}+\frac{60}{{1.03}^2}+\frac{1060}{{1.03}^3}=1,084.86$$

Price of the risky bond using credit spread = 3% benchmark rate + 1.5% = 4.5%. We will therefore discount the bond’s cash flows assuming a YTM of 4.5%

\begin{align*} \text{Price of risky bond} &=\frac{60}{1.045}+\frac{60}{{1.045}^2}+\frac{1060}{{1.045}^3}=1,041.23 \\ CVA &=1,084.86-1,041.23=43.63 \end{align*}

LOS 31 (e) Calculate the value of a bond and its credit spread, given assumptions about the credit risk parameters.

Featured Study with Us
CFA® Exam and FRM® Exam Prep Platform offered by AnalystPrep

Study Platform

Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success
Online Tutoring
Our videos feature professional educators presenting in-depth explanations of all topics introduced in the curriculum.

Video Lessons

Daniel Glyn
2021-03-24
I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
michael walshe
2021-03-18
Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.
Nyka Smith
2021-02-18
Every concept is very well explained by Nilay Arun. kudos to you man!