Binomial Option Valuation Model
One-Period Binomial Option Valuation Model In the one-period binomial model, we start today... Read More
A corporate bond yield is made up of the benchmark yield and the credit spread.
The benchmark yield is affected by macroeconomic factors, including:
The credit spread incorporates microeconomic factors related to the corporate issuer and the specific issue. These factors include
The credit valuation adjustment (CVA) is designed to capture counterparty credit risk, liquidity costs, funding costs, and taxation effects in valuing derivatives.
Credit spreads vary with the investor’s changing beliefs about the expected probability of default and recovery rates. The expectations of the economic state influence these beliefs. For example, when the economy is expected to slow down, there will be an expectation of higher defaults and lower recovery rates.
A three-year, 5% annual coupon corporate bond is currently priced at $105. The Treasury yield curve is flat at 3%. Comparable bonds have a hazard rate of 2% and a recovery rate of 30%.
Credit spread is calculated in the following steps:
VND
The bonds VND is calculated as:
$$ VND=\frac{5}{1.03}+\frac{5}{{1.03}^2}+\frac{105}{{1.03}^3}=$105.66 $$
CVA
The workings of the CVA are shown in the following table:
$$ \begin{array}{c|c|c|c|c|c|c|c} \textbf{Year} & \textbf{EE} & \textbf{LGD} & \textbf{PD} & \textbf{PS} & \textbf{EL} & \textbf{DF} & \textbf{PV} \\ & & & & & & & \textbf{of} \\ & & & & & & & \textbf{EL} \\ \hline 1 & 108.83 & 76.18 & 2.0000\% & 98.0000\% & 1.5236 & 0.9709 & 1.4792 \\ \hline 2 & 106.94 & 74.86 & 1.9600\% & 96.0400\% & 1.4672 & 0.9426 & 1.3830 \\ \hline 3 & 105.00 & 73.50 & 1.9208\% & 94.1192\% & 1.4118 & 0.9151 & 1.2920 \\ \hline & & & & & & \textbf{CVA} & \bf{4.15} \end{array} $$
$$ \begin{align*} \text{The fair value of the bond} & = VND-CVA \\ \text{Fair value} &=$105.66 – 4.15=$101.51 \end{align*} $$
Notice that the bond’s market price is $105, which is greater than its fair value ($101.51). Thus, we can conclude that the bond is overvalued.
YTM
Bond’s YTM is determined as:
$$ \begin{align*} 101.51 &=\frac{5}{1+YTM}+\frac{5}{\left(1+YTM\right)^2}+\frac{105}{\left(1+YTM\right)^3} \\ YTM & = 4.45\% \end{align*} $$
Credit spread
$$ \begin{align*} \text{Credit spread} & =\text{YTM of the risky bond}\ – \text{Benchmark YTM} \\ & = 4.45\%-3.00\%=1.45\% = 145\text{bps} \end{align*} $$
Ryker Yash, a junior credit analyst, has projected that the expected economic recovery will half the probability of default and double the recovery rate. Yash’s instinct is that the decrease in default probability will narrow the credit spread more relative to doubling the recovery rate. We can check whether his intuition is correct:
VND remains constant at $105.66
Revised CVA = 2.10
Fair value = VND – CVA
Fair value = $105.66 – $2.10 = $103.56
YTM of the bond:
$$ 103.56\ =\frac{5}{1+YTM}+\frac{5}{\left(1+YTM\right)^2}+\frac{105}{\left(1+YTM\right)^3} $$
YTM = 3.72%
Credit spread = 3.72% – 3.00% = 0.72% = 72bps
Credit spread has roughly halved as a result of decreasing the probability of default by 50%.
VND remains constant at $105.66
Revised CVA = 2.37
Fair value = VND – CVA
Fair value = $105.66 – $2.37 = $103.29
YTM of the bond:
$$ 103.29 =\frac{5}{1+YTM}+\frac{5}{\left(1+YTM\right)^2}+\frac{105}{\left(1+YTM\right)^3} $$
YTM = 3.82%
Credit spread = 3.82% – 3.00% = 0.82% = 82bps
Notice that halving the default probability decreases the credit spread to 72 bps while doubling the recovery rate decreases the credit spread to 82 bps.
Thus halving the default probability has a greater impact on the credit spread than doubling the recovery rate.
Question
Consider a fixed 7% annual coupon, a three-year corporate bond with a par value of £1,000. The risk-neutral probability of default (the hazard rate) for each date for a bond is estimated to be 1.50%, and the recovery rate is 40%. The benchmark yield curve is flat at 2.50%. Due to the expected economic deterioration, a credit analyst decides to double the bond’s default probability and half the recovery rate.
The new credit spread of the bond relative to the original spread is most likely:
- Lower.
- Higher.
- The same as it was.
Solution
The correct answer is B.
Initial credit spread
$$\begin{align*}\text{Expected return} &=\text{Yield to maturity (YTM)}+ {\Delta}\%P \\ VND & =\frac{70}{1.025}+\frac{70}{{1.025}^2}+\frac{1,070}{{1.025}^3}=£ 1,128.52\end{align*} $$
CVA
$$ \begin{array}{c|c|c|c|c|c|c|c} \textbf{Year} & \textbf{EE} & \textbf{LGD} & \textbf{PD} & \textbf{PS} & \textbf{EL} & \textbf{DF} & \textbf{PV} \\ & & & & & & & \textbf{of} \\ & & & & & & & \textbf{EL} \\ \hline 1 & 1,156.73 & 694.04 & 1.5000\% & 98.500\% & 10.4106 & 0.9756 & 10.1567 \\ \hline 2 & 1,113.90 & 668.34 & 1.4775\% & 97.023\% & 9.8747 & 0.9518 & 9.3989 \\ \hline 3 & 1,070.00 & 642.00 & 1.4553\% & 95.567\% & 9.3433 & 0.9286 & 8.6762 \\ \hline & & & & & & \textbf{CVA} & \bf{28.23} \end{array} $$
$$ \text{Fair value} = 1,128.52 – 28.23 = £1,100.29 $$
YTM is calculated such that:
$$\begin{align*} 1,100.29& =\frac{70}{1+YTM}+\frac{70}{\left(1+YTM\right)^2}+\frac{1070}{\left(1+YTM\right)^3} \\ YTM &= 3.425\%\end{align*} $$
$$\begin{align*}\text{Credit spread} & = \text{YTM of the risky bond} – \text{Benchmark YTM} \\&= 3.425\% -2.50\% =0.925\% = 92.5 \text{bps}\end{align*}$$
New credit spread
VND = £1,128.52
CVA using new default probability and recovery rate = £74.20
Fair value = VND – CVA
Fair value = £1,128.52 – £74.20 = £1,054.32
YTM is calculated such that:
$$ 1,054.32 =\frac{70}{1+YTM}+\frac{70}{\left(1+YTM\right)^2}+\frac{1070}{\left(1+YTM\right)^3} $$
YTM = 5.00%
Credit spread = 5.00% – 2.500% = 2.50% = 250bps.
The new credit spread is, therefore, higher relative to the initial credit spread. (250bps>92.5bps)
Reading 31: Credit Analysis Models
LOS 31 (f) Interpret changes in a credit spread.