 A corporate bond yield is made up of the benchmark yield and the credit spread.

1. Benchmark yield

The benchmark yield is affected by macroeconomic factors, including:

• The expected inflation rates.
• The expected real rate of return.
• Risk aversion for uncertainty regarding inflation.

The credit spread incorporates microeconomic factors related to the corporate issuer and the specific issue. These factors include

• The expected loss from default.
• Liquidity.
• Taxation.
• Risk aversion for uncertainty regarding credit risk, liquidity, and tax factors.

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%. 1. The credit spread of this bond is closest to: Credit spread is calculated in the following steps: • Determine the value of the bond given no default (VND). • Calculate the credit valuation adjustment of the bond (CVA). • Determine the fair value of the bond, where fair value = VND – CVA. • Determine the yield to maturity (YTM) of the bond that results in the fair value. • Calculate the credit spread using the relationship: • Credit spread = YTM of risky bond – YTM of benchmark bond. 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 is105, 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*} 2. 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: 1. Decreasing the default probability to 1% while maintaining the recovery rate at 30% VND remains constant at105.66

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%. 2. Doubling the default probability to 60%, with maintaining the default probability at 2%. 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:

1. Lower.
2. Higher.
3. The same as it was.

#### Solution

\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*}

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)

LOS 31 (f) Interpret changes in a credit spread.

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