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* Credit risk* is the risk of default or delay in making interest or principal payments on a loan. On the other hand,

There are four credit risk measures for fixed income securities. These include:

- Loss given default.
- Probability of default.
- Expected loss.
- The present value of the expected loss.

* Expected exposure (EE)* is the amount that an investor or bondholder stands to lose at any given point in time in case of default. It does not factor in possible recovery. The

$$ \text{Loss severity} = 1 – \text{Recovery rate.} $$

* Loss given default (LGD)* is the amount of loss to the investor if a default occurs. The loss given default is a positive function of the expected exposure and a negative recovery rate function.

$$ \begin{align*} LGD & =\text{ Loss severity}\times \text{Expected exposure} \\ LGD & =\left(1-\text{Recovery rate}\right)\times \text{Expected exposure} \end{align*} $$

* Probability of default (PD)* is the likelihood of the bond issuer not paying the interest/principal amounts when due. In other words, it is the probability of default in any given year.

* The hazard rate* is the initial probability of default. The probability of default in every subsequent year is the conditional probability of default given that default had not previously occurred.

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

Where:

\(PD_t\) is the probability of default at any given year \(t\).

\(PS_{t-1}\) is the survival probability for the previous year.

The * expected loss* varies depending on the state of the economy and other microeconomic factors. For example, in a boom phase, the value of assets is high, and the probability of default is low—further, the value of the collateral increases. Therefore, LGD decreases, and so does the expected loss.

$$ \begin{align*} \text{Expected loss} \left(\%\right) &= LGD\times PD \\ \text{Expected loss} \left($\right) &=\text{Exposure}\times PD\times LGD \end{align*} $$

* The present value of the expected loss* is obtained by discounting the expected loss by risk-neutral probabilities. This is the preferred measure of credit risk as it includes the probability of default, loss given default, time value of money, and the risk premium in its calculation.

It is worth noting that the expected losses are computed using risk-neutral probabilities, and discounting is done at the risk-free rates for the relevant maturities.

CVA can also be determined by taking the difference between the risk-free value and the value of the risky bond. It captures investors’ compensation for bearing default risk.

$$ \text{CVA} = \text{Price of a riskless bond}\ – \text{Price of the risky bond.} $$

A zero-coupon corporate bond with a par value of $100 matures in four years. The risk-neutral probability of default (hazard rate) for the bond is 1%, and the recovery rate is 40%. The benchmark spot rate curve is constant at 4%. Calculate:

- Expected exposure (EE).
- Loss given default (LGD).
- Probability of survival (PS).
- Probability of default (PD).
- Credit valuation adjustment (CVA).
- Value of the risky bond.

$$ \begin{array}{c|c|c|c|c|c|c} \text{Year} & \text{EE} & \text{LGD} & \text{PD} & \text{PS} & \text{EL}& \text{PV of EL} \\ \hline 1 & 88.900 & 53.34 & 1.0\% & 99.000\% & 0.5334 & 0.5129 \\ \hline 2 & 92.456 & 55.47 & 0.9900\% & 98.010\% & 0.5492 & 0.5078 \\ \hline 3 & 96.154 & 57.69 & 0.9801\% & 97.030\% & 0.5654 & 0.5027 \\ \hline 4 & 100.00 & 60.00 & 0.9703\% & 96.060\% & 0.5822 & 0.4976 \\ \hline & & & & & \textbf{CVA} & \bf{2.0210} \end{array} $$

**Expected exposure (EE):**$$ \begin{align*} EE_1 & =\text{Par value discounted for 3 years at }4\%=\frac{100}{\left(1.04\right)^3}=88.90 \\ EE_2 &=\text{Par value discounted for 2 years at } 4\%=\frac{100}{\left(1.04\right)^2}=92.456 \\ EE_3 &=\text{Par value discounted for 1 year at } 4\%=\frac{100}{\left(1.04\right)^1}=96.154 \\ EE_4 &=\text{Par value of the bond}=100 \end{align*} $$

**Loss given default:**$$ \begin{align*} \text{Loss given default}&=\text{Expected exposure} \times(1-\text{Recovery rate}) \\ LGD_1 &=88.90\times\left(1-40\%\right)=53.34 \\ LGD_2 &=92.456\times\left(1-40\%\right)=55.47 \\ LGD_3 &=96.154\times\left(1-40\%\right)=57.69 \\ LGD_4 &=100\times\left(1-40\%\right)=60.00 \end{align*} $$

**Probability of default and probability of survival:**$$ \begin{align*} PD_t &=PS_{t-1}\times \text{Hazard rate} \\ \\ \text{Probability of survival}& = 1-{\text{Cumulative conditional} \\ \text{probability of default}} \end{align*} $$ $$ \begin{align*} PD_1 &=PS_0\times \text{Hazard rate}=100\%\times 1\% \\ & =1\% \\ PS_1 &=1-1\% \\ & =99\% \\ PD_2 &=PS_1\times \text{Hazard rate}=99\%\times1\% \\ & =0.99\% \\ PS_2 &=1-\left(1\%+0.99\%\right) \\ & =98.01\% \\ PD_3 &=PS_2\times \text{Hazard rate}=98.01\% \times 1\% \\ & =0.9801\% \\ PS_3 &=1-\left(1\%+0.99\%+0.9801\%\right) \\ & =97.03\% \\ PD_4 &=PS_3\times \text{Hazard rate}\\ & =97.03\%\times1\%=0.9703\% \\ PS_4 &=1-\left(1\%+0.99\%+0.9801\%+0.9703\%\right)\\ & =96.06\% \end{align*} $$

**The present value of the expected loss:**$$ \begin{align*} \text{Present value of expected loss}&=\text{LGD} \times\frac{\text{PD}}{\left(1+i\right)^t} \\ \text{PVEL}_1 &=53.34\times\frac{1\%}{1.04} =0.5129 \\ \text{PVEL}_2 &=55.47\times\frac{0.99\%}{{1.04}^2} =0.5077 \\ \text{PVEL}_3 &=57.69\times\frac{0.9801\%}{{1.04}^3} =0.5027 \\ \text{PVEL}_4 &=60\times\frac{0.9703\%}{{1.04}^4} =0.4976 \end{align*} $$

**Credit valuation adjustment (CVA):**CVA is the total present value of the expected loss

Thus,

$$ \text{CVA} = 0.5129+0.5077+0.5027+0.4976=${2}.{021} $$

**Price of the risky bond:**$$ \begin{align*} \text{CVA} & = \text{Price of the riskless bond} \\ & – \text{Price of the risky bond} \\ \text{Price of the risk-free bond} & =\frac{100}{{1.04}^4}=85.48 \\ \text{Price of the risky bond}&=\text{Price of the risk-free bond} -\text{CVA} \\ & =85.48-2.021=$83.46 \end{align*} $$

## Question

A zero-coupon, $100 par corporate bond issued today matures in three years. If the bond’s annual hazard rate is 1.5%, the probability that the bond survives over the next three years is

closest to:

- 98.50%.
- 97.02%.
- 95.57%.

Solution

The correct answer is C.$$ \begin{align*} \text{Probability of} & \text{ survival } \left(PS_t\right) \\ & = 1 -{\text{Cumulative conditional probability of default}} \end{align*} $$ $$ \begin{align*} PD_1 &=PS_0\times \text{Hazard rate}=100\%\times 1.5\% \\ & =1.5\% \\ PS_1 &=1-1.5\% \\ & =98.5\% \\ PD_2 &=PS_1\times \text{Hazard rate}=98.5\%\times1.5\% \\ & =1.4775\% \\ PS_2 &=1-\left(1.5\%+1.4775\%\right) \\ &=97.0225\% \\ PD_3 &=PS_2\times \text{Hazard rate} \\ & =97.0225\%\times1.5\% \\ & =1.4553375\% \\ PS_3 &=1-\left(1.5\%+1.4775\%+1.4553375\%\right) \\ & =95.57\% \end{align*} $$

Reading 31: Credit Analysis Models

*LOS 31 (a) Explain expected exposure, the loss given default, the probability of default, and the credit valuation adjustment.*