计算VND和CVA时的概率数值

问题如下：

Ibarra performs the analysis assuming an upward-sloping yield curve and volatile interest rates. Exhibit 2 provides the data on annual payment benchmark government bonds. She uses this data to construct a binomial interest rate tree (shown in Exhibit 3) based on an assumption of future interest rate volatility of 20%.

Exhibit 3. One-Year Binomial Interest Rate Tree for 20% Volatility

As previously mentioned, Ibarra is considering a future interest rate volatility of 20% and an upward-sloping yield curve, as shown in Exhibit 2. Based on her analysis, the fair value of bond B2 is closest to:

选项：

A.

€1,101.24.

B.

€1,141.76.

C.

€1,144.63.

解释：

Correct Answer: A

The following tree shows the valuation assuming no default of bond B2, which pays a 6% annual coupon.

The scheduled year-end coupon and principal payments are placed to the right of each forward rate in the tree. For example, the Date 4 values are the principal plus the coupon of 60. The following are the four Date 3 values for the bond, shown above the interest rate at each node:

€1,060/1.080804 = €980.75

€1,060/1.054164 = €1,005.54

€1,060/1.036307 = €1,022.86

€1,060/1.024338 = €1,034.81

These are the three Date 2 values:

So, the value of the bond assuming no default (VND) is 1,144.63. This value could also have been obtained more directly using the benchmark discount factors from Exhibit 2:

€60 × 1.002506 + €60 × 0.985093 + €60 × 0.955848 + €1,060 × 0.913225 = €1,144.63.

The benefit of using the binomial interest rate tree to obtain the VND is that the same tree is used to calculate the expected exposure to default loss. The credit valuation adjustment table is now prepared following these steps:

Step 1: Compute the expected exposures as described in the following, using the binomial interest rate tree prepared earlier.

The expected exposure for Date 4 is €1,060.

The expected exposure for Date 3 is:

[(0.1250 × €980.75) + (0.3750 × €1,005.54) + (0.3750 × €1,022.86) + (0.1250 × €1,034.81)] + 60 = €1,072.60.

The expected exposure for Date 2 is:

[(0.25 × €1,008.76) + (0.50 × €1,043.43) + (0.25 × €1,067.73)] + €60 = €1,100.84.

The expected exposure for Date 1 is:

[(0.50 × €1,063.57) + (0.50 × €1,099.96)] + 60 = €1,141.76.

Step 2: LGD = Exposure × (1 – Recovery rate)

Step 3: The initial POD, also known as the hazard rate, is provided as 1.50%. For subsequent dates, POD is calculated as the hazard rate multiplied by the previous dates’ POS.

For example, to determine the Date 2 POD (1.4775%), the hazard rate (1.5000%) is multiplied by the Date 1 POS (98.5000%).

Step 4: POS is determined by subtracting the hazard rate from 100% and raising it to the power of the number of years:

(100% – 1.5000%)1 = 98.5000%

(100% – 1.5000%)2 = 97.0225%

(100% – 1.5000%)3 = 95.5672%

(100% – 1.5000%)4 = 94.1337%

Step 5: Expected loss = LGD × POD

Step 6: Discount factors (DF) in Year T are obtained from Exhibit 2.

Step 7: PV of expected loss = Expected loss × DF

Fair value of the bond considering CVA = €1,144.63 – CVA = €1,144.63 – €43.39 = €1,101.24.