Problems in Credit Risk Management

September 16, 2009

By A. V. Vedpuriswar

2

Problem

There are 10 bonds in a portfolio. The probability of default for each of the bonds over the coming year is 5%.These probabilities are independent of each other. What is the probability that exactly one bond defaults?

3

Required probability = (10)(.05)(.95)9

= .3151= 31.51%

4

Problem

A CDS portfolio consists of 5 bonds with zero default correlation. One year default probabilities are : 1%, 2%, 5%,10% and 15% respectively. What is the probability that that the protection seller will not have to pay compensation?

5

Probability of no default = (.99)(.98)(.95)(.90)(.85)= .7051= 70.51%

66

Problem If the probability of default is 6% in year 1 and 8%

in year 2, what is the cumulative probability of default during the two years?

77

Solution Probability of default not happening in both

years =(.94) (.92) = =

.8648 Required probability = 1 － .8648 =

.1352 =

13.52%

8

Problem

The 5 year cumulative probability of default for a bond is 15%. The marginal probability of default for the sixth year is 10%. What is the six year cumulative probability of default?

9

Required probability = 1- (.85)(.90)= .235= 23.5%

10

Problem

Calculate the implied probability of default if the one year T Bill yield is 9% and a 1 year zero coupon corporate bond is fetching 15.5%.

11

Solution

Let the probability of default be (1-p) Returns from corporate bond = 1.155p + (0) (1-p) Returns from treasury = 1.09

1.155p = 1.09p = 1.09/1.155 = .9437

Probability of default = .0563 = 5.63%

12

Problem

In the earlier problem, if the recovery is 80% in the case of a default, what is the default probability?

13

Solution

1.155p + (.80) (1.155) (1 － p) = 1.09 .231p = 0.166 p = .7186 1 － p = .2814

1414

Problem If 1 year and 2 year T Bills are fetching 11% and

12% and 1 year and 2 year corporate bonds are yielding 16.5% and 17%, what is the marginal probability of default for the corporate bond in the second year?

1515

Solution Yield during the 2nd year can be worked out

as follows: Corporate bonds: (1.165) (1+i) = 1.172

i = 17.5% Treasury : (1.11) (1+i) = (1.12)2

i = 13.00% p (1.175) + (1-p) (0) = 1.13 p = .9617 Default probability = 1

－ .9617 = 3.83%

16

Problem

The spread between the yield on a 3 year corporate bond and the yield on a similar risk free bond is 50 basis points. The recovery rate is 30%. What is the

cumulative probability of default over the three year period?

17

Solution

Spread = (Probability of default) (loss given default) or .005 = (p) (1-.3) or p = = .00714

= .71% per year No default over 3 years= (.9929) (.9929) (.9929)

= .9789 So cumulative probability of default = 1 – 9789 = .0211

= 2.11%

7.005.

18

Problem

The spread between the yield on a 5 year bond and that on a similar risk free bond is 80 basis points. If the loss given default is 60%, estimate the average probability of default over the 5 year period. If the spread is 70 basis points for a 3 year bond, what is the probability of default over years 4, 5?

19

Solution

Probability of default over the 5 year period

= = .0133 Probability of default over the 3 year period

= = .01167

(1-.0133)5 = (1-.01167)3 (1-p)2

or (1-p)2 = = .9688 or 1 – p = .9842 or p = .0158 = 1.58%

6.008.

6.007.

9654.9352.

20

Problem

A four year corporate bond provides a 4% semi annual coupon and yields 5% while the risk free bond, also with 4% coupon yields 3% with continuous compounding.

Defaults may take place at the end of each year. The recovery rate is 30% of the face value. What is the risk neutral default probability?

21

Solution

Risk free bondRisk free bond Corporate bondYear Cash

flowPV

factor e-(.03)t

PV PV factore-(.05)t

PV

.5 2 .9851 1.9702 .9754 1.95081.0 2 .9704 1.9408 .9512 1.90241.5 2 .9560 1.9120 .9277 1.85542.0 2 .9418 1.8836 .9048 1.80962.5 2 .9277 1.8554 .8825 1.76503.0 2 .9139 1.8278 .8607 1.72143.5 2 .9003 1.8006 .8395 1.6794.0 102 .8869 90.4638 .8187 83.51

103.6542

96.21 So expected value of losses = 103.65 – 96.21 =

7.44

Let the default probability per year = Q. The recovery rate is 30 %. So if the notional principal is 100, we can recover 30. We can work out the present value of losses assuming

the default may happen at the end of years 1, 2, 3, 4. Accordingly, we calculate the present value of the risk

free bond at the end of years 1, 2, 3, 4. Then we subtract 30 being the recovery value each year.

We then calculate the present value of the losses

using continuously compounded risk free rate.

22

23

Solution (Cont…) PV factors e-.015 = .9851 e-.030 = .9704 e-.045 = .9560 e-.060 = .9417 e-.075 = .9277 e-.09 = .9139

24

Solution (Cont…)

Time of default = 1

PV of risk free bond= 2+2e-.015+2e-.030+2e-.045+ 2e-.060+2e-075+ (102)e-.090

= 2[1+.9851+.9704+.9560+.9417+.9277]+(102)(.9139)

= 11.56 + 93.22 = 104.78Time of default = 2

PV of risk free bond = 2[1+.9851+.9704 +.9560]+(102) (.9417) = 7.823 + 96.05 = 103.88

25

Solution (Cont…)

Time of default = 3

PV of risk free bond= [1+ .9851] + (102) (.9704)

= 3.97 + 98.98 = 102.95Time of default = 4

PV of risk free bond = 102

26

Solution (Cont…)

Default point (Years)

Expected losses PV

1 (104.78 – 30)Q = 74.78Q

(74.78)Qe-.03 = 72.57Q

2 (103.88 – 30)Q = 73.78Q

(73.88)Qe-.06 = 69.58Q

3 (102.95 – 30)Q = 72.95Q

(72.95)Qe-.09 = 66.67Q

4 (104 – 30)Q = 72 Q

(72)Qe-.12 = 63.86Q

272.68 Q

So we can equate the expected losses: i.e, 7.44 = 272.68Q or Q = .0273 = 2.73%

2727

Problem A bank has made a loan commitment of $ 2,000,000

to a customer. Of this, $ 1,200,000 is outstanding. There is a 1% default probability and 40% loss given default. In case of default, drawdown is expected to be 75%. What is the expected loss?

2828

Solution Drawdown in case of default

= (2,000,000 – 1,200,000) (.75) = 600,000

Adjusted exposure = 1,200,000 + 600,000 =

1,800,000 Loss given default

= (.01) (.4) (1,800,000) = $ 7,200

29

Problem

A bank makes a $100,000,000 loan at a fixed interest rate of 8.5% per annum.

The cost of funds for the bank is 6.0%, while the operating cost is $800,000.

The economic capital needed to support the loan is $8 million which is invested in risk free instruments at 2.8%.

The expected loss for the loan is 15 basis points per year.

What is the risk adjusted return on capital?

30

Solution

Net profit = 100,000,000 (.085 - .060 - .0015) – 800,000 + (8,000,000) (.028)

= 23,50,000 – 800,000 + 224,000 = $1,774,000 Risk adjusted return on capital = 1.774/8 = .22175 =

22.175%

31

Problem

Using the Merton Model, calculate the value of the firm’s equity.

The current value of the firm’s equity is $60 million and the value of the zero coupon bond to be redeemed in 3 years is $50 million.

The annual interest rate is 5% while the volatility of the firm value is 10%.

32

Solution

Formula is: St = V x N(d) – Fe-r(T-t) x N (d-T-t) d =

V = value of firm F = face value of zero coupon debt = firm value volatility r = interest rate

tTtT

FeVnl tTr

21)(

33

Solution

S = 60 x N (d) – (50)e-(.05)(3) x N (d-(.1)3) d =

= = 2.005

S = 60 N (2.005) – (50) (.8607) N (2.005 - .17321) = 60 N (2.005) – (43.035) N (1.8318) = (60) (.9775) – (43.035) (.9665) = $17.057 million

3)1(.21

310.5060

)3)(05(.

eenl

)7321.1(21

17321.332.

34

Problem

In the earlier problem, calculate the value of the firm’s debt.

35

Solution

Dt = Fe-r(T-t) – pt

= 50e-.05(3) – pt

= 43.035 – pt

Based on put call parity pt = Ct + Fe-r(T-t) – V Or pt = 17.057 + 43.035 – 60 = .092 Dt = 43.035 - .092 = $42.943 million Alternatively, value of debt = Firm value – Equity value = 60 – 17.057 = $42.943 million

36

Consider the following figures for a company. What is the probability of default? – Book value of all liabilities : $2.4 billion– Estimated default point, D : $1.9 billion– Market value of equity : $11.3 billion– Market value of firm : $13.8 billion– Volatility of firm value : 20%

Problem

37

Distance to default (in terms of value) = 13.8 – 1.9 = $11.9

billion Standard deviation = (.20) (13.8)

= $2.76 billion

Distance to default (in terms of standard deviation) = 4.31

We now refer to the default database. If 5 out of 100 firms with distance to default = 4.31

actually defaulted, probability of default = .05

Solution

38

Given the following figures, compute the distance to default:– Book value of liabilities : $5.95 billion– Estimated default point : $4.15 billion– Market value of equity : $ 12.4 billion– Market value of firm : $18.4 billion– Volatility of firm value : 24%

Problem

39

Distance to default (in terms of value) = 18.4 – 4.15 = $14.25 billion

Standard deviation = (.24) (18.4) = $4.416 billion

Distance to default (in terms of standard deviation) = 3.23

Solution