Lecture 6

LECTURE SIX

1

a. Portfolio performance measurement

b. Hedge fund risk management

c. Credit risk management

d. Probability of default

PORTFOLIO PERFORMANCE MEASUREMENT

Part 1

2

a. Intro. Performance measurement

b. Surplus at Risk

c. The Market Line (ML) and CAPM

d. Several ratios to measure market performance

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1. Intro. Performance measurement

Portfolio management

Risk Return

Do profits reflect my risk exposure?

HOW?

• Looking at process/strategies in place, and

• Whether outcomes are in line with what was intended

or should have been achieved.

• Pure luck?

• Good strategy

• Separate effect of the market and active management

Sell side Buy side

Creation, promotion, analysis and

sale of securities.

Leverage and speculative

Examples?

Final buyers of financial assets

(Large portions of securities)

Tend to me more conservative

No leverage

Examples?

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1. Intro. Performance measurement Some definitions

Sell side Buy side

Creation, promotion, analysis and

sale of securities.

Leverage and speculative

Institutions such as

• Investment bankers

(intermediaries between issuers

and public),

• Research companies that

perform stock research and

make ratings. Ig. Roubini

• Market makers who provide

liquidity in the market.

Final buyers of financial assets

(Large portions of securities)

Tend to me more conservative

No leverage

Investing institutions such as

• mutual funds,

• pension funds and

• insurance firms

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1. Intro. Performance measurement Some definitions

Absolute and Relative risk

Absolute risk Relative Risk

With respect to the portfolio

itself

• Risk factor: σ • P: Initial portfolio value • In dollar terms

With respect to a benchmark

• Tracking error : e = RP - RB

• In dollar terms: e x P

𝜎 ∆𝑃

𝜎 ∆𝑃

𝑃∗ 𝑃

𝜎 𝑅𝑃 ∗ 𝑃

Note that there is only

the asset or portfolio

𝜎 (𝑅𝑃 − 𝑅𝐵) 𝑃

𝜎 (∆𝑃

𝑃−

∆𝐵

𝐵) 𝑃

𝜔 ∗ 𝑃

Tracking Error Volatility

Where

PBPP222 2

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Absolute risk Relative Risk

SPX: 10% return -10% return

My trade: 6% return -4% return

What would be the difference between absolute

and relative risk?

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Absolute and Relative risk

Example


2. Surplus at Risk Performance Measurement

Assets 120,00$ Liabilities 100,00$

Volatility 12% Volatility 3%

Expected R 8% Expected R 5%

0,3

Change of assets

(per period)9,60$

Change of assets

(per period)5,00$

4,60$

24,60$

190,44$

13,80$

95%

1,64

18,03$

Variance of surplus

Volatility of surplus

Confidence level

Normal deviate

VaR

Surplus 20,00$

Correlation

Expected growth of surplus

Expected surplus

Change assets - liabilities Considering the surplus

variance(a – b) = variance (a) + variance(b) - (2)cov(a,b)

Sort of relative risk measurement

Two assets, long pension assets and short pension liabilities.

The complete variance formula is: σ12P1

2+ σ22P1

2-2 σ1σ2P1P2ρ

Surplus at Risk –(expected surplus) + (volatility of surplus)*(normal deviate)

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12% of $120 5% of $100

What is my risk in the long term?

Focused on net profits

Riski

E(Ri)

M

RF

RiskM

E(RM)

ML Overvalued

Undervalued

Note: Risk is either b or

Decompose total return into a component due to market risk premia and

other factors.

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Small recap 3. The Market Line (ML) and CAPM

Riski

E(Ri)

M

RFR

RiskM

E(RM)

ML

A

B

C

D

E

Note: Risk is either b or

])([)( fMfi rRErRE b


iMiii eRR b

])([)( fMiifi rRErRE b

This is the CAPM model

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3. The Market Line (ML) and CAPM Small recap

Capital Market Line (CML)

obtained by combining the market portfolio and the riskless asset

• CML specifies the expected return for a given level of risk

• All possible combined portfolios lie on the CML, and all are Mean-Variance

efficient portfolios

• Here we have a clear relation between the risk of my portfolio and the risk

of the market. This is reflected by beta

M

Mii

RRCov2

),(

b

It measures how much an asset’s return

is driven by the market return

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If the stock has a high positive β:

• It will have large price swings driven by the market

• It will increase the risk of the investor’s portfolio(in fact, will make the

entire market more risky …)

• The investor will demand a high Er in compensation.

If the stock has a negative β : • It moves “against” the market.

• It will decrease the risk of the market portfolio

• The investor will accept a lower Er

Then the SML depicts the relation between β and the Expected Return (Er)

For the risk-free security, b = 0

For the market itself, b=1.

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Excess of return of a portfolio is a function of the excess of return of the market

W.R. a risk free rate

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4. Treynor Ratio The Treynor measure calculates the risk premium per unit of risk (bi)

Beta measures the investment volatility relative to the market volatility (systematic risk)

The Treynor Ratio is negative if

• RF > E[RP] AND β > 0 .

Manager has performed badly: failing to get performance better than

the risk free rate AND manager made a not good election of

portfolio

• RF < E[RP] AND β > 0

Manger has performed well, managing to reduce risk but getting a

return better than the risk free rate Higher Ti generally indicates better performance

)(

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P

FP

R

RRETR

b

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4. Treynor Ratio ADVANTAGE: It indicates the volatility a ASSET brings to an entire portfolio. The Treynor Ratio should be used only as a ranking mechanism for investments within the same sector. .

M

Mii

RRCov2

),(

b

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When presented with investments that have the same return, investments

with higher Treynor Ratios are less risky and better managed

5. Sharpe Ratio Describes how much excess return you are receiving for the

extra volatility that you endure for holding a riskier asset.

• Tells us whether a portfolio's returns are due to smart investment

decisions or a result of excess risk.

• The greater a portfolio's Sharpe ratio, the better its risk-adjusted

performance has been.

• A negative Sharpe ratio indicates that a risk-less asset would perform

better than the security being analysed.

)(

])([

P

FP

R

RRESR

The Sharpe measure is exactly the same as the Treynor measure, except

that the risk measure is the standard deviation:

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4-5. Sharpe V Treynor Ratio The Sharpe and Treynor measures are similar, but different:

• Sharpe uses the standard deviation, Treynor uses beta

• Sharpe is more appropriate for well diversified portfolios, Treynor for

individual assets

• Sharpe and Treynor: The ranking, not the number itself, is what is most

important

Portfolio Return RFR Beta Std. Dev. Trenor Sharpe

X 15% 5% 2.50 20% 0.0400 0.5000

Y 8% 5% 0.50 14% 0.0600 0.2143

Z 6% 5% 0.35 9% 0.0286 0.1111

Market 10% 5% 1.00 11% 0.0500 0.4545

Risk vs Return

0%

5%

10%

15%

0.00 0.50 1.00 1.50 2.00 2.50Beta

Ret

urn

MX

Y

Z

Risk vs Return

0%

5%

10%

15%

0% 5% 10% 15% 20%Std. Dev.

Ret

urn

M

X

YZ


X 15% 5% 2.50 20% 0.0400 0.5000

Y 8% 5% 0.50 14% 0.0600 0.2143

Z 6% 5% 0.35 9% 0.0286 0.1111

Market 10% 5% 1.00 11% 0.0500 0.4545

Risk vs Return

0%

5%

10%

15%

0.00 0.50 1.00 1.50 2.00 2.50Beta

Ret

urn

MX

Y

Z

Risk vs Return

0%

5%

10%

15%

0% 5% 10% 15% 20%Std. Dev.

Ret

urn

M

X

YZ


X 15% 5% 2.50 20% 0.0400 0.5000

Y 8% 5% 0.50 14% 0.0600 0.2143

Z 6% 5% 0.35 9% 0.0286 0.1111

Market 10% 5% 1.00 11% 0.0500 0.4545

Risk vs Return

0%

5%

10%

15%

0.00 0.50 1.00 1.50 2.00 2.50Beta

Retu

rn

MX

Y

Z

Risk vs Return

0%

5%

10%

15%

0% 5% 10% 15% 20%Std. Dev.

Retu

rn

M

X

YZ

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6. Sortino Ratio The Sortino ratio generalizes (focus on the downside) from the Sharpe by

using:

• In the numerator, instead of excess return (above riskfree), Sortino uses

excess above hurdle (MAR, minimum acceptable return)

• In the denominator, instead of volatility (annualized standard deviation),

Sortino uses downside deviation.

• Appears to resolve several of the issues inherent in the Sharpe ratio:

• It incorporates a relevant return target, in both the numerator and the

denominator;

• It quantifies downside volatility without penalizing upside volatility; and

because of its focus on downside risk,

• It is more applicable to distributions that are negatively skewed than

measures based on standard deviation.

)(

])([

PL

P

R

MARRESR

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5-6. Sortino Ratio and Sharpe Ratio Example

Averag montly return of P 1,821%

Average yearly return 21,851% Times 12

Month Price Returns Hurdle rate R - HUR Hurdle>R (R-Hur)^2 (R-Av)^2 Hurdle rate yield return Y 18,000% 1.5% * 12

1 663,03 -2,539% 1,50% -4,04% -4,04% 0,1631% 0,1901%

2 680,3 -9,834% 1,50% -11,33% -11,33% 1,2847% 1,3585% Rf month 1,80%

3 754,5 10,132% 1,50% 8,63% 0,6907% RF Y 21,60% Times 12

4 685,09 8,234% 1,50% 6,73% 0,4113%

5 632,97 9,120% 1,50% 7,62% 0,5327%

6 580,07 -0,136% 1,50% -1,64% -1,64% 0,0268% 0,0383% Volatility 29,06%

7 580,86 -3,966% 1,50% -5,47% -5,47% 0,2988% 0,3349% Sharpe ratio 0,865%

8 604,85 -5,675% 1,50% -7,17% -7,17% 0,5148% 0,5619%

9 641,24 3,719% 1,50% 2,22% 0,0360% Sortino 19,54%

10 618,25 6,575% 1,50% 5,07% 0,2260% Excess return 3,851% P-Hurdle

11 580,11 -10,186% 1,50% -11,69% -11,69% 1,3656% 1,4416% Montly downside VaRianc 19,71%

12 645,9 7,760% 1,50% 6,26% 0,3527%

13 599,39 1,139% 1,50% -0,36% -0,36% 0,0013% 0,0047%

14 592,64 15,067% 1,50% 13,57% 1,7545%

15 515,04 -4,791% 1,50% -6,29% -6,29% 0,3958% 0,4372%

16 540,96 -10,391% 1,50% -11,89% -11,89% 1,4140% 1,4913%

17 603,69 19,217% 1,50% 17,72% 3,0262%

18 506,38 -4,280% 1,50% -5,78% -5,78% 0,3340% 0,3722%

19 529,02 -2,772% 1,50% -4,27% -4,27% 0,1825% 0,2109%

20 544,1 -7,270% 1,50% -8,77% -8,77% 0,7692% 0,8265%

Squared difference

We take a minimun

acceptable return

Only consider the

returns below the

hurdle rateSquare difference WR

the hurdle rateSq. Difference WR the

average of returnsSumation of returns

Use the formula

Excess over Hurdle

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7. Jensen alpha

R RFR R RFRi i i M i b

Ris

k P

rem

ium

Market Risk Premium

0

> 0

= 0

< 0

Alpha = Excess of return – (Beta * (Excess of return))

Shows by much the returns of an

actively managed portfolio are

above or below market returns.

A positive Alpha means that a

portfolio has beaten the market,

while a negative value indicates

underperformance

A fund manager with a negative

alpha and a beta greater than one

has added risk to the portfolio but

has poorer performance than the

market

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Alpha = Excess of return – (Beta * (Excess of return))

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7. Jensen alpha

Portfolio

P

Portfolio

Q

Market

Beta 0.90 1.6 1.0

RM-Rf 11% 19% 10%

Alpha 2.0% 3.0% 0%


M

P M2

Exp

ecte

d R

eturn

Beta

Portfolio Q

SML

1.6 0.9

Portfolio P

19%

11%

16%

9%

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7. Jensen alpha

8. Information Ratio Measure of the risk-adjusted return of a portfolio.

Defined as expected active return divided by tracking error

• Active return : difference between the return of portfolio and

the return of a benchmark

• Tracking error is the standard deviation of the active return

• Measures the active return of the manager's (abnormal return)

portfolio per unit of risk that the manager takes relative to the

benchmark.

• The higher the information ratio, the higher the active return of

the portfolio, given the amount of risk taken, and the better the

manager.

)(

][

BP

BP

RRVAR

RREIR

Component attributable to the manager’s skill

While Sharpe consider the σ of total

returns, IF consider σ of alpha

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8. Information Ratio

)(

][

BP

BP

RRVAR

RREIR

Returns

Date Portfolio Market Excess

01/01/2010 2% 2,06% -0,49%

01/02/2010 1% -5,62% 6,64%

01/03/2010 0,61% -3,42% 4,03%

01/04/2010 0,76% 2,84% -2,08%

01/05/2010 9,69% -5,00% 14,69%

01/06/2010 1,39% 5,30% -3,91%

01/07/2010 3,10% -2,33% 5,44%

01/08/2010 0,46% 8,57% -8,12%

01/09/2010 6,11% 4,77% 1,34%

01/10/2010 9,37% 14,69% -5,32%

01/11/2010 3,88% -6,68% 10,56%

01/12/2010 9,54% 1,38% 8,16%

Mean 3,96% 1,38% 2,58%

Standard Dev 3,73% 6,40% 6,88%

Information Ratio 0,3750

Expected excess of return (Benchmark)

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9. Modigliani´s Risk Adjustment Performance • Also known as M2

• Closely related to the Sharpe Ratio

• Focuses on total volatility as a measure of risk, but its risk adjusted

measure of performance has the interpretation of a differential return

relative to the benchmark index

• The idea is to lever or de-lever a portfolio (i.e., shift it up or down the

capital market line) so that its standard deviation is identical to that

of the market portfolio.

• The formula for M2 is:

• The M2 of a portfolio is the return that this adjusted portfolio earned.

This return can then be compared directly to the market return for the

period.

ffii

M2 RRRM

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9. Modigliani´s Risk Adjustment Performance • Suppose that

• Return Ri: 35% RM: 28%

• Volatility σi: 42% σM: 30%

• Find a portfolio combination with the same level of risk than the

benchmark (market)

• Portion of the portfolio

• Portion of risk free asset

• The return of this portfolio will be

• This portfolio is -1.3% than the market return.

This is the M2

714.042

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i

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267.0]35.0*714.0[]06.0*286.0[

286.01

i

M

M

P P*

M2

Exp

ecte

d R

eturn

Volatility

Portfolio

Market

• Reduce the return of the P

• Obtain leSs volatility

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9. Modigliani´s Risk Adjustment Performance

ffii

M2 RRRM


X 15% 5% 2.50 20% 0.0400 0.5000

Y 8% 5% 0.50 14% 0.0600 0.2143

Z 6% 5% 0.35 9% 0.0286 0.1111

Market 10% 5% 1.00 11% 0.0500 0.4545

Risk vs Return

0%

5%

10%

15%

0.00 0.50 1.00 1.50 2.00 2.50Beta

Ret

urn

MX

Y

Z

Risk vs Return

0%

5%

10%

15%

0% 5% 10% 15% 20%Std. Dev.

Ret

urn

M

X

YZ

062.005.005.006.009.0

11.0M

074.005.005.008.014.0

11.0M

105.005.005.015.020.0

11.0M

2

Z

2

Y

2

X

• Recall that the market return was 0.10, so only X outperformed. This is the

same result as with the Sharpe Ratio.

• The M2 of a portfolio is the return that this adjusted portfolio earned. This

return can then be compared directly to the market return for the

period.

)(

])([

P

FP

R

RRSR

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10. Marginal Risk • Represents the change in risk due to a small increase in one of the

allocations. It is essentially a derivative that measures the rate of change in

some measure of interest given a small change in a variable.

• Beta represents the marginal contribution to the risk of the total portfolio

• Large values of beta indicates that a small addition will have a relatively large

effect on the portfolio

• Positions with large betas should be cut first to reduce risk

The portfolio risk/standard deviation is the sum of the risk contributions from

each asset.

PPi

P

Pi

i

P RRCov

wRiskM b

,

),(arg

PPiiwonToRiskContributi b ,

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10. Marginal Risk

I am running a hedge fund

Bank of America 25% . Beta=3

σp =10%,

Contribution to Risk =.25 x 3 x 10%= 7.5%

• 7,5% of my portfolio risk is going to be dictated by what happens to

Bank of America

• The risk is too concentrated in one stock. In practice, it is

desirable to spread out total risk contributions across as many stocks

or assets as you can in the most equivalent manner.

PPi

P

Pi

i

P RRCov

wRiskM b

,

),(arg

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HEDGE FUND RISK MANAGEMENT

Part 1I

30

a. Introduction

b. Strategies

1. Introduction L

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6

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s

What is a H.F.?

1. Introduction What is a H.F.?

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2. H.F. Strategies An heterogeneous group

Long positions in stocks

expected to increase.

Short positions in stocks

expected to decrease

Exploits pricing differentials

between fixed-income

securities.

Instruments whose prices

fluctuate based on the changes

in economic policies along

with the flow of capital

Long position in convertible securities.

AND

Short position in the same company’s

common stock.

Exploit pricing inefficiencies

before or after a corporate

event:

Bankruptcy, Merger, Acquisition

Holds a portfolio of other investment

funds instead of investing directly in

securities

Find “bargains” and accept risk

1000 basis points above the risk-

free rate of return

Achieve a beta as close to zero to

protect against systematic risk

CREDIT RISK MANAGEMENT

Part III

35

a. Intro

b. Drivers of Credit Risk

c. Settlement risk

d. Credit losses

1. Introduction Definition

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Ma

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The potential for loss due to failure of a borrower to meet its

contractual obligation to repay a debt in accordance with the agreed terms

• Its effect is measured by the cost of replacing cash flows if the other

party defaults

• Commonly also referred to as default risk

• Credit events include

• bankruptcy,

• failure to pay,

• loan restructuring

• loan moratorium

• Example: A homeowner stops making mortgage payments

Market Risk Credit Risk

Potential loss due to changes in

market prices or values

Potential loss due to the non

performance of a financial

contract, or financial aspects of

non performance in any contract

2. Drivers of Credit Risk

Default:

Discrete state for the counterparty (Default or not). It has associated the

Probability of Default (PD) defined as the likelihood that the

borrower will fail to make full and timely repayment of its financial

obligations

Exposure At Default (EAD)

The expected value of the loan at the time of default

Loss Given Default (LGD)

The amount of the loss if there is a default, expressed as a percentage of the

EAD

Recovery Rate (RR)

The proportion of the EAD the bank recovers

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3. Settlement risk

Settlement risk: The risk that one party will fail to deliver the terms of a

contract with another party at the time of settlement

Foreign exchange (FX) settlement risk is the risk of loss when a bank in a

foreign exchange transaction pays the currency it sold but does not receive

the currency it bought.

Settlement Risk management:

• Real time systems

• Bilateral netting agreements (two institutions)

• Multilateral netting agreements (two industries)

CLS Bank. In foreign exchange and operates the largest

multicurrency cash settlement system. It is owned by the

world's leading financial institutions

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In initial consideration

4. Credit losses (overview) Set up:

The credit losses are defined by

The Expected Credit Loss for a portfolio is:

Example

N

i iii fCEbCL1

)1(**

Where

• Random Variable bi is a bernoulli trial that takes values of 1 (Def) or 0 (non Def)

• CEi is the Credit Exposure at time of default

• fi is the recovery rate (What means 1-fi )

N

i iii fCEbECLE1

)1(**][][

N

i iii fCEp1

)1(**

Default is affected by

correlation among

assets:

Asset Exposure Prob. Def.

A $25 5%

B $30 10%

C $45 20%

$100 TT

Expected Credit Loss

)45*%20()30*%10()25*%5(

M5.13$

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Description of the complete distribution

What could be the VaR of this portfolio? Le

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Issuer Exposure P. Def P. No Def.

A 25.00£ 5% 95%

B 30.00£ 10% 90%

C 45.00£ 20% 80%

Default Loss Probability Cumulative Exp. Loss Variance

None

A

B

C

A,B

A,C

B,C

A,B,C

1. How can I find the loss

2. What is the prob. Associated to each scenario?

3. Easy

4. Expected loss of each scenario

5. Variance

4. Credit losses (overview)

PROBABILITY OF DEFAULT

Part IV

41

a. Actuarial

b. Market prices methods

Likelihood that the borrower will fail to make full and timely

repayment of its financial obligations

1. Methodologies

Actuarial methods

• Measure default rates using historical data

• Provided by external rating agencies

Market price methods

• Infer default risk from market prices of debt, equity

prices, credit derivatives

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1. Actuarial Methods Corporate Default Probabilities Increase

exponentially across Credit Grades

• Credit rating is a measure of the firm’s

credit risk

• External credit rating: Standard & Poor’s,

Moody’s, Fitch, etc.

• Probability of staying in the

same rating category is

given on the diagonal.

• Off-diagonal probabilities

present the likelihood that

the rating will change

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Factor 1

Credit Ratings

Factor 2

Prob. Of default

a.

b.

1. Actuarial Methods

• Credit migration or transition matrices use ratings migration

histories.

• One-year horizon.

• Measured using the cohort and the duration method.

• Generally, the transition matrix is affected by the business cycle:

downgrades including defaults are higher during recessions

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Factor 3

Transition Matrix


• How many companies rated ( ) defaulted in each year

• This measure is cumulative. It necessarily increases with the horizon

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Factor 4

Cumulative Default Rates


Cumulative

d1

1 - d1

1 – d2

1 – d3

d2

d3

Default

Default

Default

No Default

No Default

No Default

d1 d1+ (1- d1)d2 (1- d1)(1- d2) d3

Compute the cumulative probability of default:

• The probability of default in the first year is 5%

• The probability of default in the second year is 7%

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1. Actuarial Methods Compute the cumulative probability of default:

• The probability of default in the first year is 5%

• The probability of default in the second year is 7%

Cumulative

d1=5%

Survival rate = 95%

Survival rate 2=

95%*97% = 0.883

1 – d3

Default in 2=

95%*7%=6.65%

d3

Default

Default

Default

No Default

No Default

No Default

d1 d1+ (1- d1)d2 (1- d1)(1- d2) d3

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Recovery Rates

Amount recovered through foreclosure or bankruptcy procedures in a credit

event (default), expressed as a percentage of face value

Are function of

• The state of the economy. Higher with expansion

• The obligor’s characteristics: Higher when the borrower’s assets are

tangible and when previous rating was high

• The credit event: distressed debt has higher recovery rate than plain

default

• The status of the debtor: Higher seniority has higher recovery rates

Credit rating agencies have used two methods to calculate RECOVERY

RATES (Moody’s)

• Average issuer-weighted trading price on a sovereign's bonds 30 days

after its initial missed interest payment

• Ratio of the value of the old securities to the value of the new

securities received in exchange,

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Factor 5

Recovery Rates 1. Actuarial Methods

• Average historical sovereign recovery rate: 53%

• 67% of recovery rate according to the ratio of value

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1. Actuarial Methods Recovery Rates

Senior debt has a higher recovery rate.

• According to S&P recovery rates have averaged 51% on a discounted basis and 60%

on a nominal basis, based on a sample from 1987 to 2011.

• If measured on a dollar-weighted basis, which is the sum of all defaulted debt in the

sample divided by the sum of the dollar amount of debt recovered, the averages are

slightly lower: 48%

• Loans and revolving credit facilities, that have seniority in the capital structure and are

often secured: recovered 74% on a discounted basis and when measured on a dollar-

weighted basis, the average recovery for loans and facilities is 65%

• Bonds have lower average recoveries. The long-run discounted average recovery for

bonds is 38% Le

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1. Actuarial Methods Recovery Rates

2. Market price methods

Infer default risk from market prices of debt, equity prices, credit

derivatives

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• Infer default risk from bonds

• Merton's model (Structural Model)

Next Lecture

LECTURE SIX

52

End of the lecture

Lecture 6

Documents

Transcript of Lecture 6