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Erstes Buch Kartei 1

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Flashcards	99
Language	English
Category	Biology
Level	University
Created / Updated	13.07.2021 / 08.10.2021
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Explain Ho Lee single factor arbitrage free models

Single factor model that assumes a normal distribution for short therm rates and incorporates a drift parameter

\(r_{t+1} = r_t + \theta_t + \sigma * \epsilon_{t+1}\)

Ho lee model adds a term for time dependent mean change theta(t) which is chosen to ensure that the model fits the initial yield curve.

--> Primary advantage it is calibrated to fit the observed yield curve

--> Two disadvantages are that the model allows negative rates and it assumes simplified binominal process for bond prices

Explain and discuss the Black Derman Toy Model (BDT)

Commonly used to value FI derivatives. It remains consistent with the current yield curve while also factoring for the implied volatility on interest rate caplets. It is useful for calculating spot rates, forward rates or discounting factors.

Calibration ensures that no arbitrage opportunities exist.

Calibrating level of rates: r0 = 4.5%, r1,u = 5.03%, r1,d = 3.75% and the yield on a two year zero coupon bond is 4.45% hence two year return is (1.0445)^2 = 9.09%

The average return is 0.5*((1.045)*(1.0503)+(1.045)*(1.0375)) -1 = 9.09%

Therefore the average two period return of the two paths equals the return of the two year zero coupon bond

Calibrating the Spread of rates: \(r_u = r_d * e^{2*\sigma}\)

in the example volatility of 14.72% makes the relationship hold.

--> spot rates in the current yield curve drive level of rates - implied volatility on the interest rate caplets drive the spread between up and down rate

Explain p measure vs q measure

P measure = real world measure factors probability using historical data without making assumptions

Q measures = risk neutral measures uses risk free rate for discounting future cash flows - the bdt model uses q measure

1. Put-call parity usesexisting observations to determine how prices should be behaving under certain assumptions. Thisis a combination of:

a theoretical and a positive model

a theoretical and a normative model

an empirical and a normative model

an empirical and a positive model

2. Whichfactor is NOT considered by Vasiceks model

Standard deviation of current rates

The current rate

Mean reversion

Speed of change in interest rates

3. Ho and Lees model adds a factor:

for mean reversion

to adjust for skewness

for risk weighting

that links the modl to the current yield curve

4. The Black-Derman-Toy (BDT) model calibrates

the level of interest rates to allow for arbitrage opportunities

the mean version factor for observed trends

the spread between up rates and down rates for implied volatility

the speed of mean reversion relative to the standard deviation of returns

5. An analyst needs to factor probabilities for a financial derivative under a risk-neutral mandate.The analyst should use

the q measure

the p measure

the x factor

implied volatility

6. An analyst is trying to better understand the shape of the yield curve, which is typically upward sloping during good economic environments. Arbitrage-free models(e.g., Vasiceks model) can be used to help explain this observation. What is the connection between Vasiceks model and the shape of a normal yield curve? Explain

Vasiceks equilibrium model is based on mean reversion. As rates revert to their long-term mean, they often resolve to an upward-sloping yield curve. In other words,if current rates are below their long-term mean, then the slope of the yield curve should be upward sloping using Vasiceks framework.

7. John needs to estimate rates for a fixed-income derivative. He is considering using Vasiceks model, Ho and Lees model,or the Black-Derman-Toy (BDT)model. How should John evaluate the best model to use for his needs? Explain

Vasiceks model is an equilibrium-based model that focuses on mean reversion, which may not be the best option to pursue. Ho and Lees model is an arbitrage-free model that directly links the current rates with the current yield curve. This is a good enhancement, but the BDT model is probably the best option because it links to the current yield curve and factors implied volatility. The BDTmodel is the most sophisticated of these three models and the best choice for valuing derivatives.

What is more important than the recovery rate usually?

Calculae it given Recovery Rate is 35%, loan is 175mio borrowed for 4 years and an appropriate discount rate of 6.5%?

However that an RR of 70% does not necessarily guarantee that the lender will recover 70% of its loan at the time of default. The recovery process can be lengthy and therefore, the more relevant measure is the present value of the recovered amount

amount to be recovered = 0.35 * 175 = 61.25

PV of planned recovery = (61.25)/(1+0.065)^4 = 47.611

recovery rate = 47.611/175 = 27.21%

How is LGD, EAD, RR and PD linked?

LGD = EAD * (1-RR)

Expected Loss = LGD * PD = EAD * (1-RR) * PD

Explain the three types of credit risk modelling approaches

Structural credit risk models

Reduced form models

Empirical models

Structural: Draw explicit relationship between capital structure and default. Equity is modelled as call option with strike value equal to the face value of the bond. The bondholders have a risk free bond and a short position ina put option on the entitys assets

Reduced: View default as a random external factor

Emirical: Produce credit score that is used to rank entities by creditworthiness

Explain the Merton model including formulas

Assets are equal to Debt lus Equity.

E = max(A-K,0)

D = K - max(K-A,0)

E = A * N(d) - K * e^(-r*t) * N(d-sigma_Assets * sqrt(t))

--> The formula for d will be provided on the exam

Given that debtholders are exposed to the risk of loss from a shortfall between the value of assets and the facevalue of debt buying put option hedges this risk

P = K*e^(-rt)*N(-d+sigma_assets*sqrt(t)) - A*N(-d)

What are the shortcomings of the merton model?

What are the four important propoerties of the merton mdel?

Shortcomings are: many model paameters are not redily observable including return volatility and asset market values. The model does not explain the credit spreads on short term securities well

1. Sensitivity to maturity --> As maturity increases the PD rises.

2. Sensitivity to asset volatility -> As asset volatility i ncreases so does the PD.

3. Sensitivity to leverage --> Higher leverage increases the PD

4. Sensitivity to the risk free rate --> as risk free rate increases so does the return expectation reducing PD

Describe and provide an overview of the KMV Credit Risk model

KMV credit risk model is a structural credit risk model and is built on the Merton model but also uses estimates of the volatility and value of a firms assets and equity.

--> Instead of looking at credit risk form a lenders perspective is is used to analyze the viewpoint of the equity holder

Describe the formula of the KMV Model

sigma(equity) = Assets / Equity * Delta * sigma(asets)

Delta measures the sensitivity of firms equity to changes in asset values and is equivalent to N(d) in the Merton model.

Important difference to the Merton model is the treatment of default. KMV uses a weighted average of short and long term bonds and therefore default is calculated only when the asset values fall below the combined value of debt. Merton only uses a single zero coupon bond

Use the KMV to claculate distance to default and expected default frequency given:

Corporation is worth 100 estimated default trigger is 77 and asset volatility is 30% and assume that 750 have the same distance to default as the corporation and that 7 of them defaulted by the end of the year

DD = (110-77) / (77*0.3) = 1.43 this means that the corporation is 1.43 standard deviations away from default. If asset value drops by more than 1.43 standard deviations a firm will enter default territory

EDF = 7/750 = 0.93% One could therefore expect that 0.93% of firms with a DD equaling 1.5 would default by the end of the year

Describe reduced form models

What are the two predominant reduced form credit models?

Reduced form models consider default only as exogenous variables. They do not consider what causes default.

--> Model drivers include time to default and recovery in the event of default LGD.

--> Default intensity determines the expected time until default and the survival probability. If default intensity is 0.2 then the time until default is 1/0.2 = 5

The Jarrow Turnbull model assumes the recovery rate received at maturity regardless of the timing of default - Jarrow lando Turnbull model also takes credit ratings and migration into account

The Duffie Singleton model assumes that recovery can happen at any time

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