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# 3 altmans z score using discriminant analysis altman

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3. Altman’s Z-score Using discriminant analysis, Altman attempted to predict defaults from five accounting ratios. Z = 1.2*working capital + 1.4*Retained earnings + 3.3* EBIT + 0.6*Market value of equity + 0.999*sales, all variables are scaled by assets, except for market equity, which is scaled by book value of total liabilities. If Z-score > 3, the firm is unlikely to default . If it is between 2.7 and 3.0, we should be ‘on alert’. If it is between 1.8 and 2.7, there is a good chance of default. If it is less than 1.8, the probability of a financial embarrassment is very high. 4. Historical default probabilities - For investment-grade bonds, the probability of default in a year tends to be an increasing function of time . - For bonds with a poor credit rating, the probability of default is often a decreasing function of time . The reason is that, for a bond like this, the next year or two may be critical. If the issuer survives this period, its financial health is likely to have improved. - Default intensities (hazard rate) : λ (t) at t is defined so that λ (t) Δ t is the probability of default between time t and t + Δ t conditional on no default between time 0 and time t. Let V(t) is the cumulative probability of the firm surviving to time t (no default by time t), then V(t + Δ t) – V(t) = - λ (t) V(t) Δ t Æ 0 - ( ) ( ) ( ) ( ) V(t)=e t d t dV t t V t e dt λ τ τ λ λ = − = , where λ is the average default intensity between time zero and t. Define Q(t) as the probability of default by time t Æ Q(t) = 1 – V(t) = 1 - 0 - ( ) e t d λ τ τ = 1- t e λ Æ 1 ( ) ln[1 ( )] t t λ = − Q t , where Q(t) comes from historical data. 5. Recovery rates : the bond’s market value immediately after a default as a percent of its face value = 1 – LGD. - Recovery rates are significantly negatively correlated with default rates . - 18 -

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Study Notes: Risk Management and Financial Institutions By Zhipeng Yan - Average recovery rate = 0.52 – 6.9*average default rate. Æ a bad year for the default rate is usually doubly bad because it is accompanied by a low recovery rate . 6. Estimating default probabilities from bond prices [1 ]100 100Re 100 RfT RfT yT PD e PD e + = - An approximate calculation: Æ PD(T) = [ ] 1 1 y Rf T e R ~ spread/(1-R) - Risk-free rate: Traders usually use LIBOR/swap rates as proxies for risk-free rates when calculating default probabilities. - Credit default swaps can be used to imply the risk-free rate assumed by traders . The rate used appears to be approximately equal to the LIBOR/swap rate minus 10 basis points on average. (Credit risk in a swap rate is the credit risk from making a series of 6-month loans to AA-rated counterparties and 10 basis points is a reasonable default risk premium for an AA-rated 6-month instrument. 7. Asset swap : traders often use asset swap spreads as a way of extracting default probabilities from bond prices. This is because asset swap spreads provide a direct estimate of the spread of bond yields over the LIBOR/swap curve. Æ The present value of the asset swap spread is the amount by which the price of the corporate bond is exceeded by the price of a similar risk-free bond.
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• Spring '10
• NanLi
• Normal Distribution, ........., Risk Management and Financial Institutions, Zhipeng Yan