15

Risk-neutral default probability implied from CDS is approximately $P=1-e^\frac{-S * t}{1-R}$, where $S$ is the flat CDS spread and $R$ is the recovery rate. The CDS Spread can be solved using the inverse: $$S=\ln(1-P) \frac{R-1}{t}$$ $S$ is the spread expressed in percentage terms (not basis points) $t$ are the years to maturity $R$ is the recovery rate ...


11

I believe the answer can be further improved for all those being directed here by google after 3 years. A common way to model the default probability is by the hazard rate. As @Bob correctly mentions, a traditional requirement is for it to satisfy (see Option Futures and Other Derivatives section 23.4 in which the author discusses also other more exact ...


10

You can use the "credit triangle" which states that the (annualised) credit spread $S$ equals the annualised probability of default $p$ times the loss given default LGD which equals par minus the expected recovery amount $R$, i.e. $S=p(1-R)$. This is a "back-of-the-envelope" approximation to a full hazard rate credit model - from experience I find that the ...


7

The chapter in Hull on Credit Risk gives the same formula as emcor as a first approximation with a justification: Consider first an approximate calculation. Suppose that a bond yields 200 basis points more than a similar risk-free bond and that the expected recovery rate in the event of a default is 40%. The holder of a corporate bond must be expecting to ...


6

For an individual firm, a theoretical model of the capital structure was developed by Robert Merton in 1974. The simplest form of this model assumes the firm has zero-coupon debt maturing at some future time $T$. Default is defined as the condition where the value of the firm's assets fall below the outstanding debt. The firm equity is viewed as a call ...


5

This is what Moody's does to calculate default probabilities, but I don't believe they give a whole lot of detail on their exact methodology because they sell their models as software. I quickly found this which gives a brief overview: http://www.moodysanalytics.com/~/media/Brochures/Enterprise-Risk-Solutions/RiskCalc/RiskCalcPlus-Fact-Sheet.ashx Edit- ...


5

Firstly, the use of the logit models to estimate the PDs is particularly appreciated in some credit industries, as, for instance, the credit retail one. The logit model predicts pretty well the PD on loans, consumer credit, credit cards, ... and all concerns the retail consumer world. Mainly, those listed are the principal sub-industries in the credit ...


5

You cannot do it. It is an under-determined problem. That is to say, a whole multitude (subspace of $\mathbb{R}^{N\times N}$) of migration matrices will agree with any given table of default probabilities. Say you want to find a transition matrix for 2 states (IG, HY) plus default $$\left(\begin{matrix} p_{11} & p_{12} & p_{1D} \\ p_{21} &...


5

"Debt issuer default risk" and "counterparty risk" are very similar. From Risk magazine: Counterparty Risk The risk that a counterparty to a transaction or contract will default (fail to perform) on its obligation under the contract. Counterparty risk is not limited to credit risk (the risk that the counterparty cannot fulfill its ...


4

From this research from Deutsche Bank, : $$p_{def}=\frac{CDS_{spread}}{1-Rec}$$ $$\Leftrightarrow CDS_{spread}=p_{def}(1-Rec)$$ where $Rec$ is the recovery rate in case of default.


4

Actually, there is a practical way to do it. You can use you PoD estimates to assign a credit rating to your securities and then use a published transition matrix for your purposes. Or you can estimate transition probabilities by linear interpolation based on the PoD values that you have. Here is a publication containing transition matrices from Moody'...


4

The relationship between volatility and CDS is very interesting. Volatility in finance is synonym of risk. There are many aspects of volatility. There are 2 primary ways to find CDS premium, one is using structural model and the other is reduced form or intensity based model. Structural models use equity valuation, outstanding debt and equity volatility to ...


4

Let $\tau$ be the default time, $\lambda$ be the constant hazard rate, and $T=1.0$ be the bond maturity. The value of the defaultable zero-coupon bond is given by \begin{align*} D(0, T) &= e^{-rT}P(\tau > T). \end{align*} Then the default probability is given by \begin{align*} P(\tau \le T) &= 1- P(\tau > T)\\ &=1-D(0, T) \times e^{rT}\\ &...


4

Firstly it's good to straighten out our goal. You correctly say, that IFRS9 requires analysis of expected losses. There are two components of expected losses. 1) Expected probability of a default event 2) Expected recovery rate So, not only do we need the probability but also the recovery rate. Luckily, both are approximated by the credit spread, which ...


4

Let $\tau_{(1)} = \min(\tau_1, \ldots, \tau_K)$ be the first-to-default time. Moreover, for $1< m \le K$, let \begin{align*} \tau_{(m)} = \min\left(\tau_k: k=1, \ldots, K, \tau_{k} > \tau_{(m-1)}\right). \end{align*} be the $m^{\rm th}$-to-default time. In particular, $\tau_{(K)} = \max(\tau_1, \ldots, \tau_K)$. Note that, for $t \ge 0$, \begin{align*} ...


4

Based on your definition, they are certainly not the same. Generally, the marginal default probability is the probability that the default happens in a given time period, such as $[t, t+\Delta]$, that is, $P(t < \tau \le t+\Delta)$. Here, $\tau$ is the default time. See Chapter 10 of the book Counterparty Credit Risk and Credit Value Adjustment for ...


4

It is actually that you forgot your $1 - R$ in formula (2) :) The index survival curve is defined similarly to the tranche's : $Q\left(t\right) = 1 - \mathbb{E} \left[L\left(t\right)\right] = 1 - \left(1 - R\right)\mathbb{P}\left(\tau < t\right)$. Hence, your formula for the 0-100 tranche survival curve does coincide with the index'. That history of loss ...


4

As you see in the third equation on that Mathworks page, the Merton model postulates that the value of equity equals the value on a residual claim on a company's assets after the creditor has been repaid. Economically speaking, equity is a call option on the asset value $A$ with strike price equal to the liability $L$, the formula for which is $$ E=AN(d_1)-...


4

OK, here is a simplified demonstration: Before we consider swaps, let us consider very simple bonds. Suppose that you have a choice of two zero-coupon bonds. A riskless one costs 95 and is certain to pay 100 in 1 year. A risky one costs 90, is expected to also pay 100 in 1 year, but with some probability $p$ will default and only pay some $R<100$ on the ...


4

It helps to get some intuition on all the terms. Point-in-Time (PiT) Probability of Default (PD) is a probability that the counterparty will default in a specific time-interval. I will denote the event of default between $t_1$ and $t_2$ as $A(t_1,t_2)$ for any arbitrary time interval. If we think about it logically, given today's state of the world (i.e. ...


3

The question sounds like a conditional probability problem. However, note that, for conditional probability, people will generally say if survived to or conditional on. Here it says that survived in year one and (i.e., followed by) will default in year two. Then we should not treat this as a conditional or marginal probability. Based on the above ...


3

I'm no expert on this topic but here's my two cents. Hopefully if I'm wrong someone will correct me. From the 2 relations you wrote, we see that $$ DD_q = -N^{-1}(P) - \lambda R \sqrt{T} $$ or equivalently \begin{align} DD_q &= DD_p - \lambda R \sqrt{T} \\ &= \frac{\ln(A/D)+((\mu-\lambda \sigma R) - 0.5\sigma^2)T}{\sigma \sqrt{T}} \end{align} where ...


3

I assume that you calculate ECL in the context of IFRS9 -correct? market practice often follows the following appraoch: estimate a TTC PD/LGD (TTC = through the cycle). This corresponds to your lifetime estimate (e.g. one marginal PD value for each year of the life of your exposure) in the average of the economic cycle. But for IFRS9 provisioning you have ...


3

Let's start with the "safest" bonds in the world, and work our way down the credit quality curve. In Europe, the safest and virtually "credit-risk free" bonds are the German Bunds. If you look at the 10y yield of the German bunds, these are negative 60 bps as of this morning. The ECB deposit rate is negative 50 bps: from the fact that the ...


3

At the risk of arming you to create the next quant-apocalypse... The statement that the expected loss does not depend on correlation is typically the result of modelling a portfolio as a sum of individual exposures: X+Y+..., and then using: E[X+Y+...]=E[X]+E[y]+.... This does NOT generalize to statements like "expectations don't depend on correlation&...


3

Just to add to the above answer, if $\tau$ is the default time of an entity, we have $$P(\tau>t-1) =: SP_{t-1}$$ as the definition of survival probability beyond time $t-1$ (where $t$ and $t-1$ are some fixed period appart, say one quarter), and conditional probability of default over period $(t-1,t]$ $$ P(\tau \leq t | \tau > t-1) =: PD_t$$ as the ...


2

detailed description of the solution of this problem using Excel is in the second chapter of the book Credit Risk Modeling using Excel and VBA Gunter Löffler


2

This does not sum to 1 because you have forgotten to add the 6th scenario, the NonDefault (ND). If Ps is the probability of survival and Pd the probability of default, the ND has the probability Ps^5. This makes: Pd+ Ps*Pd+ ... Ps^4*Pd+ Ps^5= Pd*(1+Ps+...+Ps^4)+Ps^5= Pd*(1-Ps^5)/(1-Ps)+Ps^5= (1-Ps)**(1-Ps^5)/(1-Ps)+Ps^5=1.


2

I believe that your problem can be formulated as: Find PD matrix that is as close as possible to a given PD matrix (result of some previous calibration, or the matrix computed using average hazard rate, or any other "target", or the penalty on non-smoothness) subject to the following constraints: The values that are given must be matched exactly ...


2

I think the problem is that, for countries with a sizeable risk of hyperinflation, you will not have deep and mature markets to extract market expectations from. Argentina is a good example. Hyperinflation is just 'very big inflation', but you don't have inflation swaps in ARS. The CDS that you mention will pay in USD, and are therefore immune to ARS ...


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