# Quarterly Survival rate given there is a Quarterly Probability of Default

I am trying to calculate the Quarterly Marginal PD. I have calculated it as given in the below image but I am thinking about whether the Survival rate calculation is making sense or not.

The probabilities of default in "column 1" per quarter are annualized quantities. My Question is When we have Quarterly PD, Is it correct to calculate the Survival rate as (1-Quarterly PiT PD)^(1/4) or Just (1-Quarterly PiT PD). In this example, I have calculated the Survival Rate as (1-Quarterly PiT PD)^(1/4)

Additional Info: Quarterly Cumulative Data for the last row has been calculated as given in the excel formula in the last cell.

PiT PD for each quarter is derived from the below Vasicek model based on the constant TTC PD, Single macroeconomic Factor z(each quarter value), and constant asset correlation rho

PiT PD = $$\Phi \left( \frac{\phi^{-1}(PD_i)-\sqrt{\rho_i}z}{\sqrt{1-\rho_i}} \right)$$ Edit as per the answer: I tried to incorporate the modeling techniques discussed in the answer below from @Jan Stuller. My total Cumulative PD is not coming similar to the total Marginal PD. Am I missing anything here • In contrast to terms like coupon rate or volatility, I would argue that a term PD does not require any further adjustment for that term, so I would think that your second definition is sensible. But on the other hand: 10% PD per quarter implies roughly 35% PD per year… Sep 4 at 3:59
• @Kermittfrog, You mean (1-Quarterly PiT PD) is a sensible method to derive the survival rate for quarterly data. But what about if we think the first term (1-Quarterly PiT PD)^(1/4) as arrive probability of survival over the next quarter. 12% in the second quarter gives the 3% marginal PD using the first term Sep 4 at 4:07
• Yes;I would suggest you assess your data source - where do your PDs come from? Sep 4 at 5:57
• Quarterly PDs are derived from the Vasicek model using TTC PD(around 11.5%), my worry is to correctly calculate the survival rate hence it impact the marginal PD. Should it be (1-Quarterly PiT PD)^(1/4) (or) (1-Quarterly PiT PD) Sep 4 at 6:29
• How is the PD as derived from the Vasicek model defined? Is it based on some intensity model, using the integral for a quarter of a year? Sep 4 at 6:44

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. state of the world at $$t_0$$), the event $$A(t_1,t_2)$$ for any future time interval $$(t_1,t_2)$$ does not make physical sense in isolation by itself: how can we talk about the counterparty defaulting between $$(t_1,t_2)$$ in isolation, without referring to what happens between $$(t_0, t_1)$$? We cannot, it doesn't make sense!

The only way that the probability PiT $$PD(t_1,t_2)$$ makes sense is if the counterparty survives between $$t_0$$ and $$t_1$$. Therefore, to talk about PiT $$PD(t_1,t_2)$$ for any $$t_1>t_0$$, we need to make some sort of logical reference to what happens between $$t_0$$ and $$t_1$$.

Therefore, what PiT $$PD(t_1,t_2)$$ really is, is in fact the conditional probability of default between $$t_1$$ and $$t_2$$, given that there is no default between $$t_0$$ and $$t_1$$, i.e.:

$$PiT PD(t_1,t_2)=\mathbb{P}\left(A(t_1,t_2)|A'(t_0,t_1)\right)$$

In words: PiT $$PD(t_1,t_2)$$ is the probability of the event that the counterparty defaults between $$t_1$$ and $$t_2$$, conditional on the event that it does not default between $$t_0$$ and $$t_1$$.

How do we then compute the survival probabilities? For the first quarter, it is trivial, we know that PiT $$PD(t_0,t_1)=\mathbb{P}(A(t_0,t_1))$$ is 10% from your excel chart. The probability of survival is then just $$\mathbb{P}(A'(t_0,t_1))=1-\mathbb{P}(A(t_0,t_1))$$.

For the subsequent quarter, we can use Bayes law, which states that:

$$\mathbb{P}\left(A(t_1,t_2)|A'(t_0,t_1)\right)=\frac{\mathbb{P}\left(A(t_1,t_2)\cap A'(t_0,t_1)\right)}{\mathbb{P}\left(A'(t_0,t_1)\right)}$$

The second quarterly PiT PD in your excel chart is 12%, this is in fact $$\mathbb{P}\left(A(t_1,t_2)|A'(t_0,t_1)\right)$$, i.e. the probability of default in the second quarter, given no default in the first quarter. So using the Bayes formula above, we can compute the probability of the event "surviving the first quarter AND defaulting in the second quarter", i.e. $$\mathbb{P}\left(A(t_1,t_2)\cap A'(t_0,t_1)\right)$$ (i.e what you call Marginal probability of default), like so:

$$\mathbb{P}\left(A(t_1,t_2)\cap A'(t_0,t_1)\right)=\mathbb{P}\left(A(t_1,t_2)|A'(t_0,t_1)\right)*\mathbb{P}\left(A'(t_0,t_1)\right)=0.12*0.90=0.108$$

From the above two results, we can compute the probability of surviving the first two quarters, this is just:

$$\mathbb{P}(A'(t_0,t_2))=1-0.1-0.108=0.792$$

So basically it comes down to recursively using the Bayes formula to compute the survival probabilities.

These survival probabilities can then be re-used in the recursive Bayes formula to compute the cumulative Probability of Default, i.e.

$$PD(t_0,T)=\mathbb{P}(A(t_0,t_1))+\mathbb{P}(A'(t_0,t_1))\mathbb{P}(A(t_1,t_2)|A'(t_0,t_1))+\mathbb{P}(A'(t_0,t_2))\mathbb{P}(A(t_2,t_3)|A'(t_0,t_2))+...$$

In summary:

• If you assume that the PiT PD is the (conditional) probability of default per quarter, you don't need to scale it to the power of one quarter. If PiT PDs are annualized, you can scale it by simply dividing it by 4.

• To then compute the survival probabilities and cumulative probabilities of default, I'd use the recursive Bayes relationship described above.

• Could you please elaborate more on the first point in the summary, and if I am considering the PiT PD are annual in nature and reported in a quarterly frequency, so then how I would scale it. Would scaling like this (1-Quarterly PiT PD)^(1/4) suffice? Sep 6 at 14:44
• I am not from a mathematical background. Will this model gives the same output as shown in the question excel output. Sep 7 at 7:48
• My first point was to say that if your probabilities of default per quarter are annualized quantities, you can just divide them by 4, rather than taking a power of $1/4$. I.e. for the first quarter, you can just do 10% / 4 = 2.5%, and then compute the survival as (1-2.5%), rather than doing $(1-0.1)^{(1/4)}$. Sep 7 at 9:42
• Yes, the probabilities of default in "column 1" per quarter are annualized quantities. Dividing them by 4, rather than taking a power of 1/4. I can see some differences in the numbers. Will this makes any difference conceptually. Sep 7 at 10:45
• 1 ) I tried to incorporate the modeling techniques discussed in your answer. My total Cumulative PD is not coming similar to the total Marginal PD. Am I missing anything here? 2) Yes, the probabilities of default in "column 1" per quarter are annualized quantities. From my original Question, Dividing them by 4, rather than taking the power of 1/4. I can see some differences in the numbers. Will this makes any difference conceptually. Sep 7 at 18:48

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 definition of $$t$$-th PIT (point in time) probability of default (see below note too).

We then use Bayes:

$$SP_{t-1} = (1- PD_{t-1}) SP_{t-2} = (1- PD_{t-1}) (1- PD_{t-2}) SP_{t-3} =...$$ giving the survival probability in terms of PIT PD's $$SP_{t-1}= \prod_{i=1}^{t-1} (1- PD_{i}).$$

We also have: $$P(\tau\leq t) = 1- SP_{t} = 1- \prod_{i=1}^{t} (1- PD_{i}),$$

$$P(t-1<\tau \leq t) = PD_t\cdot SP_{t-1} = PD_t \prod_{i=1}^{t-1} (1- PD_{i}),$$

$$P(\tau \leq t+k | \tau >t-1) = 1- \frac{SP_{t+k}}{SP_{t-1}} = 1- \prod_{i=t}^{t+k} (1- PD_{i}), \; \;\; (*)$$

for any $$k$$ (say $$k=3$$) and (already included in answer above)

$$P(\tau \leq t) = \sum_{i=1}^t P(i-1<\tau \leq i) = \sum_{i=1}^t PD_i\cdot SP_{i-1}.$$

Note: My understanding is that, as suggested in your model formulas, to earn its full 'PIT' name, PIT PD is covering further conditioning on say a macroeconomic variable (and even some idiosyncratic ones), $$X$$, at time $$t-1$$, not just survival:

$$PD_t=PD_t(X_{t-1}):= P(\tau\leq t|\tau >t-1, X_{t-1}).$$

This allows representing the conditional PD's along a path (scenario) of $$X$$.

Note 2: Here is what you did, expressed in the notations above. You first transformed the 'annualized' PIT PD $$x$$ to true PIT PD on a specific quarter:

$$1-(1-x)^{1/4} (\approx x/4)$$

(approximation holds for small $$x$$). The rest of the columns make sense (but the titles you used are somewhat confusing, hence the need for mathematical definitions :)). Note 3: My take on 'annualized' PIT PD transformation is that one assumes that all PIT PD's (default in a specific quarter conditional on survival up to the start of the quarter) are equal, to say $$y$$:

$$PD_{t}=PD_{t+1}=PD_{t+2} = PD_{t+3} = y$$

and that $$P(\tau \leq t+3|\tau > t-1) = x$$

is given (default in a specific year conditional on survival up to the start of that year, aka the start of its first quarter). Using the penultimate equality $$(*)$$ above:

$$x = 1 - (1-y)^4$$

we get

$$y = 1- (1-x)^{1/4}.$$

• I am not from a mathematical background. Will this model gives the same output as shown in the question excel output. Sep 7 at 7:47
• If the given numbers in the first column are $PD_i$, $i=1,…,11$, PIT PD’s for the first 11 quarters, you can compute $SP_i$, survival probability beyond $i$-th quarter end, using the given formula above. That would go in your second column (if that’s what you intended to put in it). If in the last column you intended to put probability of default happening in the $i$-th quarter, you can use the formula for $P(i-1<\tau \leq i)$. If the third column is intended to have probability of the default happening before the end of the $i$-th quarter, then you can use the formula for $P(\tau \leq i)$.
– ir7
Sep 7 at 8:15
• Are these concepts applicable as it is(without any modification) if the probabilities of default in "column 1" per quarter are annualized quantities. Sep 7 at 10:48
• @ir7: either your or my Survival Probability formula is wrong. Mine is $1 -\sum_{i}\left(PD_i\right)$, yours is $\prod_{i=1}^{t-1} (1- PD_{i})$. Sep 7 at 16:39
• @JanStuller None is wrong. Yours is a first-order approximation of mine (discarding products of two or more probabilities). :) $P(\tau >1) = 1- P(\tau <1|\tau>0) = 1- PD_1$ and $P(\tau>2) = P(\tau>2)/P(\tau>1)\cdot P(\tau>1) = (P(\tau >1) - P(1<\tau <2))/P(\tau>1)\cdot P(\tau>1)=(1-PD_2)\cdot (1-PD_1)$. In your text you actually say $P(\tau >2) = 1 - PD_1 -PD_2\cdot SP_1$, where last term is a 'marginal' you computed, not a PIT PD, which is correct.
– ir7
Sep 7 at 17:13