Quarterly Survival rate given there is a Quarterly Probability of Default

Question

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

Answer 1

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.

Answer 2

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

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