Survival and marginal default probabilities · RiskAtoms Academy

By the end you will be able to

1Move between cumulative PD and survival, S(t) = 1 − PD(t), and read a default term structure.
2Compute the marginal PD S(t−1) − S(t) and the conditional PD [S(t−1) − S(t)] / S(t−1), and build survival from annual PDs and back.
3Use a constant hazard rate, S(t) = e^(−λt), with λ ≈ spread / LGD, and the regulatory marginal PD with its floor at zero.
4Write CVA as a sum of LGD × EE × discount factor × marginal PD, and work it on a netting set.

Why you need it

Halvergate Moor, one of RiskAtoms Bank's swap counterparties, trades at a credit spread of 180bp. What is the chance it defaults in year 3, rather than by year 3, or in year 3 if it is still alive at the start? Three different numbers, and the bank's CVA needs exactly one of them.

These Core Path lessons use this atom:

L4.3.10: CVA capital: the standardised approach
SA-CVA starts from regulatory CVA, built from a term structure of market-implied PDs read off credit spreads (MAR50.32).
L3.3.16: Market-implied default risk
Hazard rates and survival curves from CDS spreads, and the credit triangle. That lesson writes survival as Q(t).
L3.4.10: CVA basics
CVA as a sum over time of LGD × EE × discount factor × the PD of defaulting in each slice.
L3.3.18: Provisioning basics
Lifetime expected credit loss: the probability of first defaulting in each year of a loan's life.
L4.7.04: Measuring lifetime ECL
IFRS 9 lifetime ECL adds up a loss for each future period, each weighted by a marginal PD.

You need probability basics from M-C01: Probability basics, conditional probability from M-C12: Conditional probability and Bayes' rule and the exponential from M-A05: The number e and continuous growth. The rating-matrix PDs come from M-C13: Markov chains and transition matrices, and the sums use the notation of M-A08: Summation notation.

The idea

Alive at the start, defaulted during, alive at the end

Imagine 10,000 borrowers, all alive today. Each year some of them default and the rest carry on. Three questions about year 3 have three different answers. How many of the 10,000 are still alive at the end? That is survival. How many of the original 10,000 default during year 3? That is the marginal PD. And of those still alive when year 3 starts, how many default in it? That is the conditional PD.

Follow 10,000 borrowers for five years

Pick a year

Borrowers

Year

defaults in the year, per 10,000 at the start (marginal) per 10,000 still alive (conditional)

Alive at the start of year 39,937.6

Default during year 360.3

Still alive at the end, S(3)9,877.3

Marginal PD: 60.3 of the 10,000 we started with, 0.603%. Conditional PD: 60.3 of the 9,937.6 still alive at the start, 0.607%. Both rise each year: survivors drift down the rating scale, so the risk of a BBB borrower grows with time.

The marginal PD divides by everyone you started with; the conditional PD divides only by the survivors. The two are close while few have defaulted, and drift apart as defaults pile up. The BBB borrowers' PDs rise with time because survivors are downgraded; a constant hazard rate gives the same conditional PD every year.

The rules

Survival and cumulative PD. Start with the default time τ.

Marginal and conditional PDs. Differences of the survival curve, and ratios.

Hazard rates and spreads. A hazard rate turns a single number into a whole survival curve, and the credit triangle reads it from a spread.

CVA as a sum. Each slice of time contributes loss × discount × the chance that default falls in it.

The rules, for reference

\begin{gathered} S(t) = 1 - \text{PD}(t), \qquad S(0) = 1 \\ \text{marginal: } S(t-1) - S(t) \qquad \text{conditional: } h_t = \frac{S(t-1) - S(t)}{S(t-1)} \\ S(t) = \prod_{k=1}^{t} (1 - h_k), \qquad \sum_{k=1}^{t} \big[S(k-1) - S(k)\big] = \text{PD}(t) \end{gathered}

\begin{gathered} S(t) = e^{-\lambda t}, \qquad \lambda \approx \frac{s}{\text{LGD}} \\ \text{PD}(t_{i-1}, t_i) = \max\!\left(e^{-s_{i-1} t_{i-1}/\text{LGD}} - e^{-s_i t_i/\text{LGD}},\ 0\right) \\ \text{CVA} = \text{LGD} \sum_{i=1}^{n} \text{EE}(t_i)\,\text{DF}(t_i)\,\text{PD}(t_{i-1}, t_i) \end{gathered}

In the calculator, survival(h, t) gives e^(−ht), pddefault(h, t) gives 1 − e^(−ht) and hazard(s, R) gives s / (1 − R).

Where the regulatory formula comes from. The Basel CVA standard requires regulatory CVA to use a “term structure of market-implied probability of default (PD)”, a market-consensus expected LGD and simulated paths of discounted exposure, and says the PD term structure “must be estimated from credit spreads observed in the markets” (MAR50.32). It does not print a formula in its text. The marginal PD above is the one Basel III wrote out for the advanced CVA charge (June 2011, paragraph 98), where “the first factor within the sum represents an approximation of the market implied marginal probability of a default occurring between times t_i−1 and t_i”. The same formula is in the old Article 383(1) of the CRR, the advanced method that CRR3 replaced in the EU from 2025 and the UK replaces from 2027, and L4.3.10: CVA capital: the standardised approach uses it. Those texts average EE × DF over each slice, (EE_i−1D_i−1 + EE_iD_i)/2; on a fine grid that makes little difference.

Try it

Start from a spread curve: move the 1-year and 5-year spreads and the LGD. With equal spreads the hazard rate is constant: plot the conditional PD and it is flat, while the marginal PD slopes down. Now make the curve fall steeply, a 1-year spread far above the 5-year one, as for a name in distress, and watch the floor at zero bind. Then switch to annual conditional PDs and build survival as a product.

Try itBuild a default term structure

Start from

1-year spread180bp

5-year spread180bp

LGD60%

Linear between 1 and 5 years, flat outside. S(t) = exp(−s(t)·t / LGD); each year's marginal PD is max(S(t−1) − S(t), 0).

Plot

Marginal PD in year t

Over 10 years

Hazard at 1 year, s/LGD3.00%

Survival S(10)74.08%

Cumulative PD(10)25.92%

Sum of marginal PDs25.92%

The marginal PDs add up to the cumulative PD, as they must.

Year t	s(t)	S(t)	PD(t)	Marginal	Conditional
1	180bp	0.9704	2.96%	2.955%	2.955%
2	180bp	0.9418	5.82%	2.868%	2.955%
3	180bp	0.9139	8.61%	2.783%	2.955%
4	180bp	0.8869	11.31%	2.701%	2.955%
5	180bp	0.8607	13.93%	2.621%	2.955%
6	180bp	0.8353	16.47%	2.544%	2.955%
7	180bp	0.8106	18.94%	2.469%	2.955%
8	180bp	0.7866	21.34%	2.396%	2.955%
9	180bp	0.7634	23.66%	2.325%	2.955%
10	180bp	0.7408	25.92%	2.256%	2.955%

Worked examples

The rating matrix is Module 3.3's (illustrative). Halvergate Moor and the netting sets are RiskAtoms Bank plc's from Module 3.4, in £; CVA uses the bank's 2,000-path exposure simulation, as L4.3.10: CVA capital: the standardised approach does. SA-CVA figures are illustrative: SA-CVA needs the regulator's approval.

1. A BBB term structure, there and back

Step 1 of 4

1
Cumulative PDs from the rating matrix
The default column of Pⁿ (M-C13: Markov chains and transition matrices) gives a BBB borrower's cumulative PD for each horizon. Survival is one minus it.
Year t PD(t) S(t)
1 0.2000% 0.998000
2 0.6238% 0.993762
3 1.2267% 0.987733
4 1.9784% 0.980216
5 2.8565% 0.971435

Year t	PD(t)	S(t)
1	0.2000%	0.998000
2	0.6238%	0.993762
3	1.2267%	0.987733
4	1.9784%	0.980216
5	2.8565%	0.971435

2. From a spread: Halvergate Moor

Step 1 of 4

1
The hazard rate
$\lambda \approx \frac{s}{\text{LGD}} = \frac{0.018}{0.6} = 3.0\%\ \text{a year}$

3. CVA on the bank's netting sets

Step 1 of 4

1

One term a year for Halvergate Moor

Halvergate Moor's seven-year swap and one-year FX forward have no CSA. With EE at mid-year (as in L3.4.10: CVA basics), DF at 4% and the marginal PDs from example 2:

Year	S(start)	Marginal PD	EE, £m	DF	0.6 × EE × DF × PD
1	1.0000	2.955%	2.246	0.9806	£39.05k
2	0.9704	2.868%	2.470	0.9429	£40.08k
3	0.9418	2.783%	2.653	0.9066	£40.17k
4	0.9139	2.701%	2.557	0.8717	£36.12k
5	0.8869	2.621%	2.174	0.8382	£28.66k
6	0.8607	2.544%	1.631	0.8060	£20.06k
7	0.8353	2.469%	0.891	0.7750	£10.23k
Total		18.94%			£214.4k

Common mistakes

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Question 1 · objective 2

A borrower's cumulative PD is 2% by the end of year 1 and 5% by the end of year 2. What is its conditional PD in year 2, in %? (2 decimal places)

%

Question 2 · objective 2

The conditional PDs for years 1, 2 and 3 are 1%, 2% and 3%. What is the three-year cumulative PD, in %? (2 decimal places)

%

Question 3 · objective 3

A name's credit spread is 120bp and LGD is 60%. With a constant hazard rate, what is its five-year cumulative PD, in %? (2 decimal places)

%

Question 4 · objective 3

With a constant hazard rate, what happens to the yearly default probabilities as t grows?

AThe conditional PD stays the same; the marginal PD falls.BBoth stay the same.CThe marginal PD stays the same; the conditional PD rises.DBoth fall, because survivors are safer.

Question 5 · objective 4

In one time slice, EE is £10m, the discount factor 0.95 and the marginal PD 2%. With LGD 60%, what does the slice add to CVA, in £ thousands?

£k

Glossary

Default time, τ: The default time τ: the random date on which a borrower defaults. It may lie beyond any horizon you care about.
Cumulative PD: Cumulative PD, PD(t): the probability of default at any time up to t. It can only rise.
Survival probability: Survival probability S(t) = 1 − PD(t): the probability of no default by t. S(0) = 1 and it can only fall.
Marginal PD: Marginal (unconditional) PD for period t: the probability, seen from today, of defaulting during that period, S(t−1) − S(t).
Conditional (forward) PD: Conditional (forward) PD h_t: the probability of defaulting during period t given survival to its start, [S(t−1) − S(t)] / S(t−1).
Hazard rate: Hazard rate λ: the default intensity per year. With a constant hazard, S(t) = e^(−λt), and the chance of default in a short time dt is about λ dt.
Credit triangle: The credit triangle: spread ≈ hazard rate × LGD, so λ ≈ s / LGD. A rule of thumb for reading default risk from a credit spread.
Loss given default (LGD): Loss given default: the share of the exposure lost if the borrower defaults, 1 − recovery rate.
Expected exposure (EE): Expected exposure EE(t): the average, over simulated scenarios, of what the counterparty would owe the bank at t (zero where it owes nothing).