Probability Trees and Conditional Expectations in Credit Risk

A probability tree is a structured diagram that maps out all possible outcomes and their associated probabilities across sequential events. In fixed income analysis, probability trees are used to model multi-stage credit scenarios, where the probability of default or rating migration in a later period depends on the state reached in an earlier period. The total probability rule states that for an event $B$ with mutually exclusive and exhaustive conditioning events $A_i$: $P(B) = \sum_{i} P(B \mid A_i) \cdot P(A_i)$ The expected value given a probability tree is computed as: $E(X) = \sum_{i} x_i \cdot P(x_i)$ The conditional expected value updates expectations once new information is received. Using Bayes' theorem, the posterior probability of event $A$ given observed event $B$ is: $P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$ The variance of a random variable $X$ is: $\text{Var}(X) = E(X^2) - [E(X)]^2 = \sum_{i} P(x_i) \cdot [x_i - E(X)]^2$ And the standard deviation is simply $\sigma(X) = \sqrt{\text{Var}(X)}$. These tools allow analysts to compute expected portfolio losses, update loss estimates after observing interim credit events, and quantify uncertainty around those estimates.

Scenario: A credit analyst at a fixed income fund is evaluating a two-year corporate bond with a face value of $1,000. The bond pays no coupon (zero-coupon). The analyst uses a two-stage probability tree to model credit outcomes. Stage 1 — End of Year 1: The bond issuer either survives (no default) or defaults.

• $P(\text{Survive Year 1}) = 0.80$
\item $P(\text{Default Year 1}) = 0.20$

Stage 2 — End of Year 2 (conditional on Stage 1): If the issuer survived Year 1:

• $P(\text{Survive Year 2} \mid \text{Survive Year 1}) = 0.85$
\item $P(\text{Default Year 2} \mid \text{Survive Year 1}) = 0.15$

If the issuer defaulted in Year 1:

• $P(\text{Full Recovery} \mid \text{Default Year 1}) = 0.30$
\item $P(\text{Partial Recovery} \mid \text{Default Year 1}) = 0.70$

Recovery Assumptions:

Outcome Path	Bond Payoff at Resolution ($)
Survive Year 1, Survive Year 2	$1,000
Survive Year 1, Default Year 2	$400
Default Year 1, Full Recovery	$900
Default Year 1, Partial Recovery	$250

Additional Information: The analyst subsequently learns that the issuer did survive Year 1. Using this new information, the analyst wants to update expectations. Calculate the following:

1. The joint (unconditional) probability of each of the four terminal outcomes.
2. The unconditional expected payoff $E(X)$ of the bond at resolution.
3. The unconditional variance $\text{Var}(X)$ and standard deviation $\sigma(X)$ of the bond payoff.
4. Using Bayes' theorem, compute the posterior probability that the bond will default in Year 2 given that the issuer survived Year 1. Confirm this equals the directly stated conditional probability.
The conditional expected payoff $E(X \mid \text{Survive Year 1})$ and conditional standard deviation $\sigma(X \mid \text{Survive Year 1})$, given the updated information.

Step 1: Joint Probabilities of Each Terminal Outcome

There are four terminal paths. Joint probability = (Stage 1 probability) $\times$ (Stage 2 conditional probability). $\begin{array}{rcl} P(\text{S1, S2}) &=& P(\text{Survive Y1}) \times P(\text{Survive Y2} \mid \text{Survive Y1}) \\ &=& 0.80 \times 0.85 = 0.680 \\[6pt] P(\text{S1, D2}) &=& P(\text{Survive Y1}) \times P(\text{Default Y2} \mid \text{Survive Y1}) \\ &=& 0.80 \times 0.15 = 0.120 \\[6pt] P(\text{D1, FR}) &=& P(\text{Default Y1}) \times P(\text{Full Recovery} \mid \text{Default Y1}) \\ &=& 0.20 \times 0.30 = 0.060 \\[6pt] P(\text{D1, PR}) &=& P(\text{Default Y1}) \times P(\text{Partial Recovery} \mid \text{Default Y1}) \\ &=& 0.20 \times 0.70 = 0.140 \end{array}$ Verification: $0.680 + 0.120 + 0.060 + 0.140 = 1.000$ ✓

Step 2: Unconditional Expected Payoff

$E(X) = \sum_{i} P(x_i) \cdot x_i$ $\begin{array}{rcl} E(X) &=& (0.680)(1000) + (0.120)(400) + (0.060)(900) + (0.140)(250) \\ &=& 680 + 48 + 54 + 35 \\ &=& \$817 \end{array}$

Step 3: Unconditional Variance and Standard Deviation

First compute $E(X^2)$: $\begin{array}{rcl} E(X^2) &=& (0.680)(1000)^2 + (0.120)(400)^2 + (0.060)(900)^2 + (0.140)(250)^2 \\ &=& (0.680)(1{,}000{,}000) + (0.120)(160{,}000) + (0.060)(810{,}000) + (0.140)(62{,}500) \\ &=& 680{,}000 + 19{,}200 + 48{,}600 + 8{,}750 \\ &=& 756{,}550 \end{array}$ Now apply the variance formula: $\text{Var}(X) = E(X^2) - [E(X)]^2 = 756{,}550 - (817)^2$ $(817)^2 = 667{,}489$ $\text{Var}(X) = 756{,}550 - 667{,}489 = 89{,}061$ $\sigma(X) = \sqrt{89{,}061} \approx \$298.43$

Step 4: Bayesian Update — Posterior Probability of Default in Year 2 Given Survival in Year 1

We want $P(\text{Default Y2} \mid \text{Survive Y1})$ using Bayes' theorem. First, define the relevant events:

• Let $A = \text{Default Y2}$ (which in our tree is only reachable via Survive Y1)
Let $B = \text{Survive Y1}$

$P(\text{Survive Y1}) = P(\text{S1, S2}) + P(\text{S1, D2}) = 0.680 + 0.120 = 0.800$ $P(\text{Default Y2} \mid \text{Survive Y1}) = \frac{P(\text{S1, D2})}{P(\text{Survive Y1})} = \frac{0.120}{0.800} = 0.15$ This confirms the directly stated conditional probability of $0.15$. ✓ Similarly: $P(\text{Survive Y2} \mid \text{Survive Y1}) = \frac{0.680}{0.800} = 0.85 \checkmark$

Step 5: Conditional Expected Payoff and Conditional Standard Deviation

Given survival in Year 1, only two outcomes are possible: Survive Y2 (payoff = $1,000) or Default Y2 (payoff = $400). $E(X \mid \text{Survive Y1}) = (0.85)(1000) + (0.15)(400)$ $= 850 + 60 = \$910$ For the conditional variance, first compute $E(X^2 \mid \text{Survive Y1})$: $E(X^2 \mid \text{Survive Y1}) = (0.85)(1000)^2 + (0.15)(400)^2$ $= (0.85)(1{,}000{,}000) + (0.15)(160{,}000)$ $= 850{,}000 + 24{,}000 = 874{,}000$ Conditional variance: $\text{Var}(X \mid \text{Survive Y1}) = E(X^2 \mid \text{Survive Y1}) - [E(X \mid \text{Survive Y1})]^2$ $= 874{,}000 - (910)^2 = 874{,}000 - 828{,}100 = 45{,}900$ $\sigma(X \mid \text{Survive Y1}) = \sqrt{45{,}900} \approx \$214.24$ Summary of Results:

Metric	Unconditional	Conditional on Survive Y1
Expected Payoff	$817.00	$910.00
Variance	89,061	45,900
Standard Deviation	$298.43	$214.24

Observing that the issuer survived Year 1 increases the expected payoff by $93 and reduces the standard deviation by roughly $84, reflecting meaningfully lower uncertainty once a favorable interim outcome is observed.