Welcome to the wild jungle of credit risk, where default probabilities, exposure amounts, and loss rates form the ecosystem — and banks act as cautious explorers. 🧭 But instead of snakes and tigers, we deal with Expected Loss $($EL$)$ and Unexpected Loss $($UL$)$. Let’s begin this expedition.
🌱 Part 1: Understanding the Credit Risk Factors
Probability of Default, Exposure, and Loss Rate
Before you get lost in equations, let’s meet our main characters:
- PD $($Probability of Default$)$: Think of this as the chance your friend won’t pay you back after borrowing \$100. It’s not a big deal if they usually do — but there’s always a risk. This is expressed as a percentage, 1%.
- EA $($Exposure Amount$)$: This is the amount of money at risk — like the $100 you lent. In banking, it’s the outstanding loan or the exposure at default $($EAD$)$.
- LR $($Loss Rate$)$ or LGD $($Loss Given Default$)$: If your friend defaults, how much of your \$100 will you lose? If they return $60 after some delay, your LR is 40% and the recovery rate $($RR$)$ is 60%, since LR = 1 – RR.
🤔 Why stop here?
Because now you know the players. But what exactly is the “expected” damage if someone defaults?
🎯 Part 2: Expected Loss – What You Think You’ll Lose
Define and Calculate Expected Loss
Expected Loss EL is what the bank anticipates losing — on average — over time: $EL = EA \times PD \times LR$
🧮 Example:
A loan of \$1,800,000, PD = 1%, LR = 40%
$EL = \$1,800,000 \times 0.01 \times 0.4 = \$7,200$
💡Analogy:
It’s like insuring your mobile phone. You know, on average, a certain percentage of people break their phone each year — and you’re setting aside money accordingly.
🤔 But wait…
What if something really bad happens? What if your friend not only defaults but disappears with your $100? This is where unexpected loss enters the jungle.
⚠️ Part 3: Unexpected Loss – The Shock You Didn’t See Coming
Define and Calculate UL
EL is what you expect. UL is what surprises you. Mathematically, it’s modeled using standard deviation — measuring the variation in loss: $UL \equiv \sqrt{\text{Var}(V_H)}$
Breaking it down: $UL = EA \times \sqrt{PD \cdot \sigma^2_{LR} + LR^2 \cdot \sigma^2_{PD}}$
Where:
- $\sigma^2_{PD} = PD \cdot (1 – PD)$ $($from binomial distribution$)$
- $\sigma^2_{LR}$ = variance of loss rate $($estimated$)$
🧮 Continuing Our Example:
EA =\$1,800,000
PD = 0.01, LR = 0.4
$\sigma_{PD} = 0.1$, $\sigma_{LR} = 0.3$ $UL = \$1,800,000 \times \sqrt{0.01 \cdot 0.3^2 + 0.4^2 \cdot 0.1^2} = \$90,000$
So now, even though $EL = \$7,200$, the unexpected volatility around that number is \$90,000. 😱
🧪 Part 4: Portfolio-Level Expected and Unexpected Loss
$UL_P$ and $EL_P$
What if you have multiple loans? Enter the portfolio view:
- Portfolio Expected Loss $EL_P$ is simple: just add up the ELs.
$EL_P = \sum_{i} EA_i \cdot PD_i \cdot LR_i$
- Portfolio Unexpected Loss $UL_P$ includes correlations between loans:
$UL_P = \sqrt{ \sum_i \sum_j \rho_{ij} \cdot UL_i \cdot UL_j }$
Where $\rho_{ij}$ is the correlation between asset i and j.
🎯 Key Point:
If all loans are perfectly correlated $(\rho_{ij} = 1)$, portfolio risk is just the sum of individual ULs. But in real life, correlation < 1, so diversification reduces risk.
🔬 Part 5: Risk Contribution – Who’s Rocking the Boat?
Each asset contributes differently to the portfolio’s total risk. The Risk Contribution $(RC_i)$ tells us this: $RC_i = UL_i \cdot \frac{\sum_j {UL_{j}} {\cdot}\rho_{ij}}{UL_P}$
For Two Assets:
$RC_1 = UL_1 \cdot \frac{UL_1 + {\rho}_{1,2} \cdot UL_2}{UL_P}$
$RC_2 = UL_2 \cdot \frac{UL_2 + \rho_{1,2} \cdot UL_1}{UL_P}$
And yes, $RC_1 + RC_2 = UL_P$. Just like your pizza slices always add up to the whole pie. 🍕
🔄 Part 6: Diversification – Your Best Friend
Diversifiable Risk is the kind you can eliminate by spreading your bets across many loans. Like not putting all your eggs in one basket.
Undiversifiable Risk is what remains — driven by economic or market-wide factors. Even if you diversify across borrowers, industries, and geographies, there’s still some risk you can’t shake off.
🔄 Part 7: The Role of Correlation – Friends or Foes?
When loans are highly correlated, one bad apple spoils the bunch. If default in one loan means others likely default too $($say, due to an industry crash$)$, your portfolio risk shoots up.
Correlation is like gossip — if everyone listens to the same rumors, they panic together.
But estimating correlation is hard:
$\text{Number of correlation pairs} = \frac{n \cdot (n – 1)}{2}$
- In 20 assets: 190 pairs
- In 100 assets: 4,950 pairs
That’s a lot of gossip to track!
🧮 Final Numerical Example: Bigger Bank’s Risk
Using the formulas we learned, suppose:
- Loan A: \$8,250,000, PD = 0.5%, LR = 50%
- Loan B: \$1,800,000, PD = 1%, LR = 40%
- $\sigma_{LR} = 0.25$ for A, 0.3 for B; $\sigma_{PD}$ = 0.0705 and 0.1
- Correlation = 0.3
➡️$EL_A = \$20,625$, $EL_B = \$7,200$
➡️ $UL_A = \$325,333$, $UL_B = \$90,000$
➡️ $UL_P = \$362,642$
➡️ $RC_A = \$316,084$, $RC_B = \$46,558$
➡️ Check: $RC_A + RC_B = UL_P$ ✅
📌 Final Takeaways
- EL is predictable loss — you plan for it.
- UL is the surprise — and banks hold economic capital to cover it.
- Risk contribution tells who’s shaking the ship.
- Diversification and correlation management are your best defense against Titanic moments. 🚢