BANK3011 · Bank Financial Management
Credit Risk I: Individual Loan Risk
Individual loan risk is the Week 5 credit-risk topic in BANK3011 Bank Financial Management at the University of Sydney, where a single loan is priced for the chance it is not repaid and then accepted or rejected. The tools are the contractually promised return k (base rate plus risk premium, grossed up for fees, a compensating balance and reserves), the expected return E(1+r) = p(1+k) that discounts k for default, and RAROC — risk-adjusted return on capital — which weighs a loan's net income against the capital it puts at risk. Along the way you back a default probability out of a credit spread, meet the credit-scoring families (linear-probability, logit, Altman Z), and see why charging a higher loan rate can actually earn the bank less. Because the exam provides a formula sheet, most marks come from choosing the right formula, reading the wording (promised vs expected), and laying out clean working.
What this chapter covers
- 011. What credit risk is — the risk of default AND of a deterioration in creditworthiness (rating migration) short of default; default is only the endpoint of the scale
- 022. Borrower-specific vs market-specific factors — reputation, leverage, collateral and covenants (diversifiable) versus the business cycle and the level of interest rates (systematic)
- 033. Contractually promised gross return k — 1+k = 1 + [of + (BR + φ)] / [1 − b(1 − RR)], grossing base rate + risk premium up for the origination fee, compensating balance and reserve requirement
- 044. Expected vs promised return — E(1+r) = p(1+k) with zero recovery, or E(1+r) = (1+k)[p + (1 − p)γ] once a recovery rate γ is allowed
- 055. The backward-bending loan-rate curve — past a rate k*, a higher k drives away good borrowers and lifts the default probability faster than it lifts return, so expected return falls
- 066. Implied default probability from spreads — from p(1+k) = 1+i, p = (1+i)/(1+k) and default = 1 − p; the credit spread is C = k − i, and recovery raises the tolerable default rate
- 077. Credit-scoring families — linear-probability, logit/probit and the Altman Z-score (linear discriminant), plus market-based measures (EDF, mortality rates, spreads)
- 088. RAROC — expected one-year net income ÷ capital at risk |ΔL|, with ΔL = −D·L·ΔR/(1+R); approve the loan only if RAROC clears the bank's benchmark (its ROE / cost of equity)
RAROC accept-or-reject on a single loan
- +2Capital at risk uses the duration-based worst-case fall in loan value: ΔL = −D·L·ΔR/(1+R) = −7 × 10,000,000 × (0.045/1.11) = −$2,837,838, so |ΔL| = $2,837,838.
- +2Income lines: interest = 0.11 × 10,000,000 = $1,100,000; servicing fee = 0.003 × 10,000,000 = $30,000; cost of funds = 0.09 × 10,000,000 = $900,000.
- +1Net income = 1,100,000 + 30,000 − 900,000 = $230,000.
- +1Take the present value of the one-year net income (not the raw book figure): PV = 230,000 / 1.09 = $211,009.
- +1RAROC = PV(net income) / capital at risk = 211,009 / 2,837,838 = 0.0744 = 7.44%.
- +1Decide: 7.44% < 10% hurdle, so reject the loan as structured (or restructure it — shorten the duration, raise the fee, or widen the risk premium — and re-test).
Key terms
- Credit risk
- The risk that a borrower fails to meet the loan's terms (default) and, in the modern view, the risk of value loss from a deterioration in creditworthiness (rating migration) even before any payment is missed.
- Contractually promised return (k)
- The return a loan promises if fully repaid: base rate + risk premium, grossed up for the origination fee, compensating balance and reserve requirement via 1+k = 1 + [of + (BR + φ)] / [1 − b(1 − RR)].
- Expected return E(1+r)
- The default-adjusted return: with repayment probability p and zero recovery, E(1+r) = p(1+k); allowing a recovery rate γ gives E(1+r) = (1+k)[p + (1 − p)γ]. It is below the promised k whenever default is possible.
- Default probability (1 − p)
- The chance the borrower does not repay, where p is the repayment probability. From a spread it is 1 − p = 1 − (1+i)/(1+k).
- Loss given default (LGD)
- The fraction of exposure actually lost if default occurs, equal to 1 − recovery rate. LGD is a loss RATE, not a probability — a common exam confusion with EDF.
- Expected default frequency (EDF)
- The estimated probability that a borrower defaults over a horizon. The return on a risky loan is R = AIS − EDF·LGD, where AIS is the all-in-spread.
- Credit spread (C = k − i)
- The extra promised yield a risky loan carries over the comparable risk-free rate i. It is the market's price of the loan's default risk and pins down the implied default probability.
- RAROC
- Risk-adjusted return on capital = expected one-year net income ÷ capital at risk |ΔL|, with ΔL = −D·L·ΔR/(1+R). The bank approves a loan only if RAROC clears its benchmark return (typically its ROE).
Credit Risk I: Individual Loan Risk FAQ
Can AI help me with credit risk and individual loan risk?
Yes — ask Sia to walk through any credit risk and individual loan risk problem or concept step by step, the way University of Sydney tests it. Sia is an AI tutor that explains the method — why the compensating balance sits in the denominator of k, why the expected return is p(1+k) rather than k, and why RAROC divides by capital at risk rather than loan size — so you can reproduce the working yourself under exam conditions.
What is the difference between the promised return k and the expected return?
The contractually promised return k is what the bank earns IF the loan is fully repaid — base rate plus risk premium, grossed up for fees and the compensating balance. The expected return discounts k for the chance of default: E(1+r) = p(1+k) with zero recovery, where p is the repayment probability. They are equal only when default is impossible (p = 1); otherwise the expected return is always lower. Read the question wording carefully — it decides which one is being asked.
Why can raising the loan rate actually earn the bank less?
Because a higher rate k does two things at once: it lifts the promised return, but it also scares off the safest borrowers (adverse selection) and pushes the default probability of the remaining pool up. Past a point k*, the fall in the repayment probability p outruns the rise in k, so the expected return E(r) = p(1+k) − 1 bends back down. This backward-bending curve is why banks ration credit instead of simply charging risky borrowers ever-higher rates.
How do I back a default probability out of a credit spread?
Set the risky loan's expected return equal to the risk-free rate: p(1+k) = 1+i, so the repayment probability is p = (1+i)/(1+k) and the default probability is 1 − p. For example, with k = 9% and i = 4%, p = 1.04/1.09 = 0.9541, so default ≈ 4.6% and the credit spread is C = k − i = 5%. If the loan recovers a fraction γ on default, solve (1+k)[p + (1 − p)γ] = 1+i instead — recovery cushions the loss, so the same spread is consistent with a higher default probability.
In RAROC, what do I divide by — the loan size or something else?
Divide by the capital at RISK, not the loan size. The denominator is |ΔL|, the duration-based worst-case fall in the loan's value: ΔL = −D·L·ΔR/(1+R), where ΔR is the adverse (e.g. 99th-percentile) rise in the risk premium. Dividing net income by the loan principal L instead is the single most common RAROC error and throws the whole ratio off. Also use the present value of the one-year net income in the numerator, not raw book income.
Is the loan-pricing / RAROC formula on the BANK3011 exam formula sheet?
The University of Sydney final exam provides a formula sheet, and the available course material shows it carries RAROC and the EDF / LGD / AIS terms — so you are not expected to memorise those. It is less clear whether the gross-return grossing-up formula for k is printed, so always confirm the current formula sheet on Canvas before the exam and be ready to reproduce the k formula if it is not given. Either way, the marks reward choosing the right formula and showing clean working, not memorisation.
Studying with AI? Sia — free AI financial modeling tutor works through BANK3011 step by step.
Exam move
Treat individual loan risk as a short pipeline: promised return → expected return → priced → decided. First fix the gross-return formula for k and practise the compensating-balance denominator 1 − b(1 − RR) until it is automatic, watching that the origination fee sits on top with the interest, never in the denominator. Then always read the wording to see whether the promised return k or the expected return E(1+r) = p(1+k) is wanted — they differ by the default discount. Drill backing a default probability out of a spread both with and without recovery, and keep LGD (a loss rate) firmly separate from EDF (a probability). For RAROC, memorise the shape — PV of one-year net income over capital at risk |ΔL| = D·L·ΔR/(1+R) — and rehearse the accept/reject decision plus the three ways to rescue a marginal loan (shorten duration, raise fees, widen the risk premium). Because the exam mixes numerical problems with short-answer conceptual parts, make sure you can also explain WHY the rate curve bends backward and why a diversifiable borrower-specific risk differs from systematic market risk. When a step will not click, ask Sia to explain that exact move step by step rather than looking up a final answer.
