The Asset-Side Role of Banks
LSU Business - Department of Finance
March 2026
By the end of this lecture, you should be able to:
Part 1: The Two Information Problems
Part 2: Ex Ante — Banks as Information Producers
Part 3: Ex Post — Banks as Delegated Monitors
Part 4: The Delegation Problem and Its Solution
Part 5: Diamond (1984) — The General Model
Part 6: Diversification — The Unifying Principle
Previous lectures: the liability-side justification for banking (Diamond-Dybvig 1983)
This lecture: the asset-side justification (Diamond 1984)
The Core Question
Who monitors the borrower? And who monitors the monitor?
| Ex Ante | Ex Post | |
|---|---|---|
| When | Before lending | After lending |
| Problem | Adverse selection | Moral hazard |
| Cannot observe | Borrower type | Realized return |
| Market failure | Good borrowers excluded | Waste from liquidation |
| Solution | Screening / signaling | Monitoring |
| Key paper | Stiglitz-Weiss (1981) | Diamond (1984, 1996) |
Both solved by the same institution — the commercial bank — using the same tool: diversification.
| Symbol | Meaning | Section |
|---|---|---|
| \(\theta \in \{G, B\}\) | Borrower type (good/bad) | Ex ante |
| \(p_\theta\) | Success probability; \(p_G > p_B\) | Ex ante |
| \(y_\theta\) | Payoff if success; \(p_G y_G = p_B y_B = \mu\) | Ex ante |
| \(f\) | Face value of loan | Both |
| \(f_\theta^*\) | Type break-even: \(f_\theta^* = r/p_\theta\) | Ex ante |
| \(V \in \{H, L\}\) | Project return (high/low) | Ex post |
| \(P = 0.8\), \(H = 1.4\), \(L = 1\) | Ex post project parameters | Ex post |
| \(\mu = 1.32\), \(r = 1.05\) | Expected return, risk-free rate | Both |
| \(K\) | Information cost (screening or monitoring) | Both |
| \(S = 0.2\) | Liquidation waste | Ex post |
| \(D(n)\) | Delegation cost for \(n\)-loan bank | Ex post |
Key Connection
\(f_G^* = r/p_G = 1.05/0.8 = \mathbf{1.3125}\) — this same number reappears as Diamond’s unmonitored face value \(f^*\).
Borrowers: two types \(\theta \in \{G, B\}\), fraction \(\alpha = 0.5\) are type \(G\)
Mean-preserving spread: \[p_G > p_B, \quad y_G < y_B, \quad p_G y_G = p_B y_B = \mu = 1.32\]
Parameters:
| Type \(G\) | Type \(B\) | |
|---|---|---|
| Success prob \(p_\theta\) | 0.8 | 0.4 |
| Success payoff \(y_\theta\) | 1.65 | 3.30 |
| Break-even face value \(f_\theta^*\) | 1.3125 | 2.625 |
Key: lender cannot observe \(\theta\) — only the loan outcome
A borrower participates only if \(f \leq y_\theta\) (non-negative expected profit).
Since \(y_G < y_B\): good borrowers exit first as \(f\) rises.
The dilemma:
Proposition (Stiglitz-Weiss 1981)
Lender breaks even only at \(f_B^* = 2.625\), lending exclusively to bad borrowers. Good borrowers — who would accept \(f_G^* = 1.3125\) — are excluded entirely, despite positive-NPV projects (\(\mu - r = 0.27 > 0\)).
At \(f = y_G\): \(\Pi\) drops from \(0.99\) to \(0.66\) — a 33% fall. Both segments lie below break-even in the pooling range.
Idea: Can high-quality borrowers credibly signal their type?
Signal: entrepreneur retains equity fraction \(e\) in their own project
Why it works (single-crossing):
Result (Leland-Pyle 1977)
In the separating equilibrium, \(e^*(\theta)\) is strictly increasing in \(\theta\). Market infers type from retention; lenders offer \(f_\theta^* = r/p_\theta\). Adverse selection is eliminated for borrowers who can credibly signal.
Note: LP applies to individual entrepreneurs. The information originates with the borrower, not the bank.
LP applies to individual borrowers. RT formalize the bank-centric extension: the bank actively produces information by pooling signals from \(n\) specialists.
Each screener observes \(s_j = \theta_j + \varepsilon_j\) (independent noise). The portfolio average:
\[\bar{s} = \frac{1}{n}\sum_j s_j \quad \Rightarrow \quad \text{estimation error} = \bar{\varepsilon} \sim N\!\left(0, \frac{\sigma^2_\varepsilon}{n}\right) \to 0\]
Signaling cost of portfolio quality \(\bar{\theta}\) also vanishes: \(\frac{A}{2}(e^*)^2 \cdot \frac{\sigma^2}{n} \to 0\)
Result (Ramakrishnan-Thakor 1984)
Coalition of \(n\) screeners signals portfolio quality with precision \(\to 1\) as \(n \to \infty\). Residual uncertainty vanishes — bank funds deposits near risk-free rate. Specialization (lower \(K\) per loan) and diversification (higher precision) are complements.
Observation: Firms pay a premium for bank loans over bonds — yet value banking relationships
Why? Banks produce private information through the banking relationship:
Bank credit line as quality signal:
Key Insight (Fama 1985)
Screening and monitoring are complements — both co-locate in the bank because the banking relationship reduces costs for both functions simultaneously.
After lending, the borrower observes realized return \(V\) — the lender does not.
\[V = \begin{cases} H = 1.4 & \text{with prob } P = 0.8 \\ L = 1 & \text{with prob } 1-P = 0.2 \end{cases}\]
Expected return: \(\mu = 0.8(1.4) + 0.2(1) = 1.32 > r = 1.05\) ✓
Moral hazard: borrower can claim \(V = 0\) and pocket the proceeds. Any contract based on reported profits fails — the borrower always reports the minimum.
The Root Problem
The borrower controls the information. Equity contracts collapse. The lender needs either monitoring (verify \(V\)) or a fixed payment (debt) enforced by a liquidation threat.
CSV framework: lender can pay cost \(K\) to verify the true return
Contract design problem: which states to monitor?
Townsend’s key result: the optimal rule is deterministic — monitor in some states with certainty, not others at all. Randomized monitoring (\(K\) with probability \(\pi < 1\)) is dominated:
Which states? Only where the borrower cannot pay voluntarily — this delivers Gale-Hellwig.
Proposition (Gale-Hellwig 1985): Standard Debt is Optimal
Monitor only in default states (when borrower cannot pay). The standard debt contract with face value \(f\):
Expected monitoring cost \(= (1-P)K = 0.2 \times 0.0002 = 0.00004\) (nearly zero)
Simplification: without monitoring, lender’s only tool is liquidation — both parties get 0.
| General CSV | Diamond (1996) | |
|---|---|---|
| Bad state, no monitoring | Misreporting equilibrium | Liquidation → 0 |
| Bad state, with monitoring | Lender verifies, collects \(L - K\) | Lender accepts \(L = 1\) |
| Social waste | Incentive distortion | \(S = (1-P)L = 0.2\) |
Two benchmark contracts:
Contract: pay \(f\) or face liquidation (→ both get 0)
Borrower pays \(f\) if \(V = H\) (keeps \(H - f > 0\)), defaults if \(V = L < f\).
Lender’s break-even: \[P \cdot f + (1-P) \cdot 0 = r \quad \Rightarrow \quad 0.8f = 1.05 \quad \Rightarrow \quad \boxed{f^* = 1.3125}\]
| State | Prob | Lender gets | Borrower keeps |
|---|---|---|---|
| \(H = 1.4\) | 0.8 | 1.3125 | 0.0875 |
| \(L = 1\) | 0.2 | 0 (liquidated) | 0 |
Deadweight cost: \(S = (1-P) \cdot L = 0.2 \times 1 = \mathbf{0.2}\) destroyed per borrower
Contract: monitor in bad state; accept \(L = 1\) instead of liquidating
Lender’s break-even: \[P \cdot f + (1-P)(L - K) = r \quad \Rightarrow \quad 0.8f + 0.2(1 - 0.0002) = 1.05 \quad \Rightarrow \quad f \approx 1.0625\]
Unmonitored
Monitored
Monitoring captures the full \(S = 0.2\) at cost \(\approx 0.00004\). Enormous net gain — if monitoring is organizationally feasible.
Borrower needs \(m = 10{,}000\) investors, each contributing \(0.0001\).
If all investors monitor independently: \[\text{Total cost} = m \times K = 10{,}000 \times 0.0002 = \mathbf{2}\]
This exceeds the entire project return of 1.32. Monitoring is socially valuable (\(S = 0.2 > K = 0.0002\)) but individually prohibitive when duplicated.
Free-rider problem: if only one investor monitors, the information is a public good — everyone else free-rides. Monitoring collapses. Only unmonitored debt at \(f^* = 1.3125\) survives.
The Core Tension
\(S = 0.2 \gg K = 0.0002\) but \(S = 0.2 \ll mK = 2\). Monitoring is individually cheap but collectively ruinous when duplicated. Need: one entity monitor on behalf of many.
Structure: bank intermediates between depositors and borrower
New problem: depositors face the same moral hazard with the bank that investors faced with borrowers. The bank observes loan repayments; depositors do not.
Two-Tier Information Problem
| Private info | Misrepresentation | |
|---|---|---|
| Investor → Borrower | Borrower knows \(V\) | Claim \(V = 0\) |
| Depositor → Bank | Bank knows loan income | Claim loans returned little |
Diamond’s key insight: diversification solves the second problem without depositor monitoring.
Bank monitors one borrower, issues deposits to \(m = 10{,}000\) investors.
Bank collects \(f\) (success) or \(1\) (monitored failure). Depositors need 5% return:
\[P \cdot B = r \quad \Rightarrow \quad 0.8B = 1.05 \quad \Rightarrow \quad B = 1.3125\]
Problem: in the bad state, bank collected \(L = 1\) but owes \(B = 1.3125\). Bank fails whenever the borrower defaults — probability \(1-P = 0.2\).
No Efficiency Gain
One-loan bank fails 20% of the time — same as direct finance. Delegation provides zero efficiency gain without diversification.
Bank monitors two independent borrowers. Bank income:
| Outcome | Prob | Bank income |
|---|---|---|
| Both succeed | \(0.64\) | \(2f\) |
| One succeeds, one fails | \(0.32\) | \(f + 1\) |
| Both fail | \(0.04\) | \(2\) |
Set \(f = 1.1875\) so that \(f + 1 = 2.1875 \geq 2B\). Depositor break-even:
\[0.96 \times 2B = 2(1.05) \quad \Rightarrow \quad B = 1.09375, \quad 2B = 2.1875\]
Bank fails only when both default: probability \((1-P)^2 = \mathbf{0.04}\) (down from 20%)
Delegation cost: \[D = \underbrace{0.04 \times 2}_{\text{distress}} + \underbrace{0.0608}_{\text{control rent}} = 0.1408\] \[\boxed{K + D = 0.0002 + 0.1408 = 0.141 < S = 0.200} \quad \checkmark\]
\(n\) borrowers, i.i.d. returns \(V_j \sim G(\cdot)\), mean \(\mu\), only borrower observes \(V_j\).
Bank’s per-loan income (Gale-Hellwig CSV contract): \[R_j = \min(V_j, F) - K \cdot \mathbf{1}(V_j < F)\]
Expected per-loan income: \(\rho \equiv \mathbb{E}[R_j] > B\) (strictly above deposit rate)
Bank’s total income: \[W_n = \sum_{j=1}^n R_j \quad \xrightarrow{\text{LLN}} \quad n\rho \quad \text{as } n \to \infty\]
Since \(\rho > B\): the bank almost surely pays depositors in full as \(n\) grows. Bank failure probability \(\to 0\).
Theorem: Delegation Cost Vanishes with Diversification
Let \(R_1, R_2, \ldots\) be i.i.d. with mean \(\rho\) and variance \(\sigma^2 < \infty\), and \(B < \rho\). Then:
\[\boxed{D(n) \to 0 \quad \text{as } n \to \infty}\]
\[K + D(n) \to K \quad \text{as } n \to \infty\]
In the limit: deposit rate \(\to r\), loan rate \(\to\) CSV-optimal, full surplus \(\mu - r - K\) distributed to borrowers and depositors.
Bank fails when \(W_n/n < B\). Apply Chebyshev to the sample mean:
\[\Pr\!\left(\frac{W_n}{n} < B\right) \leq \Pr\!\left(\left|\frac{W_n}{n} - \rho\right| \geq \rho - B\right) \leq \frac{\sigma^2}{n(\rho - B)^2}\]
Binary Special Case
For \(n=2\): exact failure prob \(= (1-P)^2 = 0.04\). Chebyshev gives a conservative upper bound. As \(n \to \infty\): failure prob \(\to 0\) geometrically.
The same CSV logic (Gale-Hellwig) applies twice:
| Relationship | CSV problem | Optimal contract | Monitoring when |
|---|---|---|---|
| Bank → Borrower | Borrower hides \(V_j\) | Loan at face \(f\); monitor in default | \(V_j < f\); prob \(\approx 0.2\) |
| Depositor → Bank | Bank hides \(W_n\) | Deposit at face \(B\); liquidate on failure | \(W_n < nB\); prob \(O(1/n) \to 0\) |
The Transformation the Bank Performs
\[\underbrace{\text{Monitored loans}}_{\text{risky, individually}} \xrightarrow{\;\text{diversification}\;} \underbrace{\text{Fixed-rate deposits}}_{\text{nearly riskless}}\]
| Ex Ante: Screening | Ex Post: Monitoring | |
|---|---|---|
| Problem | Cannot observe type | Cannot observe return |
| Cost without bank | Market exclusion; signaling costly | Waste \(S = 0.2\); delegation cost \(D\) |
| Bank’s tool | Diversify signal noise | Diversify default risk |
| LLN result | \(\text{Var}(\bar{\varepsilon}) = \sigma^2/n \to 0\) | \(D(n) \to 0\) |
| Outcome | Screening cost \(\to 0\) | Delegation cost \(\to 0\) |
One principle, two applications: diversification makes the bank a scalable information producer — on both sides of the lending relationship.
Diversification is essential, not optional:
| Bank type | Failure prob | Delegation cost | Deposit rate |
|---|---|---|---|
| Undiversified (\(n\) small) | High | High | Above risk-free |
| Well-diversified (\(n\) large) | \(\approx 0\) | \(\approx 0\) | \(\approx r\) |
| Fully diversified (\(n \to \infty\)) | 0 | 0 | \(r\) |
Policy implications:
Diamond (1996)
“Diversification makes bank deposits much safer than bank loans, and in the limit of fully diversified banks with independently distributed loans, bank deposits become riskless.”
| Liability side (DD 1983) | Asset side (Diamond 1984) | |
|---|---|---|
| Core problem | Timing / liquidity | Borrower moral hazard |
| Bank’s role | Liquidity transformation | Delegated monitoring |
| Mechanism | Pool liquidity shocks | Diversify default risk |
| Why debt | Optimal with sequential service | Minimizes depositor monitoring |
| Why many loans | Less relevant | Essential: LLN reduces \(D(n)\) |
| Fragility | Coordination failure / runs | Correlated losses / undiversification |
Both theories independently predict: many loans, fixed-rate deposits, diversified portfolios.
The commercial bank is a robust solution to multiple market failures simultaneously.
Two information problems: ex ante (adverse selection, type unknown) and ex post (moral hazard, return unverifiable) — both generate market failures without intermediation
Market exclusion (Stiglitz-Weiss): pooling rate exceeds good-borrower participation constraint → good borrowers excluded despite positive NPV
Signaling (Leland-Pyle): equity retention screens types; banks reduce signaling cost via diversification (\(\sigma^2/n \to 0\))
Delegated monitoring (Diamond): bank monitors on behalf of \(m\) investors; diversification drives \(D(n) \to 0\) — deposits become nearly riskless
Unifying principle: the Law of Large Numbers makes banks efficient information producers in both the ex ante and ex post dimensions — more loans → lower information cost per loan → cheaper deposits
Debt is optimal on both sides: Gale-Hellwig applies to bank-borrower relationship (loans) and depositor-bank relationship (deposits) — same logic, same contract form
Primary:
Foundations:
FIN 7650 Banking - Lecture 5