The Encyclopedia of Economic Principals

Chapter 48

Credit, Default, Debt Overhang, and Agency

Who gets the upside, who bears the downside, and what that does to lending.

Four of the chapter's worked examples, made interactive: shareholders shifting risk onto creditors, debt overhang blocking a good project, equity priced as a call on the firm, and a bank that rations credit rather than raise its rate.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Gambling with creditors' money

Why would shareholders pick a plan that lowers the firm's value, and does an escrow stop them?

Equity keeps the upside above the debt promise and walks away from the downside, so spreading outcomes can raise equity's value while lowering the firm's. Making equity bear part of the failure state removes that private gain.

Equation, written in LaTeX: 0.55(160)+0.45(20)=97\text{ dollars}

Equation, written in LaTeX: 0.55(90)+0.45(20)=58.50\text{ dollars}

Equation, written in LaTeX: 0.55(160-90)+0.45(0)=38.50\text{ dollars}

Equation, written in LaTeX: 0.55(70)+0.45(-30)=25\text{ dollars}

Equation, written in LaTeX: 0.55(90)+0.45(20+30)=72\text{ dollars}

Scroll sideways for the whole equation

Debt promises 90 dollars. The safe plan pays 115 for sure; the risky plan pays 160 with the success probability and 20 otherwise. The escrow is returned on success and paid to creditors on failure; equity is measured net of the deposit.

Predict first. At success probability 0.55, does a 30 dollar escrow remove equity's preference for the risky plan?

Your prediction

Choose an example

Scroll sideways for the whole figure

Figure: Gambling with creditors' money. Stacked bars: safe plan creditors 90 and equity 25; risky plan with success probability 0.55 and escrow 0: creditors 58.50 and equity 38.50, total 97.00.
Success probability of risky plan: 0.55, Shareholder escrow: None
Constructed example: the chapter's hypothetical firm (debt 90, safe 115, risky 160 or 20 at 0.55, escrow 30); success probabilities 0.35, 0.45 and 0.65 and an escrow of 50 are added.

Calculated values

Risky plan enterprise value
97.00
Safe plan enterprise value
115.00
Creditors, risky plan
58.50
Equity, risky plan
38.50
Equity, safe plan
25.00
Equity gain from switching
+13.50

The risky plan is worth 0.55(160) + 0.45(20) = 97.00, which is 18.00 below the safe 115. Creditors expect 0.55(90) + 0.45(20 + 0) = 58.50 instead of 90. Against 25 under the safe plan, equity prefers the risky plan, gaining 13.50 dollars.

Worked steps

  1. Enterprise = 0.55(160) + 0.45(20) = 97.00
  2. Creditors = 0.55(90) + 0.45(20 + 0) = 58.50
  3. Equity = 0.55(70) + 0.45(0) = 38.50
  4. Equity gain = 38.50 - 25 = +13.50
  5. Value lost = 115 - 97.00 = 18.00

Use the idea

When a borrower controls risk choices, look at who bears the bad state, not only at the expected value of the plan.

Where the conclusion applies

Two outcomes, risk neutrality, no discounting and no cost of running the escrow. The escrow never exceeds the creditors' 70 dollar shortfall in the bad state.

Check your understanding: With success probability 0.45 and no escrow, does equity still prefer the risky plan?
Enterprise = 0.45(160) + 0.55(20) = 83; equity = 0.45(70) = 31.50 > 25, so yes, while the firm loses 115 - 83 = 32.

Chapter 48 source: section "Asset-substitution risk shifting".

Demonstration 2 of 4

Debt overhang blocks a positive-NPV project

Why do shareholders turn down a project that adds value, and how much debt relief changes that?

When debt exceeds what the firm can pay, new value goes first to creditors. Shareholders who pay for the project keep only the part above the face value, so good projects are refused.

Equation, written in LaTeX: 35-24=11\text{ dollars}

Equation, written in LaTeX: 10-24=-14\text{ dollars}

Equation, written in LaTeX: 105-70=35\text{ dollars}

Scroll sideways for the whole equation

Assets in place pay 70. Shareholders put in 24 for a project that adds a certain payoff. Debt is paid first, up to its face value; equity gets the rest. The discount rate is zero.

Predict first. With a project payoff of 35, which debt face lets shareholders gain from investing?

Your prediction

Choose an example

Scroll sideways for the whole figure

Figure: Debt overhang blocks a positive-NPV project. Bars of terminal payoff with debt face 95: rejecting gives creditors 70 and equity 0; investing gives creditors 95 and equity 10.
Debt face value: 95, Project payoff: 35
Constructed example: the chapter's hypothetical firm (assets 70, debt 95 written down to 70, investment 24, payoff 35); faces 80 and 105 and payoffs 30 and 40 are added.

Calculated values

Project NPV
11
Equity payoff with project
10
Equity private NPV
-14
Creditor gain
25
Shareholders invest
no

The project's NPV is 35 - 24 = 11. With it, assets are 70 + 35 = 105; debt takes min(105, 95) = 95 and equity gets 10. Without it, equity gets 0. Equity's private NPV is 10 - 0 - 24 = -14. Shareholders reject a project worth 11 to the firm, because creditors capture 25 of the 35 dollar payoff.

Worked steps

  1. Project NPV = 35 - 24 = 11
  2. Equity without = max(70 - 95, 0) = 0
  3. Equity with = max(105 - 95, 0) = 10
  4. Private NPV = 10 - 0 - 24 = -14
  5. Creditor gain = 95 - 70 = 25

Use the idea

Before blaming weak demand for low investment, check whether the borrower's existing debt would capture most of the return.

Where the conclusion applies

A certain payoff, zero discounting, no new senior financing and no renegotiation once the project is chosen.

Check your understanding: With face 80 and payoff 35, do shareholders invest?
Equity gets 105 - 80 = 25; private NPV = 25 - 24 = 1 > 0, so yes.

Chapter 48 source: section "Debt-overhang underinvestment".

Demonstration 3 of 4

Equity as a call on the firm (Merton)

What are equity and debt worth when equity is a call option on the firm's assets?

Equity holds the upside above the promise and limited liability below it, the payoff of a call. More volatility makes the call worth more and the debt worth less, with assets unchanged.

Equation, written in LaTeX: d_1=\frac{\ln(150/120)+(0.03+0.5(0.25)^2)}{0.25}\approx1.138

Equation, written in LaTeX: E_t\approx150(0.87235)-120e^{-0.03}(0.81262)\approx36.22\text{ dollars}

Equation, written in LaTeX: N(-0.888)\approx0.187

Equation, written in LaTeX: E_t=V_tN(d_1)-De^{-r\tau}N(d_2)

Scroll sideways for the whole equation

V is the firm's asset value, D = 120 the debt promise due in one year, r = 0.03 the risk-free rate and sigma the asset volatility. N is the standard normal distribution function; d2 = d1 - sigma.

Predict first. Does raising asset volatility raise or lower equity value?

Your prediction

Choose an example

Scroll sideways for the whole figure

Figure: Equity as a call on the firm (Merton). Equity payoff hockey stick with a kink at 120 and the risk-neutral density of year-end assets centred near 150; the shaded area below 120 is 18.7%. Equity is worth 36.22.
Asset value: 150, Asset volatility: 0.25
Constructed example: the chapter's hypothetical firm (V 150, D 120, r 0.03, one year, sigma 0.25); asset values 130 and 170 and volatilities 0.15 and 0.35 are added.

Calculated values

d1
1.138
d2
0.888
N(d1)
0.87235
N(d2)
0.81262
Equity value
36.22
Debt value
113.78
Risk-neutral default probability
18.7%

With V = 150 and sigma = 0.25, d1 = (ln(150/120) + (0.03 + 0.5(0.25)^2))/0.25 = 1.1376 and d2 = 1.1376 - 0.25 = 0.8876. Equity is 150(0.87235) - 120e^(-0.03)(0.81262) = 36.22, so debt is 150 - 36.22 = 113.78. The chance that assets end below 120 under the pricing measure is N(-0.888) = 18.7%; it is a pricing probability, not a forecast.

Worked steps

  1. ln(150/120) = 0.22314
  2. 0.03 + 0.5(0.25)^2 = 0.06125
  3. d1 = (0.22314 + 0.06125) / 0.25 = 1.1376
  4. d2 = 1.1376 - 0.25 = 0.8876
  5. E = 150 x 0.872351 - 116.4535 x 0.812615 = 130.8526 - 94.6318 = 36.2208
  6. Debt = 150 - 36.22 = 113.78
  7. Default probability = N(-0.8876) = 0.1874

Use the idea

Read a firm's equity and credit spread together: falling asset value or rising volatility moves both, and the model links them.

Where the conclusion applies

Assets follow a lognormal diffusion, default can happen only at the one-year maturity, there is one zero-coupon debt claim and markets are frictionless.

Check your understanding: At V = 150 and sigma = 0.35, what are equity, debt and the default probability?
d1 = (0.2231 + 0.03 + 0.06125)/0.35 = 0.898; d2 = 0.548; E = 39.84; debt = 110.16; N(-0.548) = 29.2%.

Chapter 48 source: section "Merton structural default model".

Demonstration 4 of 4

Why a bank refuses higher rates

Why would a bank turn away borrowers who offer to pay more?

A higher rate drives out safer borrowers and pushes those who remain toward riskier projects. If expected repayment falls enough, the bank earns less at the higher rate and rations credit instead of raising it.

Equation, written in LaTeX: \frac{12{,}480-12{,}000}{12{,}000}=0.04=4\%

Equation, written in LaTeX: 12{,}000(1.13)=13{,}560\text{ dollars}

Equation, written in LaTeX: \frac{12{,}240-12{,}000}{12{,}000}=0.02=2\%

Scroll sideways for the whole equation

Each loan is 12,000 dollars for a year. Expected repayment is after default and recovery. At 7 percent it is 12,480; at 13 percent it depends on who still applies and which projects they choose.

Predict first. Which expected repayment at 13 percent makes the bank indifferent between the two quotes?

Your prediction

Choose an example

Scroll sideways for the whole figure

Figure: Why a bank refuses higher rates. Two points of the bank's expected return: 4% at a 7 percent quote and 2% at 13 percent, when expected repayment at 13 percent is 12,240.
Expected repayment at 13 percent: 12,240, Quoted rate shown: 13%
Constructed example: the chapter's hypothetical bank (loan 12,000, quotes 7 and 13 percent, repayments 12,480, 12,240 and 12,840, 80 loans for 140 applicants); 12,480 at 13 percent is added as the indifference case.

Calculated values

Return at 7 percent
4%
Return at 13 percent
2%
Quoted rate shown
13%
Promised repayment at quote
13,560
Expected repayment at quote
12,240
Bank prefers
7 percent

At the 13% quote a performing loan promises 12,000(1 + 0.13) = 13,560, but the bank expects 12,240, a return of (12,240 - 12,000)/12,000 = 2%. The 7 percent quote returns (12,480 - 12,000)/12,000 = 4% and the 13 percent quote returns (12,240 - 12,000)/12,000 = 2%. The bank prefers the 7 percent quote and rations credit: 80 loans for 140 applicants, though applicants would promise 13 percent.

Worked steps

  1. Return at 7% = (12,480 - 12,000) / 12,000 = 0.04 = 4%
  2. Return at 13% = (12,240 - 12,000) / 12,000 = 0.02 = 2%
  3. Promised at 13% = 12,000 x 1.13 = 13,560

Use the idea

Unmet loan demand at a stable rate need not mean the market is failing to clear; check how repayment would change with the rate.

Where the conclusion applies

Applicants look alike to the bank, the bank funds a fixed 80 loans and expected repayments at each rate are given rather than derived.

Check your understanding: If repayment at 13 percent were 12,480, which rate does the bank prefer?
(12,480 - 12,000)/12,000 = 4%, equal to the 7 percent quote; the bank is indifferent.

Chapter 48 source: section "Stiglitz-Weiss credit rationing".