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Copyright © 2009 Pearson Prentice Hall. All rights reserved.
Chapter 6Uncertainty, Default, and Risk
Copyright © 2009 Pearson Prentice Hall. All rights reserved.6-2
Chapter 6 Outline
6.1 An Introduction to Statistics6.2 Interest Rates and Credit Risk (Default
Risk)6.3 Uncertainty in Capital Budgeting6.4 Splitting Uncertain Project Payoffs into
Debt and Equity
Copyright © 2009 Pearson Prentice Hall. All rights reserved.6-3
Uncertainty, Default, and RiskIntroduction
• What happens if we still have perfect markets, but we don’t have perfect forecasts and thus have plenty of uncertainty?
• The main impact of uncertainty is to make our decisions more challenging due to forecast errors, but our decision rule, NPV, still works best.
• With uncertainty, the quoted return may differ from the expected return. • The quoted return is also called the stated or promised return.
• Expected returns are lower than quoted returns because firms may default.
• Before we discuss firms raising capital with debt or equity issues, we have to talk about statistics.
• Wait!….don’t go……it’s basic stats….you’ll be fine…
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Uncertainty, Default, and RiskIntroduction to Statistics
• Expected Value -- the most important statistical concept• the average probability of an event• is computed over future outcomes
infinitely
• Random Variable -- the item that is yet to occur in the future• such as ‘coin toss outcome’
• Notation for Expected Outcome of a Random Variable (has a tilde)
• If a coin toss of heads pays $1 and tails pays $2, compute the expected value
• Once tossed, the outcome is known and is no longer a random variable.
(c) Expected value of random event "c"
(c) Expected value of coin toss = Prob(Heads) $1 Prob(Tails) $2
(c) $1.50
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Uncertainty, Default, and RiskHistograms
•A histogram is a graph of the distribution of possible outcomes.
FIGURE 6.1 A Histogram for a Random Variable with Two Equally Likely Outcomes, $1 and $2
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Uncertainty, Default, and RiskFair Bets
• A Fair Bet is a bet that costs its expected value.• In other words, you get what you pay for. • If the bet is made over and over, both sides come out even.
• If the cost of the bet equals its expected value, then it is fair.
• What is the expected value of a bet that has these payoffs?
$4 with a 16.7% chance $10 with a 33.3% chance $20 with a 50% chance
• You would pay $14 if you wanted to break-even in the long-term.
• Some bets are not fair. • Vegas has spent a lot of time convincing you to take less than fair bets.
(D ) Expected value of Dice Roll = Prob(1) $4 Prob(2,3) $10 Prob(4,5,6) $20
(D ) 16.7% $4 33.3% $10 50% $20
(D ) $14
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Uncertainty, Default, and RiskVariance and Standard Deviation
• Risk is the most important characteristic to know after return.
• Risk is the variability of outcomes around an expected value or mean.
• Standard deviation is the most common measure of risk. It is the square root of the average squared deviation from the mean, or sqrt(Variance).
• Looking at our $14 expected value or mean, we note the following:Outcomes $4 $10 $20Deviations -$10 -$4 +$6 (taken from $14 mean)Squared $100 $16 $36Prob weights 16.7% 33.3% 50% (investors agree here)Wt’d Squared $16.7 $5.3 $18 (sum = Variance)
Variance = sum of the weighted squares = $40.00Standard deviation is the square root of variance = $ 6.32
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Uncertainty, Default, and RiskRisk Neutrality -- A Lead into Risk Aversion
• For now, we assume risk neutral investors: they take fair bets.
• To a risk neutral investor, all fair bets are taken.• They will take a certain $1 or a 50-50 chance to earn $0 or $2.
• Risk neutral investors are motivated by the payoff they expect, not risk.
• Risk averse investors will take the certain $1 over the 50-50 chance.
• Both alternatives have an expected value of $1, but risk averse investors require a higher return than risk neutral investors to take a fair bet.
• Financial markets provide an invaluable service by spreading risks.• Individuals see a smaller level of risk (think of diversification) due to the
lower aggregate risk aversion in the market.
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Uncertainty, Default, and RiskInterest Rates and Credit Risk (Default Risk)
• Risk Neutral Investors Demand Higher Promised Rates
• When faced with the possibility of default (an uncertain cash flow), a risk neutral investor should charge a higher quoted rate or promised rate. This compensates them for the lower expected return due to default risk.
• If a borrower of $1M at a rate of 10% has a 50% chance of default and will either pay back $750,000 or $1.1M, depending on default outcome, the lender sees an expected return lower than the 10% promised return desired or needed by the lender.
Prob(Default) • Payment if Default + Prob(Solvent) • Payment if Solvent = (payout)
50% • $750,000 + 50% • $1,100,00 = $925,000 Expected Value
• The lender should not extend credit since the expected value is a loss of 7.5% on the loan. The lender needs to increase the quoted rate to raise the desired expected value to $1.1M. The quoted rate needs to be 45%!
50% • $750,000 + 50% • $1,450,00 = $1,100,000 Expected Debt Value
• The 35% return above the needed return of 10% is called the default premium.
• Expected values and returns matter, not promised returns.
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Uncertainty, Default, and RiskDefault Example with Probability Ranges: Payoff Table
• Borrower has a 98% probability of full repayment, a 1% chance of paying back 50% of the loan, and a 1% chance of paying back nothing. Assume this is a loan for $200 at a rate of 5%, what is the expected payoff?
Probability X Cash Flow = Expected Value98% $210 $205.80 1% $100 $ 1.00 1% $ 0 $ 0.00
Expected Payoff $206.80
Promised rate was 5% but payoff is only a 3.4% return. If you can buy a safe government bond that pays 5%, do that!
• What rate is needed as a quoted rate to equal a payoff of $210?
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Uncertainty, Default, and RiskDefault Example with Probability Ranges: Expected rate
• Borrower has a 98% probability of full repayment, a 1% chance of paying back 50% of loan, and a 1% chance of paying back nothing. Assume this is a loan for $200 and a safe return is 5%. What rate is needed as a quoted rate to equal a payoff of $210?
• Find the full-repayment cash flow first:
Probability X Cash Flow = Expected Value98% $ ? $209.00 1% $100 $ 1.00 1% $ 0 $ 0.00
Expected Payoff $210.00
Solving for the full-repayment cash flow, $209/.98 = $213.27.
• The promised rate will now be 6.63%, for an expected return of 5%. You can now lend to the borrower because the expected rate equals 5%.
(r) Expected rate = Prob(1) (6.63%) Prob(2) ( 50%) Prob(3) ( 100%)
(r) Expected rate = 98% (6.63%) 1% ( 50%) 1% ( 100%) 5%
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Uncertainty, Default, and RiskDeconstructing Quoted Rates of Return: Time and Default Premiums
• Earlier, the lender expected to earn 5%, but quoted 6.63%. The difference of 1.63% is the default premium for credit risk.
Promised rate = Time premium + Default premium6.63% = 5% + 1.63%
• Safe government bonds have no default premium and the quoted rate and the expected rate (time premium) are the same (5%).
• Risky corporate bonds have a risk premium for default, so the quoted rate is greater than the expected rate.
• Because lenders do not expect to earn every default premium they charge in a risk neutral setting, the expected realized default premium is 0%.
(r) Expected realized default premium = 98% (1.63%) 1% ( 55%) 1% ( 105%) 0%
Note the gains and losses are taken from a 5% return or loss of time premium.
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Uncertainty, Default, and RiskOther Debt Premiums
• In addition to the time premium and the default premium, there are:
• Liquidity premiums compensate the lender for future costs to sell bonds.It is payment for the inability to convert to cash.
• Risk premiums compensate investors for their willingness to take risk.
It is payment for risk aversion.
• These are important, but not as large as the time and default premiums.
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Uncertainty, Default, and RiskCredit Ratings and Default Rates
• Firms such as Moody’s, Fitch, Duff and Phelps, and Standard & Poor’s provide quality ratings on the credit risk of bonds.
• The usual grading scale is AAA to C ……and yes there’s grade inflation, everyone wants a high A.
• Bonds are separated into two grades or groups:
• Investment grade - high-quality borrowers0.3% chance of default in any year
• Speculative or junk - low-quality borrowers 3.5% to 5.5% chance of default in an
average year
• Junk bond default rates rise in recessions to 10% and fall in booms to 1.5%.
• The amounts recovered in default by lenders vary by bond grades.
• The amounts recovered also vary in economic boom vs. bust cycles.
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Uncertainty, Default, and RiskCredit Ratings
TABLE 6.1 Rating Categories Used by Moody’s and Standard & Poor’s
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Uncertainty, Default, and RiskCumulative Probability of Default by Original Rating
FIGURE 6.2 Cumulative Probability of Default by Original Rating
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Uncertainty, Default, and RiskBond Contract Feature: Call Risk and Early Prepayment
• Bonds have option features that allow the borrower to change the terms.
• One option feature is the ability to prepay the note before it is due.Why would you want this? To take advantage of lower rates.
Example:If you borrow at 10% and then rates drop to 5%:
You should pay back original loan early and take a new loan at 5%.
If you borrow at 10% and then rates rise to 15%:You should keep your original loan.
• For the lender this is not a good deal and thus lenders charge higher rates.
• Individuals prepay mortgages, and it is usually called refinancing.
• Firms do this with bonds: Callable bonds pay higher interest than noncallable bonds since there is an early prepayment option.
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Uncertainty, Default, and RiskDifferences in Quoted Bond Returns in 2002
TABLE 6.2 Promised Interest Rates for Some Loans in May 2002
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Uncertainty, Default, and RiskCredit Default Swaps
• The credit default swap (CDS) is an innovation in finance; it emerged in the 1990s. It allows investors to trade directly on the credit risk of a firm.
• Two counterparties bet on the credit outcome of a firm with bonds outstanding. Assume a pension fund owns $10M of bonds and is interested in protection against default on the bonds.
• A hedge fund wants to bet that the $10M in bonds does not have default risk and is the counterparty to the pension fund’s credit default swap. The hedge fund is providing insurance and collecting a fee to do so.
• Pension fund pays $130,000 to the hedge fund for credit protection.
• If the bonds default, the hedge fund owes the pension fund $10M. • If the bonds do not default, the hedge fund’s profit is $130,000.
• This is the cost of default, so it is a form of credit premium.
• By executing this swap, the pension fund collects the time premium, but not the default premium.
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Uncertainty, Default, and RiskUncertainty in Capital Budgeting: State-Contingent Payoffs
• To find the value of a project, managers construct a payoff table. It has expected discounted cash flows and uses expected rates of return.
• Example of PV with State-Contingent Payoff Tables
Expected Building Value:
Event Probability Value PV (r=10%)Tornado 20% $ 20,000 $18,181.82Sunshine 80% $100,000 $90,909.09Expected Value 20%(T) + 80%(S)= $ 84,000 $76,363.64
• If the discount rate is 10%, the PV of the expected value equals $76,363.64.
PV 20% ($18,181.82) 80% ($90,909.09) $76,363.64
or
PV 20% ($20,000) 80% ($100,000)
1.10 $76, 363.64
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Uncertainty, Default, and RiskState Dependent Rates of Return
• If you buy the building for the $76,363.64, what is your expected return?
If Sunshine:Pay $76,363.64 Value $100,000 Probability 80% Return 30.95%
If Tornado (dramatic, eh?):Pay $76,363.64 Value $20,000 Probability 20% Return -73.81%
• The expected return is the probability-weighted average return.
• The expected return of 10% is your required cost of capital: you paid $76,363.64.
• If you pay a different value than the asset’s calculated PV, you’ll change your return.
(r) Expected return = Prob(S) (30.95%) Prob(T) ( 73.81%)
(r) Expected rate = 80% (30.95%) 20% ( 73.81%) 10%
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Uncertainty, Default, and RiskSplitting the Projected Payoffs into Debt and Equity
• Debt and equity are state-contingent claims that we can value.
• Once we know the expected payoffs, we can sell the payoffs to debt and equity investors.
• We have to pay the liability (debt) owners first.
• The remaining cash flow is owned by the equity owners.
• Loans• A mortgage is a non-recourse loan: the lender can take back the building but
cannot ask the borrower for any more cash.
• This is a limited liability.
• Most financial securities offer limited liability.
• Shareholders can only lose the value of their stock, nothing more.
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Uncertainty, Default, and RiskLoans
• What if we borrow $25,000 to own the building worth $76,363.64?Now the building has two owners: a mortgage owner and the residual owner.
• The mortgage owner, the lender, has to determine an appropriate loan rate.If the lender expects to earn 10%, the quoted rate will be higher.
• To solve, find the promised payoff that will result in an expected return of 10%:
Quoted Probability Weighted Values = Expected Value80% ($Promise) + 20% ($20,000) = 25,000 + 10% 80% ($Promise) = $23,500
Promise = $23,500 / .80 = $29,375 (17.50% more than $25,000)
• If the sun shines, the promised return is 17.50% ($29,375 / $25,000 - 1).• If the tornado hits, the return is -20% ($20,000 / $25,000 - 1).• Therefore, the expected return is .80(17.50%) + .20(-20%) = 10.0%.
• The loan rate will be 17.50% to offset the loss probability and its expected rate is 10%.• If the loss or default probability were 0%, then the quoted loan rate would be 10%.
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Uncertainty, Default, and RiskLevered Equity
• What does the equity owner expect if $25,000 is borrowed?
• The equity owner has a building worth $76,363.64 and a mortgage of $25,000.• Net worth (equity) equals $51,363.64, which the owner paid in cash.• In a year the house will be worth $100,000 (Sunshine) or $20,000 (Tornado).• The equity owner will owe the lender $25,000 + $4,375 interest or the $20,000
house.• The equity owner will have either $70,625 (100,000 – 29,375) or nothing.
Owner’s Payoff Table
Expected Building Value:Event Probability Value PV (r=10%)Tornado 20% $ 0 $ 0.00Sunshine 80% $70,625 $51,363.64
Expected Value 20%(T) + 80%(S)=$ 56,500 $51,363.64
• If the appropriate rate is 10%, the owner’s expected value equals $51,363.64, which is $25,000 less than the total value of $76,363.64.
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Uncertainty, Default, and RiskLevered Equity Rate of Return
•Once we know the expected payoffs, we can find the rate of return to equity.
• The equity owner has a beginning net worth of $51,363.64, which will rise or fall:
If Sunshine, return is +37.50%: ($70,635 - $51,363.64) / $51,363.64
If Tornado, return is -100%: ($0 - $51,363.64) / $51,363.64
Since the owner also used 10% cost of capital when determining his initial purchase price, the owner expects to earn 10%. The real world could differ from expectations, of course!
(r) Expected return = Prob(S) (return if S) Prob(T) (return if T)
(r) Expected rate = 80% (37.5%) 20% ( 100%) 10%
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Uncertainty, Default, and RiskDebt and Equity Payoff Tables Summarized
TABLE 6.3 Payoff Table and Overall Values and Returns
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Uncertainty, Default, and RiskWhich is More Risky: Equity, Debt, or Full Ownership?
FIGURE 6.3 Three Probability Histograms for Project Rates of Return
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Uncertainty, Default, and RiskWhat Leverage Really Means – Financial and Operational
• Debt is often called leverage. Equity is levered ownership with debt. • Leverage increases volatility, our home owner will earn either 37.5% or -100%.• Operational leverage is a trade-off between fixed and variable costs.
High fixed costs increase the volatility of earnings.
TABLE 6.4 Financial and Real Leverage
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Uncertainty, Default, and RiskMany Possible Outcomes: Plot E(V) vs. Promised
FIGURE 6.4 Promised versus Expected Payoff for a Loan on the Project with Five Possible Payoffs
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Uncertainty, Default, and RiskMistake: Do Not Discount a Promised Payoff with a Promised Rate of Return
• We should always discount the expected payoff with the expected rate of return. If we don’t, then we will make errors.
• If a $100,000 bond promises 16% with a 50% chance of defaulting on its interest payments, do not discount the promised cash flow by the promised rate.
• If the risk-free rate is 10% and the credit premium is 2%, the promised rate is 12% and the PV of $100,000 plus 16,000 discounted at 12% is $115,195.You would incorrectly believe the NPV is a positive $3,571.
• The correct valuation is to find the PV of both $100,000 + E(Interest).If we find the probability-weighted cash flow, we can use r = 10%.
• NPV = -$100,000 + PV(100,000) + PV(50% of $16,000) = -$1,818This is a bad investment using expected values discounted by the expected rate.