copyright © 2009 pearson prentice hall. all rights reserved. chapter 6 uncertainty, default, and...

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


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…


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


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


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


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


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.


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.


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?


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%


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.


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.


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.


Uncertainty, Default, and RiskCredit Ratings

TABLE 6.1 Rating Categories Used by Moody’s and Standard & Poor’s


Uncertainty, Default, and RiskCumulative Probability of Default by Original Rating

FIGURE 6.2 Cumulative Probability of Default by Original Rating


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.


Uncertainty, Default, and RiskDifferences in Quoted Bond Returns in 2002

TABLE 6.2 Promised Interest Rates for Some Loans in May 2002


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.


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


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%


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.


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%.


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.


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%


Uncertainty, Default, and RiskDebt and Equity Payoff Tables Summarized

TABLE 6.3 Payoff Table and Overall Values and Returns


Uncertainty, Default, and RiskWhich is More Risky: Equity, Debt, or Full Ownership?

FIGURE 6.3 Three Probability Histograms for Project Rates of Return


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


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


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.

copyright © 2009 pearson prentice hall. all rights reserved. chapter 6 uncertainty, default, and...

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