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Chapter 6 Uncertainty, Default, and Risk Copyright © 2009 Pearson Prentice Hall. All rights reserved. Chapter 6 Outline 6.1 An Introduction to Statistics 6.2 Interest Rates and Credit Risk (Default Risk) 6.3 Uncertainty in Capital Budgeting 6.4 Splitting Uncertain Project Payoffs into Debt and Equity Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-2 Uncertainty, Default, and Risk Introduction • 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… Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-3 Uncertainty, Default, and Risk Introduction to Statistics • Expected Value -- the most important statistical concept • • • the average probability of an event is computed over future outcomes infinitely Random Variable -- such as ‘coin toss outcome’ • • the item that is yet to occur in the future Notation for Expected Outcome of a Random Variable (has a tilde) (c) Expected value of random event "c" • If a coin toss of heads pays $1 and tails pays $2, compute the expected value (c) Expected value of coin toss = Prob(Heads) $1 Prob(Tails) $2 (c) $1.50 • Once tossed, the outcome is known and is no longer a random variable. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-4 Uncertainty, Default, and Risk Histograms •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 Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-5 Uncertainty, Default, and Risk Fair Bets • A Fair Bet is a bet that costs its expected value. • 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? • • In other words, you get what you pay for. If the bet is made over and over, both sides come out even. $4 with a 16.7% chance $10 with a 33.3% chance $20 with a 50% chance (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 • 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. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-6 Uncertainty, Default, and Risk Variance 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 Deviations Squared Prob weights Wt’d Squared $4 -$10 $100 16.7% $16.7 $10 -$4 $16 33.3% $5.3 $20 +$6 $36 50% $18 Variance = sum of the weighted squares Standard deviation is the square root of variance Copyright © 2009 Pearson Prentice Hall. All rights reserved. (taken from $14 mean) (investors agree here) (sum = Variance) = $40.00 = $ 6.32 6-7 Uncertainty, Default, and Risk Risk 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. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-8 Uncertainty, Default, and Risk Interest 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. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-9 Uncertainty, Default, and Risk Default 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 98% 1% 1% X Cash Flow = Expected Value $210 $205.80 $100 $ 1.00 $ 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? Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-10 Uncertainty, Default, and Risk Default 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 98% 1% 1% X Cash Flow = Expected Value $ ? $209.00 $100 $ 1.00 $ 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% Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-11 Uncertainty, Default, and Risk Deconstructing 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 6.63% = Time premium + Default premium = 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. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-12 Uncertainty, Default, and Risk Other 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. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-13 Uncertainty, Default, and Risk Credit 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 borrowers 0.3% 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. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-14 Uncertainty, Default, and Risk Credit Ratings TABLE 6.1 Rating Categories Used by Moody’s and Standard & Poor’s Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-15 Uncertainty, Default, and Risk Cumulative Probability of Default by Original Rating FIGURE 6.2 Cumulative Probability of Default by Original Rating Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-16 Uncertainty, Default, and Risk Bond 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. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-17 Uncertainty, Default, and Risk Differences in Quoted Bond Returns in 2002 TABLE 6.2 Promised Interest Rates for Some Loans in May 2002 Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-18 Uncertainty, Default, and Risk Credit 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. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-19 Uncertainty, Default, and Risk Uncertainty 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 Tornado 20% Sunshine 80% Expected Value 20%(T) + 80%(S)= • Value $ 20,000 $100,000 $ 84,000 PV (r=10%) $18,181.82 $90,909.09 $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 20% ($20, 000) 80% ($100, 000) PV $76, 363.64 1.10 Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-20 Uncertainty, Default, and Risk State 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 If Tornado (dramatic, eh?): Pay $76,363.64 Value $20,000 • Probability 80% Return 30.95% Probability 20% Return -73.81% The expected return is the probability-weighted average return. (r) Expected return = Prob(S) (30.95%) Prob(T) (73.81%) (r) Expected rate = 80% (30.95%) 20% (73.81%) 10% • 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. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-21 Uncertainty, Default, and Risk Splitting 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. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-22 Uncertainty, Default, and Risk Loans • What if we borrow $25,000 to own the building worth $76,363.64? Now the building has two owners: a mortgage owner and the residual owner. • The mortgage owner, the lender, has to determine an appropriate loan rate. If the lender expects to earn 10%, the quoted rate will be higher. • To solve, find the promised payoff that will result in an expected return of 10%: Quoted Probability Weighted Values 80% ($Promise) + 20% ($20,000) 80% ($Promise) Promise = $23,500 / .80 = = = = Expected Value 25,000 + 10% $23,500 $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%. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-23 Uncertainty, Default, and Risk Levered 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 Tornado 20% Sunshine 80% Value $ 0 $70,625 Expected Value 20%(T) + 80%(S)=$ 56,500 • PV (r=10%) $ 0.00 $51,363.64 $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. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-24 Uncertainty, Default, and Risk Levered 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 (r) Expected return = Prob(S) (return if S) Prob(T) (return if T) (r) Expected rate = 80% (37.5%) 20% (100%) 10% 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! Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-25 Uncertainty, Default, and Risk Debt and Equity Payoff Tables Summarized TABLE 6.3 Payoff Table and Overall Values and Returns Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-26 Uncertainty, Default, and Risk Which is More Risky: Equity, Debt, or Full Ownership? FIGURE 6.3 Three Probability Histograms for Project Rates of Return Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-27 Uncertainty, Default, and Risk What 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 Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-28 Uncertainty, Default, and Risk Many Possible Outcomes: Plot E(V) vs. Promised FIGURE 6.4 Promised versus Expected Payoff for a Loan on the Project with Five Possible Payoffs Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-29 Uncertainty, Default, and Risk Mistake: 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) = -$1,818 This is a bad investment using expected values discounted by the expected rate. Copyright © 2009 Pearson Prentice Hall. All rights reserved. 6-30