E120 Finance Final

Final Exam E 120 NLG

E 120 SPRING SEMESTER N. L. GUNTHER INSTRUCTOR

SIN MAN CHOI & YANG WANG, GSIs

RUOYUN LI, READER

FINAL EXAMINATION, MONDAY, MAY 12, 2014

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Name: (please print)____________________________________________

SID:__________________________________________

QUESTION 1 2 3 4 5 6

POINTS 3 8 2 5 19 6

GRADE

QUESTION 7 8 9 10 11 12

POINTS 10 10 9 8 10 10

GRADE

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Any communication with other students during the exam (including showing, viewing or sharing any writing) is strictly prohibited. Any violation will result in a score of 0 points for the exam.

Clearly state all the mathematical expressions that are needed to solve the problems. Partial credit will be given for correct formulas. Show all answers to all least four significant digits.

No credit will be given to numerical answers without the proper setup, so please show your work.

Please pace yourself carefully to finish as many problems as you can within the allotted 3 hours.

Answer each of the following twelve questions in the space provided beneath the relevant question. If you need more space to show major computations you performed to obtain your answer for a particularproblem, use the back of the preceding page. If you need additional space, you may submit additional sheets that are clearly identified as yours, with your name and SID, and on which your work is clearly marked to indicate the particular problem to which it relates.

You can quote and use any result stated in class or in the main body of the textbook as well as well-known general mathematical results but no references to other sources (including textbook exercises) are allowed. In addition, you are allowed ten sheets of notes, which may be double-sided, for a total of tewemty pages of notes.

If you cannot answer a problem the result of which is used in a subsequent problem, make a reasonable assumption about the problem's answer for use in the subsequent problem, stating it clearly.

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Present your work in an organized and neat fashion.

Good luck!

QUESTION 1: LEGAL FORM: 3 POINTS

Please enter the most appropriate business organizational form in the right-hand column that best fits the statement in the left-hand column.

Description Business Organization Form that Best Fits

Best legal form for a publicly-traded business

Appropriate legal form for a start-up with someinstitutional equity investment

A small company owned and run by a single person.

Answers Points

Corporation. 1

Limited liability company or limited partnership 1

Corporation 1

QUESTION 2: FINANCIAL STATEMENT ANALYSIS: 8 POINTS

Last year, Felicia's Fancy Felafel had a net profit margin of 7%, asset turnover of 2 and a book equity multiplier of 1, while Felicia's rival, Harry's Heavenly Hummus, had total revenues of $24 million, net income of $2.4 million, total assets of $16 million, and total shareholder’s book equity of $8 million. Recall: asset turnover = Sales/Total Assets, book equity multiplier = Total Assets/Book Value of Equity

I. Use this data to compute Felicia's return on equity ('ROE'). What is the name of the formulaor identity that expresses the relationship between ROE, net profit margin, asset turnover and the book equity multiplier?

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Answers Points

Felicia’s ROE = 7% × 2 × 1 = 14.0%. The formula is the DuPont Identity.

2

II. If Felicia's managers wanted to increase Felicia's ROE by 3.5% (i.e. add 3.5% to the current ROE) solely by increasing asset turnover, what would Felicia's asset turnover need to be?

Answers Points

Using the DuPont Identity, the answer is that Felicia's new asset turnover = 17.5%/7% = 2.5 oran increase of 0.5; 10.5% = 7%*25*1

2

III. What is Harry's ROE?

Answers Points

(directly) 2.4/8 = 30% = (by DuPont) (2.4/24)*(24/16)*(16/8) = 10%*1.5* 2 2

IV.Comparing Harry's results and data with Felicia’s results and data, explain the difference between the two firms’ ROEs in terms of the formula referred to in I above. Whose stores are likely busier? Whose food is likely more expensive relative to its cost?

Answers Points

By DuPont we have: (Felicia) 7% × 2 × 1 vs (Harry) 10%*1.5*2 1

Harry's asset turnover is lower than Felicia's, so Felicia's stores are likely busier. Harry's net profit margin is greater, so Harry's food is likely more expensive relative to its cost

1

QUESTION 3: APR AND EAR: 2 POINTS

You have found a one-year deposit that pays interest at 7% APR, compounded monthly. Compute the associated EAR.

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Answers Points

(1+ 7%/12)12 – 1 = 7.23% = EAR 1 point for the formula, 1 point for thecorrect answer

QUESTION 4: ZERO PRICE COMPUTATION: 5 POINTS

The market par rates (rates on par bonds selling at face) are 25% for a 1-year maturity, and 30% for a 2-year maturity, in each case paying annually. Please compute the associated 1-year and 2-year zero coupon bond prices (“zero prices”) as a percentage of their face amounts.

I. 1- year zero price

Answer: 1/1.25 = 4/5 = 80% 2 points

II. 2-year zero price

Answer: (1 – 30%*80%)/1.3 = 58.46% 3 Points

QUESTION 5: VALUATION OF PROJECTS AND COMPANIES: 19 POINTS

I. The present value of which of the following best determines a company's total equity value?

Accounting Earnings

Dividend Payments

Total Payouts (All Dividends and Repurchases) X

Free Cash Flow

II. Which of the following does break-even analysis best determine?

The time until a project or business first earns a profit.

NPV variation as individual input levels change

NPV variation in different input level scenarios

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Check only one

Check only one

1 Point

1 Point


Input levels where NPV is zero X

III. Which of the following best describes the Profitability Index?

An index of the most profitable firms in the S&P 500

Securities ranked by profitability

Project selection subject to resource constraints X

A company's return on its equity relative to the market

IV.VALUATION BASED ON FREE CASH FLOW

Questar Corporation (“QC') is a privately held oil services company based in Sacramento, CA. QC expects total free cash flow (“FCF”) of $20 million one year from now. Analysts expect this FCF to grow at 8% per year thereafter until the end of the fourth year (again, from now.) After that, FCF growth will level off at long-run growth rate of 1% per year, remaining at that rate in perpetuity.

Assume QC has 200 million common shares outstanding, debt of $250 million and cash of $50 million. According to the discounted free cash flow model, what is the value of a share of QC common stock if the firm’s weighted average cost of capital is 6%? Show answers tothe nearest cent. Please provide your work following the outline below.

a) Draw a timeline showing FCF at times 0, 1, 2, 3 and 4 (years). 2 points

0 1 2 3 4

$20.00 $21.60 $23.33 $25.19

b) Compute the present value of the FCF over the first three years. 2 points

Answer: Student can use the annuity formula, or just compute discount factors @ 6% as follows:

94.34% 89.00% 83.96% 79.21%

PV(6%, above cash flows and discount factors) = $57.68

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Check only one

1 Point


c) Compute the present value at the end of year 3 of the long-run FCF beginning in year four, using the formula for the present value of a growing perpetuity. 2 points

Answer: $25.19/(6%-1%) = 20*$25.19 = $503.88

d) Use your answers in b) and c), and the assumptions on QC's debt, cash and number of outstanding shares, to compute the present value of a share of QC common stock today. 2 points

Answer: Total Enterprise Value = $57.68 + ($503.88*83.96%=$423.07) = $480.75

Less net debt of $200 million = $280.75 million, or $1.40 per share

V. COMPUTATION OF FREE CASH FLOW

Your company is contemplating a new business that will have, in each of the next seven years:

a) EBIT of $70 million per year, including $20 million of depreciation expense related to a required $140 million upfront investment in new equipment,

b) Inventory of $20 million, accounts receivable of $5 million and accounts payable of $15million, and

c) A facility that could otherwise be rented out for $25 million per year.

d) Your company's tax rate is 40%.

Analyze these assumptions as we did in class with the HomeNet example and on the Mid-Term,and on this basis answer the following questions:

A) What is the change in Net Working Capital in year one? 2 points

Answer: $10 million

B) What is the change in Net Working Capital in year four? 1 point

Answer: $0 million

C) What is the Free Cash Flow from the new business up front at time 0? 1 point

Answer: ($140 million)

D) What is the Free Cash Flow from the new business in year one? 2 points

Answer: ($70 - $25)*60% - $10 million +$20 million = $37 million

E) What is the Free Cash Flow from the new business in year three? 2 points

Answer: $37 million + $10 million = $47 million.

QUESTION 6: MORTGAGE AMORTIZATION: 6 POINTS

You have just purchased a home and took out a $200,000 mortgage. The mortgage has a 20-yearterm with equal monthly payments and an APR of 6%. How much will you pay in interest, and how much will you pay in principal, during the first year? Show all answers to the nearest cent.

a) First, use the formula for the PV of an annuity to find the monthly payment. Write your

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computation here: 2 points

Answer: Monthly Rate = 6%/12 = 0.50%, Number of periods = 20*12 = 240

Monthly Payment = $200,000*0.50%/(1 – 1/1.005^240) = $1,432.86

b) Use the annuity formula again to find the PV of the remaining payments at the start of the second year. This will give you the change in the principal balance over the first year, and thus the total amount of principal paid. Write your computation here: 2 points

Answer: PV(Year 2) = (Monthly Payment/0.5%)*(1 – 1/1.005^228) = $194,660.40

Thus, the principal paid over the first year = $200,00 - $194,660.40 = $5,339.60

c) The remainder of the total first year payments is interest. Write your computation here:2 points:

Answer: Interest in first year = 12* $1,432.86 - $5,339.60 = $11,854.75

NOTATION: In this question, lower case 'r' generally denotes expected return and upper-case 'R' generally denotes actual return, so r generally = E[R]. Where no confusion would result, we may drop the explicit reference to the return and just write, for example, Var(P) and Cov(A,B) for the variance of RP and the covariance of RA and RB, respectively. Thus, σ(A) denotes the standard deviation of the return RA of asset A, σ(A,P) denotes the covariance between the returns of assets A and P, and

Corr(A,P) denotes the correlation between those returns, equal toCovariance(RA , RB)

√(Var (RA))√(Var (RB)).

QUESTION 7: THEORY BEHIND MEAN-VARIANCE ANALYSIS AND THE CAPM: 10 POINTS

I. True or False: A portfolio is the efficient portfolio if it has the highest Sharpe ratio of any portfolio in the economy. Check one.

True X False

II. Let P be the efficient portfolio and A an asset with expected return rA. Write down the equation rA must satisfy in terms of rA, rP, their volatility and correlation, and the risk-free rate. (Hint: it includes A's required return with respect to P, as defined in the textbook.) Identify the beta term, referred to as βP

A.

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1 Point


Answers Points

r A>r f +σ ( A )∗Cor (r A , r P )∗(r P− r f )

σ ( P )

The coefficient of (rP – rf) is beta.

• 1 point for the formula, 1 for the correct identification of beta, for a total of 2 points

• 1 point in total for a formula with an error that's still recognizably related to the correct formula, together with the correct answer relative to the formula provided.

III. True or False: Under the CAPM assumptions, the equation you found in II above expresses the Capital Asset Pricing Model. Check one.

True X False

IV.Under the CAPM assumptions, which of the following resembles most closely P in IIabove? (Check only one.) 1 Point

Exxon-Mobil, the largestUS industrial company

A portfolio of the most successfulhigh technology companies

The S&P 500 X

V. Write the equation you found in II above in a form that relates the Sharpe Ratio of A to the Sharpe Ratio of P. (Hint: there should be a correlation term.) 2 Points

Answer: Sharpe Ratio(A) = Corr(rA,rP)*Sharpe Ratio P

VI. An application: Optima Fund has an expected return of 22% and a volatility of 10%. Assume the risk free rate is 2%. 1 Point

a) What is Optima Fund's Sharpe Ratio?

Answer: Sharpe Ratio(Optima) = 20%/10% = 2

b) Now assume Optima Fund has the highest Sharpe Ratio of any portfolio, and that AlphaPharmaceuticals has a correlation of 75% with Optima Fund. What is Alpha's Sharpe Ratio? (Based on your prior answers, this should be all the information you need.)2 Points

Answer: Sharpe Ratio(Alpha) = 75%*2 = 1.5

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1 Point


QUESTION 8: COMPUTATIONAL RISK-RETURN ANALYSIS: 10 POINTS

Suppose the risk free rate is 10% per annum, and the year-end dividends and stock prices of Acme Products Corp. (“Acme”) and DreamTech Investments (“DreamTech”), a diversified fund, are as follows. (Clarification – if needed – : while there are four times (year end 2010, year end 2011, year end 2012 and year end 2013), there are only three annual periods, measured from year-end to year-end (2010-2011,2011-2012, 2012-2013) and thus only three returns.) Using the data in the Data Table below, please answer the following questions:

a) What was Acme's average annual return over this three-year period (year-end 2010 to year-end 2013)? ½ Point

Answer: Average(4,1,1) = 200%

b) What was Acme's estimated annual variance and volatility over this period? (Hint: Use the estimation formula from the book, with denominator T-1 (not T) and first compute the differences from the mean, then the variance, and finally the volatility.) 1 Point

Answer: Variance = Sum(4,1,1)/2 = 3, volatility = √3= 1.732

DATA TABLE ACME DREAMTECH

YEAR-END STOCK PRICE DIVIDEND STOCK PRICE DIVIDEND

2010 $1 $0 $10 $0

2011 $4 $1 $20 $0

2012 $7 $1 $100 $0

2013 $12 $2 $500 $0

c) Based on this data, what is Acme's estimated Sharpe Ratio? 1 Point

Answer: Sharpe Ratio = 190%/ 1.732 = 1.097

d) What was DreamTech's average annual return over this period? ½ Point

Answer: Average(1,4,4) = 300%

e) What was DreamTech's estimated annual variance and volatility over this period? (See the hint in b above.) 1 Point

Answer: Variance = Sum(4,1,1)/2 = 3, volatility = √3= 1.732

f) Based on this data, what is DreamTech's estimated Sharpe Ratio? 1 Point

Answer: Sharpe Ratio = 290%/ 1.732 = 1.674

g) What is the covariance between DreamTech and Acme's annual returns? 1 Point

Answer: Covariance = Sum(-4,-1,-1)/2 = -3

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h) What is the correlation between DreamTech and Acme's annual returns? ½ Point

Answer: Correlation = Covariance/(Volatility*Volatility) = -1

i) What is the beta of Acme with DreamTech viewed as a portfolio, as defined in the textbook (the textbook called this the beta of investment i with Portfolio P)? 1 Point

Answer: Beta = Covariance/Variance = -3/3 = -1

j) Suppose the risk free rate is 10% per annum. What is the required return for Acme with DreamTech? 1 Point

Answer: Required Return = 10% -290% = -280%

k) If you held DreamTech as your portfolio, should you add a little Acme to your portfolio? Why? 1 Point

Answer: Yes, because its return of 200% is (much) more than the required return of -280%

l) If you add a little Acme to your DreamTech portfolio, what will happen to your Sharpe Ratio? Check all that apply. ½ Point

Decrease Increase X Stay the Same

QUESTION 9: PRACTICAL USES OF THE CAPM: 9 POINTS

I. True or False:

Under the CAPM, an asset's return is based on its risk, determined as the standard deviation of its return. 1 point

True False X

Explain your answer to I above

Answer: an asset's return is based on only the systematic portion of its risk, represented by its beta1 point

II. You have an equally-weighted portfolio with two assets, A and B with betas βA and βB. What is the beta of your portfolio? 1 point

Answer: ΒP = βA/2 + βB/2.

III. Your firm is planning to invest in a new business: building aircraft autopilots. You are to value this business. You need to find the business' cost of capital using the CAPM.

a) Wright Flight, Inc. (“Wright”) is an all equity firm that specializes in this business. Suppose Wright’s equity beta is 0.80, the risk-free rate is 2%, and the market risk premium is 4%. If your firm’s project is all equity financed, estimate its cost of capital.

Answer: rW = rf + βW*(rM – rf) = 2% + 80%*4% = 5.2% 2 points

b) You decide to look for other comparables to reduce estimation error in your cost of

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capital estimate. You find a second firm, Yaeger Design Corporation (“Yaeger”), which is also engaged in a similar line of business. Yaeger has publicly-traded stock with a price of $30 per share, of which 10 million shares are outstanding. It also has $300 million in outstanding debt, with a yield on the debt of 4 %. Yaeger’s equity beta is 1.4.

A) Assume Yaeger’s debt has a beta of zero. Estimate Yaeger’s unlevered beta. Use this unlevered beta and the CAPM to estimate Yaeger’s unlevered cost of capital.

Answer: E/(E+D) = D/(E+D) = 50%, βU = 50%*1.4 + 50%*0 = 0.7, rU = rf + 0.7*4%= 4.8% 2 points

B) If in reality Yaeger's debt is not risk-free, does this method underestimate or overestimate Yaeger's unlevered cost of capital? Explain.

Answer: This method underestimates Yaeger's unlevered cost of capital; β = 0% => r = rf= 2% < 4% = the actual interest rate on Yaeger's debt.

QUESTION 10: FIXED COSTS AND THE CAPM: 8 POINTS

Your company operates a chip manufacturing plant. Revenues from the plant are constant at$250 million per year. All of the plants costs are variable costs and are consistently 80% of revenues, including energy costs associated with powering the plant, which represent one quarter of the plant’s costs, or an average of $50 million per year. Suppose the plant has an asset beta of 1.10, the risk-free rate is 2%, and the market risk premium is 6%. The tax rate is 40%, and there are no other costs. Assume for this problem that free cash flow ('FCF') equals accounting (after-tax) net income = (Revenues – Costs) * (1 – tax rate.)

I. What is the plant's cost of capital? Use it to estimate the value of the plant today assuming no growth in FCF. 3 Points

Answer: Cost of capital = 2% + 1.1*6% = 8.6%,

Value = $30(after tax Revenue – Cost)/8.6% = $348.84

II. Suppose you enter a long-term contract which will supply all of the plant’s energy needs for a fixed cost of $40 million per year (before tax), an average savings of $10 million per year. What is the value of the plant if you take this contract? (Hint: what is the cost of capital associated with the fixed costs? What is their risk? Use this cost of capital to compute the present value of the fixed costs, and subtract from the present value of the revenues – variable costs.) 5 Points for a correct answer, 2 Points for the alternative.

Answer: This is tricky but follows Example 12.8 in the textbook and Problem 12-25, discussed in the Discussion Sections

FCF without Energy Costs = all variable costs = ($250 - $150) *60% = $60 million per year, so Value(FCF without Energy Costs) = $60/8.6% = $697.67. However, the value of the fixed costs is determined based on the risk-free rate, which is only 2%, and so equals $24/2% = $1.2 billion. In other words, the nominal cost savings are illusory and the value of the plant is now significantly negative(!) = (-$502.33). This underscores the risks associated with fixed costs, especially in a low interest-rate environment.

In the alternative, if the plant were valued solely by reference to the cost savings, the value would be:

$36 million/8.6% = $418.60 million. This is incorrect, but deserves 2 points, as does most other

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reasonable incorrect answers.

QUESTION 11: ALPHA AND EFFICIENT MARKETS: 10 POINTS

Ryan Hardy is an expert stock picker (also called an 'investment manager') at Superior Research, Inc (“SRI”). He can generate an alpha of 3%, but, unfortunately, only on a select group of stocks, namely biotechs engaged in RNA-based genetic therapy. Assume there is each year a pool (the “Pool”) of these companies with an aggregate market capitalization of $100 million that will outperform the market, and Ryan can identify successfully and buy all of these companies for his investors. With respect to other companies not in the Pool, however, Ryan is outside his specialization, has no competitive advantage and accordingly his alpha is zero.

SRI charges a fee of 1% per year on the total amount of money under management (at the beginning of each year). Assume (i) SRI wants to maximize revenue, and so requires Ryan to invest as much as investors will accept, (ii) there are always investors looking for positive alpha and (iii) no investor will invest in a fund estimated to have negative alpha, computed in each case after fees.

I. What is the maximum investment amount Ryan can manage without having negative alpha?(Hint: he can only manage $100 million while generating alpha of 3%; how much in dollars is this per annum? What are the fees? What is the net alpha investors will receive? Try three scenarios: see what happens if Ryan manages a portfolio of size $200, then $300, and finally $400 million; what is the net alpha investors receive?) 3 Points

Answer: Gross alpha of $3 million will support $3 million of fees without turning negative => $300 million total under management.

II. What is the net alpha Ryan's investors receive on the maximum amount you computed in Iabove under management? 2 Points

Answer: $0 – see I. above

III. Now, suppose Joan Watson, with the identical skills to Ryan's, becomes a stock-picker at a competing firm. Through competition Joan takes half of the Pool away from Ryan. In this event, what happens to the net alpha of Ryan's investors if Ryan continues to manage the same amount you determined in I above? (Hint: this simple model illustrates one way talented investment managers nonetheless may end up under-performing the market.)5 Points

Answer: Ryan's gross alpha falls to $1.5 million, fees stay at $3 million and thus investors net alpha falls to ($1.5 million) or -0.5% on the $300 million under management.

QUESTION 12: AN OPTION PROBLEM: 10 POINTS

You own an American call option on the stock of Gilead Sciences Inc. (GILD) with a one-year expiration date and a strike price of $76. The current price of GILD is $72 per share, the risk-free interest rate is 2%, and in one year the share price is equally likely to move up to $80 or down to $70. GILD never distributes dividends.

I. True or False: Under the call option, you will benefit if the price of GILD falls.

True False X

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1 Point


II. True or False: You should never exercise your call option prior to its expiration date.

True X False

III. Might your answer to II abovechange if GILD paid significant dividends?

Yes X No

IV.What is the expected return on GILD stock over the next year? 1 Point

Answer: E[Payoff] = 75, r = 75/72 -1 = 4.167%

V. Under the binomial model we studied in class, what is the value of the one-year call option on GILD stock? (Hint: recall that we first construct a portfolio in the stock and a riskless bond that replicates the payoffs from the option at time t=1 year. We then compute the value of the option at time t=0 as the value of this portfolio, using the Law of One Price.)

2 Points

Answer: Δ = (Cu – Cd)/(Su – Sd) = 40%, B(future) = - 40%*$70 = $28 => B(present) = $28/1.02 = -$27.45, so current option value = C(0) = 40%*$72 - $28/1.02 = $1.349

VI. What is the risk-neutral probability that GILD's stock price will move up to $80? (Hint: Recall that in class we showed how to express the formula for the value of the call you computed in V above in the form: Call Value = 1/(1+rf)*(pUp*CallUp + pDown*CallDown), with CallUp and CallDown the call option payoffs, pUp + pDown = 1 and 1≥ pU p,pDown≥ 0. This expression justified treating pUp and pDown as probabilities. The problem asks for pUp.)

2 Points

Answer: pU = (1.02*S(0) – Sd)*(Su – Sd) = 34.40%

VII. Suppose the annual volatility of GILD stock is 40% (a much higher volatility than implied from the data above.) Write the Black-Scholes formula, and then apply it to the data above to compute the value of your call option, using N(x) to denote the cumulative normal distribution function evaluated at x. Compute as many of the terms in the formula as you can, excluding the cumulative normal distributions (but do try to compute d1 and d2.)

2 Points

Answer: C(0) = S(0)*N(d1) – PV(K)*N(d2); d1 = ln(/S(0)/PV(K))/a-a/2, d2 = d1 -a , where a = σ√T, so a = 40%, PV(K) = $76/1.02 = $74.51 and d1= ln($72/$74.51)/40% - 20% = 11.43%, d2 = d1 -40% = -28.57% and we have C(0) = 72*N(0.1143) - $74.51*N(-0.2857)

CONGRATULATIONS! YOU HAVE FINISHED THE FINAL EXAM IN E 120

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1 Point

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E120 Finance Final