t = Calculate the implied interest rates and graph the term structure of interest rates. t = X t = t = 1 2 3


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1 MØA 155 PROBLEM SET: Summarizing Exercise 1. Present Value [3] You are given the following prices P t today for receiving risk free payments t periods from now. t = P t = Calculate the implied interest rates and graph the term structure of interest rates. 2. Calculate the present value of the following cash flows: t = X t = Exercise 2. Arbitrage [4] You are given the following prices P t today for receiving risk free payments t periods from now. t = P t = There are traded securities that offer $1 at any future date, available at these prices. 1. How would you make a lot of money? Exercise 3. Bonds [4] You are given the following information about three bonds. Bond Year of Coupon Yield to Bond Maturity Maturity Price A 2 10% % 1, B 2 20% % 1, C 3 8% % Coupons are paid at the end of the year (including the year of maturity). All three bonds have a face value of 1,000 at maturity. 1. Find the time zero prices, P 1, P 2, and P 3, of one dollar to be delivered in years 1, 2, and 3, respectively. 2. Find the 1, 2, and 3 year spot rates of interest r 1, r 2 and r 3. Exercise 4. Stock [4] A stock has just paid a dividend of 10. ividends are expected to grow with 10% a year for the next 2 years. After that the company is expecting a constant growth of 2% a year. The required return on the stock is 10%. etermine todays stock price. Exercise 5. Projects [3] A project costs 100 today. The project has positive cash flows of 100 in years one and two. At the end of the life of the project there are large environmental costs resulting in a negative cash flow in year 3 of 95. etermine the internal rate(s) of return for the project. Exercise 6. CAPM [2] 1
2 The current risk free interest rate is 5%. The expected return on the market portfolio is 14%. What is the expected return of a stock with a beta value of 0.5? Exercise 7. Portfolio [2] Stock A has an expected return of 10% and a standard deviation of 5%. Stock B has an expected return of 15% and a standard deviation of 20%. The correlation between the two is shares is You can invest risk free at a 5% interest rate. What is the standard deviation for a portfolio with weights 25% in A, 25% in B and 50% in the risk free asset? Exercise 8. Q [2] Equity in the company Q has an expected return of 12%, a beta of 1.4 and a standard deviation of 20%. The current risk free interest is 10%. 1. What is the current expected market return? Exercise 9. Project [3] A project with a beta of 1.5 has cash flows 100 in year 1, 200 i year 3, 500 in year 4 and 100 in year 6. The current expected market return is 10%. The risk free interest rate is 5%. What is the highest cost that makes this project worth investing in? Exercise 10. Price [2] An asset has two possible values next period, X u = 50 and X d = 500. If you are told that the state price probability in the u state is 0.4 and the risk free interest rate is 10%, what is the value of the asset? Exercise 11. XYZ option [2] The current price of an American call option with exercise price 50, written on ZXY stock is 4. The current price of one ZXY stock is 56. How would you make a lot of money? Exercise 12. ud [1] The current price of the underlying is 50. This price will next period move to either 48 or 60. Find the constants u and d. Exercise 13. Call [5] You bought a call contract three weeks ago. The expiry date of the calls is five weeks from today. On that date, the price of the underlying stock will be either 120 or 95. The two states are equally likely to occur. Currently, the stock sells for 96. The exercise price of the call is 112. Each call gives you the right to buy 100 shares at the exercise price. You are able to borrow money at 10% per annum. What is the value of your call contract? Exercise 14. Bond Covenants [3] In one or two sentences, answer the following. 1. Who benefits from the covenants in bond contracts when the firm is in financial trouble? Why? 2. Who benefits from the covenants in bond contracts when the firm is issuing debt? Why? Exercise 15. [3] The Q corporation will next period realize a project that will have value either 100 or 20. This project is the only assets that Q corporation have. Q has issued a bond with face value of 50, due next period. The risk free interest rate is 10% and the current value of equity in Q is 40. etermine the current value of the bond. Exercise 16. Frisky [4] Frisky, Inc is financed entirely by common stock which is priced according to a 15% expected return. If the company repurchases 25% of the common stock and substitutes an equal value of debt, yielding 6%, what is the expected return on the common stock after the refinancing? 2
3 Exercise 17. Leverage [6] A firm has expected net operating income (X) of $600. Its value as an unlevered firm (V U ) is $2,000. The firm is facing a tax rate of 40%. Suppose the firm changes it ratio of debt to equity ratio to equal 1. The cost of debt capital in this situation is 10%. Use the MM propositions to: 1. Calculated the after tax cost of equity capital for both the levered and the unlevered firm. 2. Calculate the after tax weighed average cost of capital for each. 3. Why is the cost of equity capital higher for the levered firm, but the weighted average cost of capital lower? Exercise 18. Bond issue [4] An firm that is currently all equity is subject to a 30% corporate tax rate. The firm s equityholders require a 20% return. The firm s initial market value is $3,500,000, and it has 175,000 shares outstanding. Suppose the firm issues $1 million of bonds at 10% and uses the proceeds to repurchase common stock. Assume there is no change in the cost of financial distress for the firm. According to MM, what is the new market value of the equity of the firm? Exercise 19. Project [3] A company is considering a project with the following aftertax cashflows: t X t 150, , , , 000 If the project is all equity financed it has a required rate of return of 15%. To finance the project the firm issues a 4 year bond with face value of 100,000 and an interest rate of 5%. Remaining investments are financed by the firm s current operations. The company is facing a tax rate of 30%. etermine the NPV of the project. Exercise 20. Option [4] The current stock price is 160. Next period the price will be one of 150 or 175. The current risk free interest rate is 6%. You buy 1 stock and issue m call options on the stock with an exercise price of 155. What must be m be for the portfolio to be risk free? Exercise 21. LRC [3] You invest $100,000 in the Liana Rope Company. To make the investment, you borrowed $75,000 from a friend at a cost of 10%. You expect your equity investment to return 20%. There are no taxes. What would your return be if you did not use leverage? Exercise 22. Negative NPV? [3] o you agree or disagree with the following statement? Explain your answer. A firm s stockholders would never want the firm to invest in projects with negative NPV. Exercise 23. [1] In the equation, NPV = Cost + PV, the term "Cost" is the A) current value of the expected future cash flows today. B) current value of the terminal cash flow. C) initial cash outflow. ) present value of the variable costs. Exercise 24. [3] 3
4 Bradley Snapp has deposited $7,000 in a guaranteed investment account with a promised rate of 6% compounded annually. He plans to leave it there for 4 full years when he will make a down payment on a car after graduation. How much of a down payment will he be able to make? A) $1, B) $2, C) $8, ) $8, E) $9, Exercise 25. [5] What is the future value of investing $9,000 for 7 years at a continuously compounded rate of 11%? A) $15, B) $18, C) $19, ) $19, Exercise 26. [1] A pure discount bond A) does not have any face value. B) pays interest annually. C) pays interest semiannually. ) does not pay a coupon. Exercise 27. [1] The coupon of a bond is A) its time period to maturity. B) its current price. C) its face value. ) its yield to maturity. E) the amount of the interest payment. Exercise 28. [1] Zerocoupon bonds A) always sell at a discount before maturity. B) always sell at a premium before maturity. C) have no face value. ) have no maturity. E) Both B and C. Exercise 29. [3] The formula P o = IV/r represents A) the present value of dividends in perpetuity. B) the value of a no growth dividend stream. C) a lower value than if a growth element was included. ) All of the above. Exercise 30. [5] A stock you are interested in paid a dividend of $1 last year. The anticipated growth rate in dividends and earnings is 25% for the next 2 years before settling down to a constant 5% growth rate. The discount rate is 12%. Calculate the expected price of the stock. 4
5 A) $15.38 B) $20.50 C) $21.04 ) $22.27 E) $26.14 Exercise 31. [1] The internal rate of return may be defined as A) the discount rate that makes the NPV cash flows equal to zero. B) the difference between the market rate of interest and the NPV. C) the market rate of interest less the riskfree rate. ) the project acceptance rate set by management. Exercise 32. [3] The two fatal flaws of the internal rate of return rule are A) arbitrary determination of a discount rate and failure to consider initial expenditures. B) arbitrary determination of a discount rate and failure to correctly analyze mutually exclusive investment projects. C) arbitrary determination of a discount rate and the multiple rate of return problem. ) failure to consider initial expenditures and failure to correctly analyze mutually exclusive investment projects. E) failure to correctly analyze mutually exclusive investment projects and the multiple rate of return problem. Exercise 33. [1] Inflation is treated properly in NPV analysis by A) discounting nominal cash flows by a nominal discount rate. B) discounting real cash flows by a real discount rate. C) discounting nominal cash flows by a real discount rate. ) discounting real cash flows by a nominal discount rate. E) Both A and B. Exercise 34. [3] Nominal cash flows should be discounted at the A) true rate of interest. B) real rate of interest. C) inflation rate plus the nominal rate of interest. ) nominal rate of interest Exercise 35. [3] Viewing capital budgeting decisions as a series of options is useful to strategic analysis because A) contingent results may provide an option to bailout of a project with subsequent poor outcomes. B) the value of the project should be considered as the NPV plus the value of the option. C) strong markets and subsequent expansion options should be considered at time 0. ) All of the above. Exercise 36. [1] When stocks with the same expected return are combined into a portfolio A) the expected return of the portfolio is less than the weighted average expected return of the stocks. B) the expected return of the portfolio is greater than the weighted average expected return of the stocks. 5
6 C) the expected return of the portfolio is equal to the weighted average expected return of the stocks. ) there is no relationship between the expected return of the portfolio and the expected return of the stocks. Exercise 37. [3] For a highly diversified equally weighted portfolio with a large number of securities, the portfolio variance is A) the average covariance. B) the average expected value. C) the average variance. ) the weighted average expected value. E) the weighted average variance. Exercise 38. [1] A welldiversified portfolio has negligible A) expected return. B) systematic risk. C) unsystematic risk. ) variance. E) Both C and. Exercise 39. [3] A typical investor is assumed to be A) a fair gambler. B) a gambler. C) a single security holder. ) risk averse. E) risk neutral. Exercise 40. [1] Total risk can be divided into A) standard deviation and variance. B) standard deviation and covariance. C) portfolio risk and beta. ) systematic risk and unsystematic risk. E) portfolio risk and covariance. Exercise 41. [1] According to the CAPM A) the expected return on a security is negatively and nonlinearly related to the security s beta. B) the expected return on a security is negatively and linearly related to the security s beta. C) the expected return on a security is positively and linearly related to the security s variance. ) the expected return on a security is positively and nonlinearly related to the security s beta. E) the expected return on a security is positively and linearly related to the security s beta. Exercise 42. [3] If investors possess homogeneous expectations over all assets in the market portfolio, when riskless lending and borrowing is allowed, the market portfolio is defined to A) be the same portfolio of risky assets chosen by all investors. B) have the securities weighted by their market value proportions. C) be a diversified portfolio. ) All of the above. Exercise 43. [3] 6
7 The WACC is used to the expected cash flows when the firm has. A) discount; debt and equity in the capital structure B) discount; short term financing on the balance sheet C) increase; debt and equity in the capital structure ) decrease; short term financing on the balance sheet Exercise 44. [3] An industry is likely to have a low beta if the A) stream of revenues is stable and less volatile than the market. B) economy is in a recession. C) market for their goods is unaffected by the market cycle. ) Both A and B. E) Both A and C. Exercise 45. [3] If the efficient market hypothesis holds, investors should expect A) to earn only a normal return. B) to receive a fair price for their securities. C) always be able to pick stocks that will outperform the market averages. ) Both A and B. E) Both B and C. Exercise 46. [1] Event studies attempt to measure A) the influence of information release to the market on returns in days other than at announcement. B) if the market is at least semistrong efficient. C) whether there is a significant reaction to public announcements. ) All of the above. Exercise 47. [1] MM Proposition I without taxes is used to illustrate A) the value of an unlevered firm equals that of a levered firm. B) that one capital structure is as good as another. C) leverage does not affect the value of the firm. ) capital structure changes have no effect stockholder s welfare. E) All of the above. Exercise 48. [1] The difference between a market value balance sheet and a book value balance sheet is that a market value balance sheet A) places assets on the right hand side. B) places liabilities on the lefthand side. C) does not equate the right hand with the lefthand side. ) lists items in terms of market values, not historical costs. E) uses the market rate of return. Exercise 49. [3] MM Proposition I with corporate taxes states that A) capital structure can affect firm value. B) by raising the debttoequity ratio, the firm can lower its taxes and thereby increase its total value. C) firm value is maximized at an all debt capital structure. 7
8 ) All of the above. Exercise 50. [3] The change in firm value in the presence of corporate taxes only is: A) positive as equityholders face a lower effective tax rate. B) positive as equityholders gain the tax shield on the debt interest. C) negative because of the increased risk of default and fewer shares outstanding. ) negative because of a reduction of equity outstanding. Exercise 51. [3] Which capital budgeting tools, if properly used, will yield the same answer? A) WACC, IRR, and APV B) NPV, IRR, and APV C) NPV, APV and Flow to ebt ) NPV, APV and WACC E) APV, WACC, and Flow to Equity Exercise 52. [1] The special contractual nature giving the owner the right to buy or sell an asset at a fixed price on or before a given date is the basis of A) a common stock. B) a capital investment. C) a futures. ) an option. Exercise 53. [3] Which of the following is not true concerning call option writers? A) Writers promise to deliver shares if exercised by the buyer. B) The writer has the option to sell shares but not an obligation. C) The writer s liability is zero if the option expires outofthemoney. ) The writer receives a cash payment from the buyer at the time the option is purchased. E) The writer has a loss if the market price rises substantially above the exercise price. Exercise 54. [3] Which of the following statements is true? A) At expiration the maximum price of a call is the greater of (S T  Exercise) or 0. B) At expiration the maximum price of a call is the greater of (Exercise  S T ) or 0. C) At expiration the maximum price of a put is the greater of (S T  Exercise) or 0. ) At expiration the maximum price of a put is the greater of (Exercise  S T ) or 0. E) Both A and. Exercise 55. [1] PutCall parity can be used to show A) how far inthemoney put options can get. B) how far inthemoney call options can get. C) the precise relationship between put and call prices given equal exercise prices and equal expiration dates. ) that the value of a call option is always twice that of a put given equal exercise prices and equal expiration dates. E) that the value of a call option is always half that of a put given equal exercise prices and equal expiration dates. 8
9 Exercise 56. [3] The higher the exercise price A) the higher the call price. B) the lower the call price. C) has no effect on call price. ) the higher the stock price. E) the lower the stock price. Exercise 57. [1] If the volatility of the underlying asset decreases, then the A) value of the put option will increase, but the value of the call option will decrease. B) value of the put option will decrease, but the value of the call option will increase. C) value of both the put and call option will increase. ) value of both the put and call option will decrease. E) value of both the put and call option will remain the same. Exercise 58. [3] In terms of relating options to the value of the firm, the equity of the firm can be viewed as A) a call option on the firm with the exercise price equal to the promised payments to the bondholders. B) a call option on the firm with the exercise price equal to the firm s aftertax cash flow. C) a put option on the firm with the exercise price equal to the promised payments to the bondholders. ) a put option on the firm with an exercise price equal to the firm s aftertax cash flow. Exercise 59. [5] Verma Violin Manufacturing Corporation has issued debt with $10 million of principal due. In terms of viewing the equity of the firm as a call option, what happens to the equity of the firm if the cash flow of the firm is greater than $10 million? A) The option is inthemoney and the stockholders earn the difference between the cash flow and the bondholder s promised payment. B) The option is inthemoney and the bondholders earn the entire cash flow. C) The option is outofthemoney, the stockholders walk away, and the bondholders receive the entire cash flow. ) The option is outofthemoney, and the stockholders make up the difference so that the bondholders receive full payment. Exercise 60. [1] Suppose a firm in financial distress is bailed out by the Federal government by guaranteeing the payment on any new debt. If the firm issues new debt, who gains? A) Existing bondholder B) New bondholders C) Stockholders ) Both A and B. E) Both A and C. Exercise 61. [3] A firm in the extraction industry whose major assets are cash, equipment and a closed facility may appear to have extraordinary value. This value can be primarily attributed to A) the potential sale of the company. B) the low exercise price held by the shareholders. C) the option to open the facility when prices rise dramatically. 9
10 ) All of the above. Exercise 62. [3] The option to abandon is A) a real option. B) usually of little value because of the cost associated with abandonment. C) irrelevant in capital budgeting analysis. ) nearly always less relevant the option to expand. E) All of the above. Exercise 63. [3] uration of a pure discount bond A) is equal to its halflife. B) is less than a zero coupon bond. C) is equal to the liabilities hedged. ) equal to its maturity. 10
11 Empirical Solutions MØA 155 PROBLEM SET: Summarizing Exercise 1. Present Value [3] 1. The implied interest rates r t are found as: 2. The present value is found as P 1 = 0.95 = r 1 r 1 = = 5.26% ( ) 2 1 P 2 = 0.9 = 1 + r 2 1 r 2 = = 5.41% ( ) 3 1 P 3 = 0.85 = 1 + r 3 r 3 = = 5.57% P V = = 265 or alternatively, using interest rates, as ( ) ( ) 2 ( ) P V = Exercise 2. Arbitrage [4] 1. This data implies an arbitrage opportunity. Note that the price of the risk free security offering $1 in period 3 is higher than the price of the risk free security offering $1 in period 2. What does this mean? It means you have to pay less today for receiving money sooner! To make a lot of money, short the risk free security for period 3, and use $0.9 of the $0.95 proceeds to buy the period 2 risk free security. The $1 you get in period 2 can be kept as money and used to cover your obligation in period 3. For each of these transactions you get $0.05 now. To get very rich, do a lot of these transactions. Exercise 3. Bonds [4] Consider the the cash flows of the bonds and let B i be the price of bond i. Period Bond
12 1. Prices of one dollar to be received at times 1, 2 and 3: We use the following system to solve for these: We first use the prices of the first two bonds: B 1 = 100P P 2 B 2 = 200P P 2 B 3 = 80P P P 3 B 1 = 100P P 2 B 2 = 200P P 2 Multiply the first equation by 2 and subtract the second: Solve for P 2 : Use this to find P 1 : P 2 = Then find P 3 using the third bond price: 2. From these we find the spot rates of interest 2B 1 B 2 = 1000P B 1 = 100P P 2 = = 100P P 1 = B 3 = 80P P P = P 3 P 3 = P 1 = = r 1 r 1 = = 9.89% ( ) 2 1 P 2 = = 1 + r 2 1 r 2 = = 7.47% ( ) 3 1 P 3 = = 1 + r 3 r 3 = = 9.92% 12
13 Exercise 4. Stock [4] Exercise 5. Projects [3] 1 = 10(1.1) = 11 2 = 10(1.1) 2 = = 2 (1.02) = = P 2 = 3 r g = = P 0 = P = = = The following picture shows the NPV as a function of the interest rate, and illustrates the fact that there are two solutions y to the problem of solving 0 = y (1 + y) (1 + y) NPV interest Exercise 6. CAPM [2] E[r] = r f + (E[r m ] r f )β = ( )0.5 = 9.5% Exercise 7. Portfolio [2] Let r A be the return on stock A and r B the return on stock B. Since the return on the risk free asset is a constant, its variance equals zero, and its covariance with other assets is also zero, and we can calculate the portfolio variance as var = var(r A ) cov(r A, r B ) var(r B ) var(r A ) = =
14 var(r B ) = = 0.04 cov(r A, r B ) = = var(r p ) = σ(r p ) = var(r p )0.545 Exercise 8. Q [2] Exercise 9. Project [3] E[r m ] = E[r] r f β E[r] = r f + (E[r m ] r f )β + r f = First find the required rate of return for the project r = ( )1.5 = 12.5% The maximal cost equals the present value of the future cash flow. P V = Thus, the maximal cost is Exercise 10. Price [2] = 11.4% (1.125) (1.125) (1.125) 6 = value = 1 ( p u X u + p d X d) ( (1 0.4) 500) = 1 + r = Exercise 11. XYZ option [2] Buy the option for 4, exercise immediately paying 50 to get the stock, sell the stock for 56. Net proceeds = 2. Repeat indefinitely. Exercise 12. ud [1] Exercise 13. Call [5] Let us find the price of a call on one share. S u = 60 = us 0 = u50 u = = 1.2 d = =
15 S u = 120 S 0 = 96 S d = 95 Find u and d: u = = 1.25 d = = Then find p u. Need to find the risk free rate for a 5 week period, approximate as Then find terminal payoffs and calculate option price p u = The call on 100 shares is then worth r (1 + r) d u d = = C u = max( , 0) = 8 C d = max(95 112, 0) = 0 = C 0 = r (pu C u + (1 p u )C d ) = 1 ( ) = r = Exercise 14. Bond Covenants [3] 1. The covenants protect bondholders from managers acting in behalf of shareholders and undertaking inefficient investment proficiencies during the time of financial distress. 2. These benefit the stockholders by allowing them to borrow from the bondholders at a reduced interest rate. Exercise 15. [3] Can use the equity to determine implied probabilities. Value of equity in the various states. max(0, ) = max(0, 20 50) = 0 15
16 40 = r (pu 50 + (1 p u )0) 40 = pu 50 p u = = Then this p u is used to determine the bond value Bond payoffs: min(50, 100) = 50 B 0 min(50, 20) = 20 B 0 = (pu 50 + (1 p u )20) = Exercise 16. Frisky [4] r = r + (r r ) E r = ( ) 1 3 = 18% Exercise 17. Leverage [6] 1. Let us use the data for the unlevered firm to find r. We are given the (beforetax) operating income X. The value of the unlevered firm is r = V U = r = aftertax income r aftertax income V U X(1 τ) 600(1 0.4) = = 18% V U 2, 000 For the unlevered firm, this is also the cost of equity capital, r E = r = 18%. Use this to find the value of the levered firm. r E = r + (r r )(1 τ) E = ( )(1 0.4) 1 = 22.8%. 16
17 2. W ACC U = 18% W ACC L = 1 2 r E (1 τ)r = (1 0.4) = 14.4% 3. The equity is riskier, hence the return on equity for the levered firm is higher. The lower W ACC reflects the tax savings from the leverage. Exercise 18. Bond issue [4] Currently, the value of the firm is $3.5 million, the same as the value of equity. The repurchase of $1 million decreases the value of equity by $1 million, to 2,500 million. But this does not account for the tax shield of the new bond issue. If the bonds are perpetual, the value of the firm increases by 1 million τ, where τ is the tax rate. In this case τ = 30%. Thus the tax shield is 300,000, and the value of equity is 2.5 mill + 300,000 = 2,800,000. Exercise 19. Project [3] Without the tax advantage NP V = Annual interest tax shield: = 1500 PV of tax shield t = C t = ( ) ( ) ( ) 3 = t = C t = Project NPV is NP V = 1500 ( ) ( ) ( ) ( ) 4 = NP V = = 7682 Exercise 20. Option [4] S d mc d = S u mc u m = S d S u = = 1.25 C u C d 20 0 m = 1.25 Exercise 21. LRC [3] 17
18 The equity returns $5,000(=20% of 25,000); the loan requires $7,500 (=10% of $75,000). Hence, the investment returns in total $12,500, which is 12.5% on $100,000. The would be the return on investment if it were totally financed by equity. Exercise 22. Negative NPV? [3] If bonds are in place (the firm has collected the money), then shareholders may have the incentive to change projects, towards a more volatile one. This may even be a negative NPV project. Convenants, warrants and conversion rights attached to the bonds will keep shareholders from doing so. Exercise 23. [1] C Exercise 24. [3] Rationale: $7,000 (1.06) 4 = $8, Exercise 25. [5] Rationale: $9, 000(e. 11(7) ) 1 = $9, 000( ) = $19, Exercise 26. [1] Exercise 27. [1] E Exercise 28. [1] A Exercise 29. [3] Exercise 30. [5] C Rationale: Price = $1.00(1.25)/ $1.25(1.25)/ [$1.5625(1.05)/( )]/ = $21.04 Exercise 31. [1] A Exercise 32. [3] E Exercise 33. [1] E Exercise 34. [3] Exercise 35. [3] Exercise 36. [1] C Exercise 37. [3] A Exercise 38. [1] 18
19 C Exercise 39. [3] Exercise 40. [1] Exercise 41. [1] E Exercise 42. [3] Exercise 43. [3] A Exercise 44. [3] E Exercise 45. [3] Exercise 46. [1] Exercise 47. [1] E Exercise 48. [1] Exercise 49. [3] Exercise 50. [3] B Exercise 51. [3] E Exercise 52. [1] Exercise 53. [3] B Exercise 54. [3] E Exercise 55. [1] C Exercise 56. [3] B Exercise 57. [1] Exercise 58. [3] A Exercise 59. [5] 19
20 A Exercise 60. [1] E Exercise 61. [3] C Exercise 62. [3] A Exercise 63. [3] 20
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