HW3 Jimmy

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Jan 9, 2024

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Foundations of Finance Prof. Toomas Laarits Spring 2023 Homework Cover Sheet Print your name: Class section: FINC-UB.0002.004, 9:30am FINC-UB.0002.005, 2:00pm FINC-UB.0002.006, 3:30pm Homework number: Students in your homework group: “I pledge my honor that I have not violated the NYU Stern Student Code of Conduct in the completion of this exam/problem set.” Signed: 1 Jimmy Li Jimmy Li HW 3 Charlie Ma

Foundations of Finance Prof. Toomas Laarits Spring ½»½¾ HW # ¾ Due: ¿/¼Ã before class on the course website. Note ¼: You are encouraged to work in groups, but everyone must submit their own work. Feel free to write up the homework any way you want. In particular, you can type it, or you can write it on paper and scan. Note ½: Please use the “Homework Cover Sheet” available on the course website as the first page of your submission. Note ¾: One of the problems is the designated “Excel problem”. You can also use other statistical software. Note too that many of the other problems are easier to solve with Excel, as they ask you to apply the same formula using multiple different inputs. No Arbitrage ¼. (2 points) Consider the following three securities: SNOW, RAIN and SUN. SNOW pays $ ¼»» if it snows on NYU graduation day. RAIN pays $ ¼»» if it rains, but doesn’t snow on NYU graduation day. SUN pays $ ¼»» if there is neither rain nor snow on graduation day. Suppose that SNOW is currently trading at $ ¼, RAIN is currently trading at $ ¿¼ and SUN is trading at a price of $ À½. (a) If you buy ¼ share of SNOW, ¼ share of RAIN and ¼ share of SUN, what is the payoff you guarantee on NYU graduation day? Since one of the three must happen on the Graduation day, the payoff is $¼»» - $¼ - $¿¼ - $À½ Ó $Á (b) According to the No Arbitrage Condition, what must be the price of a $¼»» face value zero-coupon bond that matures on NYU graduation day? The face value is ¼»»-Á Ó $Ä¿ (c) Suppose that this zero-coupon bond is trading at $ Ä». How would you set up a transaction to earn a riskless arbitrage profit? Assume no trading costs. ¼ im

I can buy a zero-coupon bond today at a price of $Ä» and sell ¼ share of rain, one share of sun, and one share of snow today at a price of Ä¿$ which will guarantee a $¼»» payment on the graudation day. This will guarantee a $¿ profit. (d) Suppose that trading zero-coupon bonds and RAIN is costless, but shorting SUN costs $ ¾ per $ ¼»» face value and shorting SNOW costs $ ¼ per $ ¼»» face value. Can you still make an arbitrage profit? Since we are shorting RAIN and SUN in the previous arbitrage, the shorirng cost will increase our cosrt by ¼ Ï¾ Ó $¿ which is equal to our arbitrage profit. The short cost will eliminate the arbitrage profit. Equity Valuation ½. (¾ points) Suppose that the consensus forecast of security analysts of NoWork Inc. is that earnings next year will be E ¼ Ó $¼» . »» per share. The company tends to plow back À»Î of its earnings and pay the rest as dividends. The CFO estimates that the company’s growth rate will be ÃÎ from now on. (a) If your estimate of the company’s required rate of return is ¼½Î, what is the equilibrium price of the stock? P Ó V» Ó $¼» * ».À/ (¼½Î - ÃÎ) Ó $¼½À (b) You observe that the stock is selling for $ ¼½».»» per share. Suppose you believe that the market price is right. What must you conclude about either ( i ) your estimate of the stock’s required rate of return, ( ii ) the CFO’s estimate of the company’s future growth rate, or ( iii ) the forecast of earnings from the analysts? If the market price is correct, my estimated price is higher than the current fair market price. Then: (i) I might have underestimate the required rate of return. (ii) The CFO might overestimate the growth rate. (iii) The future earning might be overestimated. (c) Suppose there is uncertainty about the stock’s dividend growth rate. With probability the growth rate will be ¼»Î, with probability it will be ÂÎ. What are the respective market values under the two different growth rates? $À/ (¼½Î - ¼»Î) Ó $½À» ½

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$À/ (¼½Î - ÂÎ) Ó $¼»» (d) What is the fair price of the stock given the probabilities above? $¼»» * Ҁ Ï $½À» * ⅓ Ó $ ¼À» (e) What is the expected growth rate for the stock? Given your calculations, which security is more valuable for an investor: the stock with a ÃÎ growth rate for sure or the stock described in part (c) with an uncertain growth rate. ¼» * ⅓ Ï ÂÎ * Ҁ Ó ÃÎ For investors, the stock in part c has a higher value since the uncertainty give a higher current price despite having the same expected return. ¾. Excel Question. (2 points) This question requires data collection. You can use http://finance.yahoo.com to find the numbers. Please note the date on which you downloaded the data. (date: ¿.¼Â at Àpm) The questions are about CVS Health Corporation (ticker: CVS). (a) What is the current price and the current price-earnings ratio? The current price is ÂÀ.Ã» and EPS is ¾.¼¿. Price-earnings ratio is $ÂÀ.Ã»/¾.¼¿ Ó ½¿.¼¿ (b) What is the current plow-back ratio? Plow-back ratio Ó ¼ - Payout Ratio Ó ¼ - Â».»ÁÎ Ó ½Ä.Ä¿Î (c) What is the growth rate of earnings for the next À years according to the analysts? Hint: look for annual growth rates under “Analysis”. What is the growth rate of earnings for the next year according to analysts? The growth estimate is ¿.ÄÃÎ for the next À years. The growth rate for the next year is ¾.¾»Î. (d) What is the CAPM beta of CVS? Hint: look for “Statistics.” If the risk-free rate R f is ¿Î and the market risk premium E[ R M ] − R f is ÄÎ, what is the required rate of return on CVS according to the CAPM? Beta is ».Á». The return under CAPM is Ó ».»¿Ï ».Á»*».»Ä Ó Ä.¿Î (e) Suppose the earnings and dividends of CVS will grow at a rate equal to the À year forecast from part (c) forever, meaning the Gordon Growth Model (GGM) applies. What is the price-earnings ratio according to the GGM? ¾

P»/E» Ó (¼-b)(¼Ïg)/ (R-g) Ó (¼-».¾»)(¼Ï».»¿ÄÃ)/(».»Ä¿-».»¿ÄÃ) Ó ¼Á.Á¾ (f) What growth rate does the current market price-earnings ratio imply, under our numerical assumptions regarding the value of the risk-free rate and the market risk premium. P»/E» Ó (¼-b)(¼Ïg)/ (R-g) ½¿.¼¿/(¼-».¾»)Ó (¼Ïg)/(R-g) ¾¿.¿Ä Ó (¼Ïg)/(R-g) ¾¿.¿ÄG-¾¿.¿ÄR Ó ¼Ïg g Ó (¾¿.¿Ä*».»Ä¿ - ¼)/ (¼ Ï ¾¿.¿Ä) Ó ».»Á¾½ The trailing P/E ratio is ½¿.¼¿, which implies that current market earning growth rate is Á.¾½Î, which is much higher than the company earning growth rate ¾.¾» Î. Fixed Income Securities ¿. (1.À points) Today is t Ó ». You have just bought a five-year zero-coupon Treasury bond with $ ¼»» face value. You paid $ Ã». (a) What is the annually compounded yield to maturity on the bond? ¼»» Ó Ã»/(¼ Ïy)^À, y Ó (¼»»/Ã») ^ ⅕ - ¼ Ó ».»¿ÀÁ¿ Ó ¿.ÀÁÎ (b) Suppose that yields at all maturities decrease to ½Î immediately after you have purchased the bond. Calculate the annualized holding period return if you sell the bond one year after you have purchased it, at t Ó ¼. New yield Ó½Î Price ¼ Ó ¼»»/(¼Ï½Î)^¿ Ó $Ä½.¾Ã HPRa Ó (Ä½.¾Ã - Ã») / Ã» Ó ».¼À¿Â Ó ¼À.ÀÎ (c) What is the annually compounded yield to maturity on the bond at t Ó ¼? ($¼»»/$Ä½.¾Ã)^Ë - ¼ Ó ».»½»»¼¾ Ó ½Î À. (1.À points) Suppose the yield to maturity on a one-year zero-coupon bond is ÀÎ. The yield to maturity on a two-year zero-coupon bond is ¾Î. ¿

(a) According to the Expectations Hypothesis, what is the expected one-year rate in the marketplace for year ½? (¼Ï».»¾)^½ Ó (¼Ï».»À) Ï (¼ÏE(r½)) E(r½) Ó ¼.»¾ÃÎ (b) Consider an investor who is absolutely convinced that interest rates will not change so that the yield on a one-year bond will still be ÀÎ this time next year. Which of these two bonds, the one-year zero coupon bond, or the two-year zero coupon bond, should this investor buy to maximize their one year return (under their strongly-held belief about future rates)? P Ó $¼»»/(¼Ï».»À) Ó$ ÄÀ.½¿ Pirce of two year bond Ó $¼»»/(¼Ï».»¾)^½ Ó$ Ä¿.½Á One year return Ó ($ÄÀ.½¿-$Ä¿.½Á)/$Ä¿.½Á Ó ¼.»¾ÃÎ The investor should purchase the one-year coupon bond to maximize return since it has a higher return ÀÎ while the two-year bond only has ¼.»¾ÃÎ. À

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