Fins2624 Group Assignment

Q3 (Essential to cover) Consider a coupon bond with a face value of $100, a coupon rate of 10%, a time-to-maturity of three years and a price of $119.80. What is its yield-to-maturity? (Hint: Solve numerically using a scientific calculator or Excel) Solution: The cash flow diagram of this bond is: -P = -119.80 c1 = 10 c2 = 10 C3 = 10 FV = 100 P = c 1 /(1 + y)1 + c 2 /(1 + y)2 + c 3 /(1 + y)3 + FV/(1 + y)3 (1) So if you know how to solve cubic equations, then you can directly apply the cubic equation solver to obtain the value of y, which is the yield-to-maturity under question. Alternatively, you can use a numeric solution (try-and-error) to obtain the approximate value of y. Step 1: take an initial value of y Since P > FV (traded at premium), so y < C = 10%. Let’s take the starting value of y as 9% and calculate the price when y = 9%. P(9%) = 102.53 Step2: adjust y until the calculated price crossing the actual price $119.80 Since P(9%) = 102.53 < 119.80, we should further decrease y, until the calculated price is close or below the actual price. P(3%) = 119.8003 And P(2%) = 123.07 So we conclude that the actual y should be 3% (close enough to the true value). Step3 (if necessary): Interpolation to get the answer For demonstration purpose, assume the actual price is 121.44, and you are asked to get an answer of y in percentage (rounded to first decimal). Since P(2%) = 123.07 > 121.4, and P(3%) = 119.80 < 121.4, so the actual y should lie between 2% and 3%. So we can obtain the approximate value of y by interpolation. y1 = 2% P1 = 123.07 y=? P = 121.44 y2 = 3% P2 = 119.80 Hence y ~=2%*[(121.44-119.80)/(123.07-119.80)] + 3%*[(123.07-121.44)/(123.07-119.80)] = 2.5%. As you can see, the weight on a rate (e.g., 2%) is actually the relative distance between the actual price ($121.44) and the price calculated from the other rate (P(3%) = 119.80). The idea is that the further away the price of 3% is from the actual price, the closer the actual y to 2%. Additional Note: With hindsight, this question is not well designed – you need to try too many times with hypothetical ys until the calculated price crosses the actual price. In the exam, if I decide to include YTM calculation (I have not decided yet), I will choose numbers you don’t have to try too many times. And as I indicated in the lecture note, at most I will set the maximum time periods up to T = 3. And if I require you to do the interpolation to get the answer, I will put a note and specify to what decimals the answer should be. Q9 (Essential to cover) A 2-year bond with par value of $1000 making annual coupon payments of $100 is priced at $1000. What is the yield to maturity of the bond (without any calculation, are you able to answer this question?)? What will be the realized compound yield to maturity if the 1-year interest rate next year turns out to be: a) 8%, b) 10%, c) 12%? Solution: The cash flow diagram of this bond is: -P = -1000 c1 = 100 c2 = 100 FV = 1000 P = c 1 /(1 + y)1 + c 2 /(1 + y)2 + FV/(1 + y)2 (1) [1] yield to maturity Since P = FV, so y = C = 10%. [2] Realized compound yield – let’s take 8% 1-year interest rate next year for illustration Step1: To calculate realized compound yield, you will first need to reinvest all interim cash flows (coupon payments in this question) to the maturity at the interest rate at the time of coupon payment. -P = -1000 c1 = 100 c2 = 100 FV = 1000 So the cash flow at the end of maturity (time 2) from reinvesting c1 = 100*(1+8%) = 108. Step 2: calculate the total cash flow (with reinvestment) at the end of maturity So the total cash flow at the end of maturity (time 2) of this bond = 108+100+1000 = 1208. Step 3: calculate total realized return and annualize it Total (realized) return (R) = total cash flows at maturity/price – 1 = 1208/1000 – 1 Annualize total return to annual return: r = (1+R)1/T – 1 = (1208/1000) ½ - 1 = 9.9% You can replicate the calculation above to 10% and 12% of 1-year interest rate next year. You will find that realized compound yield = 10% if the reinvestment rate = 10%, and above 10% if the reinvestment rate = 12%. So YTM (10% in this question) equals to realized compound yield if all interim cash flows could be reinvested at YTM (which is unlikely the case). Additional Note: The calculation is very simple for the two-year bonds in our example. But the same calculation process could be easily applied to more general case. For example, in the general bond example below (T-year maturity), to calculate the cash flow at the maturity (time T) from reinvesting the first coupon payment, you will need to reinvest c1 first to end of time 2 (using the 1-year return at time 1), then reinvest the proceedings at the end of time 2 further to time 3 (using the 1-year return at time 2), so on so forth, until the end of time T. Similarly, you will need to reinvest all other cash flows to the end of maturity. Finally, you sum up all reinvestment cash flows and the final payment (coupon and face value) to obtain the aggregate cash flows from this bond – based on which you can calculate total return over these T years and annualize it to realized compound yield. But don’t worry, if I really decide to test on the realized compound yield, I will set T = 2 so that you only need to reinvest one cash flow (for one year), given that the logic is the same and making T bigger just makes things more complex (rather than more challenging). -P c1 c2 ct cT FV Selected end-of-chapter questions. BKM Chapter 14 3. Zero coupon bonds provide no coupons to be reinvested. Therefore, the investor's proceeds from the bond are independent of the rate at which coupons could be reinvested (if they were paid). There is no reinvestment rate uncertainty with zeros. 4. A bond’s coupon interest payments and principal repayment are not affected by changes in market rates. Consequently, if market rates increase, bond investors in the secondary markets are not willing to pay as much for a claim on a given bond’s fixed interest and principal payments as they would if market rates were lower. This relationship is apparent from the inverse relationship between interest rates and present value. An increase in the discount rate (i.e., the market rate. decreases the present value of the future cash flows. 5. Annual coupon rate: 4.80%  $48 Coupon payments Current yield:  $48    = 4.95%  $970  8. The bond price will be lower. As time passes, the bond price, which is now above par value, will approach par. 9. Yield to maturity: Using a financial calculator, enter the following: n = 3; PV = −953.10; FV = 1000; PMT = 80; COMP i This results in: YTM = 9.88% Realized compound yield: First, find the future value (FV. of reinvested coupons and principal: FV = ($80 * 1.10 *1.12. + $80 * 1.12. + $1,080) = $1,268.16 Then find the rate (y realized . that makes the FV of the purchase price equal to $1,268.16: $953.10 × (1 + y realized .3 = $1,268.16 ⇒ y realized = 9.99% or approximately 10% Using a financial calculator, enter the following: N = 3; PV = −953.10; FV = 1,268.16; PMT = 0; COMP I. Answer is 9.99%. Note: financial calculator is not allowed in the exam. So please solve the equation with numerical method. 13. Price $400.00 500.00 500.00 385.54 463.19 400.00 Maturity (years. 20.00 20.00 10.00 10.00 10.00 11.91 Bond Equivalent YTM 4.688% 3.526 7.177 10.000 8.000 8.000 16. If the yield to maturity is greater than the current yield, then the bond offers the prospect of price appreciation as it approaches its maturity date. Therefore, the bond must be selling below par value. 17. The coupon rate is less than 9%. If coupon divided by price equals 9%, and price is less than par, then price divided by par is less than 9%. 23. The bond is selling at par value. Its yield to maturity equals the coupon rate, 10%. If the first-year coupon is reinvested at an interest rate of r percent, then total proceeds at the end of the second year will be: [$100 * (1 + r)] + $1,100 Therefore, realized compound yield to maturity is a function of r, as shown in the following table: 31. r Total proceeds Realized YTM = Proceeds/1000 – 1 8% $1,208 1208/1000 – 1 = 0.0991 = 9.91% 10% $1,210 1210/1000 – 1 = 0.1000 = 10.00% 12% $1,212 1212/1000 – 1 = 0.1009 = 10.09% a. Initial price P 0 = $705.46 [n = 20; PMT = 50; FV = 1000; i = 8] Next year's price P 1 = $793.29 [n = 19; PMT = 50; FV = 1000; i = 7] HPR = $50 + ($793.29 − $705.46) = 0.1954 = 19.54% $705.46


Problem set 9 solution Q1: c) is a violation of weak form EMH, because trading on past price/return information is able to generate abnormal returns (alphas). And if prices do not fully reflect even past price information (a sub-set of all public information), prices surely do not fully reflect all public information (thus a violation of semi-strong form EMH) and all information, both public and private (thus also a violation of strong-form EMH). e) Here fund managers are trading on private information they obtain from corporate executives and are able to generate abnormal returns (alphas). So it is a violation of strong-form EMH, but not other forms of EMH. In all other cases, it is not clear whether or not you can generate abnormal returns or alphas. For example, positive returns in a) and above than average capital gains in d) do not necessarily mean positive alphas (which means that the returns are higher than the expected returns based on some asset pricing models like CAPM). Q2 a) does not say anything about EMH, because we don’t know what are the alphas of these mutual funds. And even nearly half of mutual funds are able to generate positive alphas in a year, it does not necessarily violate EMH because it may be due to luck. b) is a violation of EMH because it implies those outperforming mutual funds could consistently generate positive alphas (and thus outperformance in a year is not due to luck). c) it does not say anything about alphas, and thus say nothing about EMH. d) violation of semi-strong form (and hence strong-from also) EMH, because you design a trading strategy buying in those stocks with increased earnings in January and holding them in February and generate superior performance. This strategy uses public information. e) implies a violation of weak form EMH (and thus semi-strong and strong forms also). Q3 a. the optimal fraction to invest in the market portfolio could be calculated from y* = 1/A*(E(rM) – rf)/σ2M With the given believes of two investors, we can easily do the calculation. y* = 1/3*(15%-3%)/(40%*40%) = 25% for rational investor and y* = 1/3*(15%-3%)/(25%*25%) = 64% for overconfident investor. b. Based on the calculated allocations to the market portfolio in a), we can calculate expected return and volatility of the complete portfolio for both investors. Rational investors: E(rc) = 0.75*rf + 0.25* E(rM) = 6% σc = 0.25* σM = 10% U = E(rc) – 1.5 σ2c = 4.5% Overconfident investors: E(rc) = 0.36*rf + 0.64* E(rM) = 10.68% σc = 0.64* σM = 25.6% U = E(rc) – 1.5 σ2c = 0.08% We can see the following points: [1] overconfident investor’s portfolio has a higher expected return and a higher volatility, due to it invests much more in the risky assets. [2] however, the utility is lower for overconfident investors. So overconfident investors are making sub-optimal investment. c&d. So updated E(rM) = 17% for both types of investors. With this new expected market return, the optimal fraction of investment in the market portfolio is y* ~= 29% for rational investors, a 4% additional weight in the market portfolio. and y* ~= 75% for overconfident investors, an 11% additional weight in the market portfolio. So we can see that though the magnitude of revision in return expectation is the same for both types of investors, overconfident investors tend to trade (i.e., rebalance their portfolios) much more aggressively. Q4. a. Based on the given information, we can calculate the alpha of asset X as: 12.5% - 4% - 0.5*(16% - 4%) = 2.5%. So both investors will deviate from the market portfolio by holding additional asset X beyond its weight in the market portfolio. Based on lecture 7, we know the optimal additional weight in asset X is given by So we need to calculate the idiosyncratic risk (the variance of residual return), σ2ε = σ2 - (beta* σM)^2, of asset X first. For rational investor, her estimate of σ2ε = 50%*50% - (0.5*20%)^2 = 24%, which results in W0A ~= 3.47% and W*A ~= 3.41%. For overconfident investor, her estimate of σ2ε = 30%*30% - (0.5*20%)^2 = 8%, which results in W0A ~= 10.42% and W*A ~= 9.9%. We can see that overconfident investors deviate much further away from the market portfolio than the rational investors and hold a much more concentrated position in the active asset, although they have the same alpha estimates. So overconfident investors tend to hold underdiversified portfolios. Selected end-of-chapter questions BKM chapter 11 1. The correlation coefficient between stock returns for two nonoverlapping periods should be zero. If not, returns from one period could be used to predict returns in later periods and make abnormal profits. 2. No. Microsoft’s continuing profitability does not imply that stock market investors who purchased Microsoft shares after its success was already evident would have earned an exceptionally high return on their investments. It simply means that Microsoft has made risky investments over the years that have paid off in the form of increased cash flows and profitability. Microsoft shareholders have benefited from the risk-expected return tradeoff, which is consistent with the EMH. 3. Expected rates of return differ because of differential risk premiums across all securities. 9. c. 14. This is a predictable pattern in returns that should not occur if the weak-form EMH is valid. d. In a semistrong-form efficient market, it is not possible to earn abnormally high profits by trading on publicly available information. Information about P/E ratios and recent price changes is publicly known. On the other hand, an investor who has advance knowledge of management improvements could earn abnormally high trading profits (unless the market is also strong-form efficient). BKM chapter 12 3. One of the major factors limiting the ability of rational investors to take advantage of any ‘pricing errors’ that result from the actions of behavioral investors is the fact that a mispricing can get worse over time. An example of this fundamental risk is the apparent ongoing overpricing of the NASDAQ index in the late 1990s. Related factors are the inherent costs and limits related to short selling, which restrict the extent to which arbitrage can force overpriced securities (or indexes) to move towards their fair values. Rational investors must also be aware of the risk that an apparent mispricing is, in fact, a consequence of model risk; that is, the perceived mispricing may not be real because the investor has used a faulty model to value the security. 6. a. Davis uses loss aversion as the basis for her decision making. She holds on to stocks that are down from the purchase price in the hopes that they will recover. She is reluctant to accept a loss. 9. a. iv b. iii c. v d. i e. ii


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