Note: There are fewer problems in the actual Final Exam!


 Scarlett Bates
 2 years ago
 Views:
Transcription
1 HEC Paris Practice Final Exam Questions Version with Solutions Financial Markets Fall 2013 Note: There are fewer problems in the actual Final Exam! Problem 1. Are the following statements True, False or Ambiguous? Provide a short justification for your answer. (Your score will primarily be determined by your justification.) (a) If CAPM is correct, it follows that the market portfolio is the portfolio with the highest Sharpe ratio of all the possible portfolios. Solution: True: We know that the tangency portfolio has the highest Sharpe ratio of all possible portfolios. CAPM implies that the market porfolio is the tangency portfolio. (b) If one introduces one more asset in the investment opportunity set, then the Sharpe ratio of the new tangency portfolio is at least as high as the Sharpe ratio of the old tangency portfolio. Solution: True. After introducing one more asset, the efficient frontier always moves to the left. This means that the tangent line to the efficient frontier also moves to the left, i.e. its slope (which is the Sharpe ratio) will be the same or higher. (c) Because of the potential for diversification if two stocks are not perfectly correlated, a portfolio that invests 50% in each stock must be less risky that a portfolio invested just in one of the stocks. Solution: False in general. Although there is a benefit to diversification, if one of the stocks is very risky then the portfolio could be riskier than an investment in the low risk stock. For example suppose that σ 1 = 0.1, σ 2 = 0.3 and ρ 1,2 = 0. The variance of the portfolio is: σ 2 P = = Hence σ P = = This has higher risk than investing entirely in stock #1. (d) You are a meanvariance investor and until recently you could not invest in commercial real estate. Today you found out that you can invest in an exchangetraded fund (ETF) which does a good job tracking a broad index of the commercial real estate sector. Then your investment opportunity set should get strictly bigger. Solution: True: It is safe to assume that commercial real estate is not a redundant asset, since it is unlikely to be perfectly correlated with some combination of assets you already own. Therefore, by being allowed to invest in this new sector, your investment opportunity set should get strictly bigger. This means that the efficient frontier moves up and to the left. Thus, for every portfolio on the old efficient frontier, you can find a better one on the new frontier: e.g., you can maintain the same risk as for the old portfolio, but increase your expected return. 1
2 (e) The duration of a (fixedrate) coupon bond is always at least as large as the duration of a zero coupon bond with the same maturity. Solution: False. The duration of a coupon bond is the valueweighted average of the durations of its individual payments, which must be smaller not larger than the duration of its last payment (which is the maturity). (f) (2 points) Consider a coupon bond with a price of 60, and a duration of 5. Then if all the interest rates increased by 1%, the price of the coupon bond would increase approximately by $3. Solution: False. If interest rates increase, then the price should decrease approximately by $3, not increase. (g) The current price of Digital stock is $44 per share. You are offered a forward price for Digital stock to be delivered in one year of $42. The forward price is lower than the spot price because the market anticipates a sharp decline in the price of Digital stock, and the contract offers a way to hedge this risk. (Assume that there is no arbitrage opportunity.) Solution: False. According to spot futures parity, the forward price for Digital should equal F 0 = S 0 (1 + r d), where d is the net benefit of holding the underlying asset (dividends in this case). So the fact that F 0 = 42 is smaller than S 0 = 44 does not mean that the stock price is expected to decline. (Remember that the spot futures parity was determined by arbitrage: if the formula were not true, you could construct an arbitrage.) Besides, if Digital was really expected to have a sharp decline in price, market efficiency would force its price to fall today! (h) The stocks of Merck and Disney are traded at the same price of $37 a share. The historical returns of Merck are more volatile than those of Disney and exhibit higher systematic risk. In addition, given the coming health reform proposal, it is generally believed that Merck may have a negative alpha in the future. It is known that neither Merck nor Disney will pay dividends in a month. Consider now forward contracts on these two stocks with one month to maturity. The forward price of 100 shares of Merck should be lower than the forward price of 100 shares of Disney. Solution: False. Essentially the same argument as in g: the forward price of both stocks should be $37 (1 + r). (i) The current spot price of gold is $300 per ounce and the oneyear futures price is $330 per ounce. (Don t argue with these facts! ) These numbers tell us that the market is expecting at most a 10% increase in the price of gold over the next year. Solution: Uncertain. The futures price does not help us figure out expectations. For example let S 0 be the current price of gold, r be the riskfree rate, and k the per ounce cost of storing gold. Then by no arbitrage, the futures price must be: F 0 = S 0 (1 + r + k) 2
3 regardless of expectations. For example suppose that r = 8%, k = 2% and S 0 = 300. Then: F 0 = 300 ( ) = 330 independent of our or the market s expectations about the price of gold in the future. (j) Suppose the current US Dollar/British Pound exchange rate is 2.00 $/, the one year riskfree rate in UK is 4%, and the 1year riskfree rate in the U.S. is 2%. You believe that the following analysis is correct: Because the economy in the UK is stronger than the economy in the US, the Pound is probably strengthening against the Dollar, i.e. the exchange rate $/ will probably increase in the near future. Then this implies that you should be able to lock in a 1year forward exchange rate larger than 2.00 $/. Solution: False. The forward rate is determined by the covered interest parity, which is a noarbitrage relationship: F $/ 0 = E $/ 0 (1 + r $ ) (1 + r ) The US 1year interest rate r $ = 1.5% is smaller than the UK interest rate r = 4%, so F $/ 0 < E $/ 0. Therefore the forward exchange rate must be smaller than 2.00 $/. (k) (2 points) The value of a European call option on a stock is always larger than the value of a futures contract on the same stock. Solution: True. The value of a European call option is always positive, since it gives the right, but not the obligation to buy the underlying stock. The futures contract has a value of zero. This is not to be confused with the futures price, which is positive and is related to the stock price via the spot futures formula. (l) You should only exercise an American put option when the underlying stock price is high enough. Solution: False. If the underlying stock price is high, you gain little from exercising. The statement is actually true if we replace high with low: you should only exercise an American put option when the underlying stock price is low enough. (m) The price of ABC stock is currently $1 per share and you know that ABC will pay no dividends in the next year. You hold an inthemoney American put option on ABC stock, with expiration in oneyear. Because the price is very low today, so it cannot fall much further in the future, you should exercise the option today. Solution: Uncertain. It is possible that exercising the option now is the best thing to do. If you exercise the put today, the advantage is that you would receive the strike 3
4 price (from selling the stock) now rather than later, and therefore you can earn interest on it. However, if you exercise now, you would lose the option value of making an extra $1 when the stock drops further in value. So it depends on whether the time value advantage is large enough to convince you to give up the option value. Typically that happens only when the price of the underlying stock is low enough (so that the option value is low enough), which is the case here. (n) (2 points) Suppose interest rates were negative: r < 0. Then it would not be optimal to exercise early an American put option. Solution: True. By exercising early, one loses both the option value and the time value of the strike price: getting K now is worse than getting K later when interest rates are K negative (in presentvalue terms, one has to compare getting K with getting > K (1+r) τ when r < 0). Also, it does not matter if the stock pays any dividends, since dividends makes the holder of a put option better off (so dividends should not make one exercise early). Problem 2. You are a mean variance investor, and you decide to choose a portfolio with the available assets. You are allowed to invest only in an asset that has a riskfree rate of return r f = 5%, and in two risky assets, GE and KO, which have the following expected return & standard deviation: Asset E( r) σ GE 10% 15% KO 12% 19% The correlation between the two stocks is ρ GE,KO = 0.4. Denote by T the tangency portfolio formed with GE and KO that you want to hold. You know that T invests 50% in GE and 50% in KO. (a) Find the expected return and standard deviation of T. Solution: Recall: If X and Y are random variables with means µ X, µ Y, standard deviations σ X, σ Y and correlation ρ XY, then a linear transformation of X and Y will have mean and standard deviation Z = αx + βy (1) µ Z = αµ X + βµ Y (2) σ Z = α 2 σx 2 + β2 σy 2 + 2αβ cov(x, Y ) (3) where cov(x, Y ) = ρ XY σ X σ Y. In this case, the tangency portfolio is a linear transformation of assets GE and KO T =.5GE +.5KO (4) 4
5 where α =.5 and β =.5. Plugging into the above formula, µ T = =.11 = 11% and σ T = =.143 = 14.3% (b) Find the Sharpe ratio of the tangency portfolio, and show that it is higher than the Sharpe ratio for both GE and KO. Solution: The Sharpe ratio of T is SR T = E T r f σ T = while the Sharpe ratios of GE and KO are: SR GE = SR GE = = = Indeed, SR T =.42 is larger than both.33 and.37. (c) Suppose your utility function is quadratic: U(E, σ) = E 1 2 A σ2 =.42 and your risk aversion coefficient is A = 7. Find your optimal portfolio formed with the riskfree asset, GE, and KO. What percentage of your wealth will you hold in each asset? Solution: Modern Portfolio Theory says that you should only invest in a combination of the riskfree asset and the tangency portfolio T. The weight that you should put on T is given by the following formula: w T = 1 A µ T r f σ 2 T = = So you should hold 42% of your wealth in T, and 58% in the riskfree asset. But T is formed by GE and KO with equal weights, so you will hold 21% in GE, 21% in KO, and 58% in the riskfree asset. Problem 3. A portfolio consists of the following 3 stocks, whose performance depends on the economic environment. Assume the bad economic outcome is twice as likely as the good one. Investment in $(mill.) Good economy Bad economy Stock % 20% Stock 2 1,250 6% 3% Stock % 2% 5
6 (a) Calculate the expected return of the portfolio. Solution: Investment in $(mill.) Good economy Bad economy Stock % 20% Stock 2 1,250 6% 3% Stock % 2% r P 2,000.25(.13) +.625(.06) +.125(.07).25(.20) +.625(.03) +.125(.02) = = So the expected return of the portfolio is (b) Calculate the variance of the portfolio. E P = 1/3( ) + 2/3( ) = Solution: The variance of the portfolio is σ 2 P = 1/3( )2 + 2/3( ) 2 = (c) What if we add to the portfolio $1,000 million of stock 4, which has a mean return of 4%, has a variance of 0.02, and is uncorrelated with the above portfolio? How will this change the expected return and variance of the total investment? Solution: Call this new portfolio Q. The new expected return and variance are 1, 000 2, 000 E Q = (0.04) + ( ) = , 3, 000 3, 000 ( 2 ( ) σ 2 = (0.02) + (0.0018) = Q 3) 3 Problem 4. The expected return of the S&P 500 index, which turns out to be the tangency portfolio of stocks in the economy, is 16%, with a standard deviation of return of 25% per year. The expected return of Microsoft is unknown, but it has a standard deviation of 20% per year, and a covariance with the S&P 500 index of Assume that the riskfree rate is 6% per year. (a) Compute Microsoft s beta and expected return. Solution: β MSFT = σ MSFT,M = 0.10 σ 2 (0.25) = 1.60, 2 M E MSFT = ( ) =
7 (b) If Intel has half the expected return of Microsoft, then what is Intel s beta? Solution: E INTC = = 0.11, CAPM = 0.11 = β INTC ( ), β INTC = (c) What is the beta and expected return of the following portfolio? 25% in Microsoft 10% in Intel 75% in the S&P 500 index 20% in GM (where β GM = 0.80) 10% in the riskfree asset Solution: β P = = 1.04, E P = ( ) = (d) Is the following statement True, False, or It Depends? If the riskfree rate drops to 5%, but the expected return on the S&P 500 index remains the same, the new expected return on Microsoft increases by 0.6%. In one or two sentences explain or show why this is true or false or why it depends. Solution: It depends, because once the riskfree asset changes, the S&P 500 may no longer be the tangency portfolio. Therefore, we don t know what the new expected return equation looks like. Remember, the S&P 500 is simply some portfolio, which happens to be the tangency portfolio given a riskfree rate of 6%, but it does not necessarily represent the market portfolio in a CAPM context. Problem 5. Assume that the CAPM holds and you have the following information about the returns of Procter & Gamble (PG) and General Motors (GM) for the next year: Asset corr( r, r M ) σ( r) E( r) PG 20% 30%? GM 50% 40%? Market (M) 100% 20% 15% (E.g., PG has a correlation with the market of 20%, and a standard deviation of 30%.) The riskfree asset has a rate of return r f = 5%. 7
8 (a) Calculate the expected returns of PG and GM. (Hint: Calculate first the betas of PG and GM.) Solution: In general, the beta of a stock i can be computed as β i = σ i,m σ 2 M = ρ i,m σ i σ M σ 2 M = ρ i,m σ i σ M. For i both PG and GM we know ρ i,m and σ i, so we can compute their betas: So their expected returns are β PG = β GM = = 0.3 = 1. E PG = r f + β PG (E M r f ) = ( ) = 8% E GM = r f + β GM (E M r f ) = ( ) = 15%. (b) You are now considering a third stock, General Electric (GE). After some analysis, you decide that next year s returns for GE depend only on two possible states for the US economy over the next year: Good and Bad. Each state is equally likely (with probability 0.5). The returns to GE in each state are as follows: State Return to GE Good 35% Bad 15% What is the implied correlation ρ GE,M between the return on GE and the market? (Hint: Calculate the expected return and standard deviation of GE, and then use CAPM to determine GE s beta.) Solution: Compute the expected return, variance, and standard deviation of GE: E GE = ( 0.15) = 10% 2 2 σ 2 GE = 1 ( )2 + 1 ( )2 = σ GE = = 25%. Use the CAPM formula to compute the beta of GE: E GE = r f + β GE (E M r f ) = β GE = E GE r f E M r f = = 0.5. Now use the formula we derived above for β i : β i = ρ i,m σ i σ M = ρ i,m = β i σ M σ i So the correlation between GE and the market is ρ GE,M = = 0.4 = 40%. 8
9 Problem 6. Your firm is evaluating a new risky capital investment. After careful consideration, you decide there are two possible outcomes or states of the world relevant to your project. State 1 represents a mild recovery, and state 2 represents a strong recovery from the recent downturn in the market. You estimate the probabilities of these alternative future states of the world to be 0.7 and 0.3, respectively. You also estimate that the market return will be 10% in the mild recovery state and 30% in the strong recovery case. The project itself has estimated returns of zero in the mild recovery case and 40% in the strong recovery scenario. Assume that the riskfree rate is 10%. (a) What are the expected returns of the market and the project, respectively? Solution: The expected returns of the project P and the market are E P =.7(0) +.3(.40) = 0.12 E M =.7(.10) +.3(.30) = 0.16 (b) What is the covariance between your project and the market? Solution: The covariance between the project and the market is σ P,M =.7(0.12)(.10.16) +.3(.40.12)(.30.16) = (c) Under the CAPM, what is your expected return on the project? Solution: Under CAPM the expected return on P is E P compute β P, compute first market variance =.10 + β P (.16.10). To σ 2 M =.7(.10.16)2 +.3(.30.16) 2 = Then This implies β P = σ P,M σ 2 M = = 2. E P = (.16.10) = (d) Should the project be accepted? Why or Why not? (Be brief.) Solution: No. Since 12% (the actual expected return of the project) < 22% (the CAPM expected return), the project does not offer a high enough expected return to compensate for its risk. Problem 7. You are a financial advisor for high net worth individuals. Your new client Karen just told you that she is willing to invest only in the following three assets: The riskfree asset, which is a TBill with interest rate r f = 5%; A bond mutual fund X, with expected return E X = 8% and standard deviation σ X = 10%. 9
10 An stock mutual fund Y, with expected return E Y σ Y = 30%; The stock and bond indexes are assumed to be uncorrelated: ρ X,Y = 0. = 14% and standard deviation Karen s current portfolio P is invested 25% in X, 25% in Y, and 50% in the TBill. Also, your quants have computed that that the tangency portfolio T is invested 75% in X and 25% in Y. (a) What are the expected return and standard deviation of the return on your client s portfolio P? What are the expected return and standard deviation of the tangency portfolio T? Solution: The portfolio expected returns are E P = 0.25 E X E Y r f = 8% E T = 0.75 E X E Y = 9.5% The standard deviations are (recall that ρ X,Y = 0) σ P = σ T = σ 2 4 X σ 2 4 Y = = 7.91% σ 2 4 X σ 2 4 Y = = 10.61% 4 4 (b) Karen is willing to accept a standard deviation no higher than her portfolio s current standard deviation. Also, she prefers to invest only in the TBill, the bond index X, and the stock index Y. You would like to find another portfolio Q with a higher expected return that Karen s current portfolio. What is the maximum expected return you can achieve? (Hint: recall that any efficient portfolio is a combination of the riskfree asset and the tangency portfolio.) Solution: There are two ways of solving this problem: Solution 1: The standard deviation of your client s current portfolio is σ P = 7.91%. You must suggest Karen an efficient portfolio, which gives you the same standard deviation, but the highest possible expected return. This efficient portfolio, Q, must be on the CML (Capital Market Line). The equation of the CML represents the expected return E of a portfolio as a function of its standard deviation. It has the intercept equals to the riskfree rate r f, and the slope equal to the Sharpe ratio of the tangency portfolio T : E = r f + SR T σ. The Sharpe ratio of the tangency portfolio is So the equation of the CML is SR T = E T r f σ T = E = σ. 10 =
11 Since the portfolio Q that you want to recommend to Karen is on the CML and has σ Q = σ P = 7.91%, the expected return of Q is E Q = = = 8.35%. Note that this is indeed higher than the expected return achieved by Karen s current portfolio: E P = 8%. Solution 2: As discussed before, any efficient portfolio Q must be on the CML, i.e., it must be a combination of the riskfree asset and the tangency portfolio T. We denote by w the weight of Q invested in T. Then the standard deviations should match: σ Q = σ P = wσ T = σ P w 10.61% = 7.91% w = = 74.54%. Then the maximum expected return achievable at the standard deviation of 7.91% is the expected returns of the above portfolio Q: E Q = w E T + (1 w) r f = 8.35%. (c) After some investigation, you decide that Karen has a quadratic utility U(E, σ) = E 1 A 2 σ2, with risk aversion coefficient A = 8. What is her optimal portfolio P formed with X, Y and the riskfree asset? Solution: The optimal portfolio P is a portfolio formed with the riskfree asset and the tangency portfolio T. Denote by w the weight of P invested in T. Then w = 1 A E T σ 2 T r f = = 0.5. The portfolio P invests 0.5 in T and 0.5 in the riskfree asset, i.e. 37.5% in X, 12.5% in Y, and 50% in the TBill. (d) How does the Sharpe ratio of the optimal portfolio P compare with the Sharpe ratio of the portfolio Q you found in (b)? (Hint: There is a qualitative answer, you don t actually have to compute the two Sharpe ratios.) Solution: Both P and Q are on the same capital allocation line between the riskfree asset and the tangency portfolio T (also called the capital market line, CML). Therefore they have the same Sharpe ratio. Problem 8. Consider two stocks: WMT and IBM with the following properties Stock E( r) σ( r) WMT 8% 13% IBM 12% 20% 11
12 The correlation of the two stock returns is and the riskfree rate is ρ WMT,IBM = 9%, r f = 1%. You are advising a client who has $1 million invested. Currently 50% of this money is in WMT and 50% is in IBM. (a) What are the expected return and standard deviation of the return on your client s portfolio? Solution: The portfolio expected return is The standard deviation is σ P = 1 E P = 50% E WMT + 50% E IBM = 10% 2 σ σ WMT σ σ ρ = = 12.41%. IBM 2 2 WMT IBM WMT,IBM (b) You are told that the tangency portfolio formed with WMT and IBM has weights w WMT = 60% and w IBM = 40%. i. What are the expected return and standard deviation of the tangency portfolio? Solution: The expected return and standard deviation of the tangency portfolio are σ T = E T = w WMT E WMT + w IBM E IBM =.6 8% % = 9.6%,.6 2 σ σ σ σ ρ = = 11.66%. WMT IBM WMT IBM WMT,IBM ii. What is its Sharpe ratio? How does it compare with the Sharpe ratio of your client s current portfolio? Explain your finding. Solution: The Sharpe ratio is SR T = E T r f σ T = 9.6% 1% 11.66% = By definition the tangency portfolio must have the highest Sharpe ratio among all the possible portfolios formed with WMT and IBM, so SR T > SR P. We can also check this directly: = SR T > SR P = 10% 1% 12.41% =
13 (c) You wish to match the expected return of your client s current portfolio using only WMT, IBM, and the riskfree security. What is the minimum standard deviation you can achieve? Solution: We can look at the problem in two different ways: Solution 1: The expected return of your client s current portfolio is 10%. Any minimumvariance portfolio P must be a combination of the riskfree security and the tangency portfolio in (b), so denote by w the weight invested in the tangency portfolio, and by 1 w the weight invested in the riskfree asset. Then the expected returns should match: w 9.6% + (1 w) 1% = 10% w 8.6% = 9% w = = %. Then the minimum standard deviation achievable at the expected return of 10% is the standard deviation of the above portfolio P : σ P = w σ T = 12.20%. Solution 2: Since we can also invest in the riskfree asset, the minimumvariance frontier is the capital allocation line from the riskfree asset to the tangency portfolio. Therefore all efficient portfolios have the same Sharpe ratio as the tangency portfolio: Denote by P the portfolio on the CAL which has the same expected return as P (10%). The Sharpe ratio of P is = (10% 1%)/σ P, therefore σ P = = 12.20%. Notice that since E P > E T, in order to achieve the same expected return as P, you want to lever up the tangency portfolio. E P E T r f $1 M = $1.0465M r f of the risky portfolio, and to do this you also need to short $1.0465M $1M = $46, 500 of the risk free security. (d) Suppose the correlation between the two stocks increased to ρ WMT,IBM = 20%. What would happen qualitatively (i.e. don t do any calculations, just tell me the general direction) to the Sharpe Ratio of the optimal portfolio? Justify your answer. Solution: When the two assets are more correlated, the efficient frontier moves to the right. (And in the extreme case when the assets are perfectly correlated, the frontier becomes the line between the two assets.) When the efficient frontier moves to the right, the tangent line (CAL) from the riskfree asset no longer intersects the frontier. This means that the Sharpe ratio must go down. 13
14 Problem 9. (a) The price of a 7year strip is given: B 7 = 73 (face value F = 100). What is its yield to maturity? Solution: ( ) 1/7 100 YTM 7 = 1 = B 7 ( ) 1/ = 4.60% 73 (b) The yield to maturity of a 10year strip is given: YTM 10 = 7%. What is its price B 10? Solution: B 10 = F 100 = = (1 + YTM 10 ) Problem 10. You are told that the current yield curve for years 1, 2 and 3 is the following: Maturity YTM 0.5% 1% 2% The 3year Treasury note with face value $1000 and annual coupon rate 5% trades at $ (a) Is the price of the bond consistent with the given list of yields? Solution: The annual coupon is 5% $1000 = $50. So the coupon bond pays $50 in years 1 and 2, and in year 3 it pays both the coupon ($50) and the face value ($1000), so $1050 in total. The fair price of the coupon bond is given by the Present Value formula: P = $ $ $ = $ Notice that the fair price is smaller than the actual price at which the bond trades: P = $ Therefore, the coupon bond is not fairly priced. (b) If the price of the coupon bond is not consistent with the given list of yields, what would you do to in order to take advantage of this situation? Be explicit in describing your strategy. You are allowed to invest in the given coupon bond, and in 1year, 2year, and 3year strips. Solution: The coupon bond trades at a higher price (P = $1090.0) than the fair price ($1088.2), thus the coupon bond is overpriced. You should do the following: Short the expensive asset, the coupon bond, for $1090.0, and Buy the cheap asset, the replicating portfolio of strips. What quantity of each strip should you buy? * * * = year strips, = year strips, and = year strips. 14
15 What are the cash flows from this strategy? Let s first calculate the price of each individual strip. For example, the price of the 3year strip is We get the following table B 3 = 100 (1 + YTM 3 ) = = Maturity Strip price So you buy 50/100 = 0.5 of 1year strips, 50/100 = 0.5 of 2year strips, and 1050/100 = 10.5 of 3year strips (we assume that you can buy and sell fractions of a strip). The cash flow at time zero you get from selling the strips is CF 0 = 0.5 $ $ $94.23 = $ Note that this is the same as the fair price P. The cash flows from buying the strips (buying the replicating portfolio) and shorting the coupon bond are: Buy Strips Time Cash Flow Short Bond Time Cash Flow So the cash flow today is positive: $ $ = $1.80, while the cash flows in the future all add up to zero. You can just pocket the difference today and the cash flows from the long position will take care of the cash flows from the short position. You have just performed what is called an arbitrage. As we saw in other examples (e.g., the red dollar / blue dollar case), we know that this type of arbitrage is not really riskfree. Problem 11. Suppose that the term structure is currently flat so that bonds of all maturities have yields to maturity of 10%. Currently a 5year coupon bond with annual coupons (with the first one due in 1 year) and face value of $1,000 is selling at par. (a) What is the current price of the 5year bond? What are the annual coupons? Solution: The price of the 5 year bond must be $1,000. Denote the annual coupons by C, ans solve for C in the asset pricing formula. P 0 = 1000 = C C C C C
16 This is one linear equation in one unknown C. The solution is easily computed: C = 100. Notice that this means that the coupon rate C equals 10%, which is exactly the yield 1000 to maturity. The result is more general, i.e. when the coupon rate equals the yield, then the bond price is equal to par value. We already mentioned this result in class. (b) A year from now interest rates will be depend on the stance of monetary policy. If monetary policy is tight the yields to maturity on all bonds will be 12%. If monetary policy is loose the yields to maturity on all bonds will be 8%. If you sell the bond a year from now when monetary policy is tight what will be the return to your investment over the year? If you sell the bond a year from now when monetary policy is loose what will be the return to your investment over the year? Solution: The price of the bond when monetary policy is tight will be: P 1,tight = = In this case the net return will be r 1,tight = P 1,tight + C P 0 P 0 = The price of the bond when monetary policy is loose will be: = 3.92%. P 1,loose = = In this case the net return will be r 1,loose = P 1,loose + C P 0 P 0 = = 16.62%. Problem 12. We see the following yield curve for discount (or zero coupon) bonds. Maturity Yield to Maturity 1 year 6% 2 years 7% 3 years 8% If the fair price for a 4year annuity paying $100 per year is $334.57, what is the yield to maturity on a four year zero coupon bond? Solution: The fair price of a 4year annuity paying 100 per year should satisfy: P = (1 + YTM 4 ) 4. 16
17 Thus the yield to maturity on a four year zero coupon bond must satisfy: or which implies = = YTM 4 = 100 (1 + YTM 4 ) 4 ( ) 1/ = 8% (1 + YTM 4 ) 4 Problem 13. You have the following information about the prices of a 1year strip and a 2year coupon bond. The 1year strip pays a face value of $100 in one year, and currently sells for $96. The 2year coupon bond has a face value of $1000 and an annual coupon of $60. The bond currently sells for a price of $1050. (a) What are the implied yields to maturity on 1 and 2year strips? Solution: We first compute YTM 2 satisfies: 1050 = YTM 1 = = 4.17% (1 + YTM 2 ) 2 = (1 + YTM 2 ) From this, we get = = (1+YTM 2, and so (1 + YTM ) 2 2 ) 2 = We solve for YTM 2 : ( ) 1/ YTM 2 = 1 = 3.35% (b) Consider a 2year annuity with annual payments of $500. What is the fair price for this annuity? Solution: The fair price of this annuity should be: =
18 Problem 14. Many institutions have fixed future liabilities (such as pension payments) to meet and they fund these future liabilities using defaultfree fixed income securities. When discount bonds of all maturities are available, these institutions can simply buy discount bonds to fund their liabilities. For example, if there is a fixed liability equal to 1 million dollars five years from now, an institution can buy a discount bond maturing in five years with a face value of 1 million dollars. Unfortunately, there may not be the right discount bonds for a fixed future liability and coupon bonds must be used. Then an institution faces reinvestment risk of the coupons. For example, suppose that the yield curve is flat at 10% and we have the following coupon bonds (with annual coupons): Bond Principal Coupon Years to Maturity A B Also, suppose we have a 1 million liability five years from now. (a) What are the prices of the two bonds? Use 4 decimal digits. Solution: At a discount rate of 10% the price of the five year is and the price of the ten year is P 5 = = P 10 = = (b) Suppose that the yield curve will remain unchanged for the following five years and you have decided to use bond A to fund the liability. That is, you want to invest in bond A and invest the coupons at the prevailing interest rates to produce a future value at the end of year five of 1 million. How much should you invest in bond A? Use 2 decimal digits. Hint: To produce a future value of $1 million in 5 years, today you should invest the present value of $1 million (at 10% interest rate). Since in (a) you computed the price of bond A, you can find out how many bonds of type A you should buy today. Solution: At a discount rate of 10% the present value of $1,000,000 in 5 years is $620, To have $1,000,000 in 5 years you need to invest $620, in bond A. This means that you will have to purchase 620,921.32/ = 5, bonds (note that in these calculations I am carrying many decimal digits). The cash flows created by this will be: 18
19 Year Cash Flow 1 78, , , , , (c) Now suppose that right after you invested in bond A, the yield curve makes a parallel move down by 1% to 9%. What is the future value five years from now of your investment? What is the future value if the yield curve moves up by 1% to 11%? Please explain why the future value changes differently depending on the direction of the change in the yield curve. Solution: At an interest rate of r the future value of the above cash flows in 5 years will be: FV = 78, (1 + r) , (1 + r) + 600, The future value at various interest rates will be: r Future Value 9% 990, % 1,000, % 1,009, (d) Part (c) shows that the future value of your investment is sensitive to interest rate fluctuations and you face the risk that your future liabilities may not be met. You should try to immunize this interest rate risk. But how? Do the following: i. Compute the duration of bonds A and B. Use 2 decimal digits. Solution: We can use the formula from class. Recall that the price of A is P A = The duration of bond A is: D A = 15/1.10 P A /1.102 P A /1.105 P A 5 = 3.95 Similarly the duration of bond B is D B = ii. You want to construct a portfolio of the two coupon bonds A and B so that the future value of the portfolio is 1 million and the duration of this portfolio is equal to 5 years, assuming that the yield curve will remain flat at 10%. Find the percentage of your wealth that you want to invest in A and B. Use 2 decimal digits (e.g., 37.54%). Solution: Let x be the proportion invested in bond A and (1 x) be the amount invested in bond B, then we need the duration of the portfolio to equal 5: x (1 x) 6.28 = 5. 19
20 Solve this equation: x = (5 6.28)/( ) = , or x = 54.94%. Hence the amount invested in bond A is $620, = $341, and the amount invested in bond B is $620, = $279, You should buy 341,134.17/ = 2, units of bond A and 279,787.15/ = 2, units of bond B. iii. Show that if immediately after you purchased this portfolio the yield curve makes a permanent parallel downward or upward move of 1%, the future value of this portfolio at the end of year 5 will still be approximately 1 million. You have immunized the portfolio of the risk associated with parallel movements of the yield curve by buying a portfolio of coupon bonds so that the duration of the portfolio matches the number of years to the payment of the fixed liability. Solution: We have the following cash flows from the portfolio: Year Cash from A Cash from B Total Cash 1 43, , , , , , , , , , , , , , , , , , , , , , , , , At the end of year 5 the accumulated value of these cash flows will be (including accumulated investment and sale of 10 year bond): Interest Rate Value 9% 1,000,339 10% 1,000,000 11% 1,000,291 Problem 15. (a) On March 3, 2008 the closing spot price of the S&P 500 index was $ , while the closing futures price for the June contract was $ Compute the dollar value of the stocks traded on one contract on the S&P 500 index (recall that the actual value comes with a multiplier of 250). If the margin requirement is 10%, how much must you deposit with your broker to trade the June contract? Solution: The closing spot price was $ The dollar value of stocks traded is: 250 $ = $332, 835 The closing futures price for the June contract was $ , which has a dollar value of: 250 $ = $333, 500 Therefore, the required margin deposit is 10% $333, 500 = $33,
21 (b) If the June futures price were to increase to $1340, by what amount would your margin account change if you entered the long side of the contract at the price given in (a)? Solution: The futures price increases by: $1340 $1334 = $6. The credit to your margin account would be: $6 $250 = $1, 500. (c) What percentage return would you earn on your net investment? Solution: The gain in the margin account is a percent gain of: $1, 500/$33, 350 = = 4.5%. Note that the futures price itself increased by only 0.45%. (d) If the June futures price falls by 1%, what is your percentage return? Solution: Following the reasoning in part (c), any change in the futures price is magnified by a ratio of (1/margin requirement). This is the leverage effect. The return will be 10%. Problem 16. Suppose the 1year futures price on the S&P500 is below the current value of the index multiplied by (1 + r), where r is the one year Tbill rate. Then investors expect the index to decline. True or false? Solution: False. The index has also benefits (dividends), which do not accrue to the futures contract owner. Therefore the futures price is F 0 = S 0 (1 + r d) (spotfutures parity with costs and benefits). Then F 0 will always be below S 0 (1+r), provided that the dividend yield is positive (which it usually is). This has nothing to do with what investors expect the future value of the index to be. Problem 17. Fill in the missing numbers in this table of stock index futures prices. Country Index Index Dividend Spot interest Futures Futures Price Name Value Yield rate Term (same unit as (annualized) (annualized) (months) index value) Japan Nikkei % 0.57 % 3??? US S&P ??? 1.77 % Solution: The formula for the futures price on the Nikkei 225 index is: F Nik 0 = S 0 (1 + r d) 3/12 = ( ) 3/12 =
22 Similarly, for the S&P 500, = F SP 0 = S 0 (1 + r d) 9/12 = ( d) 9/12. Therefore the (annualized) dividend yield on the S&P 500 is d = ( ) 12/ = = 1.38% Problem 18. Suppose it is now the end of December. The current yield curve is flat at 3%. The June futures price for gold is $988.80, whereas the next December futures price is $ Is there an arbitrage opportunity here? How would you exploit it? Solution: In what follows we will assume no costs of carry, and no convenience yield. Suppose the spot price of gold is S 0 (we don t need to know what it is). According to the parity relation, the proper prices for the June and December futures are: F June = S 0 (1 + r f ) 1/2 F Dec = S 0 (1 + r f ). If we divide the second equation by the first, we obtain F Dec = F June (1 + r f ) 1/2 = $ /2 = $ The actual futures price for December is too high relative to the June price. You should short the December contract and take a long position in the June contract. This would imply cash flows as in the following table Position t = 0 June December Long June contract $ receive gold Short December contract +$ give away gold Notice that the cash flows: $ after 6 months, and +$ after 12 months represent a long position in a forward loan. To cover this loan, one can buy a 6month strip and short a 12month strip: 22
23 Position t = 0 June December Long June contract receive gold Short December contract Buy 6month strip (F = ) /2 = deliver gold Short 12month strip (F = ) = Net You make $5.42 arbitrage profit every time you make this trade. Problem 19. Suppose oneyear interest rates are 3% in the US and 5.5% in the UK, and that the pound is currently worth $2.00. (a) What should the 12month forward exchange rate for the pound be to rule out arbitrage opportunities? Solution: By covered interest rate parity: 0 = E $/ 1 + r $ 0, 1 + r F $/ where E $/ 0 is the spot exchange rate from dollars into pound, and F $/ 0 is the pound futures price. Therefore F $/ 0 = = $1.95/. (b) Suppose you read in the paper that the 12month forward is actually $1.90/. Explain in detail how to exploit the mispricing using only the following instruments: buying or selling US or British 12month bills, and buying or selling the pound forward. Ignore transaction costs. If you could only borrow the equivalent of $100,000, how much money would you make? Solution: Since the actual forward price $1.90/ is less than the correct price $1.95/, you should buy the forward and short the pound. The strategy is as follows: Borrow in London Convert immediately to $$, lend in the US Enter into a futures contract to buy (1 + r ) in one year 23
24 Position Today 12 Months Borrow pounds in London + 50, , = 52, 750 Exchange pounds to dollars 50, 000 in the spot market +$100, 000 Invest in 12 month Tbills $100, 000 +$103, 000 Enter long forward contract + 52, , = $100, 225 Net 0 $2, 775 The only limit is the total amount you could borrow in London to set up the arbitrage (presumably you d want to borrow an infinite amount if you could). Problem 20. (a) The following data from the Wall Street Journal give you the spot and forward exchange rates for the Yen on March 8, 1995: Japan Yen spot... $ day forward... $ day forward... $ day forward... $ March 8, 1995 (Reuter)  Treasury Bill rates are 5.56% for onemonth, 5.81% for three months, 6.04% for six months. Copyright 1995, Reuters News Service Construct the zerocoupon term structure for Japanese spot interest rates. Solution: Recall covered interest rate parity: F $/ 0,T = E$/ r $,T 1 + r,t where E $/ 0 is the spot exchange rate from dollars into Yen, and F $/ 0,T is the price of the Yen future with term T. For example (interest rates are all annualized): ( ) 30/365 F $/ 1 + 0,30 days = r$,30 E$/ r,30 ( ) 30/ = r,30 ( ) 365/ r,30 = ( ) 1 = 1.159%
25 Similarly, ( ) 365/ r,90 = ( ) 1 = 1.377% ( ) 365/ r,180 = ( ) 1 = 1.367% (b) What would happen if the Japanese Central Bank raised the spot interest rates? Would that increase the forward exchange rate? Solution: No. To increase the forward exchange rate, the opposite should happen: Japanese rates go down and/or US rates go up. Problem 21. Suppose that you feel that market volatility is going to rise because of increased uncertainty due to the Federal Reserve s actions. You do not have a prediction, however, about where the market will go. Can you devise an investment strategy using options to take a bet on your feelings? Describe the strategy. Suppose instead that you think market volatility is going to fall, what strategy would you follow in this case? Can you devise this strategy in a way that you are not exposed to significant downside risk? Solution: If you feel that the market s expectation of volatility is too low you could bet on this using a straddle where you simultaneously buy a put and a call with the same strike price. Suppose, for example, that the S&P500 is currently at 1,400 and call and puts are available with strike prices of 1,400. Then the payoff and profit from these would be 200 Payoff and Profit from Straddle 150 C+P 100 Payoff and Profit Payoff Profit Stock Price Notice that you only make a profit if the index moves far enough from 1,400. If you believe that volatility is going to fall compared to the market s expectations you could write a straddle, but this exposes you to large potential losses. In addition to writing the straddle (a call and a put) you could buy a call with a strike price of 1500 (for example) and buy a put with a strike price of 1300 (for example). 25
26 100 Payoff Profit 50 Payoff and Profit Stock Price Problem 22. Consider the following option strategy: Long one call with $100 strike price, bought for $6 Long one call with $90 strike price, bought for $20 Short one call with $105 strike price, sold for $8 Short one call with $95 strike price, sold for $16 (a) Draw a picture of the payoff of the option strategy at expiration as a function of the stock price. Solution: The payoff of this option strategy at maturity is Range of stock price S T < S T < S T < S T < S T S T at expiration Payoff of long 0 S T 90 S T 90 S T 90 S T 90 Call with K = 90 Payoff of long S T 100 S T 100 Call with K = 100 Payoff of short 0 0 S T + 95 S T + 95 S T + 95 Call with K = 95 Payoff of short S T Call with K = 105 Total payoff of 0 S T 90 5 S T option strategy The payoff to this position at expiration looks like 26
27 12 Option Payoff Payoff Stock Price (b) Draw a picture of the investor s profit as a function of the stock price. Solution: The cost of this position is: = 2. Hence the investor s profit as a function of the stock price shifts down by 2. It looks like 12 Option Profit Profit Stock Price Problem 23. Stock ABC trades today at $100, and in the past 6 months its volatility was σ = 30% (annualized). The 3month Tbill rate is r = 2.75% (annualized). Consider the following table of European option prices which all expire in 90 days: Strike Call Price Put Price
28 (a) What is the price of a straddle with strike price of 100? Draw the graph of the profit of the straddle. Solution: A straddle is created by buying an ATM (atthemoney) call and an ATM put on the stock. The price therefore is $ $5.59 = $ The graph is (b) You think that the volatility of ABC will increase, but you are not completely sure and feel that buying a straddle would be too aggressive (and too expensive). Can you create an option that would do what you want? Solution: You essentially want to flatten the tails of the straddle distribution. To this end, you can sell an OTM call with strike K 1 = 110 and an OTM put with strike K 2 = 90. From the table, the price of the new option strategy is $11.85 $2.65 $1.84 = $7.36. The graph is Looks familiar? It is the profit of a short position in a butterfly option 28
29 And it makes sense, because a long position in a butterfly is a bet that volatility will decrease. Therefore a short position is a bet that volatility will increase, but it is a more moderate bet than a straddle We can also do this more formally, and write the payoff and profit tables: Range of stock price S T < S T < S T < S T S T at expiration Payoff of long 0 0 S T 100 S T 100 Call with K = 100 Payoff of long 100 S T 100 S T 0 0 Put with K = 100 Payoff of short (S T 110) Call with K = 110 Payoff of short (90 S T ) Put with K = 90 Total payoff of S T S T option strategy Range of stock price S T < S T < S T < S T S T at expiration Profit of long S T S T Call with K = 100 Profit of long S T S T Put with K = 100 Profit of short S T Call with K = 110 Profit of short S T Put with K = 90 Total profit of S T S T option strategy Problem 24. Today it is November 29, 2010, and you are contemplating the following table of April 2011 options. 29
30 Calls Strike Puts Last Chg Bid Ask Vol Open Int. Price Last Chg Bid Ask Vol Open Int , , , , , , , , , , ,649 LAST TRADE as of 29Nov :03 AM , , , , , We are told that the annualized interest riskfree rate over the life of the options is r = 0.185% per year. (a) On what date will all these options expire? Solution: On the third Friday of April 2011, which falls on April 15, (b) How much would you get if you exercised the K = 130 call today? (We are talking about the call option with strike price K = 130.) How much would you get if you sold the K = 130 call today at the market (bid) price? Recall that today is November 29, Solution: I would get the difference between today s stock price, S t = , and the strike price, K = 130, that is S t K = = If I sold the call option today, I would get (c) Should you exercise today the K = 130 call? Why or why not? Is there any reason why you might want to exercise the call option before it expires in April? Solution: I would not want to exercise the call option today, because I would get only for it, while if I sold it at the market price, I would get more for it: In general, you should not exercise a call option before it expires: in general you should get more if you sold it. Another way of thinking about that is that by exercising the 30
SAMPLE MIDTERM QUESTIONS
SAMPLE MIDTERM QUESTIONS William L. Silber HOW TO PREPARE FOR THE MID TERM: 1. Study in a group 2. Review the concept questions in the Before and After book 3. When you review the questions listed below,
More informationCHAPTER 22: FUTURES MARKETS
CHAPTER 22: FUTURES MARKETS PROBLEM SETS 1. There is little hedging or speculative demand for cement futures, since cement prices are fairly stable and predictable. The trading activity necessary to support
More informationReview for Exam 2. Instructions: Please read carefully
Review for Exam 2 Instructions: Please read carefully The exam will have 25 multiple choice questions and 5 work problems You are not responsible for any topics that are not covered in the lecture note
More information2. How is a fund manager motivated to behave with this type of renumeration package?
MØA 155 PROBLEM SET: Options Exercise 1. Arbitrage [2] In the discussions of some of the models in this course, we relied on the following type of argument: If two investment strategies have the same payoff
More informationCHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT
CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT PROBLEM SETS 1. In formulating a hedge position, a stock s beta and a bond s duration are used similarly to determine the expected percentage gain or loss
More informationChapter 2 An Introduction to Forwards and Options
Chapter 2 An Introduction to Forwards and Options Question 2.1. The payoff diagram of the stock is just a graph of the stock price as a function of the stock price: In order to obtain the profit diagram
More informationFinance 350: Problem Set 6 Alternative Solutions
Finance 350: Problem Set 6 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas
More information11 Option. Payoffs and Option Strategies. Answers to Questions and Problems
11 Option Payoffs and Option Strategies Answers to Questions and Problems 1. Consider a call option with an exercise price of $80 and a cost of $5. Graph the profits and losses at expiration for various
More informationCHAPTER 22: FUTURES MARKETS
CHAPTER 22: FUTURES MARKETS 1. a. The closing price for the spot index was 1329.78. The dollar value of stocks is thus $250 1329.78 = $332,445. The closing futures price for the March contract was 1364.00,
More informationCHAPTER 20: OPTIONS MARKETS: INTRODUCTION
CHAPTER 20: OPTIONS MARKETS: INTRODUCTION PROBLEM SETS 1. Options provide numerous opportunities to modify the risk profile of a portfolio. The simplest example of an option strategy that increases risk
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008. Options
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes describe the payoffs to European and American put and call options the socalled plain vanilla options. We consider the payoffs to these
More informationFIN 3710. Final (Practice) Exam 05/23/06
FIN 3710 Investment Analysis Spring 2006 Zicklin School of Business Baruch College Professor Rui Yao FIN 3710 Final (Practice) Exam 05/23/06 NAME: (Please print your name here) PLEDGE: (Sign your name
More informationOptions Pricing. This is sometimes referred to as the intrinsic value of the option.
Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the PutCall Parity Relationship. I. Preliminary Material Recall the payoff
More informationFactors Affecting Option Prices
Factors Affecting Option Prices 1. The current stock price S 0. 2. The option strike price K. 3. The time to expiration T. 4. The volatility of the stock price σ. 5. The riskfree interest rate r. 6. The
More informationCHAPTER 20. Financial Options. Chapter Synopsis
CHAPTER 20 Financial Options Chapter Synopsis 20.1 Option Basics A financial option gives its owner the right, but not the obligation, to buy or sell a financial asset at a fixed price on or until a specified
More informationt = 1 2 3 1. Calculate the implied interest rates and graph the term structure of interest rates. t = 1 2 3 X t = 100 100 100 t = 1 2 3
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 = 1 2 3 P t = 0.95 0.9 0.85 1. Calculate
More informationFutures Price d,f $ 0.65 = (1.05) (1.04)
24 e. Currency Futures In a currency futures contract, you enter into a contract to buy a foreign currency at a price fixed today. To see how spot and futures currency prices are related, note that holding
More informationLecture 15: Final Topics on CAPM
Lecture 15: Final Topics on CAPM Final topics on estimating and using beta: the market risk premium putting it all together Final topics on CAPM: Examples of firm and market risk Shorting Stocks and other
More informationPractice Set #4 and Solutions.
FIN469 Investments Analysis Professor Michel A. Robe Practice Set #4 and Solutions. What to do with this practice set? To help students prepare for the assignment and the exams, practice sets with solutions
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
ECON 4110: Sample Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Economists define risk as A) the difference between the return on common
More informationDerivative Users Traders of derivatives can be categorized as hedgers, speculators, or arbitrageurs.
OPTIONS THEORY Introduction The Financial Manager must be knowledgeable about derivatives in order to manage the price risk inherent in financial transactions. Price risk refers to the possibility of loss
More informationChapter 1  Introduction
Chapter 1  Introduction Derivative securities Futures contracts Forward contracts Futures and forward markets Comparison of futures and forward contracts Options contracts Options markets Comparison of
More informationFIN 432 Investment Analysis and Management Review Notes for Midterm Exam
FIN 432 Investment Analysis and Management Review Notes for Midterm Exam Chapter 1 1. Investment vs. investments 2. Real assets vs. financial assets 3. Investment process Investment policy, asset allocation,
More informationCHAPTER 7: OPTIMAL RISKY PORTFOLIOS
CHAPTER 7: OPTIMAL RIKY PORTFOLIO PROLEM ET 1. (a) and (e).. (a) and (c). After real estate is added to the portfolio, there are four asset classes in the portfolio: stocks, bonds, cash and real estate.
More informationTPPE17 Corporate Finance 1(5) SOLUTIONS REEXAMS 2014 II + III
TPPE17 Corporate Finance 1(5) SOLUTIONS REEXAMS 2014 II III Instructions 1. Only one problem should be treated on each sheet of paper and only one side of the sheet should be used. 2. The solutions folder
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the
More informationIntroduction, Forwards and Futures
Introduction, Forwards and Futures Liuren Wu Zicklin School of Business, Baruch College Fall, 2007 (Hull chapters: 1,2,3,5) Liuren Wu Introduction, Forwards & Futures Option Pricing, Fall, 2007 1 / 35
More informationUse the table for the questions 18 and 19 below.
Use the table for the questions 18 and 19 below. The following table summarizes prices of various defaultfree zerocoupon bonds (expressed as a percentage of face value): Maturity (years) 1 3 4 5 Price
More informationAFM 472. Midterm Examination. Monday Oct. 24, 2011. A. Huang
AFM 472 Midterm Examination Monday Oct. 24, 2011 A. Huang Name: Answer Key Student Number: Section (circle one): 10:00am 1:00pm 2:30pm Instructions: 1. Answer all questions in the space provided. If space
More informationFinal Exam Practice Set and Solutions
FIN469 Investments Analysis Professor Michel A. Robe Final Exam Practice Set and Solutions What to do with this practice set? To help students prepare for the final exam, three practice sets with solutions
More informationChapter 10 Forwards and Futures
Chapter 10 Forwards and Futures Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of riskadjusted discount rate. Part D Introduction to derivatives.
More informationM.I.T. Spring 1999 Sloan School of Management 15.415. First Half Summary
M.I.T. Spring 1999 Sloan School of Management 15.415 First Half Summary Present Values Basic Idea: We should discount future cash flows. The appropriate discount rate is the opportunity cost of capital.
More informationLecture 6: Arbitrage Pricing Theory
Lecture 6: Arbitrage Pricing Theory Investments FIN460Papanikolaou APT 1/ 48 Overview 1. Introduction 2. MultiFactor Models 3. The Arbitrage Pricing Theory FIN460Papanikolaou APT 2/ 48 Introduction
More informationReview for Exam 1. Instructions: Please read carefully
Review for Exam 1 Instructions: Please read carefully The exam will have 21 multiple choice questions and 5 work problems. Questions in the multiple choice section will be either concept or calculation
More informationSession IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics
Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock
More information1 Capital Asset Pricing Model (CAPM)
Copyright c 2005 by Karl Sigman 1 Capital Asset Pricing Model (CAPM) We now assume an idealized framework for an open market place, where all the risky assets refer to (say) all the tradeable stocks available
More informationwww.optionseducation.org OIC Options on ETFs
www.optionseducation.org Options on ETFs 1 The Options Industry Council For the sake of simplicity, the examples that follow do not take into consideration commissions and other transaction fees, tax considerations,
More informationLecture 12. Options Strategies
Lecture 12. Options Strategies Introduction to Options Strategies Options, Futures, Derivatives 10/15/07 back to start 1 Solutions Problem 6:23: Assume that a bank can borrow or lend money at the same
More informationCHAPTER 21: OPTION VALUATION
CHAPTER 21: OPTION VALUATION 1. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) S 0 + PV(Dividends). Given
More informationCHAPTER 22 Options and Corporate Finance
CHAPTER 22 Options and Corporate Finance Multiple Choice Questions: I. DEFINITIONS OPTIONS a 1. A financial contract that gives its owner the right, but not the obligation, to buy or sell a specified asset
More informationChapter 20 Understanding Options
Chapter 20 Understanding Options Multiple Choice Questions 1. Firms regularly use the following to reduce risk: (I) Currency options (II) Interestrate options (III) Commodity options D) I, II, and III
More informationFigure S9.1 Profit from long position in Problem 9.9
Problem 9.9 Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances
More informationTrading the Yield Curve. Copyright 19992006 Investment Analytics
Trading the Yield Curve Copyright 19992006 Investment Analytics 1 Trading the Yield Curve Repos Riding the Curve Yield Spread Trades Coupon Rolls Yield Curve Steepeners & Flatteners Butterfly Trading
More informationChapter 3 Fixed Income Securities
Chapter 3 Fixed Income Securities Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Fixedincome securities. Stocks. Real assets (capital budgeting). Part C Determination
More informationCHAPTER 10 RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM)
CHAPTER 10 RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM) Answers to Concepts Review and Critical Thinking Questions 1. Some of the risk in holding any asset is unique to the asset in question.
More informationDetermination of Forward and Futures Prices. Chapter 5
Determination of Forward and Futures Prices Chapter 5 Fundamentals of Futures and Options Markets, 8th Ed, Ch 5, Copyright John C. Hull 2013 1 Consumption vs Investment Assets Investment assets are assets
More informationIntroduction to Options. Derivatives
Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived
More informationCHAPTER 11 INTRODUCTION TO SECURITY VALUATION TRUE/FALSE QUESTIONS
1 CHAPTER 11 INTRODUCTION TO SECURITY VALUATION TRUE/FALSE QUESTIONS (f) 1 The three step valuation process consists of 1) analysis of alternative economies and markets, 2) analysis of alternative industries
More informationCHAPTER 21: OPTION VALUATION
CHAPTER 21: OPTION VALUATION PROBLEM SETS 1. The value of a put option also increases with the volatility of the stock. We see this from the putcall parity theorem as follows: P = C S + PV(X) + PV(Dividends)
More informationTrading Strategies Involving Options. Chapter 11
Trading Strategies Involving Options Chapter 11 1 Strategies to be Considered A riskfree bond and an option to create a principalprotected note A stock and an option Two or more options of the same type
More informationTest 4 Created: 3:05:28 PM CDT 1. The buyer of a call option has the choice to exercise, but the writer of the call option has: A.
Test 4 Created: 3:05:28 PM CDT 1. The buyer of a call option has the choice to exercise, but the writer of the call option has: A. The choice to offset with a put option B. The obligation to deliver the
More informationAnswers to Review Questions
Answers to Review Questions 1. The real rate of interest is the rate that creates an equilibrium between the supply of savings and demand for investment funds. The nominal rate of interest is the actual
More informationReview for Exam 2. Instructions: Please read carefully
Review for Exam Instructions: Please read carefully The exam will have 1 multiple choice questions and 5 work problems. Questions in the multiple choice section will be either concept or calculation questions.
More informationBUSM 411: Derivatives and Fixed Income
BUSM 411: Derivatives and Fixed Income 2. Forwards, Options, and Hedging This lecture covers the basic derivatives contracts: forwards (and futures), and call and put options. These basic contracts are
More informationLecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the
More information1. a. (iv) b. (ii) [6.75/(1.34) = 10.2] c. (i) Writing a call entails unlimited potential losses as the stock price rises.
1. Solutions to PS 1: 1. a. (iv) b. (ii) [6.75/(1.34) = 10.2] c. (i) Writing a call entails unlimited potential losses as the stock price rises. 7. The bill has a maturity of onehalf year, and an annualized
More informationHow credit analysts view and use the financial statements
How credit analysts view and use the financial statements Introduction Traditionally it is viewed that equity investment is high risk and bond investment low risk. Bondholders look at companies for creditworthiness,
More informationMidTerm Exam Practice Set and Solutions.
FIN469 Investments Analysis Professor Michel A. Robe MidTerm Exam Practice Set and Solutions. What to do with this practice set? To help students prepare for the midterm exam, two practice sets with
More informationChapter 5 Financial Forwards and Futures
Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment
More informationChapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.
Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of riskadjusted discount rate. Part D Introduction to derivatives. Forwards
More informationINTRODUCTION TO OPTIONS MARKETS QUESTIONS
INTRODUCTION TO OPTIONS MARKETS QUESTIONS 1. What is the difference between a put option and a call option? 2. What is the difference between an American option and a European option? 3. Why does an option
More informationGeneral Forex Glossary
General Forex Glossary A ADR American Depository Receipt Arbitrage The simultaneous buying and selling of a security at two different prices in two different markets, with the aim of creating profits without
More informationMutual Fund Investing Exam Study Guide
Mutual Fund Investing Exam Study Guide This document contains the questions that will be included in the final exam, in the order that they will be asked. When you have studied the course materials, reviewed
More informationBond valuation. Present value of a bond = present value of interest payments + present value of maturity value
Bond valuation A reading prepared by Pamela Peterson Drake O U T L I N E 1. Valuation of longterm debt securities 2. Issues 3. Summary 1. Valuation of longterm debt securities Debt securities are obligations
More informationECMC49F Options Practice Questions Suggested Solution Date: Nov 14, 2005
ECMC49F Options Practice Questions Suggested Solution Date: Nov 14, 2005 Options: General [1] Define the following terms associated with options: a. Option An option is a contract which gives the holder
More informationChapter 3: Commodity Forwards and Futures
Chapter 3: Commodity Forwards and Futures In the previous chapter we study financial forward and futures contracts and we concluded that are all alike. Each commodity forward, however, has some unique
More informationNotes for Lecture 2 (February 7)
CONTINUOUS COMPOUNDING Invest $1 for one year at interest rate r. Annual compounding: you get $(1+r). Semiannual compounding: you get $(1 + (r/2)) 2. Continuous compounding: you get $e r. Invest $1 for
More informationInstructor s Manual Chapter 12 Page 144
Chapter 12 1. Suppose that your 58yearold father works for the Ruffy Stuffed Toy Company and has contributed regularly to his companymatched savings plan for the past 15 years. Ruffy contributes $0.50
More informationCHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS
CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS PROBLEM SETS 1. (e). (b) A higher borrowing is a consequence of the risk of the borrowers default. In perfect markets with no additional
More informationChapter 21 Valuing Options
Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher
More informationA) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2%
1 Exam FM Questions Practice Exam 1 1. Consider the following yield curve: Year Spot Rate 1 5.5% 2 5.0% 3 5.0% 4 4.5% 5 4.0% Find the four year forward rate. A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2% 2.
More informationChapter 5 Option Strategies
Chapter 5 Option Strategies Chapter 4 was concerned with the basic terminology and properties of options. This chapter discusses categorizing and analyzing investment positions constructed by meshing puts
More informationOptions: Valuation and (No) Arbitrage
Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial
More informationChapter 15  Options Markets
Chapter 15  Options Markets Option contract Option trading Values of options at expiration Options vs. stock investments Option strategies Optionlike securities Option contract Options are rights to
More informationDetermination of Forward and Futures Prices
Determination of Forward and Futures Prices Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012 Short selling A popular trading (arbitrage) strategy is the shortselling or
More informationInternational Money and Banking: 12. The Term Structure of Interest Rates
International Money and Banking: 12. The Term Structure of Interest Rates Karl Whelan School of Economics, UCD Spring 2015 Karl Whelan (UCD) Term Structure of Interest Rates Spring 2015 1 / 35 Beyond Interbank
More informationFinal Exam Spring 2003
15.433 Investments Final Exam Spring 2003 Name: Result: Total: 40 points: 40 Instructions: This test has 35 questions. You can use a calculator and a cheat sheet. Each question may have multiple parts
More informationChapter. Bond Prices and Yields. McGrawHill/Irwin. Copyright 2008 by The McGrawHill Companies, Inc. All rights reserved.
Chapter Bond Prices and Yields McGrawHill/Irwin Copyright 2008 by The McGrawHill Companies, Inc. All rights reserved. Bond Prices and Yields Our goal in this chapter is to understand the relationship
More informationGlobal Financial Management
Global Financial Management Bond Valuation Copyright 999 by Alon Brav, Campbell R. Harvey, Stephen Gray and Ernst Maug. All rights reserved. No part of this lecture may be reproduced without the permission
More informationCall and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options
Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder
More informationOptions. Moty Katzman. September 19, 2014
Options Moty Katzman September 19, 2014 What are options? Options are contracts conferring certain rights regarding the buying or selling of assets. A European call option gives the owner the right to
More informationForwards, Swaps and Futures
IEOR E4706: Financial Engineering: DiscreteTime Models c 2010 by Martin Haugh Forwards, Swaps and Futures These notes 1 introduce forwards, swaps and futures, and the basic mechanics of their associated
More informationFina4500 Spring 2015 Extra Practice Problems Instructions
Extra Practice Problems Instructions: The problems are similar to the ones on your previous problem sets. All interest rates and rates of inflation given in the problems are annualized (i.e., stated as
More information9 Basics of options, including trading strategies
ECG590I Asset Pricing. Lecture 9: Basics of options, including trading strategies 1 9 Basics of options, including trading strategies Option: The option of buying (call) or selling (put) an asset. European
More informationForward Contracts and Forward Rates
Forward Contracts and Forward Rates Outline and Readings Outline Forward Contracts Forward Prices Forward Rates Information in Forward Rates Reading Veronesi, Chapters 5 and 7 Tuckman, Chapters 2 and 16
More informationInterest Rate Options
Interest Rate Options A discussion of how investors can help control interest rate exposure and make the most of the interest rate market. The Chicago Board Options Exchange (CBOE) is the world s largest
More informationFinance 436 Futures and Options Review Notes for Final Exam. Chapter 9
Finance 436 Futures and Options Review Notes for Final Exam Chapter 9 1. Options: call options vs. put options, American options vs. European options 2. Characteristics: option premium, option type, underlying
More informationLOS 56.a: Explain steps in the bond valuation process.
The following is a review of the Analysis of Fixed Income Investments principles designed to address the learning outcome statements set forth by CFA Institute. This topic is also covered in: Introduction
More informationIntroduction to Futures Contracts
Introduction to Futures Contracts September 2010 PREPARED BY Eric Przybylinski Research Analyst Gregory J. Leonberger, FSA Director of Research Abstract Futures contracts are widely utilized throughout
More informationI. Readings and Suggested Practice Problems. II. Risks Associated with DefaultFree Bonds
Prof. Alex Shapiro Lecture Notes 13 Bond Portfolio Management I. Readings and Suggested Practice Problems II. Risks Associated with DefaultFree Bonds III. Duration: Details and Examples IV. Immunization
More informationPractice set #4 and solutions
FIN465 Derivatives (3 credits) Professor Michel Robe Practice set #4 and solutions To help students with the material, seven practice sets with solutions will be handed out. They will not be graded: the
More informationMidTerm Spring 2003
MidTerm Spring 2003 1. (1 point) You want to purchase XYZ stock at $60 from your broker using as little of your own money as possible. If initial margin is 50% and you have $3000 to invest, how many shares
More informationAnswers to Concepts in Review
Answers to Concepts in Review 1. Puts and calls are negotiable options issued in bearer form that allow the holder to sell (put) or buy (call) a stipulated amount of a specific security/financial asset,
More informationPRESENT DISCOUNTED VALUE
THE BOND MARKET Bond a fixed (nominal) income asset which has a: face value (stated value of the bond)  coupon interest rate (stated interest rate)  maturity date (length of time for fixed income payments)
More informationCapital Allocation Between The Risky And The Risk Free Asset. Chapter 7
Capital Allocation Between The Risky And The Risk Free Asset Chapter 7 Investment Decisions capital allocation decision = choice of proportion to be invested in riskfree versus risky assets asset allocation
More informationExercise 6 Find the annual interest rate if the amount after 6 years is 3 times bigger than the initial investment (3 cases).
Exercise 1 At what rate of simple interest will $500 accumulate to $615 in 2.5 years? In how many years will $500 accumulate to $630 at 7.8% simple interest? (9,2%,3 1 3 years) Exercise 2 It is known that
More informationC(t) (1 + y) 4. t=1. For the 4 year bond considered above, assume that the price today is 900$. The yield to maturity will then be the y that solves
Economics 7344, Spring 2013 Bent E. Sørensen INTEREST RATE THEORY We will cover fixed income securities. The major categories of longterm fixed income securities are federal government bonds, corporate
More informationBASKET A collection of securities. The underlying securities within an ETF are often collectively referred to as a basket
Glossary: The ETF Portfolio Challenge Glossary is designed to help familiarize our participants with concepts and terminology closely associated with Exchange Traded Products. For more educational offerings,
More informationCHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES 1. Expectations hypothesis. The yields on longterm bonds are geometric averages of present and expected future short rates. An upward sloping curve is
More informationBasic Financial Tools: A Review. 3 n 1 n. PV FV 1 FV 2 FV 3 FV n 1 FV n 1 (1 i)
Chapter 28 Basic Financial Tools: A Review The building blocks of finance include the time value of money, risk and its relationship with rates of return, and stock and bond valuation models. These topics
More information