Download HOMEWORK 3 SOLUTION Chapter 8 1. Assume that your company

HOMEWORK 3 SOLUTION Chapter 8 1. Assume that your company exports to Japan and earns yen revenues, thus forecasts of the Yen/$ rate are important. Suppose two forecasters issue their predictions for the Yen/$ exchange rate. The current spot rate is Yen 90/$. Forecaster A predicts a rate of 98 next month and forecaster B predicts a rate of 88. The forward rate is at 89, reflecting the interest rate differential between the two currencies. One month later, the spot rate reaches 92 Yen/$. a. Which of the forecasters ‐‐ A, B or the forward rate ‐‐ made the most accurate forecast? b. Which one is the most useful? SOLUTIONS: a. Three forecasters (SA, SB, and SForward) forecast 98, 88 and 89 Yen/$ respectively. The future spot rate St+1 = 92 Yen/$. The forward rate was the most accurate with an error of 3/92 or 3.26%. b. Forecaster A had the most useful forecast since he correctly predicted that the yen would fall against the dollar. Forecaster B would have advised not to hedge and the firm would have incurred a foreign exchange loss on its Japanese sales. Even though the forecast is far off, Forecaster A gave the correct advice leading to a correct a hedging decision by the firm. Forward sale of Yen at 89 would have saved the firm some foreign exchange losses 2. You are the treasurer of a large multinational company. Suppose that you receive every month exchange rate forecasts for the Yen/$ exchange rate from five different forecasting institutions: A, B, C, D, and E. From your past experience, the best forecaster has been firm C, followed by A, D, B, and E. The historic RMSE (standard deviation of forecasting errors) for the five forecasters and the current predictions are shown in the following table: Forecaster RMSE Yen Forecast A 7 80 B 15 82 C 5 85 D 10 81 E 20 88 a. Calculate the composite forecast that results from evenly weighting these five forecasts. b. Calculate a composite forecast as described in the chapter that assigns greater weight to forecasts with greater accuracy. c.
Discuss the advantages and disadvantages of using the forecast in (b) versus (a). SOLUTIONS: a. The evenly weighted forecast is 0.2x80 + 0.2x82 + 0.2x85 + 0.2x81 + 0.2x88 = 83.2 b. This system assigns weights inversely proportional to each forecaster's standard deviation. These weights are wA = 25.53%, wB = 11.91%, wC = 35.74%, wD = 17.87%, and wE = 8.94%. The composite forecast: 25.53%x80 + 11.91%x82 + 35.74%x85 + 17.87%x81 + 8.94%x88 = Yen 82.92/$ c. Composite forecast (b) should be a superior forecast with a smaller RMSE and less bias since it places greater weights on forecasters that have been more accurate. The superiority of composite forecast (b), however, depends upon the individual forecasters retaining their relative accuracy ratings. If the rankings of RMSE were to change, then a simple evenly weighted composite forecast, as in (a), could be superior. 3. Suppose you are evaluating two forecasters based on the following information. Forecaster A made 30 "correct" forecasts of a total of 50 forecasts during the last year. Forecaster B made 114 "correct" forecasts of a total of 200 forecasts during the last year,. a. What is the probability that forecaster A's track record of correct forecasts was simply due to chance? (Note: you may use the normal approximation to the binomial distribution.) b. What is the probability that B's track record of correct forecasts was simply due to chance? (Note: you may use the normal approximation to the binomial distribution.) c. Discuss whether you would prefer to use the forecasts prepared by A or B. SOLUTIONS: a. b. c. A's track record is p = r/n = 30/50 = 60%. This is greater than 50%. But with only 50 observations, the standard error of the estimate is fairly large, 0.0707 = (0.5*(1‐
0.5)/50)0.5 = (p*(1‐p)/n)0.5. So the t‐value against the null hypothesis that p=0.5 [t=(0.60 ‐ 0.50)/0.0707] is 1.414, which does not fall into the reject region at α=0.025 level (tα ≈ 2.009) and we fail to reject the null hypothesis (A’s track record of correct forecasts was simply due to chance). B's track record is p = r/n = 114/200 = 57%. This is also greater than 50%. With 200 observations, the standard error of the estimate is smaller, 0.0354 = (0.5*(1‐0.5)/200)0.5 = (p*(1‐p)/n)0.5. So the t‐value against the null hypothesis that p=0.5 [t=(0.57 ‐ 0.50)/0.0354] is 1.977, which does fall into the rejection region at α=0.025 level (tα ≈ 1.960) and we reject the null hypothesis (B’s track record of correct forecasts was simply due to chance) and accept the alternative. There is a trade‐off between A and B. A appears to have a better record, but it is based on a shorter history so we have less confidence in it. B's track record is slightly less impressive, but still shows significant expertise with a track record better than 50%. 4. As a multinational firm with sales in Japan, you require forecasts of the Yen/$ exchange rate. You have been using a professional forecasting firm, Crystal Ball Associates, and now want to measure the performance of their predictions. Following is a table showing the Crystal Ball forecasts, the actual end‐of‐period rates and the one‐period ahead forward rates. Period 1 2 3 4 5 6 7 8 9 10 11 12 Crystal Ball Forecast 100 106 102 108 115 109 103 95 93 90 91 85 Actual spot rate (end of period) 101 110 108 105 110 110 98 90 91 85 88 84 1‐period forward rate 98 105 100 102 108 112 105 98 90 89 90 86 (end of period) a. Calculate the performance of the Crystal Ball forecasts and the forward rate using the MSE and RMSE method to measure their accuracy. b. Calculate the performance of the Crystal Ball forecasts and the forward rate using the percentage correct method to measure their usefulness. c. Has Crystal Ball Associates demonstrated unusual forecasting expertise according to the percentage correct method? d. Compare your results using the two methods. What do you conclude? SOLUTIONS: a. For the forward rate, MSE = 0.0022; RMSE = 0.0468 or 4.68% error in the one‐month forecast. For the Crystal Ball forecast, MSE = 0.0015; RMSE = 0.0392 or 3.92% error in the one‐month forecast. b. Crystal Ball Associates forecasts were "correct" or on the right‐side of the forward rate 10 times out of 12 months, or 83.3%. The percentage correct method cannot be used for the forward rate. c. Crystal Ball Associates track record, 83.3%, is highly unusual, signifying expertise and making the forecast very useful. The probability of getting 10, 11 or 12 correct forecasts out of 12 (assuming that the forecaster has no expertise) is (66+11+1)/4096 = 1.9%, so this is a very rare and statistically significant event. The forecaster appears to have significant expertise. [NOTE: 66 = 12!/(10!2!); 11 = 12!/(11!1!); 1 = 12!/(12!0!); and 4096 = 212 which represents the number of patterns for tossing a fair coin twelve times.] d. Crystal Ball Associates are not much more accurate than the forward rate, but they could be very useful if the firm has hedging decisions to make on a one‐month horizon. Chapter 9 3. General Motors finances itself, among other channels, by using one‐year, floating‐rate notes whose rates are re‐calculated every three months at LIBOR+1/8. A new $250,000,000 issue is planned for mid‐September 2001 with a one‐year maturity. a. Describe how GM could hedge its interest payments for the year. [For convenience, assume that CME maturity dates coincide with the firm's roll‐over dates.] b. Using Table 9.3, what is the yearly rate that GM can secure if it hedges? c. Calculate GM's total costs for the $250,000,000 issue assuming that it hedges. SOLUTIONS: a. GM could sell 250 Eurodollar interest rate futures for every maturity where its interest payments are set initially or re‐set; that is, September and December 2001 and March and June 2002. b. For the next year, GM can lock in LIBOR rates of 7.13% (Sept 2001 at 92.87); 7.17% (Dec 2001 at 92.83); 7.11% (Mar 02 at 92.89); and 7.11% (June 02 at 92.89). The annual LIBOR rate is : 7.13%/4 + 7.17%/4 + 7.11%/4 + 7.11%/4 = 7.13%/4 = 1.7825%; or (1 + 7.13%/4) x (1 + 7.17%/4) x (1 + 7.11%/4 ) x (1 + 7.11%/4) – 1 = 2.33%. c. GM will pay $250,000,000*(0.017825+0.00125) = $4,768,750. 4. The ABC firm is considering borrowing $50,000,000 for one year either at a fixed rate of 6.50% in the US domestic market or at a floating rate indexed to three‐month LIBOR+1/4 in the Eurocurrency market. Currently, 3‐month LIBOR is 5.25% and expected to remain constant for the year. a. How much would ABC save if it uses the Euromarkets and these expectations are met? [For convenience, assume that CME maturity dates coincide with the firm's roll‐over dates.] b. What are the risks in using a Euromarket loan? c. Calculate the eventual saving for ABC in the case where LIBOR increases by .50% every three months. SOLUTIONS: a. Savings are 1% of the outstanding amount for one year, or $500,000. b. The risks are interest rate risk because the Euromarket loan is on floating rate terms, and roll‐over risk if the bank has the option to refuse to renew or roll‐over the loan. If ABC has a commitment for the year, then it has no roll‐over risk as long as the bank remains in operation. c. Fixed rate costs: 6.5% of $50,000,000 = $3,250,000 Floating rate costs: (1 + 5.5%/4) * (1 + 6.0%/4) * (1 + 6.5%/4) * (1 + 7.0%/4) = 1.0625, or a cost of 6.25%. On $50,000,000 principal the interest bill will be $3,125,000; still better than a 6.5% fixed rate. 5. Suppose that three‐month Eurodollars are quoted in the interbank markets at 6.0% ‐ 6.125% by London banks, and 6.25% ‐ 6.375% by Singapore banks. a. Explain how you could attempt to make arbitrage profits in the above case. b. How large is the profit from arbitraging $1,000,000 in this case? c. What risks and/or costs do you face in attempting the arbitrage? SOLUTIONS: a. A trader would attempt to borrow dollars from a bank in London at 6.125% and then deposit them at a bank in Singapore for 6.25%. b. The potential profit is 0.00125 * $1,000,000 / 4 = $312.50. Remember, these are per annum interest rates for a three‐month period. c.
The trader carries the political risk of deposits in Singapore. If funds were blocked in Singapore, he might not be able to pay back his London loan. Time differences between London and Singapore may also increase the difficulty of this transaction. Chapter 10 1. Suppose IBM is issuing $100 million in 7‐year Eurobonds priced at U.S. Treasury minus 25 basis points. There is great demand for the issue and you are willing to bid 102 for 10% of the issue. a. If you actually get your bid executed, how much will you pay for the bond? b. A year later, the IBM Eurobonds are traded on the Luxembourg Exchange at 105. What is the value of your investment? What is your capital gain (loss)? c. You decide to sell the bond at the above price to pursue other opportunities. What amount of withholding taxes are you required to pay? SOLUTIONS: a. Price is 102% of par or 1,020 per bond; 102% * 10% * 100 million = $10.20 million for your share of the issue. b. Price is 105%*10%*100 = $10.50 million. Gain is $300,000. c. No withholding taxes apply in the Eurobond market. 2. Suppose Credit Suisse First Boston (CSFB) is the sole lead manager in a $100 million bought deal for the World Bank. CSFB decides to price the seven‐year issue at par to yield 8%. a. What will be CSFB’s position if the Fed decides to increase short‐term interest rates by 50 basis points during the offering period? b. Instead of the Fed move described in a above, suppose that international trade talks break down leading to a depreciation of the dollar on currency markets. What will be CSFB’s position in this case? c. Calculate the gain or loss for CSFB if the seven‐year Eurobond rate rises to 8.25% on the offering day. (Note: Eurobonds pay interest only once each year.) d. Suppose CSFB collects 2% in fees for lead managing the issue. Again, calculate the overall gain or loss for CSFB if the seven‐year Eurobond rate rises to 8.25% on the offering day. e. (Optional) How could CSFB hedge the risks described in (a) and (b)? SOLUTIONS: a. b. Same as in (a). To attract investors that shy away from dollar assets, CSFB will have to lower the Eurobond price to a level attractive to lenders. The yield required by the market on long‐term bonds may change in response to the 50 basis point increase in short‐term rates. If long‐term interest rates rise, then by pledging to sell the Eurobonds at par, CSFB will lose the difference between par and the new lower price of the bond. Long‐term interest rates may fall, however, if the market senses that the increase in short‐term rates will reduce longer‐run inflationary pressures. In this case, CSFB enjoys a capital gain. c. The Eurobond price falls to $987.09 per $1,000.00 face value. 7
$987.09 = ∑
T =1
80
1.0825
T
+
1000
7
1.0825
The underwriter loses 1.291% on the $100,000,000 issue or $1,291,000. d. If CSFB collects 2.0% in fees, it transfers only $980 per bond, or $98,000,000 on the entire issue to the World Bank. CSFB's net profit is then $2,000,000 ‐ $1,291,000 = $709,000. e. CSFB can hedge the increase in interest rates by selling interest rate futures.