Pace University

Prof. P.V. Viswanath

Fall 1998

Midterm Exam I

1. The following new item appeared on the Bloomberg website. Explain why the employment news report should have led to an increase in US bond prices.

New York, Oct. 2 (Bloomberg) -- U.S. bonds rose for a fourth day, rounding out their best week in more than five years, as a report showing weaker-than-expected jobs growth bolstered expectations for a slowdown and more Federal Reserve interest rate cuts. ``We're in a slowing economy,'' said Randy Bateman, who helps oversee $10 billion at Star Bank NA in Cincinnati, Ohio. ``The market recognizes that.''

The benchmark 30-year Treasury bond rose 29/32, or $9.06 per $1,000 bond, cutting its yield 5 basis point to 4.83 percent. Earlier it touched 4.82 percent, the lowest for long-term government debt since 1967. The yield on 10-year notes fell to 4.24 percent -- a level not seen since 1965.

2. Karen has borrowed $50000 on margin to buy shares in Hollywood Entertainment Corporation, which is now trading at $20.75. She has also sold short 5000 shares of AOL, currently trading at $104. The initial margin requirements for both margin purchases and short sales is 50%. Suppose the share prices of both companies are expected to grow at the same rate. By how much must share prices increase for Karen to receive a margin call, assuming that the maintenance margin is 35% for margin purchases and 40% for short sales?

3. As of Friday, Oct. 2, the following information was available. Compute the corresponding bond-equivalent yields for both bills:

T-bill maturity Banker’s Discount
12/31/98 4.10
4/01/99 4.21


1. If the employment numbers were weaker, then there would be less upward pressure on prices (assuming that the economy was close to full employment to begin with), and hence inflation estimates would drop. This, in turn, would imply a lower nominal interest rate, since the nominal interest rate = real interest rate plus the expected inflation.

2. Initial equity for Karen was $50,000 on the margin purchase (since the required initial margin is 50%, and she’s borrowing $50,000) and $260,000 (since she’s short-selling 5000 shares trading at $104, or $520,000, and the initial required margin is 50% or half of this) on the short sale. Hence total equity = $50,000 + $260,000 = $310,000.

The equity level below which Karen would be required to post margin equals the sum of the required maintenance margin levels for the margin purchase and the short sale. The required maintenance margin level for the margin purchase is 35% of the original total purchase of $100,000, i.e. $35,000. The required maintenance margin level for the short sale is 40% of the original transaction value, i.e. 40%($520,000) = $208,000. Hence, if Karen’s equity drops below a total of $243,000 (35000 + 208000), she would receive a margin call. That is, she would need to sustain a loss of 310,000 - 243000 or $67,000.

Her equity will drop by an amount equal to her losses. If prices for both shares rose by x%, her gains on the margin purchase would be (x%)($100,000) and her losses on the short sale would be (x%)($520,000). Hence her total losses would be 420,000x. Equating this to 67000, we get x = 67000/420000 = 15.95%

3. The number of days to maturity, (excluding 1 day for skip day settlement) = 89 for bill 1 and 189 for bill 2.

Hence the prices are given implicitly by the equations .041 = (100-P)*360/(89*100) and .0421 = (100-P)*360/(189*100). Solving, we get 98.9864 and 97.7898. We can compute the bond equivalent yield according to the formula [(100-98.9864)/98.9864](365/89) = 4.1995% and [(100-97.7898)/97.7898](365/189) = 4.365%

Midterm II

1. (40 points) Read the appended article and the answer the questions that follow:

  1. Use the information in the article to come up with an estimate of the beta of the Russell 2000 index vis--vis the Dow Jones. (10 points)
  2. If you believed in the framework laid out in Chapters 6 through 9, how would you answer the question asked in the first paragraph of the article? (15 points)
  3. The article seems to be making two points: one, regarding the direction in which the market is going to move; and two, regarding the increased volatility that is likely. If investors believed that the economic environment were going to become more volatile, how would they react, according to the framework developed in Chapters 6 through 9? How would their investment and portfolio strategies be affected, and what would be the effect on security prices and security returns? (15 points)

Trying to Test How False The Positives Might Be
New York Times Internet edition, Oct. 28, 1998

Is the bear market over? Or is the recent rise in stock prices just a seductive rally intended to suck investors in before the market slides once more?

That's what seasoned investors and traders wondered yesterday as they watched the Dow Jones industrial average spurt up 101 points in the first hour of trading, then lose it all and more by the end of the day. The average closed yesterday at 8,366.04, down 66.17 points, or 0.78 percent.

The move was a reversal of fortune for investors who were beginning to regain confidence that American stocks could ride out the rest of the world's economic storms. Indeed, in the last two months or so, the Dow has risen about 1,000 points, or 13 percent, from its Aug. 31 low of 7,539.07.

Blue-chip stocks were not the only ones to see an early rally fizzle. The Russell 2000, the small-capitalization stock index which has risen for two consecutive weeks, jumped 1.2 percent in the morning, only to close down 0.57 point. Investors in bigger Nasdaq stocks, happy to see technology companies reporting positive earnings, could not sustain their push either.

So where are we now? At a bull market rest stop -- or waltzing into a bear market trap?

The question may be particularly timely now because many investors are relative newcomers to stocks and have not yet experienced the trauma of a bear trap. During the two bear moves before this year's turmoil -- the decline in 1990 that took stocks down 21 percent and the 1987 crash -- there were no intermittent bear-market rallies to sucker investors. As a result, those who have been in stocks for only 10 years or so may not realize that bear markets can have deceptively sharp rallies that only serve to lure investors in, then take them down.

"The last couple of bear markets we've seen haven't lasted long enough to test people's psyches," said Charles White, president of Avatar Associates, a money management firm in New York with $3.5 billion in assets.

And while he noted that the market's breadth has improved -- the number of stocks going up has risen considerably from a month ago -- Mr. White added, "It still seems as though we're challenged to break out of some levels, which indicates that this is nothing more than a rally in a bear market."

Bear market rallies are particularly vicious because they last just long enough to lull investors back into a bullish mood. In the 1973-74 bear market, for example, when stocks fell 45 percent over 22 months, there were three meaningful rallies that -- erroneously -- led investors to believe the worst was over. And in the 36 percent decline that began in December 1968 and lasted until May 1970, investors were also treated to three such rallies.

But the biggest "gotcha" of all was in 1930 -- when stocks recovered from the brutal decline of 1929 that took the Dow average down almost 48 percent. In early 1930, stocks rose almost 50 percent, then crashed, losing 83 percent of their value in the next two years.

As in most downturns, investors today are hoping that the pain they have been through this year is all they will have to endure. Sometimes markets do go straight down and then rebound. In 1962, stocks fell 27.1 percent in six months, then bounced back into a four-year bull market that produced compound average returns of 19 percent annually.

But Alan W. Kral, a portfolio manager at Trevor, Stewart, Burton & Jacobsen in New York, is arguing against such a recovery now. "Nothing that's happened over the past four weeks has changed the longer-term outlook for the economy which will drive the market," he said.

"Over the past year we've seen enough of the props of the economy -- exports and profits -- start to be chipped away so that growth is now threatened," Mr. Kral said.

The final, and most important, prop is the consumer, who accounts for two-thirds of the nation's economic growth. Yesterday, the Conference Board said consumer confidence fell to its lowest level in nearly two years earlier this month. If consumers pull back on their purchases, the economy will slow significantly.

To be sure, investors have had reason to be optimistic. The biggest lift has come from the Federal Reserve, which has cut rates twice since the end of September. Lower interest rates are always a positive since lower borrowing costs can mean higher corporate profits.

And concentrated rate cuts are extra-bullish. Market historians note that in three of four cases when the Fed has cut rates twice in a row, stocks rise an average of 19 percent in the ensuing three months. Since the Fed last cut rates on Oct. 15, stocks are up almost 5 percent.

More support for stocks has come from a raft of upbeat earnings announcements. Companies as diverse as I.B.M., Estee Lauder, Amgen and Northern Telecom have posted better-than-expected numbers recently.

But with most third-quarter earnings announcements completed, the bounce from positive surprises is probably already built into stock prices. As a result, Mr. Kral believes that the market is at a danger point.

He believes that the nation's employment cost index, out Thursday morning, will once again focus investors on rising labor costs and declining corporate profit margins.

"G.D.P. growth slowing and labor costs going up are not good from a profit standpoint," he said. "There's a lot of risk in the market here."      

2. (30 points) You have available to you, two mutual funds, whose returns have a correlation of 0.2. Here is some information on their return probability distributions:  

  Expected Return Standard Deviation
Explorer II 23% 30%
Morningstar 15% 32%

In addition, you can also invest in a riskfree 1-year T-bill yielding 12%

  1. If you have a risk aversion coefficient of 4, and you have a total of $20,000 to invest, how much should you invest in each of the three investment vehicles (10 points)?
  2. What is the standard deviation of your optimal portfolio (10 points)?
  3. If you wanted an expected return of 50% with as low a portfolio variance as possible, what would you invest in each of the three investment vehicles? Assume for the sake of this question that you cannot borrow at the riskfree rate. (10 points)

 3. (30 points) The CAPM is often used to evaluate the performance of professional money management.  Suppose that a mutual fund has a 10-year average annual return of 14%, whereas the beta is 1.4.  The S&P 500 Index grew by 12% per year over the same period, and the average T-billyield was 5%.  The manager of this mutual fund would probably claim that it had beaten the market index by a margin of 2% per year.  Do you really believe that the mutual fund has outperformed the market if the CAPM is used to represent the risk-return relationship?

4. (Bonus question: 15 points) If the average correlation between stocks in a basket of risky stocks is 0.2, how many stocks are necessary for effective diversification? You have read in the Financial Analysts Journal that average returns drop by 0.1% for every stock included in a portfolio, due to transactions costs. The average gross return on stocks is 12% per year, and the average standard deviation of returns is 30% per year. You have figured out that your utility function is E(portfolio return) - 0.005 A (variance of portfolio returns), and you’re A-coefficient if 0.2. If you cannot get an exact answer, do as much of the computations as you can. If you cannot do the computations at all, explain how you would think about the problem. (Hint: The answer is less than 10. This is not necessarily a straightforward problem; however, we have gone over all the tools necessary to address this question.)

Solutions to Midterm II

1. a. If we assume that the Russell 2000 jumped 1.2% during approximately the same time that the Dow Jones moved up 101 points or (101/(8366.04+66.17) = 1.22%, and closed down 0.57% over the same time period that the Dow dropped 0.78%, then the beta of the Russell 2000 vis-a-vis the Dow Jones Index could be estimated as (1.2+0.57)/(1.22+.78) = 0.885

b. According to the CAPM, there is a positive relationship between expected return and risk.  Since the market portfolio is risky, it cannot have a negative expected return.  Hence, one's best estimate of the market could not be bearish.

c. If investors believed that the market were going to become more volatile, then using the formula for allocation between the riskfree and the risky portfolio (the capital allocation line), investors would move more money into the riskfree asset.

Of course, if all investors wanted to move into the riskfree asset, then the riskfree interest rate would drop.  In any case, the spread between the riskfree rate and the required rate of return on the market portfolio would widen.

2. a., b.The tangent portfolio is given by 

wmstar = 0.0521.

The variance of returns of this portfolio is .94782(30)2+.05212322+2(.0521)(.9478)(30)(32)(.2) = 830.2389, and a standard deviation of returns of 28.81. The expected return is 23(.9478) + (15)(.0521) = 22.5809.

The optimal amount to invest in this portfolio is (22.58-12)/(.01)(4)(830.2389) = 0.3186. Hence, we find that the final investment proportions are .3186(.9478) = .3020 in Explorer, .3186(.0521)=.0166 in Morning Star and .6814 or the rest in the riskfree asset.

The standard deviation of returns on the optimal portfolio is (0.3186)(28.81) = 9.18%

c. To get an expected return of 50% without shortselling the riskfree asset, the only way is to shortsell the Morning Star fund. Solve 0.5 = .15wmstar + .23(1- wmstar) to get wmstar = -3.375 and wexp = 4.375

3. The risk-adjusted required return on the mutual fund, using the CAPM is 5% + 1.4(.12-.05) = 14.8%.  Hence, on a risk-adjusted basis, the mutual fund manager has not beaten the market.

4. From Appendix A of Chapter 8 (equation 8A.4.), we see that the portfolio variance can be written as 900/n + (1-1/n)(.2)(900) = 180 + 720/n, where n is the number of securities.

Since the expected return is 12 - .1n, the value of the utility function, with n securities is 12 - 0.1n - 0.001 (180+720/n), or 12 - .18 - 0.1n - 0.72/n = 11.72 - 0.1n - 0.72/n

To choose n so that this expression is maximized, we take the first derivative and set it equal to zero. This gives us the equation: 0.1 = 0.72/n2. Solving, we find that n = 2.68. Since we must invest in either 2 or 3 securities, we must compute the utility function for 2 and for 3, to see which is greater. The utility values are 11.49 and 11.61 respectively. Hence the optimal number of securities to invest in is 3.

Alternatively, once the expression for the utility function, 11.72 - 0.1n - 0.72/n , has been obtained, it could be solved simply by trying out different values, and noting the direction of change of the expression for the utility function. Alternatively, we could note from the expression for the variance that as we increase the number of securities from n to n+1, the variance increases by -720/[n(n+1)]. As we increase the number of securities from n to n+1, the expected return increases by -0.1. Hence the value of the utility function increases by -0.1+ 0.001{720/[n(n+1)]} or -0.1 + 0.72/n(n+1). If we start out with 1 security and consider moving to 2, the utility function increases by -0.1 + 0.72/2 = 0.26; hence we would go ahead. If we start with 2 and consider switching to 3, the utility function increases by -0.1 + 0.72/6 = 0.02 > 0; hence we would go ahead. If we switch from 3 to 4, the utility function increases by -0.1 + 0.72/12 < 0; so we would stop at 3.


1. Following are Treasury bill quotes from Friday, Dec. 4, 1998 mid-afternoon, from the Wall Street Journal:

Maturity Days to Mat. Bid Asked Price Change Ask Yield
Dec 31 '98 24 4.34 4.26 +0.11 4.33
Jan 28 '99 52 4.33 4.29 +0.08 4.38
Feb 25 '99 80 4.34 4.32 +0.06 4.42
Mar 25 '99 108 4.33 4.31 +0.05 4.43
Apr 22 '99 136 4.35 4.33 +0.03 4.46
May 20 '99 164 4.36 4.34 +0.04 4.49
  1. Use the ask yields to compute the 4-week forward rates 24 days ahead, 52 days ahead, and 80 days ahead. Express the spot rate and the forward rates as effective annual yields. Keep in mind that treasury bills are quoted using the bank discount yield, rBD = (FV-P)/FV)*(360/n), where n = no. of days to maturity calculated from settlement date, FV is the face value of the bill, and P is the price of the bill. All yields are based on a one-day settlement. (15 points)
  2. Use the information from the previous question to make predictions for 4-week T-bill rates, 24 days ahead, and 52 days ahead. Assume an annualized liquidity premium of about 10 basis points for all bills upto a maturity of 6 months. (10 points)

2.  Use the information from the table in the next question to answer the following questions:

  1. Estimate the duration of the 5 3/4 Nov ’00 note as of Dec. 4, 1998. If you wish, you may use the following approximation: First compute the duration as of the end of November 1998, assuming the same price and yield information and assuming that the bond matures at the end of the month. Then subtract the number of days from the end of November to Dec. 4, 1998. (This method is exact if the price is recomputed.) (15 points)
  2. Use this information to estimate the (ask) price change for the bond, assuming a face value of $1 m. Assume that the ask yield on the bond changed by 2 basis points from the previous day. (10 points)
  3. The convexity of this bond in years is approximately 5.1707. Use this information to re-estimate the price change for the bond. (5 points)
  4. Suppose you had used the November ’00 and the November ’05 bonds yesterday to immunize a liability consisting of a single payment in November 2003. You wish to keep your money in these two bonds. How would you change the portfolio allocations to the two bonds; i.e. would you move switch money from the 7 year bond to the 2 year bond, or vice-versa, and why? (15 points)

3. Following are the quotes on Government notes from Friday, December 4, 1998, mid-afternoon. Colons in bond and note bid-and-asked quotes represent 32nds; 101:01 means 101 1/32. Net change in 32nds.

Rate Maturity (Mo/Yr) Bid Asked Price Change (in 32nds) Ask Yield
5 7/8 Nov 99 101:03 101:05 -1 4.60
5 3/4 Nov 00 102:06 102:08 -7 4.52
5 7/8 Nov 01 103:21 103:23 -9 4.53
5 3/4 Nov 02 104:05 104:07 -12 4.58
4 1/4 Nov 03 99:13 99:14 -11 4.38
7 7/8 Nov 04 116:18 116:22 -13 4.63
5 7/8 Nov 05 107:15 107:17 -15 4.59
  1. Without computing forward rates, write a brief report (several sentences) on the interest rate outlook for the next several years, using only the above data to support your point of view. (Do not use the information in the article below.) (15 points)
  1. Read the Wall Street Journal article below for the same day, and comment in several sentences on how the information in the article supports or contradicts your answer in part a., above. (15 points)

Trio of Forces Conspires To Deflate Bond Market

U.S. Treasury prices sank Friday under the weight of strengthening stocks, a strong jobs report, hedging strategies related to non-government debt paper and the usual corrective move after a heady run up.

In late trading, the price of the benchmark 30-year Treasury bond was down 5/8, or $6.25 for a bond with a face value of $1,000, at 103 5/32. The yield, which moves inversely to the price, rose to 5.04% from 5.001% late Thursday.

Prices had initially dipped following the release of November's employment report, which showed a robust nonmanufacturing sector, the economy adding 267,000 jobs and the jobless rate falling to 4.4%, down from October's 4.6%. But after a false small bounce in midsession, prices tracked lower for the remainder of the day.

"It's not such a bad idea to retrace some of our gains," said Bill Kirby, manager of government trading at Prudential Securities in New York.

This week had seen bonds flirt just under the psychologically important 5% yield -- a move not seen in six weeks -- and the contraction of at least 0.2 percentage points in overall Treasury yields from week-ago levels.

The strong jobs report reinforced expectations that the Federal Reserve would remain on hold for the rest of the year, giving market participants further reason to retreat from Treasurys.

"The Fed won't do anything and this number makes it a little less likely," said John Spinello, managing director and chief strategist for government securities at Merrill Lynch in New York.

Moreover, the report could dampen expectations for a Fed rate cut in the new year, said Ken Fan, a bond strategist at Paribas Capital Markets in New York. "We've taken out more premium of a Fed ease."

The short- to intermediate-sector of the yield curve, which is more sensitive to changes in Fed policy, pulled back more sharply relative to long bonds Friday. At the same time, the spread between the two- and 30-year yield narrowed about 0.1 percentage points from late in the previous session, making for a flatter yield curve, or difference in price between bonds of different maturities.

With the employment report seen as the last of the major economic indicators for the year, Treasurys may stay rangebound through December once the curve flattens to a comfortable level, said Joe Abate, U.S. economist at Lehman Brothers in New York.

Meanwhile, the winding and unwinding of hedges, which would have dealers first selling, then buying Treasurys, appeared to produce negligible effects on the market -- though some traders mentioned it in passing -- as prices stayed underwater even after the pricing of several corporate and agency debt paper.

Prior to pricing a debt issuance, dealers will usually buy the non-government paper and sell Treasurys to protect against interest rate risk. After the deal comes to market, the trade is reversed.

However, analysts said there is support in Treasurys despite the session's declines. Investors also may be wary of going short Treasurys over the weekend with the possibility of a flight-to-quality bid in Treasurys sparked by developments in South America -- namely Sunday's presidential elections in Venezuela.

Investors are watching as front-runner, former coup leader and known populist Hugo Chavez battles with market-favorite Henrique Salas Romer. Market participants have been fretting over the strong possibility of a victory for Mr. Chavez, which might mean the roll back of the Venezuela's free-market policies.

But with markets already seen pricing in his victory, the question will be the outlook for the country, given its role as a major oil exporter and the doldrums experienced in those commodity markets, economists said.

A win for Mr. Chavez in Venezuela, coupled with a recent rejection in Brazil's congress on key fiscal measures linked to the access of international credit lines, could rock confidence in Latin American economies and trigger a new round of flight-to-quality movement from risky emerging market investments toward the safe-haven of U.S. Treasurys, analysts said.


1. a., b.

Days to Mat. Ask Yield price 4-week forward rates Predicted spot
24 4.33 0.997113  
52 4.38 0.993673 4.608062 4.508
80 4.42 0.990178 4.700953 4.601
108 4.43 0.98671 4.679541 4.5795
136 4.46 0.983151 4.822947 4.7229
164 4.49 0.979546 4.905982 4.806

The best way to tackle this problem is to use the banker’s discount yield or the bond-equivalent yield formula to get the prices of the different bills. The ratios of the bills then give 1 plus the non-annualized forward rates. These must be annualized using compounding to obtain the forward rates in the second last column. The predicted spot rates are obtained by subtracting a tenth of a percentage point (or 10 basis points) from the forward rate.

2. a.

Date Years Flow Convexity computations Duration
May-99 0.489041 28750 18754.34 0.013437
Nov-99 0.989041 28750 49557.36 0.026574
May-00 1.489041 28750 91324.88 0.039123
Nov-00 1.989041 1028750 5127413 1.82869
      Actual Duration 1.907823

Convexity as of 12/4 = 5.170709
Convexity as of 11/30 = 5.216705

Duration computed using the approximation suggested in the problem:

approx. no. of years ahead Date Flow Duration
0.5 May-99 28750 0.013456
1 Nov-99 28750 0.026619
1.5 May-00 28750 0.039201
2 Nov-00 1028750 1.832744

1.912 - 4/365 = 1.901062

The 1.912 can also be obtained by using the duration formula for constant coupon bonds not selling at par.

2.b. DP = (-D)P(Dy)/(1+y) = (-1.901062)(102.25)(.0002)/1.0452 = -3.72 cents.

2. c. Taking convexity into account, DP = -.0372 + (0.5)(5.17)(0.0002)2(102.25) = -3.71 cents.

2.d. the yields of both bonds increased; hence their durations have decreased. Hence, I would want to move money into the longer term bond to keep the duration at the same level.

3. a. The yields are fairly constant over maturities. This indicates that the market expects future spot rates to remain quite constant as well, assuming that liquidity premiums are non-existent or very small. If liquidity premiums are not very small, then the yields imply falling rates.

b. Most of the article deals with very short-term rate changes and thus is not relevant for our purposes. However, overall, the article reinforces the idea that the yield curve is flat. Joe Abate, U.S. economist at Lehman Brothers is predicting that rates will remain within a fairly small range. This strengthens the prediction of constant rates. The last part of the article also speculates regarding a flight to US bonds if the populist and socialist Venzuelan presidential candidate Mr. Chavez wins. This suggests the possibility of interest rates dropping moderately in the near term.

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