LUBIN SCHOOL OF BUSINESS
Pace University
Fin 652 Investment Analysis
Spring 1999
Prof. P.V. Viswanath

• Practice question for Midterm I and suggested solution
• Midterm I
• Midterm I Solution
• Midterm II
• Midterm II Solution
• Finals
• Solution to Final

Practice Question for Midterm I

Japan Sets Short-Term Rates Near Zero; May Be a Step Toward Money Expansion

By Bill Spindle and Jathon Sapsford; WSJ Staff Reporters

TOKYO -- The Bank of Japan drove short-term interest rates to almost zero Wednesday, the most recent indication that it has begun a shift from conventional interest-rate policy to focus on expanding money supply.

The central bank's move came a day after Finance Ministry officials launched their own attack on interest rates, announcing they will adjust the amount of bonds the government buys and issues and saying they expect a weaker yen. The yield on Japan's benchmark long-term bond slid to 1.925% Wednesday, from 1.985% a day earlier and 2.135% before Tuesday's policy offensive. The yen also weakened on the rate-cut news, to more than 118 yen to the dollar in Tokyo trading, compared with about 117 yen a day earlier and 114 at the start of Tuesday trading.

The central bank's decision to push the interest rate it most directly controls to 0.08% -- and Gov. Masaru Hayami's telling reporters this week that even a rate of zero would be acceptable -- may be one of the first steps in a policy shift toward aggressively printing money to break a deflationary spiral, according to analysts and financial-industry officials.

"They'll be forced to do it if the economy continues this way," said a senior executive in charge of capital markets at a major Japanese financial institution.

Virtually Uncharted Waters

The move Wednesday bolstered perceptions in the financial industry that the government has exhausted conventional means of boosting the economy and is now headed into waters virtually uncharted for a half-century. In just two weeks, the idea of Japan revving up the yen printing presses has moved from the realm of arcane academic discussion to the center of public-policy debate. But the risks of such a policy, which some business leaders and many analysts view as likely sometime this year, are huge.

Some economists say that if Japan aggressively expands its money supply, the yen will depreciate significantly, perhaps even to 200 yen to the dollar. That could put pressure on other currencies in Asia, potentially sparking another round of global financial turmoil.

The Bank of Japan, now facing a major challenge to its independence, could also lose control of monetary policy to politicians who already have exhausted their own fiscal-policy options. Indeed, a small but vocal group of politicians is urging the central bank to finance government spending directly by printing money. The last time the central bank tried that was to pay the military expense of World War II; the result was hyperinflation. Germany also went down the money-printing path, in the post-World War I Weimar Republic, and rendered its currency worthless.

"You don't want to write an open check, especially to the Liberal Democratic Party," said Ron Bevacqua, an economist at Merrill Lynch, referring to Japan's ruling political party.

Formerly Esoteric Arguments

Mr. Hayami has vowed that the central bank will "never" directly underwrite government spending, which is currently against the law. And the Bank of Japan's policy board, its top policy-making organ, last week rebuffed calls to buy more government bonds in the open market.

Still, powerful economic and political forces have pushed what were once esoteric monetary arguments into mainstream debate. On one prime-time television news show this week, the anchors donned placards labeling themselves "Bank of Japan" and "Government," and then passed around oversized copies of government bonds to explain a debate that raged last week in Parliament. Even politicians have dutifully attended what amounts to tutorials on how a central bank works.

Driving the debate over monetary policy is a rise in interest rates since December. The yields on Japan's 10-year government bonds have tripled as government-borrowing plans proliferated and a government agency that was a huge bond buyer ceased purchases.

Incentives to Act

Interest rates of 2% may seem low by global standards, but with wholesale prices falling, some economists worry that the real cost of borrowing has become high enough to threaten an economic recovery. Higher Japanese rates have also rattled the U.S. bond market, sparking fears that Japan will lure capital away from the U.S.

All of these are powerful incentives for the Bank of Japan to act, but the trick may be finding a way to do so without appearing to cave in to political pressure.

Already, the debate over expanding money supply has raged within the central bank for almost a year, Bank of Japan officials say privately. Since 1990, as the government launched one massive fiscal-stimulus package after another -- to little effect -- the central bank has slashed interest rates to break the deflationary spiral that has accompanied Japan's recession.

With rates approaching zero, some members of the bank's top policy board began debating more-radical approaches. One is expanding money supply rapidly with the aim of creating inflation. That might not only break the deflationary trend but also ease the burden of indebted banks, construction companies and the government -- all of which could pay back what they owe in cheaper yen, economists say.

An academic paper by Massachusetts Institute of Technology Prof. Paul Krugman plugging this approach -- heresy to central bankers long schooled to prevent inflation -- was translated, circulated among bank staff last May and discussed at key meetings. "It was mind-boggling for us," said a member of the bank's bureaucratic rank and file.

Questions:

1. There are two possible effects of the Government printing more money: what are they, and which one is the Government hoping will dominate?
2. Relate the drop in the long-term Japanese bond yield to the drop in the yen rate against the dollar.
3. If the government prints a lot of money, it might break the deflationary spiral as desired by the government. However, the inflation would clearly raise nominal yen interest rates. This would seem to be in appropriate for a government that is worried over currently high interest rates. Explain why there need not be a contradiction.

Answers:

1. The two possible effects of the Government printing more money are: i) higher inflation if the economy is at capacity, and ii) higher growth if the economy is experiencing liquidity problems and is not at capacity. The Japanese government is hoping that the second effect will dominate.
2. The drop in the long-term bond yield moves money out of the Japanese bond market and into the US bond market; and decreases demand for the yen, leading to a drop in the yen rate against the dollar. (We didn’t really discuss exchange rate determination, but this indicates the substitution effect of US bonds for Japanese bonds).
3. Printing a lot of money would probably increase inflation. Nevertheless, if it jump starts the economy by transferring purchasing power to businesses that are currently heavily indebted, and thus increases productivity, the risk premium demanded by the market on equities might drop. The nominal required rate on equities would then rise less than the inflation rate, which would mean that the real rate of borrowing for businesses would drop. Ultimately, the rate that drives growth is the real rate, not the nominal rate.

Midterm I

• The midterm is closed book. However, you are allowed to bring in one 8.5"x11" sheet, containing anything you wish.
• Time allowed is 2 hours.
• Explain all your steps; correct answers without explanations may not be given any credit at all.
• Show all your computations and formulae. Make all your assumptions explicit. If your approach is correct, you will get some credit, even if your arithmetic answer is wrong. So concentrate on getting your logic right.
• If you answer a question, I have the discretion to award you some points, even if you are completely wrong. If you don't attempt the question at all, I can give you no points! So attempt every question.

1. Read the following article from the Wall Street Journal and answer these questions:

1. Although some sectors of the stock market didn’t fare as well, overall the stock market rose. At the same time, bond prices fell. How can you reconcile these two facts?
2. What evidence in the article is consistent with a drop in bond prices? Explain.
3. What evidence in the article contradicts, at least on the face of it, the drop in bond prices?

Stocks Are Mixed As Tech Steady; Treasuries Slide
By Terri Cullen, Interactive WSJ; Thursday, February 18, 1999

Stocks were mixed Thursday, as technology shares struggled higher after a shaky start to trading. Bond prices fell as investors sorted through the latest batch of economic data, while the dollar rose against the yen.

The Dow Jones Industrial Average, which had lost 101.56, or 1.1% the day before, was up 43 points to 9239 in midday trading. The Standard & Poor's 500-stock index rose 5.60 to 1229.60 and the New York Stock Exchange Composite Index climbed 3.20 to 583.50.

(part of article deleted)

Meanwhile, the dollar rose to a two-month high against the yen, continuing its rally as the Bank of Japan signaled it is clearly sanctioning a weaker Japanese currency. Japan's Vice Minister for International Affairs Eisuke Sakakibara reiterated Thursday that it was "natural for the yen to weaken as a result of the very dramatic easing of monetary policy."

The Bank of Japan drove short-term interest rates in that country to almost zero Wednesday in the most recent indication that it has begun a shift from conventional interest-rate policy to focus on boosting the economy. The U.S. currency also posted slight gains against the euro.

Bond prices fell following the release of some mixed inflation data. Traders said early signs of stability on Wall Street also lured some money away from bonds.

The producer price index jumped to its biggest gain in 27 months in January, climbing 0.5% amid a surge in food and energy prices, the Labor Department reported. But the core index, which excludes the volatile food and energy sectors, actually lost 0.1% in the month.

The market's reaction to the report was muted, traders said, since labor productivity has continued to outpace the increase in labor costs, which has helped to keep inflation in check. Inflation erodes the value of fixed-income holdings such as bonds.

World-wide, stocks rose in dollar terms. The Dow Jones World Stock Index was up 0.36 to 199.24 as of noon EST.

2. You shortsold 300 shares of Transamerica on February 17, 1999, when it was trading at 57 5/8. Your broker demands a 50% initial margin on all short sales, and a 40% maintenance margin.

1. How much could the price increase before you would have to put up additional margin?
2. Unfortunately for you, on February 18, 1999, the Dutch insurance company Aegon agreed to acquire Transamerica, causing Transamerica to shoot up to 72 7/8. How much additional margin do you have to put up?

3. Disney split 3-for-1 after trading ceased on July 9, 1998. The closing price that day was $111.00. The split adjusted price was $37.00. If the Dow closed at 9089.78 on July 9, 1998, and the post-split divisor was 0.24275214, what was the pre-split divisor? Assume that the post-split divisor was computed so that the value of the index would not change between the end of trading on July 9, before the split, and the beginning of trading of July 10, when the stock was trading on a split basis.

Solutions to Midterm I:

1.

1. The high labor productivity mentioned in the article might have caused stock prices to move higher, in spite of the higher long term interest rates. Also, the stability in the stock market might have caused the equities risk premium to drop, triggering the reverse of the "flight to safe havens" effect.
2. The increase in inflation is consistent with the drop in bond prices.
3. The drop in Japanese interest rates seems to be inconsistent with the rise in US interest rates, since it seems to violate the law of one price.

2. a. Solve the equation 0.4 = (Mkt. Value of assets - loan)/loan. The amount of the loan in this case is simply the market value of the securities borrowed for short-selling, i.e. 300P. The market value of assets equals the original sale price of the securities plus the margin originally advanced. This equals (1.5) x 300 x (57 5/8) =25,931.25. Solving the resulting equation (0.4) 300P = 25,931.25 - 300P for P, we find P = 61.74.

2. b. If we assume that the additional money to be brought in is z, the equation to solve is (25931+z-300(72 7/8))/300(72 7/8) = 0.4. Solving, we find z = 4676.25.

Alternatively, the old margin is 0.5(300x57 5/8), the new margin is 0.4(300x72 7/8), and the difference is $101.25. However, a loss of $4575 has been incurred [(72 7/8 - 57 5/8)x300], which has to be made up as well. This gives us a total amount of 4575 + 101.25 = $4676.25.

3. We know that the formula for computing the average is (Sum of prices of all stocks in Index)/Divisor. Hence, if we denote by Sum, the sum of the prices of all stocks except Disney, then we find that at the end of July 9, (Sum + 111)/Pre = 9089.78, where Pre is the pre-split divisor. Similarly, (Sum+37)/ 0.24275214 = 9089.78. Solving the second equation, we find Sum = 2169.5632. Substituting into the first equation, we find that Pre = (2169.5632 + 111)/9089.78 = 0.2508931.

Midterm II

• The quiz is closed book.
• Time allowed is 2 hours.
• Explain all your steps; correct answers without explanations may not be given any credit at all.
• Show all your computations and formulae. Make all your assumptions explicit. If your approach is correct, you will get some credit, even if your arithmetic answer is wrong. So concentrate on getting your logic right.
• If you answer a question, I have the discretion to award you some points, even if you are completely wrong. If you don't attempt the question at all, I can give you no points! So attempt every question.

1. Read the following article by Jonathan Clements in the Wall Street Journal of Feb 23, 1999, and answer the following questions:

1. (20 points) The article first presents a point of view that argues that diversification has been disproved by the performance of the market in the past few years; the proof being that investing in small companies and foreign shares would have hurt you, rather than helped. Dan Wheeler, an investment advisor disagrees. Which point of view is correct? (Use no more than one side of one page.)
2. (20 points) Suppose William Bernstein is correct in that older investors are more risk-tolerant than younger investors; suppose further that the investing population in the US is getting older. Would you expect the stock market to grow at a higher rate now than 20 years ago, or less? Explain your answer. (Hint: Don’t confuse the possibly hypothetical situation envisaged in the question with what might have actually happened over the last twenty years.)

Some Investing Ideas With a Fresh Twist

Old dogs, new tricks.

If you have been investing for a while, you might think you have heard it all. After more than a decade of writing about personal finance, I sure feel that way.

But every so often, I trip across an insight that makes me think about investing in a new way. Here are a few such ideas:

Investors are often advised to load up on stocks when they are young and then gradually scale back as they grow older. But in the real world, investors become more risk tolerant with age, reckons William Bernstein, a financial adviser in North Bend, Ore., who runs a Web site devoted to portfolio theory (http://www.efficientfrontier.com).

"The average 25-year-old is absolutely devastated when he loses 5% or 10% of his capital," he says. "As people grow older, they become more risk tolerant. In 1932 and 1974, the only people buying stocks were the old gray heads who knew the world wasn't coming to an end."

The past few years are seen by many as a repudiation of stock-market diversification. If you ventured beyond the bluechip stocks in Standard & Poor's 500-stock index and bought small companies and foreign shares, you ended up hurting your results.

But in fact, this is clear evidence that diversification works, argues Dan Wheeler, director of financial-adviser services at Dimensional Fund Advisors in Santa Monica, Calif. "Diversification is doing exactly what you would expect it to do," he says.

Stock-market diversification smooths out a portfolio's returns, by making the performance valleys less deep and the peaks less high. Right now, those who built globally diversified portfolios might not like this smoothing.

But at another time, when the S&P 500's performance is less stellar, this smoothing may be a great comfort. As Mr. Wheeler puts it, "The problem with diversification is it works whether you want it to or not."

-- To further diversify, many investors have combined an S&P 500-index fund with a separate small-company stock-index fund.

"If you're locked into big gains on an S&P 500 fund, it makes sense to add a small-stock index fund," says George Sauter, a managing director at Vanguard Group Inc., the Malvern, Pa., fund group. "But if you're starting from scratch, the most tax-efficient strategy is to buy a total stock-market portfolio. We believe that's the best core holding for an investor."

With these broad-based index funds, which track the Wilshire 5000 index of all regularly traded U.S. stocks, you avoid the trading costs and tax bills generated as stocks migrate between the S&P 500 and the small-stock indexes. Fidelity Investments, T. Rowe Price Associates Inc., Wilshire Associates and Vanguard all offer Wilshire 5000-index funds.

Value stocks, those companies that are cheap compared with current earnings or corporate assets, have emerged as the top choice of some investment advisers. These advisers, drawing on academic literature, see value stocks as offering both higher returns and less downside risk.

Dan Goldie, a financial planner in Portola Valley, Calif., believes the case is overstated. Yes, value stocks may offer higher returns than growth stocks, despite the more-impressive sales and profit increases boasted by growth companies. But value stocks are also more risky.

"Some advisers believe value stocks decline less than growth stocks, because they look only at selective data since the early 1970s," he says. "When you include data from 1926, including both the 1990 and 1998 market declines, it is clear that value is not a safe haven in down markets. Value stocks are not a free lunch. They have more financial risk than growth stocks and are equally volatile in down markets."

Just as folks prefer paying for goods rather than services, so they would rather pay for services than information.

That puts brokers and financial planners, who are in the business of peddling investment advice, in a tough spot. Their response? They pretend the information is free and instead charge for services such as purchasing stocks and buying mutual funds.

"Talk to physicians," says Meir Statman, a finance professor at Santa Clara University in California. "They'll tell you the same thing. It's a lot easier to get paid for procedures than for a consultation."

Is it easy to make money in stocks? Depends whom you ask. For proof, consider two letters I received from readers last year.

The first letter pointed to the huge gap between the 52-week high and low on most stocks, and pronounced it easy to make money. The next day, I received a second letter, pointing to the same spread. My second correspondent's conclusion? The gap was clear evidence of how easily money is lost.

2. The website http://www.morningstar.net/news/Ms/LostinShuffle/990312lost.msnhtml has the following information on Salomon Brothers Capital fund (SACPX)

Portfolio Analysis as of 12-31-98

Note: The beta listed for Fine Host 144A Cv 5%, which refers to Fine Host Corporation’s 5% convertible bonds is actually that of the stock itself. The true beta of the convertible should be lower than 0.90.

1. (20 points) Assuming that the remainder of the portfolio has the same characteristics as the 31.25% actually listed above, can you estimate the beta of SACPX?
2. (10 points) Here’s some information on the fund’s performance over two different time periods. Using this information, can you obtain another estimate of the fund’s beta?

Performance and Risk Analysis 2-28-99

3. (10 points) Using the following data provided on the Morningstar site, I computed the standard deviation of annual returns to be 17.5%. If I put one third of my money in another mutual fund portfolio that had a standard deviation of 20%, what would the standard deviation of returns on my entire portfolio be, assuming a correlation coefficient of 0.7 between returns on this other portfolio and SACPX?

4. (10 points) Here is some information on the fund’s sector weightings relative to the S&P 500. Would you call this fund diversified? Why or why not?

5. (10 points) The S&P 500 has increased from about 550 in January 1995 to approximately 1220 at the end of February, when the above report was issued. This implies an annual rate of return on the S&P 500 of (1220/550)(1/4.167) = 21% p.a. over the 4.167 years. Very short-term US treasury securities earned about 5% p.a. over that period. What rate of return would investors expect from the fund, taking its beta risk into account?

Bonus questions:

1. (5 points) Suppose you decide to put all your money SACPX. Based on your answer to question d. above, would you require a higher rate of return than you computed in your answer to e.?
2. (5 points) Do you agree with the site’s evaluation of SACPX in the following quote about Margolies, the fund manager, from the Morningstar site? Why or why not?

Margolies is well on his way to building an outstanding longer-term record here. Since he took over the fund in January 1995, the fund has earned an annualized 26.9%, versus 18.6% for the mid-cap blend category--so overall it has beaten its average peer by a cumulative 66 percentage points. Margolies has also helped guide other Salomon offerings to strong returns. With less than $250 million combined in this fund's several share classes, he should be able to stick to his flexible--and successful--investment strategy.

Solutions:

1.a. Dan Wheeler is correct. Diversification means smoothing out. You can’t look at portfolio performance over a short period of time to evaluate performance.

b. If risk tolerance is greater, the required rate of return is lower. Hence the stock market will grow at a slower rate.

2. a. Taking a weighted average of the betas of the top ten holdings, we find that the fund has a beta of about 0.93.

b. I assume that the return information for year-to-date is non-annualized. Annualizing it, we get a return of (1.0094)⁶ - 1 = .0577 or 5.77% for the S&P 500. Similarly, for the SACPX, we get (1-.0448)⁶ - 1 = -0.24 or -24%. Consequently, my beta estimate is (9.66-(-24))/(19.74-5.77) = 2.41.
If the data is already annualized, then my beta estimate is (9.66-(-4.48))/(19.74-0.94) = 14.14/18.8 = 0.75.

c. The variance of returns on the portfolio is given by (17.5)²(2/3)² + (20)²(1/3)² + 2(1/3)(2/3)(0.7)(17.5)(20) = 289.44. Hence the standard deviation = (289.44)^0.5 = 17.

d. I would call this fund diversified, since it’s sector weights are fairly similar to that of the S&P. On the other hand, it does have different sector weightings. For example, it is more heavily weighted in technology stocks, relative to the S&P 500; similarly, for health stocks.

e. The required rate of return according to the CAPM, and using the beta estimate for part a. is 0.05 + 0.93(0.21-0.05) = 19.88.

Bonus:
i) Because of the lack of diversification detected in part c. above, I might require additional compensation for the non-diversified non-systematic risk, over and above the % computed in the previous answer

ii) The actual performance still seems to be commendable: 26.9 versus 19.88.

Final

•  The quiz is closed book.

• Time allowed is 2 hours.

• Explain all your steps; correct answers without explanations may not be given any credit at all.

• Show all your computations and formulae.  Make all your assumptions explicit.  If your approach is correct, you will get some credit, even if your arithmetic answer is wrong.  So concentrate on getting your logic right.

• If your answer is not clear and legible, I will deduct points.

 1. (20 points) Read the following piece, and estimate the duration of the 30-year Treasury bond using the information on the price movement of the bond. 

Treasurys Suffer Sharp Losses, Hurt by Strong Economic Data

Wall Street Journal, April 30, 1999; By STEVEN VAMES 

 NEW YORK -- U.S. Treasurys sank Friday after early economic data turned the market bearish, and reports of large sellers extended the negative sentiment.  In late trading, the price of the 30-year Treasury dropped 1 28/32, or $18.75 for a bond with a $1,000 face value, to 94 3/32. Its yield, which moves inversely to its price, soared to 5.657% from 5.522% late Thursday in New York.

 2. Here is some information from Bloomberg.com as of Fri, 30 Apr 1999, 11:48pm EDT.  Assume that these rates are available as of May 1, 1999.  The Federal Funds rate is the rate that banks charge each other on their borrowings. 

 a.       (10 points) Convert these annual rates into monthly rates.  (Hint: the correct way to convert annual rates into monthly rates is not to divide by twelve.)

b.      (20 points) Using these rates, estimate the market’s expectation of the one-month Federal Funds rate for June and July.  Also estimate the market’s expectation of the 3-month Federal Funds rate as of August 1.  Assume that the unbiased expectations hypothesis holds. 

c.       (20 points) Banc First of Boston, Ma. wishes to obtain a commitment from Banc Deuxieme of Paris, Ky. for a loan of two successive tranches of $1m. to be provided by Deuxieme on June 1 and July 1, respectively with a repayment date of August 1.  How much will Banc First have to repay Banc Deuxieme on August 1?

 3. You can buy a 2 year bond paying annual coupons of 5% at a price of $91.322 per $100 of face value, and a 3 year bond paying annual coupons of 12% at a price of $104.974 per $100 of face value. 

a.       (10 points) Compute the duration of the two bonds.

b.      (10 points) If you have to pay $1000 to a client in 2.5 years, how much should you invest in each bond, in order to be fully immunized?  Assume that only parallel shifts of the yield curve are likely. 

c.       (10 points) If you have the option to buy a zero coupon bond yielding 10% and maturing in 2.5 years, would you prefer to invest your money in that bond or in a portfolio of the two- and three-year bonds?

 4. Bonus: Read the following (partially reproduced) article and answer these questions in no more than three sentences each. (3 points each for correct answers; no partial credit)

a)      The article makes the following point early on: “Rising interest rates typically wreak havoc with earnings of financial stocks, especially thrifts. That's because the interest rates they pay on deposits and borrowings goes up as short-term rates rise.”  What does this imply about the relative duration of thrifts’ assets and liabilities?

b)      Can you rely upon duration alone as a measure of interest rate risk in a world where the yield curve is likely to steepen or to become more flat? Why or why not?

c)      If the yield curve is becoming steeper, what can you say about the market’s expectations regarding future interest rates?  Assume that the expectations hypothesis does not hold, but that liquidity premiums remain the same, even as the yield curve changes.

 Taking a Ride on the Steepening Yield Curve Of Perelman's Thrift Entices Some Investors

Heard on the Street column, WSJ New York; Mar 5, 1999; By Gregory Zuckerman

 Bond prices are getting whacked. An interest-rate boost could be coming. So why are some people excited about Golden State Bancorp., the California thrift controlled by financier Ronald Perelman?

Rising interest rates typically wreak havoc with earnings of financial stocks, especially thrifts. That's because the interest rates they pay on deposits and borrowings goes up as short-term rates rise. Climbing rates also weigh on the economy, slowing lending activity.

But longer-term rates have been rising much more quickly than short-term rates lately, a so-called steepening of the yield curve. This is good news for Golden State, because it can now charge higher rates on home mortgages, the thrift's biggest product and about 60% of the company's revenue. A steeper yield curve also makes adjustable-rate mortgages, which comprise 78% of Golden State's loan portfolio, more popular. And as long as rates stay at these higher levels, mortgage refinancings should slow, alleviating pressure on the thrift's balance sheet.

Of course, the yield curve could reverse itself. Critics cite concern about the thrift's relatively high debt level. And there are doubts about how well Golden State will meld a growing stable of thrift franchises.

 Solutions to Final:

 1. Duration is defined as –[the percentage change in bond price/percentage change in (1+yield)] = (18.75/959.6875)/(.00135/1.05522) = 15.27 years, the price dropped $18.75 from an initial price of 959.6875 and the yield rose from 5.522 to 5.657.

 2.

 The given rates are annualized rates.  The monthly rates are computed using the formula: (1+ monthly rate)¹² = 1+annualized rate.
The 1 month forward rate, one month ahead is computed as (1.004376)²/(1.00417) - 1
The 1 month forward rate, two months ahead is computed as (1.003923)³/(1.004376)² – 1
The 3 month forward rate, three months ahead is computed as (1.004273)⁶/(1.003923)³ – 1 = 0.0039336 for three months (i.e. not a monthly rate), which can be converted into a monthly rate by taking the third root of 1.0039336, i.e. (1.0039336)^1/3, which works out to 0.4624%. [Note, if you look at the algebra, this can also be done as (1.004273)²/(1.003923) – 1 = 0.4624%].
The resulting monthly rates are then annualized using the formula above.

 Since the expectations hypothesis holds, the forward rates are also the market’s expectation of the corresponding future spot rates.

 d.      Banc First is looking for a forward commitment of $1m. one month ahead, and another of $1m. two months ahead.  Hence to figure out the repayment amount, we simply use the forward rates: $1m.(1.004583)(1.003018) + $1m.(1.003016) = $2.0106308m.

 3.  a. 2 year, 5% coupon bond

 3 year, 12% coupon bond

 The yield on the bond has to be computed by trial and error, first.  However, since the 2 yr bond is selling at a discount, we know that the yield is higher than 5%.  The 3^rd and 4^th columns show the price computation at yields of 8% and 10%.  Obviously, the yield is 10%, given the price.  The last column computes the duration.  A similar procedure is done for the 3 year bond as well. 

b.      The objective is to create a portfolio that has a duration of 2.5 years.  Since the two bonds both yield 10%, the duration of a combination of the two bonds is simply the weighted average of the durations of the two bonds.  Hence, if the amount to be invested in the 2 yr bond is y, we have 1.95y + 2.7(1-y) = 2.5.  Solving, we find y = 26.67%; 1-y = 73.73%

 c.       I’d prefer the portfolio of two- and three-year bonds, because it has greater dispersion and convexity properties.  However, with the single bond, I’d have lower portfolio rebalancing costs to maintain duration.

 4.a.       Their assets have longer duration than their liabilities.  When interest rates rise, the returns on their assets are locked, and do not change, while they have to pay higher rates on their borrowings.

b.      No, because there is more than one factor affecting interest rates: the level of interest rates and the steepness of the yield curve.

c.       Forward rates are increasing and therefore the markets expectation of future short rates is going up.

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