BA 517 Test
By how many basis points does the interest rate change if it increases from 9% to 12%?
1% = 100 basis points, so 300 basis points
What is the annuity formula?
C1*{1 - [1/(1 + r)]^T}/r
If the first-year interest rate is 2% and the second year interest is 3%, what is the two-year total interest rate?
r0,2 = (1 + r0,1) · (1 + r1,2) - 1 = 1.02 · 1.03 - 1 = 5.06%
An eternal patent contract states that the patentee will pay the patentor a fee of $1.5 million next year. The contract terms state a fee growth with the inflation rate, which runs at 2% per annum. The appropriate cost of capital is 14%. What is the value of this patenting contract?
$1.5 million/(14% - 2%) = $12.5 million
A bond promises to pay $150 in 12 months. The bank quotes you an interest rate of 5% per annum, compounded daily. What is the bond's price today?
$150/(1+(5%/365))^365 = $142.68
A bond promises to pay $150 in 12 months. The annual true interest rate is 5% per annum. What is the bond's price today?
$150/1.05 = $142.86
If you invest $2,000 today and it earns 25% per year, how much will you have in 15 years?
$2,000*1.25^15 = $56,843
If the 5-year holding rate of return is 100% and interest rates are constant, what is the (compounding) annual interest rate?
(1 + 100%)^(1/5) - 1 = 14.87%
If you earn an (effective) interest rate of 12% per annum, how many basis points do you earn in interest on a typical calendar day? (Assume a year has 365.25 days)
(1 + r)^365.25 = 1.12 Therefore, 1.12^(1/365.25) - 1 = 0.0310% or 3.1 basis points
What is the quarterly interest rate if the annual interest rate is 50%?
(1 + r-quarter)^4 = (1 + r). Thus, r-quarter = 4throot(1 + r) - 1 = 1.5^(1/4) - 1 = 10.67%
From memory, write down the relationship between nominal rates of return (rnominal), real rates of return (rreal), and the inflation rate (π)
(1 + rnominal) = (1 + rreal) · (1 + π)
A ten-year and a 1-year zero-bond both offer an interest rate of 8% per annum. 1. How does an increase of 1 basis point in the prevailing interest rate change the value of the 1-year bond? (Use 5 decimals in your calculation.) 2. How does an increase of 1 basis point in the prevailing interest rate change the value of the ten-year bond? 3. What is the ratio of the value change over the interest change? (In calculus, this would be called the derivative of the value with respect to interest rate changes.) Which derivative is larger?
(1) For the 1-year bond, the value of a $100 bond changes from $100/1.0800 ≈ $92.59259 to $100/1.0801 ≈ $92.58402. This is about a -0.009% change. (2) For the ten-year bond, the value of a $100 bond changes from $100/1.0810 ≈ $46.31935 to $100/1.080110 ≈ $46.27648. This is a -0.09% change—ten times that of the 1-year bond. (3) The derivative of the 1-year bond is -0.009/0.01 = -0.9 ≈ -1. The derivative of the ten-year bond is -0.09/0.01 ≈ -9. The derivative of the ten-year bond is about nine times more negative
You can invest in a project with diminishing returns. Specifically, the formula relating next year's payoff to your investment today is C1 = p -C0 , where C0 and C1 are measured in millions of dollars. For example, if you invest $100,000 in the project today, it will return p $0.1 ≈ $0.316 million next year. The prevailing interest rate is 5% per annum. Use a spreadsheet to answer the following two questions: 1. What is the IRR-maximizing investment choice? What is the NPV at this choice? 2. What is the NPV-maximizing investment choice? What is the IRR at this choice?
(1) The IRR-maximizing investment choice of C0 is an epsilon. The IRR is then close to infinity. The NPV is 0. (2) The NPV-maximizing (and best) choice is an investment of $226,757. This also happens to be the project's NPV. The IRR is 110%
A project costs $200 and will provide cash flows of +$100, +$300, and +$500 in consecutive years. The annualized interest rate is 3% per annum over one year, 4% per annum over two years, and 4.5% per annum over three years. What is this project's NPV
- $200 + $100/1.03 + $300/1.04^2 + $500/1.045^3 ≈ $612.60
What is the IRR of a project that costs $1,000 now and produces $1,000 next year?
-$1, 000 + $1, 000/(1 + IRR) = 0 IRR=0%
What is the IRR of a project that costs $1,000 now and produces $500 next year and $500 the year after?
-$1, 000 + $500/(1 + IRR) + $500/(1 + IRR)^2 = 0 IRR = 0%
What is the IRR of a project that costs $1,000 now and produces $600 next year and $600 the year after?
-$1,000 + $600/(1 + IRR) + $600/(1 + IRR)^2 = 0 IRR = 13.07%
What is the IRR of a project that costs $1,000 now and produces $900 next year and $900 the year after?
-$1,000 + $900/(1 + IRR) + $900/(1 + IRR)^2 = 0 IRR = 50%
A new product may be a dud (20% probability), an average seller (70% probability), or dynamite (10% probability). If it is a dud, the payoff will be $20,000; if it is an average seller, the payoff will be $40,000; if it is dynamite, the payoff will be $80,000. The appropriate expected rate of return is 6% per year. If a loan promises to pay off $40,000, what are the promised and expected rates of return?
.22 With 20% probability, the loan will pay off $20,000; with 80% probability, the loan will pay off the full promised $40,000. Therefore, the loan's expected payoff is 20% · $20,000 + 80% · $40,000 = $36,000. The loan's price is $36,000/1.06 ≈ $33,962. Therefore, the promised rate of return is $40,000/$33,962 - 1 ≈ 17.8%. The expected rate of return was given 6%.
From memory, write down the equation that defines IRR
0 = C0 + C1 /(1 + IRR) + C2/(1 + IRR)^2 + C3 (1 + IRR)^3 + ··
If the risk-free rate of return is 4% per annum, how big is the difference between the arithmetic and geometric average rate of return?
0, because the risk-free rate has no standard deviation
What is the annualized holding rate of return and average rate of return for an asset that returns -10% and 20% in alternate years?
0.9*1.20-1=8% So, annualized rate of return is sqrt(1.08)-1=3.92% Average rate of return is 5%
What is the market beta of the market?
1-it becomes the slope of the 45 degree diagonal line
What is the annualized holding rate of return and average rate of return for an asset that returns 0% and 10% in alternate years?
1.00*1.10-1=10% So, annualized rate of return is sqrt(1.1)-1=4.88% Average rate of return is 5%
If you earn a rate of return of 5% over 4 months, what is the annualized rate of return?
1.0512/4 ≈ 15.76%
The nominal interest rate is 20%. Inflation is 5%. What is the real interest rate?
1.20/1.05 ≈ 1.1429. The real interest rate is 14.29%
If the 1-year rate of return is 20% and interest rates are constant, what is the 5-year holding rate of return?
1.20^5 - 1 = 148.83%
If the cost of capital is 5% per annum, what is the discount factor for a cash flow in two years?
1/[(1.05)(1.05)] = 0.9070
If the bank quotes an interest rate of 12% per annum, and there are 52.2 weeks, how much interest do you earn on a deposit of $1,000 over 1 week?
12%/365 = 0.032877% per day. Weekly rate of return is (1 + 0.032877%)^7 - 1 = 0.23036%. You earn more money when the 12% is the quoted rate than when it is the effective rate.
A new product may be a dud (20% probability), an average seller (70% probability), or dynamite (10% probability). If it is a dud, the payoff will be $20,000; if it is an average seller, the payoff will be $40,000; and if it is dynamite, the payoff will be $80,000. 1. What is the expected payoff of the project? 2. The appropriate expected rate of return for such payoffs is 8%. What is the PV of the payoff? 3. If the project is bought for the appropriate present value, what will be the rates of return in each of the three outcomes? 4. Confirm the expected rate of return when computed from the individual outcome-specific rates of return.
14 For the dynamite/dud project: 1. The expected payoff is E (P) = 20%·$20,000+70%·$40,000+ 10% · $80,000 = $40,000. 2. The present value of the expected payoff is $40,000/1.08 ≈ $37,037. 3. The three rate of return outcomes are $20,000/$37,037 - 1 ≈ -46%, $40,000/$37,037 - 1 ≈ +8%, $80,000/$37,037 - 1 ≈ +116%. 4. The expected rate of return is 20% · (-46%) + 70% · (+8%) + 10% · (+116%) ≈ 8%
What changes have to be made to the NPV formula to handle an uncertain future?
2 The actual cash flow is replaced by the expected cash flow, and the actual rate of return is replaced by the expected rate of return.
If an interest rate of 10% decreases by 20 basis points, what is the new interest rate?
20 basis points are 0.2%, so interest rate declined from 10% to 9.8%
A factory can be worth $500,000 or $1,000,000 in two years, depending on product demand, each with equal probability. The appropriate cost of capital is 6% per year. What is the present value of the factory?
3 The factory's expected value is E Value at Time 2 = [0.5 · $500,000 + 0.5 · $1,000,000] = $750,000. Its present value is therefore $750,000/1.062 ≈ $667,497.33
What is the annualized holding rate of return and average rate of return for an asset that returns 5% each year?
5% for both
Are investors more risk-averse for small bets or for large bets? Should "small" be defined relative to investor wealth?
6 Investors are more risk-averse for large bets relative to their wealth
You are considering moving into a building for three years, for which you have to make one payment now, one in a year, and a final one in two years. A) Would you rather have a lease, paying $1,000,000 upfront, then $500,000 each in the following two years; or would you rather pay $700,000 rent each year? B) If the interest rate is 10%, what equal payment amount (rather than $700,000) would leave you indifferent?
A) Rent vs lease preference depends on the interest rate. If interest rate is zero, then you would prefer $2M sum lease payments to the $2.1 million sum rent payments. If interest rate is less than 21.5% it is better to lease. B) At 10% interest, total net present cost of lease is $1 + 0.5/1.1 + 0.5/1.1^2 = 1.868M. x + x/(1.1) + x/(1.1^2) = 1.868M — so the equivalent rental cost would be $682.864
The common shares of networking giant Cisco Systems (CSCO) recently traded on NASDAQ for $22.64 per share. You have employee stock options to purchase 1,000 CSCO shares for $22/share. The options expire in three years. The annualized volatility of CSCO stock according to Robert's Historical Stock Volatilities in a recent month was 31.41 percent. The company's dividend yield is 3.0 percent, and the interest rate is 2.5 percent. The options are European options that may only be exercised at the maturity date. A. Is this option a call or a put? B. Using a calculator, estimate the value of your CSCO options (www.intrepid.com/robertl/option-pricer1.html) C. What is the estimated value of the options if their maturity date is five months instead of three years? Why does the value of the options decline as the maturity declines? D. What is the estimated value of the options if their maturity is three years, but CSCO's volatility is 45%? Why does the value of the options increase as volatility increases?
A. Call option, you have the option to buy B. Value of the option per share is $4.55. With 1,000 options, they are worth $4,550 C. $2.09 x 1,000 options = $2,090. Much better chance CSCO stock will rise to high levels in three years than in five D. $6.37 x 1,000 options = $6,370. The more volatile CSCO stock, the more likely it will rise to high levels during the life of the option.
What is the NPV capital budgeting rule?
Accept if NPV is positive. Reject if NPV is negative
Is a deposit into a savings account more like a long-term bond investment or a series of short-term bond investments?
An investment in a series of short-term bonds
If the bank quotes an interest rate of 12% per annum (not as an effective interest rate), how many basis points do you earn in interest on a typical day?
Bank pays 12%/365.25 = 3.28 basis points per day
If the bank quotes an interest of 6% per annum, what does a deposit of $100 in the bank come to after one year?
Bank quote of 8% means you have to pay an interest rate of 6%/365 = 0.0164% per day. This earns an actual interest rate of (1 + 0.0164%)^365 - 1 = 6.18% per annum. Each invested $100 grows to $106.18, thus earning $6.18 over the year
If the bank quotes a loan APR rate of 8% per annum, compounded monthly, and without fees, what do you have to pay back in one year if you borrow $100 from the bank?
Bank quote of 8% means you have to pay an interest rate of 8%/12 = 0.667% per month. This earns an actual interest rate of (1 + 0.667%)^12 - 1 = 8.30% per annum. You will have to pay $108.30 in repayment for every $100 borrowed.
What are the three types of Treasuries? How do they differ?
Bills, notes, and bonds. T-bills have maturities of less than 1 year. T-notes have maturities from 1 to 10 years. T-bonds have maturities greater than 10 years.
Does the historical evidence show that lower-grade borrowers default more often or that they pay less upon default?
Both. The historical evidence is that lower-grade borrowers both default more often and pay less upon default.
What are the two main functions of brokerage firms?
Brokers execute orders and keep track of investors' portfolios. This facilitates purchasing on margin.
Rental agreements are not much different from mortgages. For example, what would your rate of return be if you rented your $500,000 warehouse for 10 years at a monthly lease payment of $5,000? If you can earn 5% per annum elsewhere, would you rent out your warehouse?
C1 (1 - [1/(1 + r)]^T)/r = $5 (1 - [1/(1 + 0.005)]^360)/0.005 ≈ $833.9
From memory, write down the growing perpetuity formula
C1/(r - g)
Write down the perpetuity formula.
C1/r. The first cash flow occurs next period, not this period
What can you see in a compound return graph that is not in the time-series graph?
Compound return graph shows how a time series of rates of return interacts to produce long-run returns. (You can see whether a lon-run investment would have made or lost money)
How does a crossing system differ from an electronic exchange?
Crossing system does not execute trades unless there is a counterparty. Tries to cross orders a few times a day.
In computing the cost of your M.B.A., should you take into account the loss of salary while going to school? What are non monetary benefits you reap as a student? Try to attach a monetary value to them.
Definitely yes, forgone salary can be estimated. Nonmonetary benefits include reputation, education, pleasure from excessive beer consumption.
How do shares disappear from the stock exchange?
Delisting or a repurchase
How easy is it to value a painting?
Depends. Warhol painted many similar works and the law of one price could work. Other paintings, like the Mona Lisa, are difficult because Da Vinci painted a lot of pieces but Mona Lisa is unique.
You buy a stock for $40 per share today. It will pay a dividend of $1 next month. If you can sell it for $45 right after the dividend is paid, what would be its dividend yield, what would be its capital gain, and what would be its total rate of return?
Dividend yield would be $1/$40 = 2.5% Capital gain is $45 - $40 = $5 Capital gain yield is $5/$40 = 12.5% Total rate of return would be ($45 - $40)/$40 = 15%
How do shares disappear from the public financial markets back into the pockets of investors?
Dividends and share repurchases
What is the formula for a promised loan payoff between $80 and $90?
E Payoff( $80 ≤ Loan Promise = x ≤ $90) = $60 + $8 + $6 + 40% · (x - $80)
If the U.S. stock market is efficient, how do you explain the fact that some people make very high returns? Would it be more difficult to reconcile very high returns with efficient markets if the same people made extraordinary returns year after year?
Earning high returns in an efficient market is like winning at roulette. In any random process, there will be winners and losers, and some winners might win big. Earning consistently high returns over time is also possible in an efficient market, just like a gambler on a lucky streak might win repeatedly at roulette. The relevant questions are whether the very high returns or the length of the winning streak is inconsistent with blind luck or not. The continued investment success of Warren Buffett and his associated value investors does pose a challenge to market efficiency. The argument that this success is just luck stretches credulity. A more likely explanation is that high intelligence, extreme emotional discipline, and driven dedication do enable some people to earn superior market returns. At the same time, evidence tells us that these individuals are extremely few and far between, and that it is virtually impossible to identify these individuals in advance with any reliability. It should also be noted that Warren Buffett is not a passive stock picker, that much like a private equity firm, Berkshire Hathaway, adds considerable value to acquired companies via changes in operations, management, incentives, and governance practices.
How easy is it to value and envelope containing foreign currency?
Easy. Many foreign currencies and exchange rates.
How easy is it to value foreign stamps?
Easy. Stamp collectors know and usually publish the prices.
For what kind of bonds are expected and promised interest rates the same?
Expected and promised rates are the same only for riskfree (i.e., government) bonds. Most other bonds have some kind of default risk—though even the U.S. Treasury is now rated to have some credit risk
Work out the present value of your tuition payments for the next two years. Assume that the tuition is $30,000 per year, payable at the start of the year. Your first tuition payment will occur in 6 months, and your second tuition payment will occur in 18 months. You can borrow capital at an effective interest rate of 6% per annum.
First tuition payment is worth $30,000/(1.06)^0.5 = $29,139. Second tuition payment is worth $30,000/(1.06)^(3/2) = $27,489. Thus, total present value is $56,628
Why do you suppose that smaller firms tend to rely on bank financing while larger companies are more apt to sell bonds in financial markets?
Flotation costs from issuing bonds are more expensive than bank borrowing. There is a longer procedure for issuing bonds than bank borrowing Less interest in small firm bonds from investors Bonds have longer maturity periods, interest payable for a longer period on bonds
Give an example of a problem that has multiple IRR solutions.
For example, C0 = -$100,C1 = +$120,C2 = -$140,C3 = +$160,C4 = -$20. (The solutions are IRR ≈ -85.96% and IRR ≈ +$9.96%. The important aspect is that your example has multiple inflows and multiple outflows.)
Give an example of a project that has no IRR.
For example, C0 = -$100,C1 = -$200,C2 = -$50. No interest rate can make their present value equal to zero, because all cash flows are negative. This project should never be taken, regardless of cost of capital
Would it be good or bad for you, in terms of the present value of your liabilities, if your opportunity cost of capital increased?
Good. Your future payments would be worth less in today's money.
What is the holding period return on a bond with a par value of $1,000 and a coupon rate of 6 percent if its price at the beginning of the year was $1,050 and its price at the end was $940? (Interest paid annually)
HPR = [($1,000 X 6%) - ($940 - $1,050)]/$940 = -4.76%
Is your rate of return higher if you short a stock in the perfect world or in the real world? Why?
Higher in the perfect world because you earn interest on the proceeds. Real world brokers often take the interest.
Why has the average annual rate or return on common stocks exceeded the return on government bonds in the US?
Higher risk in common stock yields higher return. Fixed rate of interest on bonds, so they do not have risk of losing their interest earned on investments. No risk and low fixed rate of interest for government bonds. Lower interest for short term investments. Short term investments are made in the money market. Money market instruments carry lower rate of interest than capital market instruments. So, the short term investments and consumer price index hav lower rate of return than long term investments and common stocks.
What can you see in a histogram that is difficult to see in a time-series graph?
Histogram makes it easier to see how frequent different types of outcomes are—and thus, where the distibution is centered and how spread out it is.
Some refer to common stock in a company with debt outstanding as an option on a company's assets. Do you see any logic to this statement? What is the logic, if any?
I see the logic. Equity is basically a call option on the company's assets with a strike price equal to the value of debt outstanding. When the value of company assets is low, equity holders call option is out of the money. If desired they can walk away, leaving their option unexercised and firm assets in the hands of creditors. When the value of company assets exceeds value of debt, the owners' call option is in the money. They can exercise their option by paying the value of the debt to creditors and owning the assets free and clear.
What are the main mechanisms by which money flows from investors into firms?
IPOs and SEOs Also reverse mergers, which are then sold off to investors
A project has cash flows of -$1,000, -$2,000, -$3,000, +$4,000, and +$5,000 in consecutive years. Your cost of capital is 20% per annum. Use the IRR rule to determine whether you should take this project. Confirm your recommendation using the NPV rule.
IRR = 19.73%, lower than 20% cost of capital, so reject NPV is -$23.92, negative so reject Same recommendation -- reject
A project has cash flows of -$1,000, -$2,000, +$3,000, and +$4,000 in consecutive years. Your cost of capital is 30% per annum. Use the IRR rule to determine whether you should take this project. Does the NPV rule recommend the same action?
IRR = 56.16%, higher than cost of capital, so accept NPV = $1,057.35, positive so accept Same recommendation -- accept
A project has cash flows of +$200, -$180, -$40 in consecutive years. The prevailing interest rate is 5%. Should you take this project?
IRR = 8.44%. This is above the prevailing interest rate. However, the cash flows are like that of a financing project. This means that it is a negative NPV project of -$7.71. You should not take it.
In June, 2016, an inflation-adjusted 30-year Treasury bond offered a real yield of about 0.7% per year. The equivalent non-inflation-adjusted bond offered 2.25% per year. In what inflation scenario would you be better off buying one or the other? (The most recent historical inflation rate was 1% per year.)
If the inflation rate will increase to more than 1.0225/1.007 - 1 ≈ 1.5% per year, the inflation-adjusted bond will be better. Otherwise, the non-inflation adjusted bond will be better
For an ordinary die, assume that the random variable is the number on the die times two. Say the die throw came up with a "six" yesterday. What was its expected outcome before the throw? What was its realization?
If the random variable is the number of dots on the die times two, then the expected outcome is 1/6 · (2)+1/6 · (4)+1/6 · (6)+ 1/6 · (8) + 1/6 · (10) + 1/6 · (12) = 7. The realization was 12
Could it be that the expected value of a bet is a random variable?
If you do not know the exact bet, you may not know the expected value, which means that even the expected value is unknown. This may be the case for stocks, where you are often forced to guess what the expected rate of return will be (unlike for a die, for which you know the underlying physical process, which assures an expected value of 3.5). However, almost all finance theories assume that you know the expected value. Fortunately, even if you do not know the expected value, finance theories hope you still often have a pretty good idea
You have $500 and really, really want to go to the Superbowl tonight (which would consume all your cash). You cannot wait until your project completes: This project would cost $400 and offer a rate of return of 15%, although equivalent interest rates are only 10%. If the market is perfect, what should you do?
If you invest $400, the project will give you $400*1.15 = $460 next period. The capital markets will value the project at $460/1.10 ≈ $418.18. You should take the project and immediately sell it for $418.18. Thereby, you will end up being able to consume $500 - $400 + $418.18 = $518.18
What should happen if the holdings of an open-end fund are worth much more than what the shares of the fund are trading for? What should happen in a closed-end fund?
In an open-end fund, you should buy fund shares and request redemption (consider shorting the underlying holdings while you wait for the redemption to not suffer price risk). In a closed-ended fund, you would have to oust the management to allow you to redeem your shares.
If you buy a house and live in it, what are your inflows and outflows?
Inflows: Value of implicit rent, capital gain if house appreciates Outflows: Maintenance costs, transaction costs, mortgage costs, real estate tax, uninsured potential losses, capital loss if house depreciates, etc.
Suppose the realized rate of return on government bonds exceeded the return on common stocks one year. How would you interpret this result?
It is not evidence that investors are willing to settle for lower returns on stocks than bonds. It means that investors' expectations were not met. By taking on more risk, investors need additional EXPECTED return, but this is different than a realized return. Stock returns will fluctuate form year to year, but the expected returns on common stock will always be higher than the expected returns on government bonds.
Interpret the meaning of the discount factor.
It is today's value in dollars for 1 future dollar, that is, at a specific point in time in the future.
How easy is it to value the U.S. Presidency?
Kind of. We know how much candidates spend to win the election, however it can vary quite a bit.
A project lost one-third of its value each year for 5 years. What was its total holding rate of return? How much is left if the original investment was $20,000?
Losing one-third is a ROR of -33%. So holding rate of return = (1 + (-1/3)^5) - 1 = -86.83% About (1 - 86.83%)($20,000) = $2,633.74 remains
Rank the following asset categories in terms of risk and reward: cash (money market), long-term bonds, the stock market, and a typical individual stock.
Lowest for cash, then bonds, then stock market portfolio, then individual stocks. Average reward increases with greater risk (though not usually for individual stock)
How easy is it to value Manhattan?
Many individual buildings in Manhattn have sold, so you have decent comparables. However, nobody has done that before so there is no historical data for a purchase that large.
What is the main objective of corporate managers that this book assumes?
Maximizing the value of the firm
Describe some alternatives to trading on the main stock exchanges.
Most alternatives are electronic and they rely on matching trades, so they don't execute trades unless they can match them. Most of these are called Electronic Communication Networks Another alternative is over-the-counter (OTC) market, which is a network of geographically dispersed dealers who make markets in various securities.
Write down the NPV formula from memory.
NPV = SUM[(cash flows)/(1 + i)^n]
What are the perfect market assumptions?
No taxes, no transaction costs, no differences in opinions, and no large buyers or sellers
In the example, the building was worth $75, the mortgage was worth $70, and the equity was worth $5. The mortgage thus financed about 93.3% of the cost of the building, and the equity financed 6.7%. Is the arrangement identical to one in which two partners purchase the building together—one puts in $70 and owns 93.3% of the building, and the other puts in $5 and owns 6.7
No! Partners would share payoffs proportionally, not according to "debt comes first." For example, if it rains, the 6.7% partner would still receive $4, and not $0 that the levered equity owner would receive.
Is the expected outcome (value) of a die throw a random variable?
No! The expected outcome (value) is assumed to be known—at least for an untampered die throw. The following is almost philosophy and beyond what you are supposed to know or answer here: It might, however, be that the expected value of an investment is not really known. In this case, it, too, could be a random variable in one sense—although you are assumed to be able to form an expectation (opinion) over anything, so in this sense, it would not be a random variable, either
You see an article in the newspaper that details the performance of mutual funds over the last five years. You see that, out of 5,600 actively managed mutual funds in the study, 104 outperformed the market in each of the last five years. The author of the article argues that these mutual funds are examples of market inefficiency. "If markets are efficient, you would expect to see mutual funds outperforming the market for short periods of time. But when more than 100 mutual funds are able to outperform the market in each of the last five years, you can no longer suppose that markets are truly efficient. Obviously, these 100 fund managers have figured out a way to beat the market every year." Do you think that this is evidence that markets are not efficient?
No, mutual funds involve larger portfolio and are random. Almost half of the mutual funds can outperform the market.
Is the expected default premium positive?
No, the expected default premium is zero by definition
One month ago, a firm suffered a large court award against it that will force it to pay compensatory damages of $100M next January 1. Are shares in this firm a bad buy until January 2?
No, the market price will have already taken the compensatory damages into account in the share price a month ago, just after the information had become public.
How easy is it to value the Washington Monument?
Not super easy because it is worth more than other buildings around it.
What is an OTC market?
OTC is not really a market. It means that traders handle transactions on a one-on-one basis.
Give two reasons the price of a bond might fall over a year.
One reason is the increase of interest rates, which would lead to a decrease in the price of a bond. Another reason is because of a default premium on a bond. Increases in default premium causes the expected return on the bond to increase and decrease the price of the bond.
The price of a bond that offers a safe promise of $100 in one year is $95. What is the implied interest rate? If the bond's interest rate suddenly jumped up by 150 basis points, what would the bond price be? How much would an investor gain/lose if she held the bond while the interest rate jumped up by these 150 basis points?
Original interest rate is $100/$95 - 1 = 5.26%. Increasing the interest rate by 150 basis points is 6.76%. This means that the price should be $100/(1.0676) = $92.67. A price change from $95 to $93.67 is a rate of return of $93.67/$95 - 1 = -1.40%
In Britain, there are Consol bonds that are perpetuity bonds. (In the United States, the IRS does not allow companies to deduct the interest payments on perpetual bonds, so U.S. corporations do not issue Consol bonds.) What is the value of a Consol bond that promises to pay $2,000 per year if the prevailing interest rate is 4%?
PV = $2,000/4% = $50,000
What is the PV of a perpetuity paying $5 each month, beginning next month, if the monthly interest rate is a constant 0.5%/month?
PV = C1/r = $5/0.005 = $1,000
A company wants to raise $500 million in a new stock issue. It's investment banker indicates that the sale of a new stock will require 8 percent underpricing and a 7 percent spread. A. Assuming the company's stock price does not change from its current price of $75/share, how many shares must the company sell and at what price to the public? B. How much money will the investment banking syndicates earn on the sale? C. Is the 8 percent underpricing a cash flow? Is it a cost? If so, to whom?
Part A) Stock price: $75 Underpricing 8%: 75*0.08 = $6 Issue price: $75 - $6 = $69 7 percent spread: $69*0.07 = $4.83 Sale Price: $69 - $4.83 = $64.17 Number of shares needed: $500,000,000/$64.17 = 7,791,803 shares Part B) Investment banking earnings = (#Shares issued)(Spread) IBE = (7,791,803)($4.83) IBE = $37,634,408 Part C) Underpricing is cost and not cash flow. It is opportunity cost for owner which indicates that company can issue higher number of shares to raise its required capital.
How easy is it to value the Chrysler Building in New York?
Pretty easy. There are many similar buildings that have been sold in the last few years.
What is most important to investors: the number of a company's shares they own, the price of the company's stock, or the value of their shareholding's in the company?
Price of a company's stock is most important to investors—price to earnings ratio and price to book ratio are derived with this price. Stock price influences company's true value. This dictates dividend amounts and therefore it is more important than the number of shares they own and the percentage of the company's equity.
An investment costs $1,000 and pays a net return of $25. What is the rate of return?
R = $25/$1,000 = 2.5%
What are the units on rates of return, discount factors, future values, and present values?
Rate of return and additional factors are unit-less. The latter two are in dollars
How do you graph a "market beta"? What is an individual data point?
Rate of return on the market on x-axis, rate of return on the investment on the y-axis. A data point is the two rates of return from the same time period (i.e. over a year). Market beta becomes the slope of the best fitting line.
Under what interest rates would you prefer a perpetuity that pays $2 million per year beginning next year to a one-time payment of $40 million?
Rearrange P = C1/r into r = C1/P = $2/$40 = 5%. At a 5% interest rate, you are indifferent. If the interest rate is above 5%, the immediate one-time payment is better, because future cash flows are less valuable. If the interest rate is below 5%, the perpetuity payment is better, because future cash flows are more valuable.
Although a two-year project had returned 22% in its first year, overall it lost half of its value. What was the project's rate of return after the first year?
Solve (1 + x) · (1 + 22%) = (1 - 50%), so the project had a rate of return of -59.00%
What is a specialist? What is a market maker? When trading, what advantage do the two have over you?
Specialist is a monopolist who makes the market on the NYSE, selling and buying from their own inventory (hence "making a market). Market makers are the equibalent on NASDAQ, but there are usually many and they compete. They both have the advantage of seeing limit orders placed by other investors.
If you want to borrow $65, what do you have to promise?
The $65 today requires an expected payoff of 1.2 · $65 = $78. This is on the final line segment. The formula is E Payoff( $90 ≤ Loan Promise = x ≤ $100) = $60 + $8 + $6 + $4 + 20% · (x - $90) = $78 + 20% · (x - $90) Thus, x = $90
Read the Bureau of Labor Statistics' website descriptions of the CPI and the PPI. How does the CPI differ conceptually from the PPI? Are the two official rates different right now?
The CPI is the average price change to the consumer for a specific basket of goods. The PPI measures the price that producers are paying. Taxes, distribution costs, government subsidies, and basket composition drive a wedge between these two inflation measures
The prevailing interest rate is 5% over the first year and 10% over the second year. That is, over two years, your compounded interest rate is (1 + 5%) · (1 + 10%) - 1 = 15.5%. Your project costs $1,000 and will pay $600 in the first year and $500 in the second year. What does the IRR rule recommend? What does the NPV rule recommend?
The IRR is 6.81%. This is between the one-year 5% and the two-year 10% interest rates. Therefore, the IRR capitalbudgeting rule cannot be applied. The NPV rule gives you -$1, 000+ $600/1.05 + $500/1.155 ≈ $4.33, so this is a good project that you should take
What is the YTM of a 5-year zero-bond that costs $1,000 today and promises to pay $1,611?
The YTM is 10%, because -$1, 000 + $1, 611/1.10^5 ≈ 0
Assume that the two-year holding rate of return is 40%. The average (arithmetic) rate of return is therefore 20% per year. What is the annualized (geometric) rate of return? Is the annualized rate the same as the average rate?
The annualized rate of return is sqrt(1.4) - 1 ≈ 18.32%. It is therefore lower than the 20% average rate of return
Is the compounded rate of return higher or lower than the sum of the individual rates of return? Is the annualized rate of return higher or lower than the average of the individual rates of return? Why?
The compounded rate of return is always higher than the sum, because you earn interest on interest. The annualized rate of return is lower than the average rate of return, again because you earn interest on the interest. For example, an investment of $100 that turns into an investment of $200 in two years has a total holding rate of return of 100%—which is an average rate of return of 100%/2 = 50% and an annualized rate of return of sqrt(1 + 100%) - 1 ≈ 41.42%. Investing $100 at 41% per annum would yield $200, which is lower than 50% per annum.
What is the YTM of an x% annual level-coupon bond whose price is equal to the principal paid at maturity? For example, take a 5-year bond that costs $1,000 today, pays 5% coupon ($50 per year) for 4 years, and finally repays $1,050 in principal and interest in year 5
The coupon bond's YTM is 5%, because -$1, 000 + $50/1.05 + $50/1.05^2 + $50/1.05^3 + $50/1.05^4 + $1, 050/1.05^5 = 0. The YTM of such a bond (annual coupons) is equal to the coupon rate when a bond is selling for its face value.
How does a prime broker differ from a retail broker?
The difference is prime brokers are used by larger investors because they allow investors to employ their own traders to execute trades. Prime brokers then provide the portfolio accounting, margin, and securities borrowing.
A stock that has the following probability distribution (outcome P+1 ) costs $50. Is an investment in this stock a fair bet?
The expected value of the stock investment is 5% · ($41) + 10%· ($42)+20%· ($45)+30%· ($48)+20%· ($58)+10%· ($70)+ 5% · ($75) = $52. Therefore, buying the stock at $50 is not a fair bet, but it is a good bet
What is the main assumption that allows you to consider investment (project) choices without regard to when you need wealth (or how much money you currently have at hand)?
The fact that you can use capital markets to shift money back and forth without costs allows you to consider investment and consumption choices independently
From the closing of December 31, 2009 to December 31, 2015, Vanguard's S&P 500 fund (which received and paid dividends on the underlying constituent stocks to its fund investors, but charged administration fees) returned the following annual rates of return: 2010 2011 2012 2013 2014 2015 15.0% 2.1% 16.0% 32.3% 13.7% 1.3% What was the rate of return over the first 3 years, and what was it over the second 3 years? What was the rate of return over the whole 6 years? Was the realized rate of return time-varying?
The first three-year compounded rate of return was r2010,2012 ≈ (1 + 0.150) · (1 + 0.021) · (1 + 0.16) - 1 ≈ +36.2%. (The notation is a bit ambiguous when month and day are omitted, because the first return is from the end of 2009 to the end of 2010.) The second three-year rate was r2013,2015 ≈ +52.42%. The full six-year compounded rate of return was thus r2010,2015 ≈ (1+.362%)·(1+.524%)-1 ≈ +107.6%. Although these were very fat years for stock investors. the realized rate was indeed time-varying
What is the PV of a perpetuity paying $15 each month, beginning next month, if the effective annual interest rate is a constant 12.68% per year?
The interest rate is 1.1268(1/12) -1 ≈ 1% per month. Thus, PV = C1/r ≈ $15/0.01 ≈ $1,500
Assume that the 3% level-coupon bond discussed in this chapter has not just 5 years with 10 payments, but 20 years with 40 payments. Also, assume that the interest rate is not 5% per annum, but 10.25% per annum. What are the bond payment patterns and the bond's value?
The interest rate is 5% per half-year. Be my guest if you want to add 40 terms. I prefer the annuity method. The coupons are worth PV Coupons = C1 (1 - [1/(1 + r)]^T)/r = $1,500 (1 - [1/(1.05)]^40)/0.05 ≈ $25,738.63 The final payment is worth PV(Principal Repayment) = $100,000/(1.05)^40 ≈ $14,204.57. Therefore, the bond is worth about $39,943.20 today
If the real interest is 3% per annum and the inflation rate is 8% per annum, then what is the present value of a $500,000 nominal payment next year?
The nominal interest rate is 1.03 · 1.08 - 1 = 11.24%. Therefore, the cash flow is worth about $500,000/1.1124 ≈ $449,479
Magenta Corporations wants to raise $50 million in a seasoned equity offering, net of all fees. Magenta stock currently sells for $10/share. The underwriters will require a spread of $0.50/share, and indicate that the issue must be underpriced by 5 percent. In addition to the underwriter's fee, the firm will incur $1,000,000 in legal, accounting, and other costs. How many shares must Magenta sell?
The price will be set at 5% below the current price, or at 0.95 × 10 = $9.50. The underwriters will take $0.50 per share, leaving $9.00/share for Magenta. Magenta needs to receive $51 million (in order to have $50 million net of all fees), and it gets $9/share, so it must sell $51 million/$9=5.667 million shares
What are the problems with the IRR computation and criterion?
The problems are (a) you need to get the sign right to determine whether you should accept the project above or below its hurdle rate; (b) you need to make sure you have only one unique IRR (or work with a more complicated version of IRR, which we have not done); (c) you cannot use it to compare different projects that have different scales; and (d) you must know your cost of capital.
What is the expected payoff if the promised payoff is $72?
The relevant line segment (and numeric answer) are E = $68 + 60% · ($72 - $70) = $69.20.
Define holding period return
The return an investor earns on a bond over a period of time [(interest income +/- change in the bond's price)/(beginning bond price)
A project lost one-third of its value the first year, then gained fifty percent of its value, then lost two-thirds of its value, and finally doubled in value. What was the average rate of return? What was the investment's overall four-year rate of return? If one is positive, is the other, too?
The returns were (-33%,+50%, -67%,+100%). Thus the average rate was 12.5% and the overall rate of return was -33.33%. It is always true that the compound rate of return is always less than the average rate of return. The example shows that the two can differ in sign.
If the total holding interest rate is 50% for a five-year investment, what is the annualized rate of return?
The six-year holding rate of return was 107.6%. Thus, the annualized rate of return was r8 = 6root(1 + 107.6%) - 1 ≈ 12.9%.
A project has cash flows of -$100, $55, and $70 in consecutive years. Use a spreadsheet to find the IRR
The spreadsheet function is called IRR(). The answer pops out as 15.5696%. Check: -$100 + $55/1.16 + $70/1.16^2 ≈ 0
Reconsider the stock investment from Question 6.4. What is its risk—that is, what is the standard deviation of its outcome P+1
The variance of the P+1 stock investment is Var P+1 = 5%·($41-$52)^2+10%·($42-$52)^2+20%·($45-$52)^2+30%·($48- $52)^2 + 20% · ($58 - $52)^2 + 10% · ($70 - $52)^2 + 5% · ($75 - $52)^2 = 5% · $121 + 10% · $100 + 20% · $49 + 30% · $16 + 20% · $36+10%·$324+5%·$529 = $96.70. Therefore, the standard deviation (risk) is Sdv P+1 = sqrt($96.70) ≈ $9.83
What is the PV of a perpetuity paying $8 each month, beginning this month (in 1 second), if the monthly interest rate is a constant 0.5%/ month (6.2%/year) and the cash flows will grow at a rate of 0.8%/month (10%/year)?
This is a nonsensical question, because the value would be infinite if g>/=r
What can you see in a time-series graph that is not in a histogram?
Time-series graph shows how individual years matter.
What is the holding rate of return for a 20-year investment that earns 5% per year each year? What would a $200 investment grow into?
Total holding rate of return = 1.05^20 - 1 = 165.33% You end up with $200(1 + 165.33%) = $530.66
How easy is it to value yourself?
Tough. Are you a collection of cash flows? Insurance companies attach value to life. You also attach value to life in the how often you risk it (crossing the street, snowboarding, motorcycling, etc). Complicated.
At a constant rate of return of 6% per annum, how many years does it take you to triple your money?
Tripling is equivalent to earning a rate of return of 200% (1 + 6%)^X = (1 + 200%), or Xlog(1.06) = log(3.00) X = 18.85 years
What are the three main types of investment companies as defined by the SEC? Which is the best deal in a perfect market?
UITs, open-ended funds (mutual funds), and closed-ended investment funds. In a perfect market, none is the best deal. You always get what you pay for.
Is the average individual stock safer or riskier than the stock market?
Usually individual stocks are riskier
How easy is it to value the species Chimpanzee?
Very difficult. Governments spend a lot to protects certain species. Yangtze river dolphin just went extinct. Nothing to compare that to.
If there were infinitely many possible outcomes (e.g., if the building value followed a statistical normal distribution), what would the graph of expected payoffs of the loan as a function of promised payoffs look like?
With infinitely many possible outcomes, the function of expected payoffs would be a smooth increasing function. For the mathematical nitpickers: [a] We really should not allow a normal distribution, because the value of the building cannot be negative; [b] The function would increase monotonically, but it would asymptote to an upper bound.
Recompute the example from the text, but assume now that the probability of receiving full payment in one year on a $200 investment of $210 is only 95%, the probability of receiving $100 is 1%, and the probability of receiving absolutely no payment is 4%. 1. At the promised interest rate of 5%, what is the expected interest rate? 2. What interest rate is required as a promise to ensure an expected interest rate o
With the revised probabilities: 1. The expected payoff is now 95% · $210 + 1% · $100 + 4% · $0 = $200.50. Therefore, the expected rate of return is $200.50/$200 = 0.25%. 2. You require an expected payoff of $210 to expect to end up with 5%. Therefore, you must solve for a promised payment 95%·P+1%·$100+4%·$0 = $210 ⇒ P = $209/0.95 = $220. On a loan of $200, this is a 10% promised interest rat
How easy is it to value love?
Yeesh.
Is it possible for an investment to have a positive average rate of return, but still lose you every penny?
Yes (see UAL in Exhibit 7.6 if you want proof)
Can you use the law of one price in your decision of whether to take or reject projects?
Yes most of the time. The law of one price is the foundation which all project choice is based.
Is 10 the same as 1,000%?
Yes, 10 = 1,000%
Is the distance between annualized and average rate of returns larger when there is more risk?
Yes, because it is greater in a more volatile market
Can the aggregate financial market be less risk-averse than each of its individual investors?
Yes, individual investors are typically more risk-averse than investors in the aggregate. This can even be the case for all investo
Compute the three-year holding rate of return on December 31, 2015. Then, using the two-year holding rate of return on December 31, 2015, and your calculated three-year holding rate of return, compute the forward interest rate for a 1-year investment beginning on December 31, 2017, and ending on December 31, 2018. Are these the numbers in Exhibit 5.3?
Yes. The answers are right in the table. The threeyear rate of return is 1.01313 - 1 ≈ 3.98%. The forward rate is 1.0398/(1.0065 · 1.0147) - 1 ≈ 1.81%
Two 20-year bonds are identical in all respects except that one allows the issuer to call the bond in return for $1,000 cash at any time after five years while the other contains no call provisions. Will the yield to maturity on the two bonds differ? If so, which will be higher and why?
Yield on bond with the call provision will be higher. The call option gives the issuer a valuable option and investors will demand compensation for this option in the form of a higher yield. For example, suppose the interest rate declines in year six. The price of the non-callable bond will rise as the holder earns an above market current yield, while the price of the callable bond will remain at $1,000 in anticipation that the issuer will call the bond and refinance at the new lower rate. Holders of the callable bonds will demand a higher yield to compensate for such eventualities
Compute the yield-to-maturity of a two-year bond that costs $25,000 today and pays $1,000 at the end of each of the 2 years. At the end of the second year, it also repays $25,000. What is the bond's YTM?
You are seeking the solution to -$25, 000+ $1, 000/(1 + YTM)^1 + $1, 000/(1 + YTM)^2 + $25, 000/(1 + YTM)^2 = 0. It is YTM = 4%
What is the PV of a perpetuity paying $5 each month, beginning this month (in 1 second), if the monthly interest rate is a constant 0.5%/ month (6.2%/year) and the cash flows will grow at a rate of 0.1%/month (1.2%/year)?
You get C0 = $5 today, and next month you will receive a payment of C1 = (1 + g) · C0 = 1.001 · $5 = $5.005. The growing perpetuity is worth PV = C1/(r - g) = $5.005/(0.5% - 0.1%) = $1,251.25. The total value is $1,256.25
if you received the dividend at the beginning of the period instead of the end of the period, could this change your effective rate of return? Why?
Your stock costs $100 today, pays $5 in dividends at the end of the period, and then sells for $98. What is your rate of return?
In many defined-contribution pension plans, the employer provides a fixed-percentage contribution to the employee's retirement. Assume that the employer must contribute $4,000 per annum beginning next year (time 1), growing annually with the inflation rate of 2% per year. What is the present value of the pension cost of hiring a 25-year-old who will stay with the company for 35 years? Assume a discount rate of 8% per year. Note: Please look up the growing annuity formula to solve this problem
[$4,000/(0.08 - 0.02)] (1 - (1.02^35/1.08^35) ≈ $57,649.2
An investment costs $1,000 and pays a return of $1,050. What is the rate of return?
r = ($1,050 - $1,000)/$1,000 = 5%
If the bank states an effective interest rate of 12% per annum, and there are 52.2 weeks per year, how much interest do you earn on a deposit of $1,000 over 1 week? On a deposit of $100,000?
r = (1 + 12%)^(1/52.2) - 1 = 0.00217 For $1,000, end up with $1,002.17 For $100,000, end up with $102,170
A project with a cost of capital of 30% pays off $250. What should it cost today?
r = 30% = ($250 - x)/x x = $250/1.30 = $192.31
Repeat the calculation with the five-year annualized rate of return of 1.76%. That is, what is the five-year holding rate of return, and how can you compute the annualized forward interest rate for a two-year investment beginning on December 31, 2018, and ending on December 31, 2020?
r0,5 = 1.01765 - 1 ≈ 9.12%. Therefore, 1 + r3,5 = 1.01765/1.01313 - 1 ≈ 4.94%. This is a two-year forward holding rate of return. Thus, it is 1.0494^(1/2)-1 ≈ 2.44% in annualized terms
If the per-year interest rate is 10% for each of the next 5 years, what is the annualized five-year rate of return?
r0,5 = 50% (1 + r 5 )^5 = 1.50 ⇒ r5 = 1.50^(1/5) - 1 ≈ 8.45%
If the per-year interest rate is 5%, what is the 10-year total interest rate?
r10 = (1 + r1)^10 - 1 = 1.05^10 - 1 = 62.89%
If the per-year interest rate is 5%, what is the 100-year total interest rate? How does tis compare to 100 times 5%?
r100 = (1 + r1)^100 - 1 = 1.05^100 - 1 = 13,050%
If the per-year interest rate is 5% what is the two-year total interest rate?
r2 =(1 + r0,1)(1 + r1,2) - 1 = (1.05)(1.05) - 1 = 10.25%
