34 Chapter - Financial Economics
Chapter 34 Financial Economics
QUESTIONS
1. Suppose that the city of New York issues bonds to raise money to pay for a new tunnel linking
New Jersey and Manhattan. An investor named Susan buys one of the bonds on the same day that
the city of New York pays a contractor for completing the first stage of construction. Is Susan
making an economic or a financial investment? What about the city of New York? LO1
Answer: New York is making an economic investment. Recall that an economic
investment refers either to paying for new additions to the capital stock or new
replacements for capital stock that has worn out. The issuance of bonds is financing the
new tunnel, which is an addition to society's capital stock.
Susan is making a financial investment. Recall that a financial investment refers to the
purchase of an asset using an existing asset. Here Susan exchanges one asset for another,
say income out of her checking account for the bond. Susan has only changed her
portfolio of assets.
2. What is compound interest? How does it relate to the formula: X dollars today = (1 + i )tX
dollars in t years? What is present value? How does it relate to the formula: X/(1 + i)t dollars
today = X dollars in t years? LO1
Answer: Compound interest describes how quickly an investment increases in value
when interest is paid, or compounded, not only on the original amount invested but also
on all interest payments that have been previously made.
This concept relates to the formula (1+i)tX through the variables i, the interest rate, and
t, the amount of years (time) X dollars is invested. The first year X dollars are invested the
payoff is (1+i)X. If we allow this investment to 'roll-over' another year and invest (1+i)X
we will have (1+i)(1+i)X = (1+i)2X at the end of year two. That is we earn interest on
the principal and interest from the previous year. After t years we have (1+i)tX .
The present value model simply rearranges the equation above to make it easier to
transform future amounts of money into present amounts of money. Instead of using the
formula (1+i)tX to calculate the 'future value' of X dollars today we can write the
formula as X/(1 + i)t to calculate how much X dollars in the future is worth to us today.
For example, assume I offer you $1100 a year from now or $1000 today that you can't
spend for a year (you must save the $1000). Also, assume the current interest rate is 10%.
Which would you choose? Your answer should be it doesn't matter which I give you. If
you take the $1000 today it is worth $1100 a year from now. Thus, the offer of $1100 in
the future is equivalent to $1000 (= X/(1 + i) = $1100/1.1).
3. How do stocks and bonds differ in terms of the future payments that they are expected to
make? Which type of investment (stocks or bonds) is considered to be more risky? Given what
you know, which investment (stocks or bonds) do you think commonly goes by the nickname
fixed income? LO2
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Answer: Stocks pay dividends out of profits to the shareholders that own them, with the
percentage of the total dividend received being based on the percentage of stocks owned.
While many corporations pay regular dividends on their stocks, there is no requirement to
do so, and many corporations pay no dividends or pay them on an irregular basis. Stocks
also return capital gains when their owners sell their shares, assuming that the price they
receive for the shares is greater than what they paid for them. Bonds pay a predetermined
amount of interest at regular intervals.
Stocks are considered more risky. Stock prices and profits are highly variable, so the
capital gains and dividends tied to them can also fluctuate greatly. There is some risk
with bonds, particularly with bonds issued by corporations that are struggling financially,
but on average they are less risky than stocks, in part because government bonds (U.S.
Federal government bonds in particular) carry virtually no risk of losing the amount
invested.
Bonds are frequently referred to as fixed income investments because of the regularity
in size and timing of the payments to their owners.
4. What are mutual funds? What different types of mutual funds are there? And why do you think
they are so popular with investors? LO2
Answer: Mutual funds pool investors money and buy a collection of stocks or bonds.
Some are narrowly defined, focusing on a particular sector of the economy (technology,
health care) or type of asset (small cap stock funds; long-term government bonds); others
are more broadly defined (growth, income funds); others have a social agenda (ethical
investment). There are both actively managed funds (with frequent buying and selling of
assets in the fund) and passively managed funds (such as index funds where the
allocation is tied to an index rather than varying according to a managers discretion).
Investing in a specific stock or bond can be risky, and some of this risk (diversifiable or
idiosyncratic risk) can be reduced by buying many different stocks and/or bonds. By
pooling the funds of many investors, the mutual fund can invest in many different
financial assets simultaneously, diversifying the risk. Mutual funds are also popular
because they dont require the investor to closely monitor the portfolio or have expertise
in the many stocks and bonds in the portfolio.
5. Corporations often distribute profits to their shareholders in the form of dividends, which are
simply checks mailed out to shareholders. Suppose that you have the chance to buy a share in a
fashion company called Rogue Designs for $35 and that the company will pay dividends of $2
per year on that share every year. What is the annual percentage rate of return? Next, suppose that
you and other investors could get a 12 percent per year rate of return by owning the stocks of
other very similar fashion companies. If investors care only about rates of return, what should
happen to the share price of Rogue Designs? (Hint: This is an arbitrage situation.) LO3
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Answer: A yearly dividend of $2 on a $35 share of stock equals a 5.71% annual rate of
return ($2/$35 = .0571 = 5.71%).
If stocks returning 12 percent annual rates of return became available, investors would
sell shares of Rogue Designs and buy shares in companies earning the 12% return. This
would cause the price of Rogue Designs stock to fall. Note that as the stock price falls,
the annual percentage return from owning Rogue Designs stock will rise, assuming the $2
annual dividend continues. Eventually the rates of return of the two (or more) similar
companies should equalize.
6. Why is it reasonable to ignore diversifiable risk and care only about nondiversifiable risk?
What about investors who put all their money into only a single risky stock? Can they properly
ignore diversifiable risk? LO4
Answer: It is reasonable to ignore diversifiable risk if the investors portfolio is already
diversified. By investing in different types of assets, diversifiable (idiosyncratic) risk can
be reduced or eliminated, so that for future investments the only concerns are the nondiversifiable (systemic) risks.
An investor who puts all of his money into a single risky stock cannot ignore
diversifiable risk because risk it is still present (failure to diversify). Thus the investor
still faces both that and non-diversifiable risk.
7. If we compare the betas of various investment opportunities, why do the assets that have higher
betas also have higher average expected rates of return? LO5
Answer: The assets with the highest betas are the most risky. Potential investors in
assets with higher risk will only purchase these assets if they are appropriately
compensated for taking on that expected risk. Investors will insist on paying lower prices
for riskier assets, and lower prices mean higher rates of return.
8. In this chapter we discussed short-term U.S. government bonds. But the U.S. government also
issues longer-term bonds with horizons of up to 30 years. Why do 20-year bonds issued by the
U.S. government have lower rates of return than 20-year bonds issued by corporations? And
which would you consider more likely, that longer-term U.S. government bonds have a higher
interest rate than short-term U.S. government bonds, or vice versa? Explain. LO5
Answer: U.S. government bonds have lower rates than bonds issued by corporations
because the risk of default is lower with the Federal government. Federal government
bonds in particular are low-risk or risk-free because, if all else fails, the government can
print more money to pay its debts.
A longer-term U.S. government bond would be expected to have a higher interest rate
than a short-term U.S. government bond. Time preference would suggest that investors
would need to be compensated for waiting longer for their return. To the extent there is
risk with U.S. government bonds, that risk is greater over longer periods of time (the
future is more uncertain), so that would also justify a (slightly) higher rate on long-term
U.S. government bonds.
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9. What determines the vertical intercept of the Security Market Line (SML)? What determines
its slope? And what will happen to an assets price if it initially plots onto a point above the
SML? LO5
Answer: The vertical intercept is the risk-free interest rate (the rate on short-term U.S.
government bonds) that is determined by Federal Reserve monetary policy.
The slope of the SML depends on investor feelings about risk and the compensation they
require for assuming that risk. If investors strongly dislike risk and require much greater
compensation, the SML will be steeper than if investors are less concerned about risk.
An asset plotting above the SML will be earning a higher rate than the average expected
rate of return for similar investments. Investors will be attracted to that investment, and
through the process of arbitrage, demand for the investment will increase, raising the
price of asset and lowering its expected rate of return.
10. Suppose that the Federal Reserve thinks that a stock market bubble is occurring and wants to
reduce stock prices. What should it do to interest rates? LO5
Answer: If the Fed wants to decrease stock prices it should raise interest rates.
Increasing the risk-free interest rate will make risk-free investments more attractive and
riskier investments (like stocks) relatively less attractive. As investors purchase their
risk-free assets and sell more stocks, stock prices will fall.
11. Consider another situation involving the SML. Suppose that the risk-free interest rate stays
the same, but that investors dislike of risk grows more intense. Given this change, will average
expected rates of return rise or fall? Next, compare what will happen to the rates of return on lowrisk and high-risk investments. Which will have a larger increase in average expected rates of
return, investments with high betas or investments with low betas? And will high-beta or lowbeta investments show larger percentage changes in their prices? LO5
Answer: Average expected rates of return will rise on all assets except those paying the
risk-free interest rates. Higher beta investments will see a greater increase in their rates
of return relative to low beta investments, as greater investment dislike of risk will cause
them to require even greater compensation for the riskier investments. High-beta
investments will show larger percentages in their rates of return, and hence larger
percentage changes in their prices.
12. LAST WORD Why is it so hard for actively managed funds to generate higher rates of return
than passively managed index funds having similar levels of risk? Is there a simple way for an
actively managed fund to increase its average expected rate of return?
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Answer: Actively managed funds have difficulty generating higher rates of return than
passively managed index funds for two reasons. First, arbitrage causes rates of return to
equalize across similar assets, so even if an asset paying a higher return is found, the
many investors pursuing higher returns will drive up its price and lower the returns.
Second, actively managed funds incur greater expenses. The pursuit of higher rates of
returns requires greater research expenditures and greater costs from the more frequent
buying and selling of assets. A simple way for an actively managed fund to increase its
average expected rate of return is to cut down on trading activities that generate costs.
The dilemma that, is if taken to the extreme, that actively managed fund becomes a
passively managed funds, and may become undesirable to investors trying to beat the
market.
PROBLEMS
1. Suppose that you invest $100 today in a risk-free investment and let the 4 percent annual
interest rate compound. Rounded to full dollars, what will be the value of your investment 4 years
from now? LO1
Answer: $117.
Feedback: Consider the following example. Suppose that you invest $100 today in a
risk-free investment and let the 4 percent annual interest rate compound. Rounded to full
dollars, what will be the value of your investment 4 years from now?
After the first year you will have $104 ( = (1+0.04)$100 = $100 (principal) + $4
(interest)).
Carrying this $104 forward another year will give you $108.16 ( = (1+0.04)$104 =
$104 (principal) + $4.16 (interest)) two years from now.
Carrying this $108.16 forward another year will give you $112.4864 ( =
(1+0.04)$108.16) three years from now.
Carrying this $112.4864 forward another year will give you $116.98585 ( =
(1+0.04)$112.4864) four years from now.
Rounding to the nearest dollar gives us $117.
An easier way to calculate this value is to recognize that we multiply each consecutive
value by the (1+0.04).
Which implies we have the value four years from now of:
Value = (1+0.04)(1+0.04)(1+0.04)(1+0.04)$100 = (1+0.04) 4$100 = $117 after rounding.
In general, we have the formula
Value = (1+i)tX where i is the interest rate, t is the number of years of compounding , and
X is the initial investment.
2. Suppose that you desire to get a lump sum payment of $100,000 two years from now. Rounded
to full dollars, how many current dollars will you have to invest today at a 10 percent interest to
accomplish your goal? LO1
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Answer: $82, 645.
Feedback: Consider the following example. Suppose that you desire to get a lump sum
payment of $100,000 two years from now. Rounded to full dollars, how many current
dollars will you have to invest today at a 10 percent interest to accomplish your goal?
To answer this question we start at the end of the problem.
Two years from now we want to have $100,000. How much do I need to save one year
from now to achieve this goal given the interest rate of 10%?
$100,000 (future value) = (1+0.10) Saving
Rearranging, we have,
Saving = $100,000/(1+0.10) = $90,909.09.
This implies that next year I need to have $90,909.09. How do much I need to save today
to achieve this goal given the interest rate of 10%?
$90,909.09 (future value) = (1+0.10) Saving
Rearranging, we have,
Saving = $90,909.09/(1+0.10) = $82,644.627.
Thus, if I want to have $100,000 two years from now I need to save $82,645 after
rounding.
In effect, what we have found is the present value of $100,000 two years from now.
Present Value = $100,000/(1+0.10)2
In general, we can use the following formula.
Present Value = X/(1+i)t where i is the interest rate, t is the number of years of in the
future , and X is the desired future value.
3. Suppose that a risk-free investment will make three future payments of $100 in one year, $100
in two years, and $100 in three years. If the Federal Reserve has set the risk-free interest rate at 8
percent, what is the proper current price of this investment? What is the price of this investment if
the Federal Reserve raises the risk-free interest rate to 10 percent? LO1
Answers: $257.58; $248.68.
Feedback: Consider the following example. Suppose that a risk-free investment will
make three future payments of $100 in one year, $100 in two years, and $100 in three
years. If the Federal Reserve has set the risk-free interest rate at 8 percent, what is the
proper current price of this investment? What is the price of this investment if the Federal
Reserve raises the risk-free interest rate to 10 percent?
Here we need to use the concept of present value.
How much is $100 one year from now worth today at an interest rate of 8%.
Present Value (one year from now) = $100/(1.08) = $92.59
How much is $100 two years from now worth today at an interest rate of 8%.
Present Value (one year from now) = $100/(1.08)2 = $85.73
How much is $100 three years from now worth today at an interest rate of 8%.
Present Value (one year from now) = $100/(1.08)3 = $79.26
Since we receive all of the payments above, the present value of this payment stream
equals;
present value = $100/(1.08) + $100/(1.08)2 + $100/(1.08)3 = $92.59 + $85.73 + $79.26 =
$257.58
We would pay $257.58 for a three year payment stream of $100.
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If the interest rate were 10% we use the same procedure.
present value = $100/(1.1) + $100/(1.1)2 + $100/(1.1)3 = $90.91 + $82.64 + $75.13=
$248.68
If we had additional years we would just add the present value of these payments to the
value above.
4. Consider an asset that costs $120 today. You are going to hold it for 1 year and then sell it.
Suppose that there is a 25 percent chance that it will be worth $100 in a year, a 25 percent chance
that it will be worth $115 in a year, and a 50 percent chance that it will be worth $140 in a year.
What is its average expected rate of return? Next, figure out what the investments average
expected rate of return would be if its current price were $130 today. Does the increase in the
current price increase or decrease the assets average expected rate of return? At what price
would the asset have a zero average expected rate of return? LO3
Answers: 3.125%; -4.808%; The increase in price reduces the expected return on the asset;
$123.75.
Feedback: Consider the following example. Consider an asset that costs $120 today.
You are going to hold it for 1 year and then sell it. Suppose that there is a 25 percent
chance that it will be worth $100 in a year, a 25 percent chance that it will be worth $115
in a year, and a 50 percent chance that it will be worth $140 in a year. What is its average
expected rate of return? Next, figure out what the investments average expected rate of
return would be if its current price were $130 today. Does the increase in the current
price increase or decrease the assets average expected rate of return? At what price
would the asset have a zero average expected rate of return?
The first exercise is to calculate the expected payoff for this asset. To do this, multiply
the probability (decimal representation of percentage) for each payoff (state) by the actual
payoff.
For the value above.
Expected payout = 0.25x$100 + 0.25x$115 + 0.5x$140 = $25 + $28.75 + $70 = $123.75.
If the current price is $120, the expected return equals 3.125% (= ((123.75 120)/120)x100).
If the current price is $130, the expected return equals -4.808% (= ((123.75 130)/130)x100).
The increase in price reduces the expected return on the asset, as shown above.
To calculate the price where the return equals zero we use the following.
return = 0 = (123.75-price)/price, which implies the price = $123.75 (the expected
payoff).
5. Suppose initially that two assets, A and B, will each make a single guaranteed payment of $100
in 1 year. But asset A has a current price of $80 while asset B has a current price of $90. LO3
a. What are the rates of return of assets A and B at their current prices? Given these rates of
return, which asset should investors buy and which asset should they sell?
b. Assume that arbitrage continues until A and B have the same expected rate of return. When
arbitrage ends, will A and B have the same price?
Next, consider another pair of assets, C and D. Asset C will make a single payment of $150 in
one year while D will make a single payment of $200 in one year. Assume that the current price
of C is $120 and that the current price of D is $180.
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c. What are the rates of return of assets C and D at their current prices? Given these rates of
return, which asset should investors buy and which asset should they sell?
d. Assume that arbitrage continues until C and D have the same expected rate of return. When
arbitrage ends, will C and D have the same price?
Compare your answers to questions a through d before answering question e.
e. We know that arbitrage will equalize rates of return. Does it also guarantee to equalize prices?
In what situations will it equalize prices?
Answer: (a) Return on asset A = 25%; Return on asset B = 11.11%; Individuals will buy
asset A and sell asset B.
(b) Yes, assets A and B will have the same price because the payoff is the same.
(c) Return on asset C = 25%; Return on asset D = 11.11%; Individuals will buy asset C and
sell asset D.
(d) No, assets C and D will not have the same price because the payoffs are different.
(e) No, we do not necessarily observe price equalization with equal returns. Only when the
payoffs are the same do we observer equal prices.
Feedback: Consider the following example. Suppose initially that two assets, A and B,
will each make a single guaranteed payment of $100 in 1 year. But asset A has a current
price of $80 while asset B has a current price of $90.
Part a:
What are the rates of the return of assets A and B at their current prices? Given these
rates of return, which asset should investors buy and which asset should they sell?
return on asset A = ($100-$80)/$80 =0.25 (25%)
return on asset B = ($100-$90)/$90 =0.1111 (11.11%)
Individuals will buy asset A because it has the higher return. This implies individuals will
sell asset B.
Part b:
Assume that arbitrage continues until A and B have the same expected rate of return.
When arbitrage ends, will A and B have the same price?
Yes, the assets will have the same price because the payoff is the same ($100).
return asset A =(Payoff - price A)/price A = (Payoff - price B)/price B = return asset B
or,
(Payoff / price A) - 1 = (Payoff /price B) - 1 and (Payoff / price A) =(Payoff /price B)
then,
(1 / price A) =(1 /price B) and price A = price B
Next, consider another pair of assets, C and D. Asset C will make a single payment of
$150 in one year while D will make a single payment of $200 in one year. Assume that
the current price of C is $120 and that the current price of D is $180.
Part c:
c. What are the rates of the return of assets C and D at their current prices? Given these
rates of return, which asset should investors buy and which asset should they sell?
return on asset C = ($150-$120)/$120 =0.25 (25%)
return on asset D = ($200-$180)/$180 =0.1111 (11.11%)
Individuals will buy asset C because it has the higher return. This implies individuals will
sell asset D.
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Part d:
d. Assume that arbitrage continues until C and D have the same expected rate of return.
When arbitrage ends, will C and D have the same price?
No, because the payoffs are different. See logic above with different payoff structure.
Part e:
Compare your answers to questions a through d before answering question e.
e. We know that arbitrage will equalize rates of return. Does it also guarantee to equalize
prices? In what situations will it equalize prices?
No, we do not necessarily observe price equalization with equal returns. Only when the
payoffs are the same do we observer equal prices.
6. Advanced Analysis Suppose that the equation for the SLM is Y = 0.05 + 0.04X, where Y is the
average expected rate of return, 0.05 is the vertical intercept, 0.04 is the slope, and X is the risk
level as measured by beta. What is the risk-free interest rate for this SML? What is the average
expected rate of return at a beta of 1.5? What is the value of beta at an average expected rate of
return is 7 percent? LO5
Answers: 5 percent; 11 percent; 0.5.
Feedback: Consider the following example. Advanced Analysis Suppose that the
equation for the SLM is Y = 0.05 + 0.04X, where Y is the average expected rate of
return, 0.05 is the vertical intercept, 0.04 is the slope, and X is the risk level as measured
by beta. What is the risk-free interest rate for this SML? What is the average expected
rate of return at a beta of 1.5? What is the value of beta at an average expected rate of
return is 7 percent?
The risk free is the intercept of SLM line, which is 0.05. Thus, the risk free rate is 5%.
(This represents a beta of zero, which implies no risk).
The average expected rate of return at a beta of 1.5 is 0.11 (or 11%). To find this value
substitute the given beta value into the SLM line Y = 0.05 + 0.04x1.5 = 0.11.
The value of beta at an average expected rate of return is 7% is 0.5. To find this value use
the SLM line given the average expected rate of return 0.07 = 0.05 + 0.04xbeta.
Solving for beta, we have beta = 0.02/0.04 = 0.5.
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