Interest Rates and the Time Value of Money
The Interest Rate: An exchange rate across time
o The rate at which we can exchange money today for money in the future is determined by the current
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interest rate
Risk free interest rate is the interest rate at which money can be borrowed or lent without risk
Interest rate factor = 1 + r
Discount Factor = 1 / (1 + r)
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Making Consumption Choices over Time
+Consider a person who has an income of $ 50,000 this year and an income of $60,000 next year
o The market allows him not only to consume $50,000 worth of goods this year and $60,000 next year but
also to borrow and lend at the equilibrium interest rate
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Figure below shows all of the consumption possibilities
open to the person through borrowing or lending, and the
shaded area contains all of the feasible choices.
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Slope *
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571 HD
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By borrowing and lending different amounts, the person can achieve any point on line AB . Interest = 0.1
o Point C:
Consumption this year: $50,000 $10,000 = $40,000
Consumption next year: $60,000 + [$10,000 x (1 + r)] = $71,000
o Point D :
Consumption this year: $50,000 + $10,000 = $60,000
Consumption next year: $60,000 [$10,000 x (1 + r)] = $49,000
Line has a slope of (1 + r), meaning that for each dollar that is added to the x coordinate along the line, (1 + r)
dollars are subtracted from the y coordinate
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Present Value and the NPV Decision Rule
The NPV of a project or investment is the difference between the present value of its benefits and the present
value of its costs
o NPV = PV (Benefits) PV (Costs)
o NPV = PV(A11 project cash flows)
The NPV Decision Rule:
o When making an investment decision, take the alternative with the highest NPV
o Choosing this alternative is equivalent to receiving its NPV in cash today
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Accenting or Rejecting a Project:
o
o
Accept those projects with positive NPV
Reject those projects with negative NPV
Solution
If you lake the offer, the benefit is that you won’t have to pay $9500 today, which is already
in PV terms. The cost, however, is $10,000 in one year. We therefore convert the cost to a
present value at the risk-free interest rate:
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PV { Co» ) = ( $10,000 inoneyear ) + ( 1.07 Sin one vcar /$ today ) = $9345.79 today
The NPVof the promotional offer is the difference between the benefits and the costs:
NPV = $9500 - $9345.79 = $154.21 today
The NPV is positive, so the investment is a good deal. It is equivalent to getting a cash
discount today of $154.21. and only paying $9345.79 today for the copier. To confirm our
calculation , suppose you take the offer and invest $9345.79 in a bank paving 7% interest.
With interest, this amount will grow to $9345.79 X 1.07 = $10,000 in one year, which you
can use to pay for the copier.
and the Law
Arbitrage
of One Price
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Arbitrage: The
practice of buying and selling equivalent goods in different markets to take advantage of a price
difference
o An arbitrage opportunity occurs when it is possible to make a profit without taking any risk or making an
investment
Normal Market: A competitive market in which there are no arbitrage opportunities
Law of One Price: If equivalent investment opportunities trade simultaneously in different competitive markets,
then they must trade for the same price in both markets
No-Arbitrage and Security Prices
Valuing a security with the Law of One Price
o Assume a security promises the risk-free payment of S1000 in one year. If the risk-free interest rate is 5%,
what can we conclude about the price of this bond to the normal market?
PV($1000 in one year) = (SI 000 in one year) / (SI .05 in one year $1 today) = $952.38 today
Price(Bond ) = $952.38
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• Valuing a Security (cont'd )
Valuing a Security (cont' d )
- What if the price of the bond is not $952.38 ?
• Assume the price is $940.
• Assume the price is $960.
hET CASH FLOWS FROM SELLING THE BOND AND INVESTING
NET CASH FLOWS FROM BUYING THE BOND AND BORROWING
.
lo Ny ( SI
In One Year ( S)
Buy the bond
-940.00
+ 1000.00
Borrow train the bank
+ 952.38
+ 12.38
lOOOOO
Net cadi flow
- What if the price of the bond is not $952.38 ?
0.00
• The opportunity for arbitrage will force the price of the
bond to rise until it is equal to $952.39.
Sell the bond
Invest at the bank
Net cad) Hew
lodav ( 5 )
+ 960.00
- 952.38
+ 7.62
In One Year ( S )
- 1000.00
+ 1000.00
0.00
• The opportunity for arbitrage will force the price of the bond
to fall until it is equal to $952.38.
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Determining the No Arbitrage Price:
o Unless the price of the security equals the present value of the security’s cash flows, an arbitrage
opportunity will appear
o No Arbitrage Price of a Security
Price(Security) = PV(A11 cash flows paid by the security)
Problem
Consider a security that pays its owner $100 today and $ 100 in one year, without any risk.
Suppose the risk -free interest rate is 10%. What is the no-arbitrage price of the security today
( before the first $100 is paid )? If the security Is trading for $195, what arbitrage opportunity
is available?
Solution
We need to compute the present value of the security’s cash flows. In this case there are two
cash flow's: $100 today, which is already in present value terms, and $100 in one year. The
present value of the second cash flow is
$100 in one year + ( 1.10 $ in one year/ S codav )
= $90.91 today
Therefore, the total present value of the cash flows is $100 + $90.91 = $190.91 today,
which Is the no-arhitrage price of the security.
If the security is trading for $19$, wc an exploit its overpricing by selling it for $195. Wc an
then use $ 100 of the sale proceeds to replace the $ 100 wc would have received from the securin'
today and invest $90.91 of the sale proceeds at 10% to replace the SKK) wc would have received
in one year. The remaining $ 195 - $ 100 - $90.91 = $4.09 is an arbitrage profit.
Determining the interest rate from bond price
o If we know the price of a risk free bond, we can use: Price(Security) = PV(All cash flow's paid by the
security)
o Can use to determine what the risk free interest rate must be if there are no arbitrage opportunities
The bond pays $1000 in one year, currently priced $929.80
1000e929.80 = 1.0755 = 1 + r... r = 0.0755 = 7.55%
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The NPV of Trading Securities and the Optimal Investment Decision
o In a normal market, the NPV of buying or selling a security is zero
NPV(Buy Security) = PV(A11 cash flows paid by the security) Price(Security) = 0
NPV(Sell Security) = Price(Security) PV(A11 cash flows paid by the security) = 0
o Separation of the Investment and Financing Decisions:
Security transactions in a normal market neither create nor destroy value on their own
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We can evaluate the NPV of investment decisions separately from the decision the firm makes
Problem
Your firm is considering a project that will require an upfront investment of $ 10 million today
and will produce $ 12 million in cash flow for the firm in one year without risk. Rather than
pay for the $ 10 million investment entirely using its own cash , the firm is considering raising
additional funds by issuing a security that will pay investors $ 5.5 million in one year. Suppose
the risk-free interest rate is 10%. Is pursuing this project a good decision without issuing the
new security? Is it a eood decision with the new security?
me
an
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auuiuoiiai
million to take
project
To compute the projects NPV in this case, note that in one year the firm will receive the
$12 million payout of the project , but owe $5.5 million to the investors in the new security,
leaving $6.5 million for the firm. This amount has a present value of
$6.5 million in one year + ( 1.10 $ in one year/ $ today )
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= $5.91 million today
Thus, the project has NPV = $5.91 million $5 million = $0.91 million today, as before.
In either case, we get the same result for the NPV. The separation principle indicates that
we will get the same result for any choice of financing lor the firm that occurs in a normal
market. We can therefore evaluate the project without explicitly considering the different
financing possibilities the firm might choose.
regarding how to finance the investment or any other security transactions the firm is considering
Consider 2 securities. A and B. Suppose C has the same cash flows as A and B combined. In this case. C
is equivalent to a portfolio, or combination, of the securities A and B
Price(C) = Price(A+B) = Price(A) + Price(B)
o Value Additivity and Firm Value:
To maximize the value of the entire firm, managers should make decisions that maximize NPV
The NPV of the decision represents its contribution to the overall value of the firm
o
Problem
Horton Holdings is a publicly traded company with only two assets: It owns 60% of Tim’s
Donuts chain and 100% of the Caribou hockey team. Suppose the market value of Horton
Holdings is $ 160 million , and the market value of the entire Tim's Donuts chain ( which is
also publicly traded) Is $ 120 million . What is the market value of the (Caribou hockey team?
The
Risk
Solution
We can think of Horton is a portfolio consisting of a 60% stake in Tim’s Donuts and 100%
of the Caribou hockey team. By value additivity, the sum of the value of the stake in Tim’s
Donuts and the Caribou hockey team must equal the $ 160 million market value of Horton.
Because the 60% stake in Tim's Donuts is worth 60% X $ 120 million = $72 million , the
Caribou hockev team has a value of $ 160 million - $72 million = $88 million.
Price of
Risk:
Market Price Today
Security
Risk- free bond
Security A
Market index
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Cash Flow in One Year
Weak Economy
Strong Economy
769.23
unknown
800
800
0
600
1000
800
1400
Risk premiums depend on the risk
o Given more variable returns, security A pays investors a higher risk premium
o Given an initial price of $230.77 and expected payoff of 300. security A has an expected return of (300
230.77)/230.77 = 30%
o Security A’s risk premium is 30% 4% = 26%
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Risk is Relative to the Overall Market
o The risk of a security must be evaluated in relation to fluctuations of other investments in the economy
o A security’s risk premium will be higher the more its returns tend to vary with the overall economy and
the market index
o If the security’s returns vary in the opposite direction of the market index, it offers insurance and will
have a negative risk premium
Problem
Solut
Suppose security B pays $600 if the economy is weak and $0 if the economy is strong. What
arc its no-arhitrage price, expected return, and risk premium?
11 we combine the market index and security B together in a portfolio, we earn the same
payoff as a risk- free bond that pays § 1400, as shown here (amounts in $ ):
Market Price
Today
Security
Market index
Security B
Risk-free bond
10(H )
unknown
1346.15
Cash Flow in One Year
Weak
Strong
Economy
Economy
8( H )
600
14( H)
1400
1400
0
Because the market price of the risk-free bond is S 1400 / 1.04 = $ 1346.15 today, we can
conclude from the Law of One Price that security B must have a market price of $ 1346.15
1000 = S346.15 today.
If the economy is weak , security B pays a return of (600 346.15 ) / 346.15 = 600 / 346.15 ~
1 = 0.733 = 73.3%. If the economy is strong, security B pays nothing, for a return of
100%. The expected return of security B is therefore 1 ( 73.3% ) + * ( 100% ) = 133%.
Its risk premium is 13.3% 4% = 17.3%; that is, security B pays investors 17.3% less
on average than the risk - free interest rate.
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The
of Risk
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Price
Risk, Return, and Market Prices
The discount rate, rs, includes a risk premium appropriate for the investment’s risk
o
rs = rf + (risk premium for investment s)
Example:
USING THE RISK PREMIUM TO COMPUTE A PRICE
Problem
Consider a risky bond with a ask flaw of $1100 when the economy is strong and $1000
wken die economy is weak. Suppose a 1% risk premium is appropriate for this bond. If the
risk-free interest rate is 4%, what it the price of the bond today?
Solution
the appropriate discount race for the bond is
r*
= r/ + ( Risk ptemiim for thebond ) = 4% + 1% - 5%
The expected ash flow of the bond it ' ( SI 100) + ',( $1000 )
the pike of the bond today is
= $1050 in one year Thus,
Bond Price = ( Avenge ash flow in one year ) + ( l + r S in one ycar/$ today )
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= ( $1050 in one year ) + ( 1.05
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$ in one vear/T today )
SI 000 today
Given this price, the bond's return is 10% when the economy is strong, and 0% when the
economy it weak. ( Note that the difference in the tenant is 10% which it at variable as
die market index Correspondingly, the risk premium of the bond is I that of the market
index as well.)
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Arbitrage with Transaction Costs
Commissions paid to your broker
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Bid ask spread
o The difference of the price you receive when you sell (bid price) and when you buy (ask price) a security
These costs are small for most markets
Example:
THE NO- ARBITRAGE PRICE RANGE
Problem
Consider a bond that pays SI 000 at the end of the year. Suppose the market interest rate for
deposits is 6%, but the market interest rate for borrowing is 6.5%. What is the no-arbitrage
price range for the bond? That is, what is the highest and lowest price the bond could trade
for without creating an arbitrage opportunity?
Solution
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The no arbitrage price for the bond equals the present value of the cash (lows. In this case, we
can use either of two interest rates to compute the present value, depending on whether we
are borrowing or lending. For example, the amount we would need to put in the (sank today
to receive $ I ( KM ) in one year is
( SlOOOinoneyear ) + ( 1.06 S in oneyear / S today )
=
$943.40 today
where we have used the 6% interest rate that we will earn on our deposit . The
we can borrow today if we plan to repay $1000 in one year is
( $ 1000 in oncyear ) + ( 1.065 $ in one year/S today )
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amount
that
$938.97 today
where we have used the higher 6.5% rate that we will have to pay if we borrow.
Suppose the hond price /’, exceeded $943.40. Ilien you coukl profit by short selling the bond at
its current price and investing $943.40 of the proceeds at the 6% interest rate. The $1000
received from your investment would be used to exit the short bond posirion netting you $0
in one year, but you would get to keep the difference HP
943.40) today. This arbitrage
opportunity will keep the price of the hond from going higher than S943.40.
Alternatively, suppose the bond price
were less than $938.97. Then you could borrow
S938.97 at 6.5% and use P of it to buy the bond. This would leave you with $( 938.97 l1)
today, and no obligation in the future because you can use the $ I (MM) bond payoff to repay the
loan. This arbitrage opportunity will keep the price of the bond from falling below $938.97.
If the hond price. P is between $938.97 and $943.40 then both of the preceding strategies will lose money, and there is no arbitrage opportunity. Thus no arbitrage implies a narrow
range of possible prices for the bond ( $938.97 to $943.40) rather than an exact price
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