Notes - Corporate Finance:
Module 1 - Investors, Firms and Financial Decision Making:
Chapter 3 - Financial Decision Making and the Law of One Price
“Does the cash value today of its benefits exceed the cash value today of its costs?”

NPV = The net amount by which the decision will increase wealth.
3.1 - Valuing decisions:

Whenever a good trade in a competitive market - by which we mean a market in
which it can be bought and sold at the same price - that price determines the cash
value of the good. So, the value of a good doesn´t depend on the decision maker´s
views or preferences in a competitive market. If you can buy gold for 9000 kr, then the
maximum amount you can sell it for is also 9000 kr. You can´t sell it for less either.
Valuation Principle:

The value of an asset to the firm or its investors is determined by its competitive
market price. The benefits and costs of a decision should be evaluated using these
market prices, and when the value of the benefits exceeds the value of the costs, the
decision will increase the market value of the firm.
When competitive market prices are not available:
When competitive market prices are not available, preferences and views determine the worth
of an offer. If you for example in a campaign would get an Ipad worth 399 dollars, the value
of the offer would be 399 dollars if your intention would have been to buy an Ipad anyways.
If you´re not interested in getting an Ipad, however, the value of the Ipad to you would be the
price you could sell it for. If you could sell it for 300 dollars, that would be the value of the
offer for you.
3.2 - Interest rates and the Time value of money:
The interest rate - and exchange rate over time:




Interest rate = the rate at which we can exchange money today for money in the future.
An interest rate is like an exchange rate, except that it converts money across time and
not from one currency to another.
Risk-free interest rate = the interest rate at which money can be borrowed or lent
without risk over that period.
Interest rate factor for risk free cash flows = (1+rf)
At the risk free interest rate  the supply of savings equals the demand for borrowing.
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1
Discount factor = (1+𝑟)𝑛
3.3 - Present value and the NPV decision rule:






NPV = PV(BENEFITS) - PV(COSTS)
NPV = PV (ALL PROJECTS CASH FLOWS)
NPV > 0  Invest
If you can only choose one investment  choose the one with the highest NPV since
this alternative is equivalent to receiving its NPV in cash today.
As long as you can borrow and lend at the same rate, your preferences of when you
want the cash flow doesn´t matter. Always take the alternative with the highest NPV.
“Regardless of our preferences for cash today versus cash in the future, we should
always maximize NPV first. We can then borrow or lend to shift cash flows through
time and find our most preferred pattern of cash flows.”.
3.4 – Arbitrage and the Law of One Price
Arbitrage = the practice of buying and selling equivalent goods in different markets to take
advantage of a price difference.
Arbitrage opportunity = a possibility to make a profit without taking any risk or making any
investment.
Whenever an arbitrage opportunity appears in the market, investors will race to take
advantage of it. Those investors who spot the opportunity first and who can trade quickly will
have the ability to exploit it. Once they place their trades, prices will respond, causing
arbitrage opportunity to evaporate.
Arbitrage opportunities are like money lying the street; once spotted, they will quickly
disappear. Thus, the normal state of affairs in markets should be that no arbitrage
opportunities exist.
We call a competitive market in which there are no arbitrage opportunities a normal market
Law of one price
If equivalent investment opportunities trade simultaneously in different competitive markets ´,
then they must trade for the same price in all markets.
The law of one price makes it possible to use any competitive price to determine cash value,
without checking the price in all possible markets.
3.5 – No arbitrage and Security Prices
Financial security (or more simply, a security) = an investment opportunity that trades in a
financial market.
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Valuing a security with the law of one price:
Suppose a security that promises a one-time payment to its owner of 1000 dollars. Suppose
there is no risk that the payment will not be made. One example of this type of security is a
bond, a security sold by the government and corporations to raise money from investors
today in exchange for the promised future payment.
To calculate how much the bond should cost we can look at an alternative investment that
would generate the same cash flow in one year. If we would put money in the bank at a 5%
risk-free interest rate, we would need to calculate the PV to see how much money we need to
put in the bank to have 1000 dollars in one year.
PV(1000 dollars in one year) = 1000/1,05 = 952,38 today.
Because the transaction of putting money in the bank and buying a bond should generate the
same cash flow of 1000 in the future, they must, in a normal market with The Law of one
Price, have the same price. Therefore,
Price(bond) = 952,38 dollars.
Identifying arbitrage opportunities with securities:
If the bond would have another price there would be an arbitrage opportunity.


If the price is lower than the market price we should buy the bond and borrow money
from the bank. Then our cash flow will increase today and be unchanged in the future.
If the price is higher than the market price, we should sell the bond and invest at the
bank. Then our cash flow will also increase today and be unchanged in the future.
As people begin selling or buying these bonds however, the arbitrage opportunity will be
exploited and the price will fall/rise to the market price.
See page 106 and 107 (Table 3.3 and 3.4).
Short sale:
In a financial market, it is possible to sell a security without owning it by doing a short-sell.
In a short-sell the person who intends to sell a security first borrows it from someone who
already own it. Later, that person must either return the security by buying it back or pay the
owner the cash flows he or she would have received. For example, we could short sell the
bond in the example effectively promising to repay the current owner 1000 dollars in one
year. By executing a short sale, it is possible to exploit arbitrage opportunity when the bond is
overpriced even if you do not own it.
No-arbitrage price = the price of a bond in a normal market
General process for pricing other securities:
1. Identify the cash flows that will be paid by the security
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2. Determine the “do-it-yourself” cost of replicating those cash flows on our own; that is,
the present value of the security´s cash flows.
Unless the price equals the present value, there is an arbitrage opportunity.
Price(security) = PV(All cash flows paid by the security).
Determining the interest rate from bond prices:
The risk-free interest rate equals the percentage gain that you earn from investing in the bond,
which is called the bond´s return:
Return = Gain at the end of Year/Initial cost = (1000 - 929,80)/929,80 = 1000/929,80 – 1 =
7,55% = the risk-free interest rate.
If the bond offered a higher return than the risk-free interest rate, the investors would earn a
profit by borrowing at the risk-free interest rate and investing in the bond. If the bond had a
lower return than the risk-free interest rate, investors would sell the bond and invest the
proceeds at the risk-free interest rate.
The NPV of trading securities and Firm decision making:
When there is no-arbitrage prices:


NPV(buy security) = PV(all cash flows paid by the security) – Price(Security) = 0
NPV(sell security) = Price(security) – PV(all cash flows paid by the security) = 0
Financial transactions are not sources of value but instead serve to adjust the timing and risk
of the cash flows to best suit the needs of the firm or investors.
Separation principle:
Security transactions in a normal market neither create nor destroy value on their own.
Therefore, we can evaluate the NPV of an investment decision separately from the decision
the firm makes regarding how to finance the investment or any other security transactions the
firm is considering.
Valuing a portfolio:
Value additivity:
Price (C ) = Price (A+B) = Price (A) + Price (B)
Value additivity implies that the value of a portfolio is equal to the sum of the values of its
parts.
Value additivity and Firm value:
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The cash flows of the Firm = the total cash flows of all projects and investments within the
firm.
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.
Chapter 4 - The time value of money:
4.1 - The timeline


When you´re doing a time line, point 1, 2, 3…n means the end of each year. Point 1 is
the end of the first year. Similarly, date 1 is the beginning of the second year.
Approach every problem with drawing a timeline. It will make it a lot easier!
4.2 - The three rules of time travel
Rule 1: Comparing and combining values
 It is only possible to compare or combine values at the same point in time.
Rule 2: Moving cash flows forward in time
 Moving cash flow forward in time is known as compounding
 To move a cash flow forward in time you must compound it
 The difference in value between money today and money in the future represents the
time value of money
 The effect of earning “interest on interest” is known as compound interest


𝐹𝑉𝑛 = 𝐶 × (1 + 𝑟) × (1 + 𝑟) × … × (1 + 𝑟) = 𝐶 × (1 + 𝑟)𝑛
The balance in the account (with simple interest, that is without “interest on interest”)
is given by:
𝐹𝑉 = 𝑃𝑉 × 𝑟 × 𝑛 + 𝑃𝑉
“Interest on interest” = FV(Compounded interest) – FV(simple interest)
Rule 3: Moving cash flows back in time:
 Finding the equivalent value today of a future cashflow = discounting
𝐶
 𝑃𝑉 = (1+𝑟)𝑛

To move a cash flow back in time we have to discount it
4.3 - Valuing a stream of cash flows:
𝐶1
𝐶2
 𝑃𝑉 = 𝐶0 + (1+𝑟)
+ (1+𝑟)
2 + ⋯+

𝐶𝑛
(1+𝑟)𝑛
𝐹𝑉𝑛 = 𝑃𝑉 × (1 + 𝑟)𝑛
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4.4 - Calculating the net Present Value:

NPV = PV(benefits) - PV(costs) = PV(benefits - costs)
4.5 - Perpetuities and Annuities
Perpetuities:







Perpetuities = a stream of cash flows that occur at regular intervals and lasts forever.
One example is the British government bond called consol (or perpetual bond).
Consol bonds promise the owner a fixed cash flow every year, forever.
The first cash flow does not occur immediately; it arrives at the end of the first period.
The timing is sometimes referred to as payment in arrears and is a standard
convention that we adopt throughout this text.
PV = C/(1+r) + C/(1+r)^2 + C/(1+r)^3 +…sum of all cash flows, where C is the
payment and r is the interest rate.
Cn = C  because a cash flow for a perpetuity is constant.
C0 = 0 since the first cash flow occurs in the first period.
The Law of One Price  the value of the perpetuity must be the same as the cost we
would incur to create it ourselves.
Suppose we invest an amount P in the bank. Every year we can withdraw the interest we have
earned, C = r * P, leaving the principle, P, in the bank. The present value of receiving C in
perpetuity is therefore the upfront cost P = C/r. Therefore,
Present value of perpetuity:
𝑃𝑉 (𝐶 𝑖𝑛 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦) =
𝐶
𝑟
Where C is the annual cash flow, and r is the interest rate
Present value of a perpetuity that doesn´t start at date 1:
If the Perpetuity doesn´t start at date one, you combine the formula for the PV of a perpetuity,
and the PV of a future amount.
𝑃𝑉 =
1
𝐶
×
(1 + 𝑟) 𝑟
See example 4.7 at page 145
Remember - the PV formula for the perpetuity already discounts the cash flows to one period
prior to the first cash flow.
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Annuities:



An annuity is a stream of N equal cash flows paid at regular intervals.
Most careloans, mortgages, and some bonds are annuities.
The first payment takes place at date 1, one period from today.
PV = C/(1+r) + C/(1+r)^2 + C/(1+r)^3 + …+ C/(1+r)^N
First, we invest P in the bank, and withdraw only the interest C = r* P each period. After N
periods, we close the account. Thus, for an initial investment of P, we will receive an Nperiod annuity of C per period, plus we will get back our original P at the end. P is the total
present value of the two sets of cash flows, or
P = PV(annuity of C for N periods) + PV(P in period N)
PV(annuity of C for N periods) = P - PV(P in period N)
= P - (P/(1+r)^N)) = P(1-(1/(1+r)^N))
Present value of an annuity:
1
1
PV(annuity of C for N periods with interest rate r) = 𝐶 × 𝑟 (1 − (1+𝑟)𝑁 )
Annuity due = if the payment begins immediately, and not in one period from now. If there
will be 30 payments and the first payments occur today, the last payment will occur in 29
years.
Future value of an annuity:
𝐹𝑉(𝑎𝑛𝑛𝑢𝑖𝑡𝑦) = 𝑃𝑉 × (1 + 𝑟)𝑁 =
𝐶
1
𝟏
×1−
× (1 + 𝑟)𝑁 = 𝑪 × ((𝟏 + 𝒓)𝑵 − 𝟏)
𝑁
(1 + 𝑟)
𝑟
𝒓
PV of an annuity due:
1
PV(annuity due of C for N periods) = 𝐶 × 𝑟 (1 −
1
)
(1+𝑟)𝑁
× (1 + 𝑟)
Growing cash flows:

A cash flow that is expected to grow at a constant rate in ach period.
Growing perpetuity
= a stream of cash flows that occur at regular intervals and grow at a constant rate forever.
The first payment does not include growth, even though it is one period away!
PV = C/(1+r) + C(1+g)/(1+r)^2 + C(1+g)^3/(1+r)^3+…(C(1+g)^n-1)/(1+r)^n
Suppose g>r
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An infinite cash flow means that no matter how much money you start with, it is impossible
to sustain a growth rate of g forever and reproduce those cash flows on your own.
A perpetuity of this sort doesn´t exist in reality (see page 150).
Consequently, we assume that g < r for growing perpetuities.
Present value of a growing perpetuity:
𝐶
PV (growing perpetuity) = 𝑟−𝑔
C = the first payment
r-g = the difference between the interest rate and the growth rate.
(se sista stycket på sidan 151 för förklaring till formeln).
Growing annuity:
= a stream of N growing cash flows, paid at regular intervals. It is a growing perpetuity that
eventually comes to an end.
1. The first cash flow arrives at the end of the first period.
2. The first cash flow does not grow.
Present value of a Growing annuity:
1
1+𝑔 𝑁
𝑃𝑉 = 𝐶 ×
(1 − (
) )
𝑟−𝑔
1+𝑟
Because the annuity has only a finite number of terms, the equation also works when g>r.
The formula for a growing annuity when N = infinity:
PV = C/(r-g)
If r = g, the formula does not hold. Then you need to calculate the present values of all the
payments and add them up.
Present value of a growing annuity when r = g, and the first payment is at date 1
𝑃𝑉 =
𝐶
×𝑛
(1 + 𝑟)
Present value of a growing annuity when r = g, and the first payment is at date 0
𝑃𝑉 = 𝐶 × 𝑛
See question 12 and 13 at Chapter 4.5 at MyFinanceLab
4.9 - The internal rate of return
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Interest rate = internal rate of return = IRR  defined as the interest rate that sets the NPV of
the cash flows equal to zero.
The Internal rate of Return (IRR) is the r that solves:
𝑃𝑉 =
𝐹𝑉1
(1 + 𝑟)
𝐹𝑉 = (1 + 𝐼𝑅𝑅)𝑁
1
𝐹𝑉 𝑛−1
𝐼𝑅𝑅 = ( )
𝑃
𝐶
IRR of growing perpetuity = 𝑃 + 𝑔
When facing a problem, always think that you are going to calculate it so that NPV = 0.
Chapter 5 – Interest rates
Interest rates can vary depending on when the interest is paid, it can depend on investment
horizon, and it can vary due to risk or tax consequences.
5.1 – Interest rate quotes and Adjustments
We must use a discount rate that matches the time period of our cash flows; this discount rate
should reflect the actual return we could earn over that time period.
The effective annual rate:
Effective annual rate = EAR  indicates the actual amount of interest that will be earned at
the end of one year.
EAR can be calculated as:
1
𝑓 𝑛
𝐸𝐴𝑅 = ( ) − 1
𝑝
where EAR is the equivalent annual rate, f is the future value, p is the present value and n is
the number of years.
We can use this formula if we for example know that we invest 100 dollars today, and that it
grows to 133.93 dollars in 5 years, and they want to know what the APR I received was if the
interest was compounded semiannually.
To get the APR we convert the EAR to an APR
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𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑛 − 𝑝𝑒𝑟𝑖𝑜𝑑 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑟𝑎𝑡𝑒 = (1 + 𝑟)𝑛 − 1
In this formula, n can be larger than 1 (to compute a rate over more than one period) or
smaller than 1 (to compute a rate over a fraction of a period).
Example:
A 6% EAR is equivalent to earning (1,06)^1/12 – 1 = 0,4868% per month.
Annual percentage rates:
Annual Percentage rates = APR  indicates the amount of simple interest earned in one year,
that is, the amount of interest earned without the effect of compounding.
Because it does not include the effect of compounding, the APR quote is typically less than
the actual amount of interest that you will earn. To compute the actual amount that you will
earn in one year, we must first convert the APR to an effective annual rate.
It is important to remember that because the APR does not reflect the true amount you will
earn over one year, we cannot use the APR itself as a discount rate. Instead, the APR with k
compounding periods is a way of quoting the actual interest earned each compounding period:
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 =
𝐴𝑃𝑅
𝑘 𝑝𝑒𝑟𝑖𝑜𝑑𝑠/𝑦𝑒𝑎𝑟
Converting an APR to an EAR
𝐴𝑃𝑅 𝑘
1 + 𝐸𝐴𝑅 = 1 + (
)
𝑘
𝐴𝑃𝑅 = ((1 + 𝐸𝐴𝑅)1/𝑘 − 1) × 𝑘
The EAR increases with the frequency of compounding because of the ability to earn interest
on interest sooner.
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5.2 – Application: Discount rates and Loans:
Many loans are amortizing loans where you pay an interest on the loan plus some part of the
loan balance every month.
Typical terms of a new car loan could be: “6,75% APR for 60 months”
This quote means that the loan will be repaid with 60 equal monthly payments, computed
using a 6,75% APR with monthly compounding.
The 6,75% APR with monthly compounding corresponds to a one-month discount rate of
6,75%/12 = 0,5625%.
Than use this rate in the annuity formula to compute the PV of the loan payments.
See page 180
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Formula for computing a loan payment is the annuity formula rearranged:
𝐶=
𝑃𝑉 × 𝑟
1
1
(1
−
)
𝑟
(1 + 𝑟)𝑛
Outstanding loan balance = The PV of the remaining future loan payments.
See example on page 181.
5.3 – The determinants of Interest rates
Inflation and Real versus Nominal Rates:
Nominal interest rates = indicate the rate at which your money will grow if invested for a
certain period. However, the nominal interest rate does not reflect the actual purchasing
power if the money also grows because of inflation.
Real interest rates = the growth of your purchasing power, adjusted for inflation.
r = nominal interest rate
i = the rate of inflation
r(real) = real interest rate
𝑇ℎ𝑒 𝑟𝑒𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒: 𝑟(𝑟𝑒𝑎𝑙) =
𝑟−𝑖
1+𝑖
If inflation rates are low, the real interest rate is approximately equal to the nominal interest
rate minus the rate of inflation.
Growth of purchasing power = 1 + real rate
Interest rates tend to be high when inflation is high.
Nominal interest rates = can never be negative because by holding cash, an investor can
always earn 0% on their investments.
Real interests rate = can be negative whenever the rate of inflation exceeds the nominal
interest rate.
Investment and Interest rate policy, page 183-184:
All else equal, higher interest rates will tend to shrink the set of positive-NPV investments
available to firms.
The yield curve and discount rates:
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The relationship between the investment term and the interest rate is called the term structure
of interest rates. We can plot this relationship on a graph called the yield curve.
The interest rate that banks offer on investments or charge on loans depends on the horizon,
or term, of the investment or loan.
We can use the term structure to compute the present and future values of a risk-free cash
flow over different investment horizons. We can apply the same logic when computing the
PV of cash flows with different maturities. A risk-free cash flow received in two years should
be discounted at the two-year interest rate, and cash flow received in ten years should be
discounted at the ten-year interest rate. In general, a risk-free cash flow of Cn received in n
years has a present value of:
𝑃𝑉 =
𝐶𝑛
(1 + 𝑟𝑛 )𝑛
Where rn is the risk-free interest rate (expressed as an EAR) for an n-year term. In other
words, when computing a present value we must match the term of the cash flow and term of
the discount rate.
We must use a different discount rate for each cash flow, based on the rate from the yield
curve with the same term.

When the yield curve is relatively flat, this distinction Is relatively minor and is often
ignored by discounting using a single “average” interest rate r.

An upward sloping yield curve is often a sign that investors expect interest rates to rise
in the future. Rising interest rates are no guarantee that investments will yield higher
returns in the future.

Warning! All the formulas for annuity and perpetuity are based on discounting all of
the cash flows at the same rate. They cannot be used in situations in which cash flows
need to be discounted at different rates.
SEE “COMMOM MISTAKES” ON PAGE 186 – IMPORTANT!
5.5 – The Opportunity Cost of Capital
The opportunity cost of capital (or more simply, the cost of capital) provides the benchmark at
which the cash flows of the new investment should be evaluated.
There are many different discount rates to choose from. Therefore, going forward, we will
base the discount rate that we use to evaluate cash flows on the investor´s opportunity cost of
capital, which is the best expected return offered in the market on an investment of
comparable risk and term to the cash flow being discounted.
For a risk-free project the opportunity cost of capital correspond to the interest rate on U.S
Treasury securities with a similar term. The cost of capital for risky projects will often exceed
this amount, depending on the nature and magnitude of the risk.
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Chapter 6 – Valuing Bonds
6.1 – Bond Cash flows, Prices and Yield:
Coupons = the promised interest payments of a bond
Principal (or face value) = the notional amount we use to compute the interest payments.
Usually the face value is repaid at maturity.
Coupon payment:
𝐶𝑃𝑁 =
𝐶𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 × 𝐹𝑎𝑐𝑒 𝑣𝑎𝑙𝑢𝑒
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟
Example:
A “1000 dollar bond with a 10% coupon rate and semiannual payments” will pay coupon
payment of 1000*10%/2 = 50 dollars every six months.
Zero-coupon bonds:
A zero-coupon bond does not make coupon payments. The only cash payment the investor
receives is the face value of the bond on the maturity date.
Treasury Bills = US government bonds = zero-coupon bond.
The price of a zero-coupon bond is less than its face value, so zero-coupon bonds trades at a
discount (a price lower than the face value), so they are also called pure discount bonds.
IRR = the discount rate at which the NPV of the cash flows of the investment opportunity is
equal to zero. So, the IRR of an investment in a zero-coupon bond is the rate of return that
investors will earn on their money if they buy the bond at its current price and hold it to
maturity.
The IRR of an investment in a bond = yield to maturity.
The yield to maturity of a bond is the discount rate that sets the present value of the promised
bond payments equal to the current market price of the bond.
Formula:
𝑃=
𝐹𝑉
(1 + 𝑌𝑇𝑀𝑛 )𝑛
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Yield to maturity of an n-Year Zero-Coupon bond:
𝐹𝑉 1/𝑛
𝑌𝑇𝑀𝑛 = 𝐸𝐴𝑅 = ( ) − 1
𝑃
The yield to maturity is the period rate of return for holding the bond from today until
maturity on date n.
Risk-free interest rate:
Because a default-free zero-coupon bond that matures on date n provides a risk-free return
over the same period, the Law of One Price guarantees that the risk-free interest rate equals
the yield to maturity on such a bond!
r(n) = YTM(n)
Some professionals also use the term spot interest rates to refer to these default-free, zerocoupon yields.
Coupon bonds:
Coupon bonds = pay the investors their face value at maturity. In addition, these bonds make
regular coupon interest payments.


Treasury notes = have original maturities from one to 10 years.
Treasury bonds = have original maturities of more than 10 years.
Yield to maturity of a coupon bond:
𝑃 = 𝐶𝑃𝑁 ×
1
1
𝐹𝑉
(1 −
)
+
(1 + 𝑦)𝑁
𝑦
(1 + 𝑦)𝑁
Unfortunately, unlike in the case of a zero-coupon bond, there is no simple formula to solve
for the yield to maturity directly. Instead, we need to use either trial-and-error or the annuity
spreadsheet we introduced in chapter 4, or Excel´s IRR function.
When we calculate a bond´s yield to maturity by solving this equation, the yield we compute
will be a rate per coupon interval. This yield is typically stated as an annual rate by
multiplying it by the number of coupons per year, thereby converting it to an APR with the
same compounding interval as the coupon rate.
When prices are quoted in the bond market, they are conventionally quoted as a percentage of
their face value.

The price of a bond right before the first coupon payment is made is a 9-year bond
plus the coupon payment that is about to be paid (if it is a 10-year bond from the
beginning).
15

The price of a bond right after the first payment is made is just a 9-year bond (if it is a
10-year bond from the beginning).
6.2 – Dynamic Behaviour of Bond Prices
Bonds can trade at a discount, par or premium:
Discount:
 If a bond trades at a discount, an investor who buys the bond will earn a return both
from receiving the coupons and from receiving a face value that exceeds the price paid
for the bond. As a result, its yield to maturity will exceed its coupon rate.
 Coupon rate < Yield to maturity
Premium:
 An investor´s return from the coupons is diminished by receiving a face value less
than the price paid for the bond. Thus, a bond trades at a premium whenever its yield
to maturity is less than its coupon rate.
 Coupon rate > Yield to maturity
Par:
 A bond trades at par when its price is equal to its face value. The coupon rate is equal
to its yield to maturity.
 Coupon rate = Yield to maturity
Time and Bond Prices:
If a bond´s yield to maturity has not changed, then the IRR of an investment in the bond
equals its yield to maturity even if you sell the bond early.
IRR = YTM if the yield to maturity has not changed when you sell the bond early
IRR of holding a bond to maturity = YTM when you purchased the bond
If the yield to maturity has changed when you sell the bond early
If you for example buy a 30-year, zero-coupon bond, with a yield to maturity of 4%. You sell
the bond after 5 years. Then,
𝑃5 1/5
𝐼𝑅𝑅 = ( ) − 1
𝑃0
The effect of Time on the Price of a coupon bond: See example 6.6 on page 213
16
Interest Rate Changes and Bond Prices:
As interest rates in the economy fluctuate, the yields that investors demand to invest in bonds
will also change.
A higher yield to maturity implies a higher discount rate for a bond´s remaining cash flows,
reducing their present value and hence the bond´s price. Therefore, as interest rates and bond
yields rise, bond prices will fall, and vice versa.
The sensitivity of a bond´s price to changes in interest rates depends on the timing of its cash
flows. Because it is discounted over a shorter period, the present value of a cash flow that will
be received in the near future is less dramatically affected by interest rates than a cash flow in
the distant future. Thus, shorter-maturity zero-coupon bonds are less sensitive to changes
in interest than are longer-term zero-coupon bonds. Similarly, bonds with higher coupon
rates – because they pay higher cash flows upfront – are less sensitive to interest rates
changes than otherwise identical bonds with lower coupon rates. The sensitivity of a bond´s
price to changes in interest rates is measured by the bond´s duration. Bonds with higher
durations are highly sensitive to interest rate changes.
Calculate the change in price when interest rates changes (when the yield to maturity
changes)
1. Calculate the old price (with the old interest rate)
2. Calculate the new price (with the new interest rate)
3. OBS! Make sure to use the right Price-formula depending on if it´s a zero-coupon
bond or a coupon-bond
𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑏𝑜𝑛𝑑 𝑤𝑖𝑡ℎ 𝑛𝑒𝑤 𝑌𝑇𝑀−𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝐵𝑜𝑛𝑑 𝑤𝑖𝑡ℎ 𝑜𝑙𝑑 𝑌𝑇𝑀
4. Calculate the change in price =
𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑏𝑜𝑛𝑑 𝑤𝑖𝑡ℎ 𝑜𝑙𝑑 𝑌𝑇𝑀
6.3 – The yield curve and bond arbitrage
Using the Law of One Price, we can show that given the spot interest rates, which are the
yields of default-free zero-coupon bonds, we can determine the price and yield of any other
default-free bonds.
If the coupon bond cash flows are identical, the Law of One Price states that the price of the
portfolio of zero-coupon bonds must be the same as the price of the coupon bond.
Yield to maturity of a zero-coupon bond = the competitive market interest rate for a risk-free
investment with a term equal to the term of the zero-coupon bond. Therefore, the price of a
coupon bond must equal the present value of its coupon payments and face value discounted
at the competitive market interest rates.
Price of a Coupon bond:
17
𝑃 = 𝑃𝑉(𝐵𝑜𝑛𝑑 𝐶𝑎𝑠ℎ 𝐹𝑙𝑜𝑤𝑠) =
𝐶𝑃𝑁
𝐶𝑃𝑁
𝐶𝑃𝑁 + 𝐹𝑉
+
+⋯+
1
2
(1 + 𝑌𝑇𝑀1 )
(1 + 𝑌𝑇𝑀2 )
(1 + 𝑌𝑇𝑀𝑛 )𝑛
where CPN is the bond coupon payment, YTM(n) is the yield to maturity of a zero-coupon
bond that matures at the same time as the nth coupon payment, and FV is the face value of the
bond.
We can determine the no-arbitrage price of a coupon bond by discounting its cash flows using
the zero-coupon yields. In other words, the information in the zero-coupon yield curve is
sufficient to price all other risk-free bonds.
Coupon bond yields:
If we know the price of a coupon bond we can calculate the yield to maturity of this bond.
The YTM is the rate, y, that satisfies the price. If the price for example is 1153, the YTM is:
P = 1153 = 100/(1+y) + 100/(1+y)^2 + (100+1000)/(1+y)^3
By using excel we can calculate that the YTM is 4,44%.
See page 219.
Coupon bonds with the same maturity can have different yields depending on their coupon
rates. As the coupon increases, earlier cash flows become relatively more important than later
cash flows in the calculation of the present value. If the yield curve is upward sloping, the
resulting yield to maturity decreases with the coupon rate of the bond. Alternatively, when the
zero-coupon yield is downward sloping, the yield to maturity will increase with the coupon
rate. When the yield curve is flat, all zero-coupon and coupon-paying bonds will have the
same yield, independent of their maturities and coupon rates.
Treasury Yield Curves:
Coupon-paying yield curve = The plot of the yields of coupon bonds of different maturities.
Using similar methods to those employed in this section, we can apply the Law of One Price
to determine the zero-coupon bond yields using the coupon-paying yield curve. Thus, either
type of yield curve provides enough information to value all other risk-free bonds.
6.4 – Corporate Bonds
So far, we have focused on default-free bonds such as US Treasury Bonds, for which the cash
flows are known with certainty. For other bonds, such as Corporate Bonds (bonds issued by
corporations), the issuer may default – that is, it might not pay back the full amount promised
in the bond prospectus.
Credit risk = the risk of default of the bond (means that the bond´s cash flows are not known
with certainty.
18
Corporate bond yields:
Investors pay less for bonds with credit risk than they would for an otherwise identical
default-free bond. Because the yield to maturity for a bond is calculated using the promised
cash flows, the yield of bonds with credit risk will be higher than that of otherwise identical
default-free bonds. The prospect of default lowers the cash flow investors expect to receive
and hence the price they are willing to pay.
Price of no-default bond:
P = 1000/1+YTM(1) = 1000/1,04 = 961,51 dollars
Price of Certain-default bond:
If investors are certain they are only getting 90% of its outstanding obligations, we calculate
the price as:
P = 900/(1+YTM(1) = 900/1,04 = 865,38 dollars.
Given the bond´s price, we can compute the bond´s yield to maturity. When computing this
yield, we use the promised rather than the actual cash flows. Thus,
YTM = FV/P – 1 = 1000/865,38 – 1 = 15,56%.
The 15,56% yield to maturity of Avant´s bond is much higher than the yield to maturity of the
default-free Treasury Bill. However, this doesn´t mean that investors who buy the bond will
earn 15,56% return. Because Avant will default, the expected return of the bond equals its 4%
cost of capital:
900/865,38 = 1,04.
Note that the yield to maturity of a defaultable bond exceeds the expected return of investing
in the bond. Because we calculate the yield to maturity using the promised cash flows rather
than the expected cash flows, the yield will always be higher than the expected return of
investing in the bond.
To determine the price of a bond where there is 50% chance that the bond will default and
you will receive 900 dollars and 50% chance that the bond will repay its face value in full, we
must discount the expected cash flow using a cost of capital equal to the expected return of
other securities with equivalent risk. If a firm (like most firms) is more likely to default if the
economy is weak than if the economy is strong, then investors will demand a risk premium to
invest in this bond. That is, the firm´s debt cost of capital, which is the expected return the
firm´s debt holders will require to compensate them for the risk of the bond´s cash flows, will
be higher than the risk-free interest rate.
Let´s suppose investors demand a risk-premium of 1,1% for this bond, so that the appropriate
cost of capital is 5,1 instead of the risk-free interest rate of 4%. Then present value of the
bond´s cash flow is
19
With the 50% chance of default and the 50% chance of getting the full payment, you will on
average receive 950 dollars (In between 900 and 1000)
P = 950/1,051 = 903,90 dollars
YTM = FV/P – 1 = 1000/903,90 – 1 = 10,63%.
Note that, the bond´s prices decrease, and its yield to maturity increases, with a greater
likelihood of default.
Conversely, the bond´s expected return, which is equal to the firm´s debt cost of capital, is
less than the yield to maturity if there is a risk of default. Moreover, a higher yield to maturity
does not necessarily imply that a bond´s expected return is higher.
Price of a default-bond:
𝑃=
𝐹𝑉 × (1 − 𝑑) + 𝑑 × 𝑟
(1 + 𝑦)𝑛
where FV is the face value, d is the risk of default, r is the amount investors expect to receive
in case of default, y is the expected return, and n is the number of periods.
Yield to maturity
𝐹𝑉 1/𝑛
𝑌𝑇𝑀 = ( ) − 1
𝑃
Bond rating:
There are many different rating classes of bonds – ranging from AAA to C/C,D.


Bonds in the top four categories = investment-grade bonds because the low default
risk.
Bonds in the bottom five categories = speculative bonds, junk bonds, or high-yield
bonds because their likelihood of default is high.
The difference between the yield of the corporate bonds and the Treasury yields as the
default spread or credit spread.
Credit spread is high for bonds with low ratings.
6.5 – Sovereign Bonds
Sovereign bonds = bonds issued by national government.
History has showed that these bonds can be risky – for example the 2008 financial crisis.
Sovereign bond yields reflect investor expectations of inflation, currency and default risk.
Hence, higher yield means higher risk of default.
20
Chapter 6 – Appendix.
Forward interest rates:
An interest rate forward contract (also called a forward rate agreement) is a contract
today that fixes the interest rate for a loan or investment in the future.
Computing Forward rates:
Forward rate = an interest rate that we can guarantee today for a loan or investment that will
occur in the future.
The forward rate for year one is the rate on an investment that starts today and is repaid in one
year; it is equivalent to an investment in a one-year, zero-coupon bond. Therefore, by The
Law of One Price, these rates must coincide:
f(1) = YTM(1)
In general, we can compute the forward rate for year n by comparing an investment in n-year,
zero-coupon bond to an investment in an (n-1) year, zero-coupon bond, with the interest rate
earned in the nth year being guaranteed through an interest rate forward contract. Because
both strategies are risk-free, they must have the same payoff or else an arbitrage opportunity
would be available. Comparing the payoffs of these strategies we have:
(1+YTM(n))^n = (1+YTM(n-1))^n-1)(1+fn)
General formula for the forward interest rate:
𝑓𝑛 =
(1 + 𝑌𝑇𝑀𝑛 )𝑛
−1
(1 + 𝑌𝑇𝑀𝑛−1 )𝑛−1
See example on page 239.
When YTMn > YTMn-1  forward rate is higher than the zero-coupon yield, fn > YTM
(when the yield curve is increasing)
When the yield curve is flat, the forward rate equals the zero-coupon yield.
Hence, when the yield curve is flat, the forward rate equals the spot rate.
Computing Bond Yields from Forward rates:
In the above we computed forward rates using the zero-coupon yields. It is also possible to
compute the zero-coupon yields from the forward interest rates.
See page 239.
Forward rates and Future Interest rates:
21
Expected future spot interest rate = Forward Interest rate + Risk premium.
The risk premium can be either positive or negative depending on investor´s preferences. As a
result, forward rates tend not to be ideal predictors of future spot rates.
Chapter 10 – Capital Markets and the Pricing of Risk
10.1 – Risk and Return: Insights from 89 years of Investor History
CPI = consumer price index (investments can be measured by the CPI)
There are many different ways to invest your money. Small stocks often give the highest
return, but they also have the highest risk. Small stocks have the highest return on average
over time, but during some periods you can also make big losses on small stocks.
The stocks that have the highest returns also have the most variable returns.
As time horizon lengthens, the relative performance of the stock portfolio improves.
In chapter 3 we explained why investors are averse to fluctuations in the value of their
investments, and that investments that are more likely to suffer losses in downturns must
compensate investors for this risk with higher expected returns. In this chapter, we want to
explain how much investors demand in terms of compensation (how much higher expected
return) to bear a given level of risk.
10.2 – Common Measures of Risk and Return
Here, we begin with reviewing the standard ways to define and measure risks.
Probability distribution:
To make different investments comparable we express their performance in terms of their
returns. The return indicates the percentage increase in the value of an investment per dollar
initially invested in the security. We summarize the likelihood of a certain return to occur
with a probability distribution. It assigns a probability, pR, that each possible return, Rm
will occur.
See table 10.1 on page 354.
Expected return:
We calculate the expected (or mean) return as a weighted average of the possible returns
where the weights correspond to the probabilities.
Expected (mean) return
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 = 𝐸(𝑅) = ∑ 𝑝𝑅 × 𝑅
𝑅
22
The expected return is the return we would earn on average if we could repeat the investment
many times, drawing the return from the same distribution each time.
Variance and Standard deviation:


Variance = expected squared deviation from the mean
Standard deviation = square root of the variance
Variance and Standard Deviation of the Return Distribution
𝑉𝑎𝑟(𝑅) = 𝐸[(𝑅 − 𝐸[𝑅])2 ] = ∑ 𝑝𝑅 × (𝑅 − 𝐸[𝑅])2
𝑅
𝑆𝐷(𝑅) = √𝑉𝑎𝑟(𝑅)
If the return is risk-free and never deviates from its mean the variance is zero.
In finance, we refer to the standard deviation of return as its volatility
Two stocks can have the same expected return, but different variance and volatility. See page
356 – figure 10.4
10.3 – Historical Returns of Stocks and Bonds.
The distribution of past returns can be helpful when we seek to estimate the distribution of
returns investors may expect in the future.
Computing historical returns:
Realized return = the return that actually occurs over a particular time period.
Realized return = dividend Yield + Capital Gain rate (expressed as a percentage of the initial
stock price).
Realized Return over one period:
𝑅𝑡+1 =
𝐷𝑖𝑣𝑡+1 + 𝑃𝑡+1
𝐷𝑖𝑣𝑖+1 𝑃𝑡+1 − 𝑃𝑡
−1=
+
𝑃𝑡
𝑃𝑡
𝑃𝑡
Calculating realized annual returns:
1 + 𝑅𝑎𝑛𝑛𝑢𝑎𝑙 = (1 + 𝑅𝑄1 )(1 + 𝑅𝑄2 ) … (1 + 𝑅𝑄𝑛 )
𝑅1 =
𝑃2 + 𝐷2
−1
𝑃1
See formula 10.5 at page 357
23
Empirical distribution = when we plot the probability distribution as a histogram using
historical data.
See figure 10.5 on page 360
Average annual returns:
The average annual return of an investment during some historical period is simply the
average of the realized returns for each year. You can use the average return over the period
as the estimate of the monthly or annual expected return.
That is, if Rt is the realized return of a security in year t, then the average annual return for
years 1 through T is:
Average annual return of a Security:
𝑇
1
∑ 𝑅𝑡
𝑇
𝑡−1
𝑅𝑡 is the realized return of a security in year t, and the calculation above is for the average
annual return for years 1 through T.
Notice that the average annual return is the balancing point of the empirical distribution – in
this case, the probability of a return occurring in a particular range is measured by the number
of times the realized return falls in that range. Therefore, if the probability distribution of the
returns is the same over time, the average return provides an estimate of the expected return.
The Variance and Volatility of Returns:
Variance Estimate Using Realized Returns:
𝑇
1
𝑉𝑎𝑟(𝑅) =
∑(𝑅𝑡 − 𝑅̅ )^2
𝑇−1
𝑡−1
𝑅̅ = 𝑡ℎ𝑒 𝑎𝑣𝑎𝑟𝑎𝑔𝑒 𝑎𝑛𝑛𝑢𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛
Standard deviation or volatility = the square root of the variance.
Estimation Error: Using Past Returns to predict the Future
To estimate the cost of capital for an investment, we need to determine the expected return
that investors will require to compensate them for that investment´s risk. If the distribution of
past returns and the distribution of future returns are the same, we could look at the return
investors expected to earn in the past on the similar investments, and assume they will require
the same return in the future. However, there are two difficulties with this approach.
24
1. We do not know what investors expected in the past; we can only observe the actual
returns that were realized.
2. The average return is just an estimate of the true expected return, and is subject to
estimation error.
Standard error:
We measure the estimation error of a statistical estimate by its standard error. The standard
error is the standard deviation of the estimated value of the mean of the actual distribution
around its true value; that is, it is the standard deviation of the average return. The standard
error provides an indication of how far the sample average might deviate from the expected
return.
Standard Error of the Estimate of the Expected Return
SD(Average of Independent, Identical Risks) =
𝑆𝐷(𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑅𝑖𝑠𝑘)
√𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑂𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
The 95% confidence Interval for the expected return is:
Historical Average return ± (2 x Standard Error)
Limitations of Expected Return Estimates.
First, we will consider how to measure a security´s risk, and then we will use the relationship
between risk and return – which we must still determine – to estimate its expected return.
Compound Annual Return
𝐹𝑖𝑛𝑎𝑙 𝑉𝑎𝑙𝑢𝑒
1/𝑇
(𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡)
1
− 1 = [(1 + 𝑅1 ) × (1 + 𝑅2 ) × … × (1 + 𝑅𝑇 )]𝑇 − 1 =
The compound annual return will always be below the average return, and the difference
grows with the volatility of the annual returns.
The compound annual return is a better description of the long-run historical performance of
an investment. It describes the equivalent risk-free return that would be required to duplicate
the investments performance over the same time period. The average return is a better
measure of the expected return next year.
10.4 – The Historical Trade-Off between Risk and Return
In Chapter 3, we discussed the idea that investors are risk-averse: The benefit they receive
from an increase in income is smaller than the personal cost of an equivalent decrease in
income. This idea suggests that investors would not choose to hold a portfolio that is more
volatile unless they expected to earn a higher return.
The Returns of Large Portfolios:
25
Excess return = the difference between the average return for the investment and the average
return for Treasury bills, a risk-free investment, and measures the average risk premium
investors earned for bearing the risk of the investment.
Investments with higher volatility have historically rewarded investors with higher average
returns. Riskier investments must offer investors higher average returns to compensate them
for the extra risk they are taking on.
The Returns of Individual Stocks:
Investments with higher volatility should have a higher risk premium and therefore higher
returns.
If we look at the volatility and return of individual stocks, we do not see any clear relationship
between them.
From figure 10.7 on page 366 we can make several observations.
1. There is a relationship between size and risk. Larger stocks have lower volatility
overall.
2. Even the largest stocks are typically more volatile than a portfolio of large stocks, the
S&P 500.
3. There is no clear relationship between volatility and return.
Thus, while volatility is perhaps a reasonable measure of risk when evaluating large portfolio,
it is not adequate to explain the returns of individual securities.
10.5 – Common versus Independent Risk
In this section, we explain why the risk of an individual security differs from the risk of a
portfolio composed of similar securities.
Theft Versus Earthquake Insurance: An example (page 367)
1% chance that the Home will be robbed and 1% chance that the home will be damaged by an
earthquake.
We know that the risks of the individual policies are similar, but are the risks of the portfolios
of policies similar?
 The expected numbers of claims may be the same (1%), but earthquake and theft
insurance lead to portfolios of very different risk characteristics. For earthquake
insurance, the number of claims is very risky. It will most likely be zero, but there is
1% chance that the insurance company will have to pay all the policies it wrote. In the
theft sector the risk is more spread. If the insurance company writes 100 000 policies
of each type for homeowners, they would expect 1% of the 100 000 homes to
experience a robbery per year – the number of claims per year will be about 1000, so
if the insurance company are sufficient for 1200 they should be safe. When it comes to
the earthquake however, all homes are likely to be affected if an earthquake occur and
the insurance company has to expect as many as 100 000 claims. In this case, the risk
of the portfolio of insurance policies is no different from the risk of any single policy –
it is still all or nothing. Conversely, for theft insurance, the number of claims in a
26
given year is quite predictable. The portfolio of theft insurance policies has almost no
risk, because year in and year out, it will be very close to 1% of the total number of
policies, 1000 claims.
Types of risk:



Earthquake = Common risk = risk that is perfectly correlated (affects all
simultaneously – if one home is affected by an earthquake, probably all other homes
are too affected.)
Theft = Independent risk = risk that is independent and uncorrelated between, for
example homes. When risks are independent, some individual home-owners are
unlucky and others are lucky, but overall the number of claims is quite predictable.
Diversification = the averaging out of independent risks in a large portfolio
The role of diversification:
At the beginning of the year, the homeowner expects a 1% chance of placing a claim for
either type of insurance. But at the end of the year, the home-owner will have a filed claim
(100%) or not (0%).
SD(claim) = √𝑉𝑎𝑟(𝐶𝑙𝑎𝑖𝑚) = √0.99 × (0 − 0.01)2 + 0.01 × (1 − 0.01)2 = 9.95%
For the home-owners, this standard deviation is the same for a loss from earthquake or theft.
The case for the insurance company:
The percentage of claims is either 100% or 0% when it comes to the earthquake – the
common risk. Thus, the percentage of claims received by the earthquake insurer is also 1% on
average, with a 9.95% standard deviation.
While the theft insurer also receives 1% of claims on average, because the risk of theft is
independent across households, the portfolio is much less risky. To quantify the difference,
let´s calculate the standard deviation of the average claim using the below equation. Recall
that when risks are independent and identical, the standard deviation of the average is known
as the standard error, which declines with the square root of the number of observations.
Therefore,
SD(Percentage Theft Claims) =
𝑆𝐷(𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝐶𝑙𝑎𝑖𝑚)
√𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑂𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
=
9.95%
√100 000
= 0.03%
Thus, there is almost no risk for the theft insurer. The principle of diversification is routinely
used in the insurance industry, as well as many other industries. Even in the case of
earthquake insurance, insurers can achieve some diversification by selling policies in different
geographical regions or combining different types of policies.
The risk of a portfolio of insurance contracts depends on whether the individual risks within it
are common or independent. Independent risks are diversified in a large portfolio, whereas
common risks are not.
27
10.6 – Diversification in Stock Portfolios
Firm-Specific Versus Systematic risk.
Over any given time period, the risk of holding a stock is that the dividends plus the final
stock price will be higher or lower than expected, which makes the realized return risky.
Usually, stock prices and dividends fluctuate due to two types of news:
1. Firm-specific news is good or bad news about the company itself. For example, a firm
might announce that it has been successful in gaining market share within its industry.
2. Market-wide news is news about the economy as a whole and therefore affects all
stocks. For instance, the Federal Reserve might announce that it will lower interest
rates to boost the economy.
Fluctuations of stock´s return that are due to firm-specific news  Independent risk
 Also referred to as firm-specific, idiosyncratic, unique, or diversifiable risk.
Fluctuations of stock´s return that are due to market-wide news  Common risk
 Also referred to as systematic, undiversifiable, or market risk.
o When we combine many stocks in a large portfolio, the firm-specific risks for each
stock will average out and be diversified. Good news will affect some stocks, and bad
news will affect others, but the amount of good or bad news overall will be relatively
constant.
o The systematic risk, however, will affect all firms – and therefore the entire portfolio –
and will not be diversified.
If we have stocks with both diversifiable risk and systematic risk in a portfolio, only the firmspecific risk will be diversified when we combine many firm´s stocks into a portfolio. The
volatility will therefore decline until only the systematic risk, which affects all firms, remains.
Therefore, the portfolio as a whole has lower volatility than each of the stocks within it.
No Arbitrage and the Risk Premium:
Because each Individual type I firm (firms that only contain firm-specific risk) is risky,
should investors expect to earn a risk-premium when investing in type I-firms?
 In a competitive market, the answer is no. Because a large portfolio of type I firms has
no risk, it must earn the risk-free interest rate. If the diversifiable risk of stocks were
compensated with and additional risk premium, the investors could buy stocks, earn
additional premium, and simultaneously diversify and eliminate the risk. This noarbitrage argument suggests the following more general principle:
 The risk premium for diversifiable risk is zero, so investors are not compensated for
holding firm-specific risk.
 However, investors will demand a risk premium for holding systematic risk; otherwise
they would be better off selling their stocks and investing in risk-free bonds. Because
investors can eliminate firm-specific risk for free by diversifying, whereas systematic
risk can be eliminated only by sacrificing expected returns, it is a security´s systematic
risk that determines the risk premium investors require to hold it. This fact leads to as
second key principle:
28

The risk premium of a security is determined by its systematic risk and does not
depend on its diversifiable risk.
Although both type S firms and I firms have the same volatility, type S firms have an
expected return of 10% and type I-firms have an expected return of 5%. The difference in
expected returns derives from the difference in the kind of risk each firm bears. Type I firms
have only firm-specific risk, which does not require a risk premium, so the expected return of
5% for type I firms equals the risk-free interest rate. Type S firms have only systematic risk.
Because investors will require compensation for taking on this risk, the expected return of
10% for type S firms provides investors with a 5% risk-premium above the risk-free interest
rate.
While volatility might be a reasonable measure of risk for a well-diversified portfolio, it is not
an appropriate metric for an individual security.
10.7 – Measuring Systematic Risk
To determine the additional term, or risk premium, investors require to undertake an
investment, we first need to measure the investment´s systematic risk.
Identifying Systematic Risk: The Market Portfolio
We would like to know how sensitive the stock is to systematic shocks that affect the
economy as a whole.
The first step is to find a portfolio that only contains systematic risk. Changes in the price of
this portfolio will correspond to systematic shocks of the economy. We call such a portfolio
an efficient portfolio. A natural candidate for an efficient portfolio is the market portfolio,
which is a portfolio of all stocks and securities traded in the capital markets. Because it is
difficult to find data for the returns of many bonds and small stocks, it is common in practice
to use the S&P 500 portfolio as an approximation for the market portfolio, under the
assumption that the S&P500 is large enough to be essentially fully diversified.
Sensitivity to Systematic Risk: Beta
We can measure the systematic risk of a security by calculating the sensitivity of the
security´s return to the return of the market portfolio, known as beta of the security. More
precisely,
The beta of a security is the expected % change in its return given a 1% change in the return
of the market portfolio.
10.8 – Beta and the Cost of Capital
Cost of capital = the best expected return available on alternative investments in the market
with comparable risk and term.
29
For risk-free investments, this cost of capital corresponds to the risk-free interest rate, plus an
appropriate risk premium.
Estimating the risk premium:
The size of the risk premium investors will require to make a risky investment depends on
their risk aversion. Rather than attempt to measure this risk aversion directly, we can measure
it indirectly by looking at the risk premium investors´ demand for investing in systematic, or
market, risk.
The market risk premium:
The risk premium investors earn by holding a market risk is the difference between the
market portfolio´s expected return and the risk-free interest rate.
Market risk Premium = 𝐸[𝑅𝑀𝑘𝑡 ] − 𝑟𝑓
In the same way that the market interest rate reflects investors´patience and determines the
time value of money, the market risk premium reflects investors´risk tolerance and determines
the market price of risk in the economy.
Adjusting for Beta:
Beta of 1 = the risk of the market portfolio itself
Beta of 2 = the investment carries twice as much systematic risk as an investment in the
market portfolio. That is, for each dollar we invest in the opportunity, we could invest twice
as much in the market portfolio and be exposed to exactly the same amount of systematic risk.
Because it has twice as much systematic risk, investors will require twice the risk premium to
invest in an opportunity with a beta of 2.
Estimating the cost of capital of an investment from Its Beta:
𝑟𝑖 = 𝑅𝑖𝑠𝑘 − 𝑓𝑟𝑒𝑒 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 + 𝛽𝐼 ×= 𝑀𝑎𝑟𝑘𝑒𝑡 𝑟𝑖𝑠𝑘 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
= 𝑟𝑓 + 𝛽𝐼 × (𝐸[𝑅𝑀𝑘𝑡 ] − 𝑟𝑓 )
This is equal to the expected return of the stock according to CAPM.
When the expected returns match the actual expected return we say that the CAPM holds.
Negative Beta:
If a stock has a negative Beta, it would have a negative risk-premium – it would have an
expected return below the risk-free rate. A stock with a negative beta will tend to do well in
bad times, so owning it will provide insurance against systematic risk of other stocks in the
portfolio. Risk-averse investors are willing to pay for this insurance.
Chapter 11 – Optimal Portfolio Choice and the Capital Asset Pricing Model
30
11.1 – The Expected Return of a Portfolio
To find an optimal portfolio, we need a method to define a portfolio and analyse its return.
We can describe a portfolio by its portfolio weights, the fraction of the total investment in the
portfolio held in each individual investment in the portfolio:
𝑥𝑖 =
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑖
𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜
These portfolio weights add up to 1, so that they represent the way we have divided our
money between the different individual investments in the portfolio.
The return of a portfolio:
𝑅𝑝 = 𝑥1 𝑅1 + 𝑥2 𝑅2 + ⋯ + 𝑥𝑛 𝑅𝑛 = ∑ 𝑥𝑖 𝑅𝑖
𝑖
Expected Return of a Portfolio:
𝐸(𝑅𝑝 ) = 𝐸 [∑ 𝑥𝑖 𝑅𝑖 ] = ∑ 𝐸[𝑥𝑖 𝑅𝑖 ] = ∑ 𝑥𝑖 𝐸(𝑅𝑖 )
𝑖
𝑖
𝑖
That is, the expected return of a portfolio is simply the weighted average of the expected
returns of the investments within it, using the portfolio weights.
11.2 – The Volatility of a Two-Stock Portfolio
In this section, we describe the statistical tools that we can use to determine the risk stocks
have in common – the Common risk – and determine the volatility of a portfolio.
Combining Risks
1. By combining stocks into a portfolio, we reduce risk through diversification. Because
the prices of the stocks do not move identically, some of the risk is averaged out in a
portfolio. As a result, both portfolios have lower risk than the individual stocks.
2. The amount of risk that is eliminated in a portfolio depends on the degree to which the
stocks face common risks and their prices move together.
Determining Covariance and Correlation
To fins the risk of a portfolio, we need to know more than the risk and return of the
component stocks: We need to know the degree to which the stocks face common risks and
their returns move together. In this section, we introduce covariance and correlation, that
allow us to measure the co-movement of returns.
Covariance = the expected product of the deviations of two returns from their means.
Covariance between Returns Ri and Rj
31
𝐶𝑜𝑣(𝑅𝑖 , 𝑅𝑗 ) = 𝐸[(𝑅𝑖 − 𝐸[𝑅𝑖 ])(𝑅𝑗 − 𝐸[𝑅𝑗 ])]
When estimating the covariance from historical data, we use the formula:
Estimate of the Covariance from Historical data
𝐶𝑜𝑣(𝑅𝑖 , 𝑅𝑗 ) =
1
∑ (𝑅 − 𝑅̅𝑖 )(𝑅𝑗,𝑡 − 𝑅̅𝑗 )
𝑇 − 1 𝑡 𝑖,𝑡
Intuitively, if two stocks move together, their returns will tend to be above or below average
at the same time, and the covariance will be positive. If the two stocks move in opposite
directions, one will tend to be above the average when the other is below average, and the
covariance will be negative.
Correlation = the covariance of the returns divided by the standard deviation of each return.
We calculate the correlation in order to control for the volatility of each stock and quantify the
strength of the relationship between them:
𝐶𝑜𝑟𝑟(𝑅𝑖 , 𝑅𝑗 ) =
𝐶𝑜𝑣( 𝑅𝑖 , 𝑅𝑗 )
𝑆𝐷(𝑅𝑖 )𝑆𝐷(𝑅𝑗 )
Correlation is always between -1 and +1, which allows us to gauge the strength of the
relationship between the stocks.
-1  Always move Oppositely
< 0  Tend to move oppositely
0  No tendency
>0  Tend to move together
+1  Always move together
The closer the correlation is to +1, the more the returns tend to move together as a result of
common risk.
Independent risks are uncorrelated.
Stocks in the same industry tend to have more highly correlated returns than stocks in
different industries.
Computing the Covariance from the Correlation:
𝐶𝑜𝑣(𝑅𝑖 , 𝑅𝑗 ) = 𝐶𝑜𝑟𝑟(𝑅𝑖 , 𝑅𝑗 )𝑆𝐷(𝑅𝑖 )𝑆𝐷(𝑅𝑗 )
Computing a Portfolio´s Variance and Volatility:
The Variance of a two-stock Portfolio:
𝑉𝑎𝑟(𝑅𝑝 ) = 𝑥12 𝑉𝑎𝑟(𝑅1 ) + 𝑥22 𝑉𝑎𝑟(𝑅2 ) + 2𝑥1 𝑥2 𝐶𝑜𝑣(𝑅1 , 𝑅2 )
32
Volatility, 𝑆𝐷(𝑅𝑝) = √𝑉𝑎𝑟(𝑅𝑝 )
The variance of the portfolio depends on the variance of the individual stocks and on the
covariance between them.
With a positive amount invested in each stock, the more stocks move together and the higher
their covariance or correlation, the more variable the portfolio will be.
LÄR DIG PQ-FORMELN!!
11.3 – The Volatility of a Large Portfolio
The variance of a portfolio is equal to the weighted average covariance of each stock with the
portfolio:
𝑉𝑎𝑟(𝑅𝑝 ) = ∑ 𝑥𝑖 𝐶𝑜𝑣(𝑅𝑖 , 𝑅𝑝 ) = ∑ 𝑥𝑖 𝐶𝑜𝑣(𝑅𝑖 , ∑ 𝑥𝑗 𝑅𝑗 ) = ∑ ∑ 𝑥𝑖 𝑥𝑗 𝐶𝑜𝑣 (𝑅𝑖 , 𝑅𝑗 )
𝑖
𝑖
𝑗
𝑖
𝑗
This formula says that the variance of a portfolio is equal to the sum of the covariance’s of the
returns of all pairs of stocks in the portfolio multiplied by each of their portfolio weights. That
is, the overall variability of the portfolio depends on the total co-movement of the stocks
within it.
Diversification with an equally weighted Portfolio
Equally weighted portfolio = a portfolio in which the same amount is invested in each stock.’
Variance of an Equally Weighted Portfolio of n Stocks:
𝑉𝑎𝑟(𝑅𝑝 ) =
1
(𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑆𝑡𝑜𝑐𝑘𝑠)
𝑛
1
+ (1 − ) (𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑆𝑡𝑜𝑐𝑘𝑠)
𝑛
This equation shows that as the number of stocks, n, grows large, the variance of the portfolio
is determined primarily by the average covariance among the stocks.
The benefit of diversification is most dramatically initially: almost all of the benefit of
diversification can be achieved with about 30 stocks.
Volatility when risks are independent:
If risks are independent, they are uncorrelated and their covariance is zero. The volatility of
an equally weighted portfolio of the risks is:
𝑆𝐷(𝑅𝑃 ) = √𝑉𝑎𝑟(𝑅𝑃 =
𝑆𝐷(𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑅𝑖𝑠𝑘)
Diversification with General Portfolios:
33
√𝑛
Volatility of a Portfolio with Arbitrary weights
𝑆𝐷(𝑅𝑝 ) = ∑ 𝑥𝑖 × 𝑆𝐷(𝑅𝑖 ) × 𝐶𝑜𝑟𝑟(𝑅𝑖 , 𝑅𝑝 )
𝑖
𝑀𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝐶𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑡𝑜 𝑟𝑖𝑠𝑘 = 𝑆𝐷(𝑅𝑖 ) × 𝐶𝑜𝑟𝑟(𝑅𝑖 , 𝑅𝑝 )
See equation 11.13 on page 401.
Volatility of a large portfolio:
𝑆𝐷(𝑅𝑝 ) = √𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
Module 2.05
11.4 – Risk Versus Return: Choosing an Efficient Portfolio
Efficient Portfolios with Two Stocks:
See figure 11.2 on page 403!!


Inefficient portfolio = whenever it is possible to find another portfolio that is better in
terms of both expected return and volatility
Efficient portfolio = when there is no other portfolio of the two stocks that offers a
higher expected return with lower volatility. Investors choose among portfolios based
on their own preferences for return versus risk.
The Effect of Correlation:





Correlation has no effect on the expected return of a portfolio.
However, the volatility of the portfolio will differ depending on the correlation. In
particular, the lower the correlation, the lower the volatility we can obtain.
If two stocks are perfectly correlated there will be a straight line between them. In this
extreme case, the volatility of the portfolio is equal to the weighted average volatility
of the two stocks – there is no diversification. When the correlation is less than 1,
however, the volatility of the portfolios is reduced due to diversification, and the curve
bends to the left. The reduction in risk becomes greater as the correlation decreases.
The lower the correlation, the more will the curve bend to the left. When the two
stocks are perfectly negatively correlated (when the correlation is -1), it becomes
possible to hold a portfolio that bears absolutely no risk.
See figure 11.4 on page 404!!
Short sales:
Long position in the security = when we make a positive investment
34
Short position = when we invest a negative amount in a stock (by engaging in a short sale, a
transaction in which you sell a stock today that you do not own, with the obligation to buy it
back in the future).
Short selling is profitable is you expect a stock´s price to decline in the future. But short
selling can be advantageous even if you expect the stock´s price to rise, as long as you invest
the proceeds in another stock with an even higher expected return. However, short selling can
greatly increase the risk of the portfolio. Short selling leads to higher expected return, but also
higher volatility.
Efficient Portfolios with Many Stocks:
Combining more stocks allows for more options and different combinations in a portfolio. It
can increase the expected return and lower the risk.
When the investment opportunities increase from two to three stocks, the efficient frontier
improves. In general, adding new investment opportunities allows for greater diversification
and improves the efficient frontier.
Thus, to arrive at the best possible set of risk and return opportunities, we should keep adding
stocks until all investment opportunities are represented. Ultimately, based on our estimates of
returns, volatilities, and correlations, we can construct the efficient frontier for all available
risky investments showing the best possible risk and return combinations that we can obtain
by optimal diversification. The efficient frontier expands as new investments are added.
See page 408 and 409!!
11.5 – Risk-free Saving and Borrowing:
There is another way besides diversification to reduce risk that we have not yet considered:
We can keep some of our money in a safe, no-risk investment like Treasury Bills.
In this section, we will see that the ability to choose the amount to invest in risky versus riskfree securities allows us to determine the optimal portfolio of risky securities for an investor.
Investing in risk-free securities:
Consider an arbitrary risky portfolio with returns Rp. Let´s look at the effect on risk and
return of putting a fraction x of our money in the portfolio, while leaving the remaining
fraction (1-x) in risk-free Treasury Bills with a yield of 𝑟𝑓 .
Return = 𝑅𝑥𝑝
𝐸[𝑅𝑥𝑝 ] = (1 − 𝑥)𝑟𝑓 + 𝑥𝐸[𝑅𝑝 ] = 𝑟𝑓 + 𝑥(𝐸[𝑅𝑝 ] − 𝑟𝑓 )
The expected return is the weighted average of the expected returns of Treasury Bills and the
Portfolio.
35
We can see in the next equation that the expected return is equal to the risk-free rate plus a
fraction of the portfolio´s risk premium.
Next, let´s compute the volatility. Because the risk-free interest rate 𝑟𝑓 is fixed and does not
move with (or against) our portfolio, its volatility and covariance with the portfolio <re both
zero.
Thus,
𝑆𝐷(𝑅𝑥𝑃 ) = √(1 − 𝑥)2 𝑉𝑎𝑟(𝑟𝑓 ) + 𝑥 2 𝑉𝑎𝑟(𝑅𝑝 ) + 2(1 − 𝑥)𝑥𝐶𝑜𝑣(𝑟𝑓 , 𝑅𝑝 ) = √𝑥 2 𝑉𝑎𝑟(𝑅𝑝 )
= 𝑥𝑆𝐷(𝑅𝑝 )
That is, the volatility is only a fraction of the volatility of the portfolio, based on the amount
we invest in it.
Borrowing and Buying Stocks on Margin:
Buying stocks on margin or using leverage = Borrowing money to invest in stocks.
A portfolio that consists of a short position in the risk-free investment is known as a levered
portfolio. Margin investing is a risky investment strategy.
See graph at page 410. Note that the region of the blue line in figure 11.9 with x > 100% has
higher risk than the portfolio P itself. At the same time, margin investing can provide higher
expected returns than investing in P using only the funds we have available.
𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 =
𝐶𝑎𝑠ℎ + 𝐵𝑜𝑟𝑟𝑜𝑤𝑖𝑛𝑔
𝐶𝑎𝑠ℎ
Expected return of a leveraged portfolio:
E[RxQ ] = rf + x ∗ (E[RJ ] − rf )
x = leverage
𝑆𝐷(𝑅𝑥𝑄 ) = 𝑥𝑆𝐷(𝑅𝑄 )
Identifying the tangent portfolio:
To earn the highest possible expected return for any level of volatility we must find the
portfolio that generates the steepest possible line (in figure 11.9) when combined with the
risk-free investment.
Sharpe ratio = the slope of the line through a given portfolio P.
𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 =
𝐸[𝑅𝑝 ] − 𝑟𝑓
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛
=
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦
𝑆𝐷(𝑅𝑝 )
36
The sharpe ratio measures the ratio of reward-to-volatility provided by a portfolio. The
optimal portfolio to combine with the risk-free asset will be the one with the highest Sharpe
Ratio, where the line with the risk-free investment just touches, and so is tangent to, the
efficient frontier of risky investments. The portfolio that generates this tangent is known as
the tangent portfolio. All other portfolios of risky assets lie below this line. Because the
tangent portfolio has the highest Sharpe ratio of any portfolio in the economy, the tangent
portfolio provides the biggest reward per unit of volatility of any portfolio available.
As is evident from Figure 11.10 (page 412), combinations of the risk-free asset and the
tangent portfolio provide the best risk and return trade-off available to an investor. This
observation has a striking consequence: The tangent portfolio is efficient and, once we
include the risk-free investment, all efficient portfolios are combinations of the risk-free
investment and the tangent portfolio. Therefore, the optimal portfolio of risky investments no
longer depends on how conservative or aggressive the investor is; every investor should
invest in the tangent portfolio independent of his or her taste of risk. The investor´s
preferences will determine only how much to invest in the tangent portfolio versus the riskfree investment.
The efficient portfolio is the tangent portfolio, the portfolio with the highest Sharpe-ratio in
the economy.
See example 11.12 at page 413.
11.6 – The Efficient Portfolio and Required Returns
We know turn to the implications of the result that the tangent or efficient portfolio is the best
one to invest in, for a firms cost of capital. After all, if a firm wants to raise new capital,
investors must find it attractive to increase their investment in it. In this section we derive a
condition to determine whether we can improve a portfolio by adding more for a given
security, and use it to calculate an investor´s required return for holding an investment.
Portfolio Improvement: Beta and the Required Return:
Increasing the amount invested in i will increase the Sharpe Ratio of portfolio P if its expected
return E(Ri) exceeds its required return given portfolio P, defined as:
𝑟𝑖 = 𝑟𝑓 + 𝛽𝐼𝑃 × (𝐸[𝑅𝑝 ] − 𝑟𝑓 )
Beta of investment i with portfolio P:
𝑆𝐷(𝑅𝑖 ) × 𝐶𝑜𝑟𝑟(𝑅𝑖 , 𝑅𝑝 )
𝛽𝑖𝑃 =
𝑆𝐷(𝑅𝑝 )
The required return is the expected return that is necessary to compensate for the risk
investment i will contribute to the portfolio. The required return for an investment i is equal to
the risk-free interest rate plus the risk premium of the current portfolio, P, scaled by i´s
sensitivity to P, 𝛽𝑖𝑃
If i´s expected return exceeds its required return, then adding more of it will improve the
performance of the portfolio.
37
See example 11.13 at page 415!
Expected Returns and the Efficient Portfolio:
A portfolio is efficient if and only if the expected return of every available security equals its
required return.
This result implies the following relationship between the expected return of any security and
its beta with the efficient portfolio:
Expected return of a Security (equation 11.21)
𝑒𝑓𝑓
𝐸[𝑅𝑖 ] = 𝑟𝑖 = 𝑟𝑓 + 𝛽𝑖
× (𝐸[𝑅𝑒𝑓𝑓 ] − 𝑟𝑓 )
where R(eff) is the return of the efficient portfolio.
We can determine the appropriate risk premium for an investment from its beta with the
efficient portfolio.
To understand the connection between the market portfolio and the efficient portfolio, we
must consider the implications of the collective investment decisions of all investors, which
we turn to next.
11.7 – The Capital Asset Pricing Model
To implement the approach of equation 11.21, we face an important practical problem: To
identify the efficient portfolio we must know the expected returns, volatilities, and
correlations between investments. These quantities are difficult to forecast. Under these
circumstances, how do we put theory into practice?
 We revisit the Capital Asset Pricing Model (CAPM). This model allows us to identify
the efficient portfolio of risky assets without having any knowledge of the expected
return of each security.
The CAPM assumptions:
1. Investors can buy and sell all securities at competitive market prices (without
incurring taxes or transactions costs) and can borrow and lend at the risk-free interest
rate.
2. Investors hold only efficient portfolios of traded securities – portfolios that yield the
maximum expected return for a given level of volatility.
3. Investors have homogenous expectations regarding the volatilities, correlations, and
expected returns of securities.
Homogenous expectations = All investors have the same estimates concerning future
investments (this is an approximation we can do since all investors base their valuation on the
same historical data).
38
Supply, Demand, and the Efficiency of the Market Portfolio:
If all investors have homogenous expectations, then each investor will identify the same
portfolio as having the highest Sharpe ratio in the economy. Thus, all investors will demand
the same efficient portfolio of risky securities – the tangent portfolio.
The efficient, tangent portfolio of risky securities (the portfolio that all investors hold) must
equal the market portfolio).
The insight that the market portfolio is the efficient is really just the statement that demand
must equal supply. All investors demand the efficient portfolio, and the supply of securities is
the market portfolio; hence the two must coincide.
See example 11.15 at page 418!
Optimal investing: The Capital Market Line:
When the tangent line goes through the market portfolio it is called the capital market line
(CML). According to the CAPM, all investors should choose a portfolio on the capital market
line, by holding some combination of the risk-free security and the market portfolio.
11.8 – Determining the risk premium
Under the CAPM assumptions, we can identify the efficient portfolio: It is equal to the market
portfolio. Thus, if we don´t know the expected return of a security or the cost of capital of an
investment, we can use the CAPM to find it by using the market portfolio as a benchmark.
Market Risk and Beta
The CAPM Equation for the Expected Return:
𝐸[𝑅𝑖 ] = 𝑟𝑖 = 𝑟𝑓 + 𝛽𝑖 × (𝐸[𝑅𝑀𝑘𝑡 ] − 𝑟𝑓 )
where 𝛽𝑖 is the beta of the security with respect to the market portfolio, defined as:
𝛽𝑖 =
𝑆𝐷(𝑅𝑖 ) × 𝐶𝑜𝑟𝑟(𝑅𝑖 , 𝑅𝑀𝑘𝑡 )
𝐶𝑜𝑣(𝑅𝑖 , 𝑅𝑀𝑘𝑡 )
=
𝑆𝐷(𝑅𝑀𝑘𝑡 )
𝑉𝑎𝑟(𝑅𝑀𝑘𝑡 )
The beta of a security measures its volatility due to market risk relative to the market as a
whole, and thus captures the security´s sensitivity to market risk.
See example 11.16 at page 420!
To hold a security with a negative beta is not good if you have only that stock, but it can be
good to hold it in combination with other securities as part of a well-diversified portfolio. The
39
stock with the negative beta tend to rise when the market and most other securities fall, so it
provides a “recession insurance”.
See example 11.17 at page 421!!
The Security Market Line:
There is a linear relationship between a stock´s beta and its expected return.
Security market line (SML) = a line through the risk-free investment (with a beta of 0) and
the market (with a beta of 1)
Under the CAPM assumptions, the security market line (SML) is the line along which
individual securities should lie when plotted according to their expected return and beta, as it
is showed on page 423.
The beta of a portfolio is the weighted average beta of the securities in the portfolio.
See equation 11.24 at page 423!
Chapter 12 – Estimating the Cost of Capital
The cost of capital should include a risk-premium that compensates investors for taking on the
risk of the new project. In the last two chapters, we learned a model to calculate the risk
premium and the cost of capital – the Capital Asset Pricing Model.
In this chapter, we will apply this knowledge to compute the cost of capital for an investment
opportunity.
12.1 – The Equity Cost of Capital
Cost of capital = the best expected return available in the market on investments with similar
risk and term.
The CAPM model provides a practical way to identify an investment with similar risk.
Under the CAPM, the market portfolio is a well-diversified, efficient portfolio representing
the non-diversifiable risk in the economy. Therefore, investments have similar risk if they
have the same sensitivity to market risk, as measured by their beta with the market portfolio.
So, the cost of capital of any investment opportunity equals the expected return of available
investments with the same beta. The estimate is provided by the Security Market Line
equation of the CAPM, which states that, given the Beta of the investment opportunity, its
cost of capital is:
The CAPM Equation for the cost of capital (Security Market Line)
𝑟𝑖 = 𝑟𝑓 + 𝛽𝑖 × (𝐸[𝑅𝑀𝑘𝑡 ] − 𝑟𝑓 )
40
𝛽𝑖 × (𝐸[𝑅𝑀𝑘𝑡 ] − 𝑟𝑓 ) = 𝑅𝑖𝑠𝑘 𝑃𝑟𝑒𝑚𝑖𝑢𝑚 𝑓𝑜𝑟 𝑠𝑒𝑐𝑢𝑟𝑖𝑡𝑦 𝑖
In other words, investors will require a risk premium comparable to what they would earn
taking the same market risk through an investment in the market portfolio.


-
Volatility = measures total risk
Beta = measures systematic risk/market risk
If the beta is for example 1.25, we expect the price of the stock to move 1.25% for
every 1% move of the market. The risk-premium will then be 1.25 times the risk
premium of the market
Because market risk cannot be diversified, it is market risk that determines the cost of capital.
Computing the equity cost of capital:


Construct the market portfolio, and determine its expected excess return over the riskfree interest rate.
Estimate the stock´s beta, or sensitivity to the market portfolio.
12.2 – The Market Portfolio
To apply the CAPM, we must identify the market portfolio. In this section we examine how
the market portfolio is constructed, common proxies that are used to represent the market
portfolio, and how we can estimate the market risk premium.
Constructing the market portfolio:
The market portfolio contains more of the largest stocks and less of the smallest stocks.
Specifically, the investment in each security i is proportional to its market capitalization,
which is the total market value of its outstanding shares:
𝑀𝑉𝑖 = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆ℎ𝑎𝑟𝑒𝑠 𝑖 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔) × (𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑖 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒)
We then calculate the portfolio weights of each security as follows:
𝑥𝑖 =
𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖
𝑀𝑉𝑖
=
𝑇𝑜𝑡𝑎𝑙 𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐴𝑙𝑙 𝑆𝑒𝑐𝑢𝑟𝑖𝑡𝑖𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 ∑𝑗 𝑀𝑉𝑗
Value weighted portfolio = a portfolio where in which each security is held in proportion to
its market capitalization.
A value-weighted portfolio is also an equal-ownership portfolio: We hold an equal fraction
of the total number of shares outstanding of each security in the portfolio. This last
observation implies that even when market prices change, to maintain a value-weighted
portfolio, we do not need to trade unless the number of shares outstanding of some securities
changes. Because very little trade is required to maintain it, a value-weighted portfolio is also
a passive portfolio.
41
Market Indexes:
A market index reports the value of a particular portfolio of securities. The S&P 500 is an
index that represents a value-weighted portfolio of 500 of the largest U.S stocks. This is the
standard portfolio used to represent “the market portfolio” when using the CAPM in practice.
The Wilshire 5000, provide a value-weighted index of all U.S stocks listed on the major
stock exchanges. The returns are very similar between the S&P 500 and the Wilshire 5000.
The correlation between their weekly returns were almost 99% between 1990 and 2015.
Price weighted portfolio = holds an equal number of shares of each stock, independent of
their size. One example of such a U.S stock index is the Dow Jones Industrial Average
(DIJA). It´s non-representative.
Index fund = invests in Market Index such as the S&P 500 and the Wilshire 5000. They are
easy to invest in.
Exchange traded fund = a security that trades directly on an exchange, like a stock, but
represents ownership in a portfolio of stocks.
By investing in an index or exchange-traded fund, an individual investor with only a small
amount to invest can easily achieve the benefits of broad diversification.
Market Proxy = a portfolio whose return investors believe closely tracks the true market
portfolio – like the S&P 500. How well the model works depends on how closely the market
proxy actually tracks the true market portfolio.
The Market risk premium:
Risk premium = the expected excess return of the market portfolio: 𝐸[𝑅𝑀𝑘𝑡 ] − 𝑟𝑓
The market risk premium provides the benchmark by which we assess investors´ willingness
to hold market risk.
Determining the Risk-free Rate:
Risk-free interest rate in the CAPM model = the risk-free rate at which investors can both
borrow and save.
Even if a loan is essentially risk-free, the market premium compensates lenders for the
difference in liquidity compared with an investment in Treasuries. As a result, practitioners
sometimes rates from the highest quality corporate bonds in place of Treasury rates.
Different factors have reduced the risk of holding stocks over time, so therefore the premium
investors require have declined.
We can use historical data to estimate the market premium, but it´s best to use recent data
because the risk nowadays isn´t the same as long before.
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12.3 – Beta Estimation
Beta = measures the sensitivity of the security´s returns to those of the market. Because beta
captures the market risk of a security, as opposed to its diversifiable risk, it is the appropriate
measure of risk for a well-diversified investor.
Using Historical Returns:
Ideally, we would like to know a stock´s beta in the future; that is, how sensitive will its
future returns be to market risk. In practice, we estimate beta based on the stock´s historical
sensitivity. This approach makes sense if a stock´s beta remains relatively stable over time,
which appears to be the case for most firms.
The differences in betas between firms reflect the sensitivity of each firm´s profits to the
general health of the economy.
Firms like Apple have very high betas because the demand for their products usually varies
with the business cycle. Companies and consumers tend to invest in technology when times
are good and the opposite when times are bad. The demand for personal and household
products, however, has very little relation to the state of the economy. These firms have very
low betas.
If a firm has a beta that is larger than one, it varies with the economy but with a greater
amplitude. For example, if a 10% change in the market´s return corresponds to 15% change in
the firms return.
Identifying the Best-fitting Line:
Beta corresponds to the slope of the best-fitting line in the plot of the security´s excess returns
versus the market excess return.
Deviations from the best-fitting line result from risk that is not related to the market as a
whole. They represent firm-specific risk that is diversifiable and that averages out in a large
portfolio. On average the deviation is equal to zero.
See figure 12.1 at page 446
Using Linear Regression:
Linear Regression = the statistical technique that identifies the best-fitting line through a set
of points.
Alpha = measures the historical performance of a security relative to the expected return
predicted by the security market line – it is the distance the stock´s average return is above or
below the SML.
𝐸[𝑅𝑖 ] = 𝑟𝑓 + 𝛽𝑖 (𝐸[𝑅𝑀𝑘𝑡 ] − 𝑟𝑓 ) + 𝛼𝑖
𝛼𝑖 = 𝑡ℎ𝑒 𝑠𝑡𝑜𝑐𝑘´𝑠 𝑎𝑙𝑝ℎ𝑎
43
𝛼𝑖 = 𝐸[𝑅𝑖 − 𝑟𝑓 ] − 𝛽𝑖 × 𝐸[𝑅𝑀𝑘𝑡 − 𝑟𝑓 ]
Chapter 20 – Financial Options
20.1 – Option Basics
A financial option contract gives its owner the right (but not the obligation) to purchase or
sell an asset at a fixed price at some future date.
Two different options:
 Call option = gives the owner the right to buy the asset
 Put option = gives the owner the right to sell the asset
Because an option is a contract between two parties, for every owner of a financial option,
there is also an option writer, the person who takes the other side of the contract.
The most commonly encountered option contracts are options on shares of stock. A stock
option gives the holder of the option the right to buy or sell a share of stock on or before a
given date for a given price.
Understanding Option Contracts:




-
Exercising an option = when a holder of an option enforces the agreement and buys
or sells a share of stock at the agreed-upon price.
Strike price or exercise price = the price at which the holder buys or sells the share
of stock when the option is exercised.
American options = the most common kind, allow their holders to exercise the option
on any date up to and including a final date called the expiration date.
European Options = allow their holders to exercise the option only on the expiration
date – holders cannot exercise before the expiration date.
The names American and European have nothing to do with the location where the
options are traded: Both types are traded worldwide.
An option contract is a contract between two parties. The option buyer, also called the option
holder, holds the right to exercise the option and has a long position in the contract.
The option seller, also called the option writer, sells (or writes) the option and has a short
position in the contract.
Because the long side has the option to exercise, the short side has an
obligation to fulfil the contract.
Investors exercise options only when they stand to gain something. Consequently, whenever
an option is exercised, the person holding the short position funds the gain. That is, the
obligation will be costly. Why then, do people write options? The answer is that when you
sell an option you get paid for it – options always have positive prices. The market price of
44
the option is also called the option premium. This upfront payment compensates the seller
for the risk of loss in the event that the option holder chooses to exercise the option.
Interpreting Stock Option Quotations:
Stock Options are traded on organized exchanges. The oldest and largest is the Chicago Board
Options Exchange (CBOE).
Open interest = the total number of outstanding contracts of that option.
When the exercise price of an option is equal to the current price of the stock, the option is
said to be at-the-money.
Stock option contracts are always written on 100 shares of stock.
For each expiration date, call options with lower strike prices have higher market prices – the
right to buy the stock at a lower price is more valuable than the right to buy it for a higher
price.
Conversely, because the put option gives the holder the right to sell the stock at the strike
price, for the same expiration date, puts with higher strikes are more valuable.
On the other hand, holding fixed the strike price, both calls and puts are more expensive for a
longer time to expiration. Because these options are America-style options that can be
exercised at any time, having the right to buy or sell for a longer period is more valuable.
If the payoff from exercising an option immediately is positive, the option is said to be in-themoney.
-
Call options with strike prices below the current stock price  in-the-money
put options with strike prices above the current stock price  in-the-money.
Conversely, if the payoff from exercising the option immediately is negative, the option is
out-of-the-money.
-
Call options with strike prices above the current stock price  out-of-the-money
Put options with strike prices below the current stock price  out-of-the-money
Of course, a holder would not exercise an out-of-the-money option.
Deep in-the-money or deep out-of-the-money = options where the strike price and the stock
price are very far apart.
See example 20.1 at page 759!
Options on Other Financial Securities.
Hedging = Using an option to reduce risk.
45
20.2 – Option Payoffs at Expiration:
From The Law of One Price, the value of any security is determined by the future cash flows
an investor receives from owning it. Therefore, before we can assess what an option is worth,
we must determine an option´s payoff at the time of expiration.
Long Position in an Option Contract:
The difference between the strike price and the stock price = what the option is worth.
Call Value at Expiration:
C = max(S – K, 0)
S = the stock price at expiration
K = the exercise price
C = the value of the call option
Max = the maximum of the two quantities in the parentheses
The call´s value is the maximum of the difference between the stock price and the strike , S –
K, and zero.
See figure 20.1 at page 760!
Put Price at Expiration
P = max(K – S, 0)
See example 20.2 at page 761!
Short Position in an Option Contract
The short position´s cash flows are the negative of the long position´s cash flows.
See figure 20.2 and example 20.3 at page 762!
Because the stock price cannot fall below zero, the downside for a short position in a put
option is limited to the strike price of the option. A short position in a call, however, has no
limit on the downside.
Profits from Holding an Option to Expiration:
Although payouts on a long position in an option contract are never negative, the profit from
purchasing an option and holding it to expiration could well be negative because the payout at
expiration might be less than the initial cost of the option.
See page 763 and 764 (hard to explain)!
46
Returns for Holding an Option to Expiration:
We can also compare options based on their potential returns.
Call options:
If a stock had a positive beta, call options written on the stock will have even higher betas and
expected returns than the stock itself.
Put options:
The put position has a higher return with low stock prices; that is, if the stock has a positive
beta, the put has a negative beta. Hence, put options on positive beta stocks have lower
expected returns than the underlying stock.
The deeper out-of-the-money the put option is, the more negative its
beta, and the lower its expected return. As a result, put options are
generally not held as an investment, but rather as insurance to hedge
other risk in a portfolio.
Combinations of Options:
Sometimes investors combine option positions by holding a portfolio of options.
Straddle = a combination of a long position in a put and a call with the same strike price and
expiration date. A straddle provides a positive payoff so long as the stock price does not equal
the strike price. You will receive cash so long as the options do not expire at-the-money. The
farther away from the money the options are, the more money you will make.
After deducting the cost of the options, the profit is negative for stock prices close to the
strike price and positive elsewhere.
See figure 20.5 at page 765!
This strategy is sometimes used by investors who expect the stock to be very volatile and
move up or down a large amount, but who do not necessarily have a view on which direction
the stock will move. Conversely, investors who expect the stock to end up near the strike
price may choose to sell a straddle.
Strangle = if you have a combination of options where you do not receive money of the stock
price is between the two strike prices. The combination of options makes money when the
stock and strike prices are far apart.
See example 20.5 at page 766!
Butterfly Spread = a combination of options that is the opposite from a straddle. This one
pays off when the stock price is close to the strike price. Because the payoff of the butterfly
spread is positive, it must have a positive initial cost. (Otherwise it would be an arbitrage
opportunity).
See page 766 and 767!
47
Portfolio insurance:
Protection put = you can purchase a put option to insure the stock against the possibility of a
price decline. You can do the same thing by using put options on the portfolio of stocks as a
whole  portfolio insurance.
You can also get the same effect by purchasing a bond and a call option.
See figure 20.7 at page 768!
20.3 – Put-call Parity
Two different ways to construct portfolio insurance.
1. Purchase the stock and a put
2. Purchase a bond and a call
Because both positions provide exactly the same payoff, the Law of One Price requires that
they must have the same price.
Let’s write this concept more formally. Let K be the strike price of the option (the price we
want to ensure that the stock will not drop below), C be the call price, P be the put price, and
S be the stock price. Then, if both positions have the same price,
S + P = PV(K) + C
The left side of the equation is the cost of buying the stock and a put (with a strike price of
K); the right side is the price of buying a zero-coupon bond with a face value K and a call
option (with a strike price of K).
The price of a zero-coupon bond is the present value of its face value, which we have denoted
by PV(K). Rearranging terms gives an expression for the price of a European call option for a
non-dividend-paying stock:
C = P + S – PV(K)
Put-call parity = this relation between the value of the stock, the bond, and call and put
option.
It says that the price of a European call equals the price of the stock plus an otherwise
identical put minus the price of a bond that matures on the exercise date of the option. In other
words, you can think of a call as a combination of a levered position in the stock, S - PV(K ),
plus insurance against a drop in the stock price, the put P.
What happens if the stock pays a dividend? In that case, the two different ways to construct
portfolio insurance do not have the same payout because the stock will pay a dividend while
48
the zero-coupon bond will not. Thus, the two strategies will cost the same to implement only
if we add the present value of future dividends to the combination of the bond and the call:
S + P = PV(K) + PV(Div) + C
The left side of this equation is the value of the stock and a put; the right side is the value of a
zero-coupon bond, a call option, and the future dividends paid by the stock during the life of
the options, denoted by Div. Rearranging terms gives the general put-call parity formula:
Put-call Parity with dividends:
C = P + S – PV(Div) – PV(K)
Put-call Parity without dividends:
𝐶 =𝑃+𝑆−
𝐾
1+𝑟
C = Price of the call
P = price of the put
S = Market value of the stock
K= strike price
R = one-year risk-free interest rate
In this case, the call is equivalent to having a levered position in the stock without dividends
plus insurance against a fall in the stock price.
See example 20.7 at page 770!
20.4 – Factors Affecting Option Prices
Put-call parity gives the price of a European call option in terms of the price of a European
put, the underlying stock, and a zero-coupon bond. Therefore, to compute the price of a call
using put-call parity, you have to know the price of the put.
Strike Price and Stock Price:
As we noted earlier for the Amazon option quotes in Table 20.1, the value of an otherwise
identical call option is higher if the strike price the holder must pay to buy the stock is lower.
Because a put is the right to sell the stock, puts with a lower strike price are less valuable.
For a given strike price, the value of a call option is higher if the current price of the stock is
higher, as there is a greater likelihood the option will end up in-the-money. Conversely, put
options increase in value as the stock price falls.
Arbitrage Bounds on Option Price:


An American option cannot be worth less than its European counterpart.
The maximum payoff for a put option occurs if the stock becomes worthless (if, say,
the company files for bankruptcy). In that case, the put’s payoff is equal to the strike
49
price. Because this payoff is the highest possible, a put option cannot be worth more
than its strike price.
For a call option, the lower the strike price, the more valuable the call option. If the call
option had a strike price of zero, the holder would always exercise the option and receive
the stock at no cost. This observation gives us an upper bound on the call price: A call
option cannot be worth more than the stock itself.


-
Intrinsic value of an option = the value it would have if it expired immediately.
An American option cannot be worth less than its intrinsic value.
Time value of an option = the difference between the current option price and its
intrinsic value.
Because an American option cannot be worth less than its intrinsic value, it cannot
have a negative time value.
Option prices and the Exercise date:
For American options, the longer the time to the exercise date, the more valuable the option.
-
An American option with a later exercise date cannot be worth less than an identical
American option with an earlier exercise date.
The same argument will not work for European options, because a one-year European option
cannot be exercised early at six months. As a consequence, a European option with a later
exercise date may potentially trade for less than an otherwise identical option with an earlier
exercise date.
Option Prices and Volatility:
Example 20.8 – Option Value and volatility (page 772):
Problem
Two European call options with a strike price of $50 are written on two different stocks.
Suppose that tomorrow, the low-volatility stock will have a price of $50 for certain. The highvolatility stock will be worth either $60 or $40, with each price having equal probability. If
the exercise date of both options is tomorrow, which option will be worth more today?
Solution
The expected value of both stocks tomorrow is $50—the low-volatility stock will be worth
this amount for sure, and the high-volatility stock has an expected value of 1 ($40) + 1 ($60)
= $50.
22 However, the options have very different values. The option on the low-volatility stock is
worth nothing because there is no chance it will expire in-the-money (the low-volatility stock
will be worth $50 and the strike price is $50). The option on the high-volatility stock is worth
a positive amount because there is a 50% chance that it will be worth $60 - $50 = $10, and a
50% chance that it will be worthless. The value today of a 50% chance of a positive payoff
50
(with no chance of a loss) is positive.
Conclusion:
-
The value of an option generally increases with the volatility of the stock.
The intuition for this result is that an increase in volatility increases the likelihood of very
high and very low returns for the stock. The holder of a call option benefits from a higher
payoff when the stock goes up and the option is in-the- money, but earns the same (zero)
payoff no matter how far the stock drops once the option is out-of-the-money. Because of this
asymmetry of the option’s payoff, an option holder gains from an increase in volatility.
20.5 – Exercising Options Early
One might guess that the ability to exercise the American Option early would make an
American option more valuable than an equivalent European Option. Surprisingly, this is not
always the case – sometimes, they have equal value.
Non-Dividend-Paying stocks:
Put-call parity for an option that do not pay any dividends prior to the expiration date:
C = P + S – PV(K)
Price of the zero-coupon bond: PV(K) = K – dis(K), where dis(K) is the amount of the
discount from face value to account for interest. Substituting this expression into put-call
parity gives:
C = S – K + dis(K) + P
S – K = Intrinsic value
Dis(K) + P = Time Value
In this case, both terms that make up the time value of the call option are positive before the
expiration date: As long as interest rates remain positive, the discount on a zero-coupon bond
before the maturity date is positive, and the put price is also positive, so a European call
always has a positive time value. Because an American option is worth at least as much as a
European option, it must also have a positive time value before expiration.
Hence,
The price of any call option on a non-dividend-paying stock always exceeds its intrinsic
value.
This result implies that it is never optimal to exercise a call option on a non-dividend-paying
stock early – you are always better off just selling the option.
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It is straightforward to see why. When you exercise an option, you get its intrinsic value. But
as we have just seen, the price of a call option on a non-dividend-paying stock always exceeds
its intrinsic value. Thus, if you want to liquidate your position in a call on a non-dividendpaying stock, you will get a higher price if you sell it rather than exercise it.
Because it is never optimal to exercise an American call on a non-dividend-paying stock
early, the right to exercise the call early is worthless. For this reason,
An American call on a non-dividend-paying stock has the same price as its European
counterpart.
Two benefits to delaying the exercise of a call option:
1. The holder delays paying the strike price
2. By retaining the right not to exercise, the holder´s downside is limited.
A European option can sell for less than its intrinsic value. An American option, however,
cannot sell for less than its intrinsic value (because otherwise arbitrage profits would be
possible by immediately exercising it), which implies that the American option can be worth
more than an otherwise identical European option. Because the only difference between the
two options is the right to exercise the option early, this right must be valuable – there must
be states in which it is optimal to exercise the American put early.
Let’s examine an extreme case to illustrate when it is optimal to exercise an American put
early: Suppose the firm goes bankrupt and the stock is worth nothing. In such a case, the
value of the put equals its upper bound—the strike price—so its price cannot go any higher.
Thus, no future appreciation is possible. However, if you exercise the put early, you can get
the strike price today and earn interest on the proceeds in the interim. Hence it makes sense to
exercise this option early. Although this example is extreme, it illustrates that it can be
optimal to exercise deep in-the-money put options early.
Exercising early can be good for the deep in-the-money put options.
Dividend-paying Stocks:
When stocks pay dividends, the right to exercise an option on them early is generally valuable
for both calls and puts. To see why, let’s write out the put-call parity relationship for a
dividend-paying stock:
C = S – K + dis(K) + P – PV(Div)
S – K = Intrinsic Value
dis(K) + P – PV(Div) = Time Value
If PV(Div) is large enough, the time value of a European call option can be negative, implying
that its price could be less than its intrinsic value. Because an American option can never be
worth less than its intrinsic value, the price of the American option can exceed the price of a
European option.
To understand when it is optimal to exercise the American call option early, note that when a
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company pays a dividend, investors expect the price of the stock to drop to reflect the cash
paid out. This price drop hurts the owner of a call option because the stock price falls, but
unlike the owner of the stock, the option holder does not get the dividend as compensation.
However, by exercising early, the owner of the call option can capture the value of the
dividend. Thus, the decision to exercise early trades off the benefits of wait- ing to exercise
the call option versus the loss of the dividend. Because a call should only be exercised early to
capture the dividend, it will only be optimal to do so just before the stock’s ex-dividend date.
Dividends have the opposite effect on the time value of a put option. Again, the put-call parity
relation, we can write the put value as:
P = K – S + C – dis(K) + PV(Div)
K – S = Intrinsic value
C – dis(K) + PV(Div) = Time value
Intuitively, when a stock pays dividends, the holder of a put option will benefit by waiting for
the stock price to drop after it goes ex-dividend before exercising. Thus, it is less likely that a
put option on a dividend-paying stock will be exercised early.
Chapter 21 – Option Valuation
Black-Scholes Option Pricing Model = a formula to calculate the price of an option. Robert
Merton, Myron Scholes and Fischer Black won the Nobel Prize in economics year 1997 for
their discovery in 1973.
The great insight that Black, Scholes, and Merton brought to economics is that in the case of
options it is not necessary to model preferences. Their work demonstrated how to apply the
Law of One Price to value a vast new range of financial securities based on the current market
prices of stocks and bonds.
21.1 – The Binomial Option Pricing Model
This model prices Options by making the simplifying assumption that at the end of the next
period, the stock price has only two possible values. Under this assumption, we demonstrate
the key insight of Black and Scholes – that option payoffs can be replicated exactly by
constructing a portfolio out of a risk-free bond and the underlying stock.
A Two-State Single-Period Model:
We start by calculating the price of a single-period option in a very simple world. We will
value the option by first constructing a replicating portfolio.
Replicating portfolio = a portfolio of other securities that has exactly the same value in one
period as the option.
These have the same payoffs, and hence the Law of One Price implies that the current value
of the call and the replicating portfolio must be equal.
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Consider a European call option that expires in one period and has an exercise price of $50.
Assume that the stock price today is equal to $50. Assume here and throughout the chapter
that the stock pays no dividends (unless explicitly indicated). In one period, the stock price
will either rise by $10 or fall by $10. The one-period risk-free rate is 6%. We can summarize
this information in a binominal tree.
Binomial tree = a timeline with two branches at every date representing the possible events
that could happen at those times:
The binominal tree contains all the information we currently know: the value of stock, bond,
and call options in each state in one period, as well as the price of the stock and bond today.
Up state = the state in which the stock price goes up (to 60 dollars)
Down state = the state in which the stock price goes down (to 40 dollars)
In order to determine the value of the option using the Law of One Price, we must show that
we can replicate its payoffs using a portfolio of the stock and the bond. Let ∆ be the number
of shares of stock we purchase, and let B be our initial investment in bonds. To create a call
option using the stock and the bond, the value of the portfolio consisting of the stock and
bond must match the value of the option in every possible state. Thus, in the up state, the
value of the portfolio must be $10 (the value of the call in that state):
60∆ + 1.06B = 10
In the down state, the value of the portfolio must be zero (the value of the call in that state):
40∆ + 1.06B = 0
∆ = 0,5
B = - 18,8679
A portfolio that is long 0.5 share of stock and short approximately $18.87 worth of bonds
(i.e., we have borrowed $18.87 at a 6% interest rate) will have a value in one period that
matches the value of the call exactly. Let’s verify this explicitly:
60 * 0.5 – 1,06 * 18,87 = 10
40 * 0,5 – 1,06 * 18,87 = 0
Therefore, by the Law of One Price, the price of the call option today must equal the current
market value of the replicating portfolio. The value of the portfolio today is the value of 0.5
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shares at the current share price of $50, less the amount borrowed:
50∆ − 𝐵 = 50(0.5) − 18.87 = 6.13
Thus, the price of the call today is 6.13 dollars.
The secret of the binomial model is that while the option and the replicating portfolio do not
have the same payoffs in general, they have the same payoffs given the only two outcomes we
have assumed possible for the stock price; 40 dollars and 60 dollars.
Note that by using the Law of One Price, we are able to solve for the price of the option
without knowing the probabilities of the states in the binomial tree. That is, we did not need to
specify the likelihood that the stock would go up versus down.
The preceding argument shows that we do not need to know these probabilities to value
options. It also means that we do not need to know the expected return of the stock, which
will depend on these probabilities.
The Binomial Pricing Formula:
Replicating Portfolio in the Binomial Model:
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∆=
𝐶𝑢 − 𝐶𝑑
𝐶𝑑 − 𝑆𝑑 ∆
𝑎𝑛𝑑 𝐵 =
𝑆𝑢 − 𝑆𝑑
1 + 𝑟𝑓
Note that the formula for ∆ in the equation above, can be interpreted as the sensitivity of the
option’s value to changes in the stock price. It is equal to the slope of the line showing the
payoff of the replicating portfolio in Figure 21.1.
Once we know the replicating portfolio, we can calculate the value C of the option today as
the cost of this portfolio:
Option Price in the Binomial Model:
𝐶 = 𝑆∆ + 𝐵
By applying these two models in different ways, we will see that they are quite powerful. For
one thing, they do not require that the option we are valuing is a call option – we can use them
to value any security whose payoff depends on the stock price. We can for example use them
to price a put:
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A Multiperiod Model:
The problem with the simple two-state example is that there are many more than two possible
outcomes for the stock price in the real world. To make the model more realistic, we must
allow for the possibility of many states and periods.
Two-period binomial tree:
The key property of the binomial model is that in each period, there are only two possible
outcomes – the stock either goes up or down.
To calculate the value of an option in a multiperiod binomial tree, we start at the end of the
tree and work backward.
At time 2, the option expires, so its value is equal to its intrinsic value. In this case, the call
will be worth $10 if the stock price goes up to $60, and will be worth zero otherwise.
What is the value of the option if the stock price has gone up to $50 at time 1?
What if the stock price has dropped to $30 at time 1?
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The option is worthless in both states at time 2, so the value of the option in the down state at
time 1 must also be zero (and the replicating portfolio is simply ∆ = 0 and B = 0). Given the
value of the call option in either state at time 1, we can now work backward and determine the
value of the call at time 0. In that case, we can write the binomial tree over the next period as
follows:
See page 795 (hard to explain)!
Dynamic trading strategy = the idea that you can replicate the option payoff by dynamically
trading in a portfolio of the underlying stock and a risk-free bond.
Making the Model Realistic:
By decreasing the length of each period and increasing the number of each periods in the
stock price tree, we can construct a realistic model for the stock price.
21.1 – The Black-Scholes Option Pricing Model
Although Fischer Black and Myron Scholes did not originally derive it that way, the BlackScholes Option Pricing Model can be derived from the Binomial Option Pricing Model by
making the length of each period, and the movement of the stock price per period, shrink to
zero and letting the number of periods grow infinitely large. Rather than derive the formula
here, we will state it and focus on its applications.
The Black-Scholes Formula:
The value at time t, of a call option on a stock that does not pay dividends prior to the option´s
expiration date:
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Black-Scholes Price of a Call Option on a Non-Dividend-Paying Stock
𝐶 = 𝑆 × 𝑁(𝑑1 ) − 𝑃𝑉(𝐾) × 𝑁(𝑑2)
S = the current price of the stock
T = the number of years left to expiration
K = the exercise price
𝜎 = 𝑡ℎ𝑒 𝑎𝑛𝑛𝑢𝑎𝑙 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 (𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛) of the stock´s return.
PV(K) = the present value (price) of a risk-free zero-coupon bond that pays K on the
expiration date of the option.
N(d) = the cumulative normal distribution – that is, the probability that a normally
distributed variable is less than d – and
𝑑1 =
𝑙𝑛[𝑆/𝑃𝑉(𝐾)]
𝜎√𝑇
+
𝜎√𝑇
𝑎𝑛𝑑 𝑑2 = 𝑑1 − 𝜎√𝑇
2
We only need five input parameters to price the call:
- the stock price
- the strike price
- the exercise date
- the risk-free interest rate (to compute the present value of the strike price)
- the volatility of the stock.
We do not need to know the expected return of the stock to calculate the option price in the
Black-Scholes Option Pricing Model
Indeed, the only parameter in the Black-Scholes formula that we need to forecast is the
stock’s volatility. Because a stock’s volatility is much easier to measure (and forecast) than its
expected return, the Black-Scholes formula can be very precise.
The Black-Scholes formula is derived assuming that the call is a European option. Recall
from Chapter 20 that an American call option on a non-dividend-paying stock always has the
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same price as its European counterpart. Thus, the Black-Scholes formula can be used to price
American or European call options on non-dividend-paying stocks.
Notice how the value of the option always lies above its intrinsic value.
European Put Options. We can use the Black-Scholes formula to compute the price of a
European put option on a non-dividend-paying stock by using the put-call parity formula we
derived in Chapter 20 (see Eq. 20.3). The price of a European put from put-call parity is
P = C – S + PV(K)
Substituting for C using the Black-Scholes formula gives
Black-Scholes Price of a European Put Option on a Non-Dividend-Paying Stock
𝑃 = 𝑃𝑉(𝐾)[1 − 𝑁(𝑑2 )] − 𝑆[1 − 𝑁(𝑑1 )]
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Dividend-paying stocks:
Because a European call option is the right to buy the stock without these dividends, we can
evaluate it using the Black-Scholes formula with Sx in place of S.
A useful special case is when the stock will pay a dividend that is proportional to its stock
price at the time the dividend is paid. If q is the stock’s (compounded) dividend yield until the
6
expiration date, then
𝑆𝑥 =
𝑆
1+𝑞
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Implied Volatility:
Of the five required inputs in the Black-Scholes formula, four are directly observable: S, K, T,
and the risk-free interest rate. Only one parameter, s, the volatility of the stock price, is not
directly observable.
Two strategies to estimate the value of the volatility:
1. Use historical data on daily stock returns to estimate the volatility of the stock over the
past several months. Because volatility tends to be persistent, such estimates can
provide a reasonable forecast for the stock´s volatility in the near future.
2. Use current market prices of traded options to “back out” the volatility that is
consistent with these prices based on the Black-Scholes formula.
Implied Volatility = an estimate of a stock´s volatility that is implied by an option´s price
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The Replicating Portfolio:
The Price of a call option is given by the price of the replicating portfolio:
𝐶 = 𝑆∆ + 𝐵
Comparing this expression to the Black-Scholes formula from Eq. 21.7 gives the shares of
stock and amount in bonds in the Black-Scholes replicating portfolio:
Black Scholes Replicating Portfolio of a Call Option
∆ = 𝑁(𝑑1 )
𝐵 = −𝑃𝑉(𝐾)𝑁(𝑑2 )
Recall that N(d ) is the cumulative normal distribution function; that is, it has a minimum
value of 0 and a maximum value of 1. So, ∆ is between 0 and 1, and B is between -K and 0.
The option delta, ∆, has a natural interpretation: It is the change in the price of the option
given a 1 dollar change in the price of the stock.
Because ∆ is always less than 1, the change in the call price is always less than the change in
the stock price.
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Black-Scholes Replicating Portfolio of a Put Option
∆ = − [1 − 𝑁(𝑑1 )]
𝐵 = 𝑃𝑉(𝐾)[1 − 𝑁(𝑑2 )]
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