Lecture Notes
FM 423: Asset Markets
Lecture 1
Time Value of Money
Igor Makarov
London School of Economics
Page 2 of Lecture 1
1.1
Big Picture
FM423
Asset(Pricing
Financial'markets
Firms
Corporate(
Finance
FM422
Page 3 of Lecture 1
Why Do We Need Financial Markets?
1. One of the most important functions of financial markets is allocation of
resources to their most productive use
Capital
Households
Firms
Goods
Labor
• Which goods should be produced, how should they be produced, and how
should they be distributed?
? Challenges: people might have very di↵erent preferences, views, abilities,
and information
Page 4 of Lecture 1
Allocation of Resources: Two Main Approaches
• Central planner economy: production decisions are delegated to a central
authority
? Historically, not very viable
⇤ Worked well during “disaster” times (e.g., wars) when goals were clear
⇤ Many spectacular failures during “normal” times: overproduction of
some goods and shortage of others
? The main reason for failures: insufficient production of information
• Market economy: allow people to trade goods and services and choose which
goods and services to produce
? Production decisions are guided by the price signals created by the forces
of supply and demand
? Higher(lower) demand leads to higher(lower) prices, which in turn lead to
adjustment in supply
? Prices carry important informational role — Hayek (1945), “The use of
knowledge in society”
Page 5 of Lecture 1
• Three main markets: goods, labor, and capital
• Finance is interested in the market for capital
Page 6 of Lecture 1
2. Another important function of financial markets is to provide means for
moving funds from those who have a surplus to those who have a shortage
? People with resources are frequently not the same people who have
profitable investment opportunities
Financial Intermediaries
(e.g., banks)
Lenders
Borrowers
Capital Markets
(debt and equity)
• Debt — A contractual agreement by the borrower to pay the holder of the
instrument a fixed amount at predetermined set of dates
• Equity — a claim to share in the net income and the assets of a business
trillion in global equity market cap, or $48 trillion; this is 3.6x
the next largest market, China.
Other EM
9.2%
Page 7 of Lecture 1
Fixed Income: U.S. fixed income markets comprise 39% of
EU 11.2%
China 11.4%
the $123 trillion securities outstanding across the globe,
Global
Capital
Markets
or $48 trillion;
this is 1.9x
the next largest market, the EU
Are
(excluding the U.K.).
Canada 2.6%
UK 3.1%
Other DM 1.1%
Other EM 1.4%
Australia 1.5%
Australia 2.0%
Singapore 0.6%
Canada 3.3%
HK 4.8%
HK 0.5%
Singapore 0.4%
UK 5.7%
Japan
5.9%
rld and
nd most
he $118
this is 3.6x
se 39% of
Japan 11.1%
Other DM
8.9%
Global Equity
$118T
Other EM
9.2%
US 38.9%
China 15.5%
EU 11.2%
EU 20.1%
China 11.4%
e globe,
the EU
Global FI
$123T
US 40.8%
Source: World Federation of Exchanges (as of September 2021), Bank for International Settlements (as of March 2021)
Source: World Federation of Exchanges (as of September 2021),
Note: Equity = market cap, FI =fixed income, includes structured products
= securities outstanding. EU = 27 member states, excluding
Other EM 1.4%
Bank for International Settlements (as of March 2021)
Other DM 1.1%
Note: =
Equity
= market markets
cap, FI =fixed income, includes structured
the UK; FI = fixed
income; EM = emerging
markets; HK = Hong Kong; DM
developed
HK 0.5%
Australia 2.0%
products = securities outstanding. EU = 27 member states,
Canada 3.3%
Singapore 0.4%
excluding the UK; FI = fixed income; EM = emerging markets; HK =
Hong Kong; DM = developed markets
UK 5.7%
Japan 11.1%
7
Global FI
$123T
US 38.9%
INVESTING ESSENTIALS
Page 8 of Lecture 1
Another
otson 3.SBBI
®
®
important function of financial markets is to provide means for
funds 1926–2021
across time
Bonds, Bills,moving
and Inflation
?
$100k
$56,034
Compound annual return
$14,086
Small stocks
12.1%
Large stocks
10.5
Government bonds 5.5
Treasury bills
3.3
Inflation
2.9
10k
1k
$177
100
$22
$16
10
1926
1936
1946
1956
1966
1976
1986
1996
2006
Trump impeached
by the House
Brexit Referendum
Emergency Economic
Stabili ation Act
Wall St. Reform Act
U.S. Credit Downgrade
Detroit Bankruptcy
Gramm-Leach Bliley Act
September 11
Start of Gulf War
Stock Market Crash
Start of low in ationary period
Gold window closed
Arab oil embargo
May Day, the deregulation
of brokerage fees
Tet offensive in Vietnam
JFK assassinated
Sputnik launched
U.S. Treasury-Federal
Reserve Accord
General Agreement
on Tariffs and Trade
Pearl Harbor
0
Glass-Steagall Act
Securities Exchange Act
1
Stock Market Crash
inancial goals,
ecure retirement
r a college
nvesting makes
ou can see here
h of $1 over the
a a
a to
bonds, and
s should all
e in a properly
n t
strategy.
2016
Past performance is no guarantee of future results.
Past performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 1926. Assumes reinvestment of income and no transaction costs or taxes. This is
for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar 2022 and Precision Information, dba Financial
Fitness Group 2022. All Rights Reserved.
Page 9 of Lecture 1
4. Still another important function of financial markets is to provide means for
for risk sharing and risk management
? The future is uncertain. Who should bear risk and at what price?
? How to mitigate the impact of risk?
Page 10 of Lecture 1
• To e↵ectively function markets have to solve a number of problems:
? Transaction costs
? Solve various agency problems arising because people have di↵erent
objectives, expertise, beliefs, and information
? Facilitate information aggregation across various investors to deal with
uncertainty about the future
? Provide liquidity to market participants
• Financial intermediaries are often necessary to alleviate the above problems
• Since this is your first course in finance, as a starting point, we often assume
many of the above problems away and cosider the so called perfect markets
• Once we understand how perfect markets work we study the impact of various
market imperfections
Page 11 of Lecture 1
• In this course we will study three major financial markets:
? Fixed-income (debt) markets: Lectures 2–4
? Stock markets: Lectures 5–13
? Derivative markets: Lectures 14–19
• We will focus on the design of these markets, their functions, and valuation
of financial assets there
Page 12 of Lecture 1
Financial Assets
• Financial markets are named after assets which are traded there
• Debt and equity are examples of financial assets
• A financial asset is an asset that promises some cash flows to its owner
C0
C1
C2
0
1
2
u
u
u
CT
···
u
-
T
• For financial markets to work we need to know how to value financial assets,
that is, how much future cash flows are worth today
• Two fundamental challenges in valuing financial assets:
? Cash flows can be at di↵erent points in time
? Cash flows can be random: take di↵erent values
• Intuitively cash flows that are certain and realize sooner are more valuable
than risky ones and ones that realize later
Page 13 of Lecture 1
• We will start with a particularly simple case of certain cash flows. In this
case, we only need to worry about the time value of money, which is the
di↵erence in value between money today and money in the future
Page 14 of Lecture 1
1.2
Time Value of Money
• Suppose there is a market where we can trade future dollars for today’s
dollars
• Suppose the price of a dollar in year t is dt (in today’s dollars)
• The price dt is called the discount factor for year t
• It is customary to write dt as
1
dt =
(1 + rt)t
)
✓
1
rt =
dt
◆1
t
1
where rt is called the discount rate for year t or t-year interest rate or
t-year spot rate
Page 15 of Lecture 1
• To understand the logic behind this notation suppose that the interest rate
is time-independent and that you can lend and borrow in each period at the
annual interest rate r
• Consider investing $100 at the annual interest rate r = 10%. How much
will your investment be worth in 3 years?
? After one year, you’ll have: $100 + $100 ⇥ 0.1 = $110 (100 ⇥ 1.1)
? After two years, you’ll have: $110 + $110 ⇥ 0.1 = $121 (100 ⇥ 1.12)
? After three years, you’ll have: $121 + $121 ⇥ 0.1 = $133.1 (100 ⇥ 1.13)
? This is di↵erent than $100 + $30 = $130
? You have to account for the interest on interest
• More generally, investing C dollars at the annual interest rate r gives
C(1 + r)t dollars after t years. Hence, to have C dollars in t-years one needs
to have C/(1 + r)t dollars today
• Let us show that in this world dt should be equal to
1
(1+r)t
Page 16 of Lecture 1
• Suppose that dt 6=
any risk?
one
dollar
1
.
(1+r)t
Can we find a way to make a profit without taking
grate
1
(1+r)t
) Yes, if dt >
we can sell one dollar at time t for dt dollars today and
1
lend (1+r)
t dollars today to get one dollar at time t. Then, at time t we
1
have zero cash flows and today we pocket dt (1+r)
t > 0.
1
If dt < (1+r)
t we can complete the reverse transaction. That is, we can
1
borrow (1+r)
t dollars today to repay one dollar at time t and buy one dollar
at time t for dt dollars today. Then, at time t we have zero cash flows and
1
today we pocket (1+r)
dt > 0
t
mere
d
dd
ENT
M
NI
• A situation in which it is possible to make a profit without taking any risk
or making any investment is known as an arbitrage opportunity, and the
practice of buying and selling equivalent goods in di↵erent markets to take
advantage of a price di↵erence is known as arbitrage
more money to lend
• Normally, we should not expect an arbitrage opportunity to exist for a long
time since once it appears investors will race to take advantage of it
Page 17 of Lecture 1
• The presence of an arbitrage opportunity is usually an indication of some
financial frictions
• In the absence of financial frictions, we expect the Law of One Price to
hold
Equivalent goods in di↵erent competitive markets
must trade for the same price in both markets
Page 18 of Lecture 1
Case Study: Bitcoin Kimchi Premium
Source: Makarov and Schoar, “Trading and arbitrage in cryptocurrency markets”, Journal of Financial Economics (2020)
Implementation of Arbitrage
• Kimchi premium — the spread between bitcoin’s price on Korean and US
• In a frictionless world if prices are different across exchanges there is
exchanges. In a frictionless
world if prices are di↵erent across exchanges
a riskless arbitrage:
there is a riskless arbitrage:
Exch 1:
P1 = 100
Exch 2:
P2 = 200
B1
B1
$100
$200
Makarov and Schoar, Trading and Arbitrage in Cryptocurrency Markets
21
Page 19 of Lecture 1
Bitcoin Kimchi Premium (cont.)
• An arbitrage trade would involve buying bitcoins on a US exchange, transferring and selling them on a Korean exchange, and repatriating proceeds back
to the US
• South Korea’s cross-border capital controls and regulations limit foreign
investors from trading on domestic exchanges and limit the amount of capital
nationals of Korea can take out of the country
• Capital controls thus prevent efficient use of arbitrage capital
• As a result, when arbitrage capital is scarce the bitcoin price on Korean
exchanges can deviate from the bitcoin price on US exchanges
Page 20 of Lecture 1
1.2.1 Present value
• If we know the discount rates we can compute the value of future cash flows
in terms of cash today. This value is called the present value (PV)
• The present value of the cash flow Ct in year t is
P V = Ct ⇥ d t =
Ct
(1 + rt)t
• Suppose that the interest rate for years 1, 10, and 100 is the same and equal
to 5%. What is the PV of the cash flow 100 in years 1, 10, and 100?
) Year 1: P V = 100/1.05 = 95.23
) Year 10: P V = 100/1.0510 = 61.39
) Year 100: P V = 100/1.05100 = 0.76
Page 21 of Lecture 1
• If the discount rate for year t is rt, the present value of a yearly cash flow
stream C0, C1, .., CT , is
P V = C0 +
C1
CT
+ ... +
1 + r1
(1 + rT )T
• The present value formula applies to risky cash flows as well but with
appropriately modified discount rates
Page 22 of Lecture 1
• The Guardian, Aug 24, 2017: A Massachusetts hospital worker, Mavis
Wanczyk, has won the largest single-ticket prize in US lottery history, a
$758.7M Powerball jackpot. Wanczyk chose to take a lump-sum payment of
$480M
Page 23 of Lecture 1
• The lump-sum payment is substantially less than the nominal Powerball
jackpot
• This is because a Powerball jackpot is paid over 29 years: an immediate
initial payment followed by 29 annual payments. Each payment is 5 percent
larger than the previous one. The sum of 30 payments is equal to the jackpot
• Exercise (Excel): Compute the PV of a $758.7M Powerball jackpot if all
interest rates are equal to 2.75%
Page 24 of Lecture 1
Solution: We first find the initial payment C . It solves
⇣
2
C ⇥ 1 + (1 + 5%) + (1 + 5%) + · · · + (1 + 5%)
= $758.7M
)
C = $11.42M
29
⌘
= C ⇥ 66.44
The PV is
0
$11.42M ⇥ @1 +
!
1 + 5%
1 + 5%
+
1 + 2.75%
1 + 2.75%
!2
+ ··· +
= $477.3M
• $477.3M < $758.7M ) the time value of money is important!
1 + 5%
1 + 2.75%
!291
A
Page 25 of Lecture 1
• We can also compute the future value of the today’s cash flow at year T
F V = C0 ⇥ (1 + rT )T
• What is the value of the cash flow Ct at year T ?
)
(1 + rT )T
Ct ⇥
(1 + rt)t
Proof:
Ct
(1+rt )t
u
0
u
1
Ct
u
t
Ct
(1+rt )t
? We can exchange Ct dollars at time t for
? And then exchange
Ct
(1+rt )t
CT
···
Ct
(1+rt )t
u
-
T
Ct ⇥
-
(1+rT )T
(1+rt )t
dollars today
T
T)
dollars today for Ct ⇥ (1+r
dollars at time T
(1+r )t
t
Page 26 of Lecture 1
1.2.2
Forward Rates
Ct
(1+rt )t
u
Ct
u
0
(1 + t fT
u
1
···
t
Ct
(1+rt )t
• Define a rate tfT
(1 + tfT t)T
t
t
t)
T t
-
-
CT
u
T
Ct ⇥
-
(1+rT )T
(1+rt )t
as
T
=
(1 + rT )
(1 + rt)t
T
)
t fT
t
=
(1 + rT )
(1 + rt)t
!
1
T t
1
• The rate tfT t is called the forward rate between years t and T . This is the
rate that we can guarantee today for investing in the future from year t to
year T
• Check yourself: show that if the forward rate (1 + tfT t)T
(1+rT )T
then there is an arbitrage
(1+r )t
t
t
does not equal
Page 27 of Lecture 1
Obtaining Forward Rates from Spot Rates: An Example
• Suppose that the 1-year spot rate is 5%, the 2-year spot rate is 4.5%, and
the 3-year spot rate is 4%. Which forward rates can you compute based on
this information?
) Forward rate between years 1 and 2:
(1 + r2)2
) 1f1 = 4.0%.
1 + 1 f1 =
1 + r1
) Forward rate between years 2 and 3:
(1 + r3)3
1 + 2 f1 =
) 2f1 = 3.0%.
(1 + r2)2
a
) Forward rate between years 1 and 3:
(1 + r3)3
(1 + 1f2) =
) 1f2 = 3.5%.
1 + r1
2
Page 28 of Lecture 1
1.3
Interest rate quotes
So far, we have considered yearly cash flows. How does the PV rule apply when
cash flows are monthly, quarterly, semiannual, etc?
) In exactly the same way but we need to make sure that we use the correct
discount rates that correspond to the frequency of the cash flows. To compute
the correct discount rates we need to know how interest rates are quoted
• There are two common ways to quote interest rates
? As an E↵ective Annual Rate (EAR), which indicates the total amount
of interest that will be earned at the end of the year
? As an Annual Percentage Rate (APR), which indicates the amount of
simple interest earned in one year. When interest rates are quoted in this
way, the compounding period is specified in the quote, e.g., daily, monthly,
quarterly, semiannual, or annual. APR does not reflect the true amount
you will earn over one year. Therefore, APR itself cannot be used as a
discount rate
Page 29 of Lecture 1
Suppose that an EAR is 5%
• What is the semiannual rate? ) Let x denote the semiannual rate. We have
1
(1 + x)2 = 1 + 5% ) 1 + x = 1.05 2 ) x = 2.47%
• What is the quaterly rate? ) Let x denote the quaterly rate. We have
1
(1 + x)4 = 1 + 5% ) 1 + x = 1.05 4 ) x = 1.22%
• What is the monthly rate? ) Let x denote the monthly rate. We have
1
(1 + x)12 = 1 + 5% ) 1 + x = 1.05 12 ) x = 0.407%
Page 30 of Lecture 1
Suppose that your bank o↵ers a 1-year loan with a 5% APR
• What is the semiannual rate if compounding is semiannual? )
• What is the quaterly rate if compounding is quaterly? )
• What is the monthly rate if compounding is montly? )
5%
4
5%
12
• What is the EAR with
? annual compounding?
) 1.05
⇣
? semiannual compounding? ) 1 +
? quarterly compounding?
? monthly compounding?
? daily compounding?
⇣
) 1+
⇣
) 1+
⇣
) 1+
⌘2
5%
= 1.0506
2
⌘4
5%
= 1.0509
4
⌘12
5%
= 1.05116
12
⌘365
5%
= 1.051268
365
5%
2
Page 31 of Lecture 1
1.4
Perpetuities and Annuities
• Perpetuity: Stream of constant cash flows, starting one period from today,
and lasting forever
u
Time 0
C
C
1
2
u
u
C
···
u
T
-
···
• Suppose that all interest rates rt are the same and equal to r. Then
PV =
• This is
C
C
+
+ ...
2
1+r
(1 + r)
✓
◆
C
1
C
1
C
1+
+ ... =
=
1
1+r
1+r
1 + r 1 1+r
r
Page 32 of Lecture 1
Example: Suppose you want to endow an annual graduation party at LSE.
You want the event to be a memorable one, so you budget £10, 000 per year
forever for the party. Suppose that the first party in one year’s time and all
interest rates are the same and equal to 5%. How much will you need to donate
to endow the party?
) This is a standard perpetuity of £10, 000 per year. The funding you would
need to give LSE in perpetuity is the PV of this cashflow stream:
PV =
C
£10, 000
=
= £200, 000
r
5%
Page 33 of Lecture 1
• Growing perpetuity: Stream of constant cash flows, starting one period
from today, growing at constant rate, and lasting forever
u
Time 0
C
u
1
C(1 + g)
u
2
···
C(1 + g)T
u
T
1
-
···
• Suppose that all interest rates rt are the same and equal to r > g . Then
C
C(1 + g)
C
PV =
+
+ ... =
1+r
(1 + r)2
r g
Page 34 of Lecture 1
Example: Suppose you want to endow an annual graduation party at LSE. You
want the event to be a memorable one, so you finance all future parties. As
before, you want to budget £10, 000 for the first party, which is in one year’s
time. However, now you want to account for the e↵ect of inflation on the cost
of the party in future years. Your estimate that the party’s cost will rise by 2.5%
per year. Assuming that all interest rates are the same and equal to 5%, how
much will you need to donate to endow the party?
)
PV =
C
r
g
=
£10, 000
= £400, 000
5% 2.5%
di
s's
4
c
c
c
c
c
c
c
c
s
c
c
Cccc
cc
Page 35 of Lecture 1
• Annuity: Stream of constant cash flows, starting one period from today, and
lasting for T periods
u
C
C
1
2
u
Time 0
u
C
···
u
-
T
• We can view an annuity as the di↵erence between two perpetuities
? First perpetuity: cash flows start one period from today
? Second perpetuity: cash flows start T + 1 periods from today
• The di↵erence is:
PV =
C
r
"
1
C
C
=
1
T
(1 + r) r
r
1
(1 + r)T
#
Page 36 of Lecture 1
• Warning: When computing the value of second perpetuity we implicitly used
the fact that all forward rates are same and equal to r. This is only true
when all spot rates are equal to r
• Exercise: You have the following information. Today’s spot rates are 2%
until year 3 and 3% from year 4 onward. Compute the value of perpetuity
that pays $1 each year starting from year 4
Solution: It might be tempting to compute the value of the perpetuity as
P V0 =
1
$1
(1 + 2%)3 3%
arguing that the PV of the perpetuity’s cash flows in year 3 is
the PV in year 0 is
1
$1
P V0 =
(1 + 2%)3 3%
However, this is incorrect
$1
.
3%
Hence,
Page 37 of Lecture 1
The above argument is wrong because to compute the present value of future
cash flows in year 3 one needs to use the corresponding forward rates:
$1
$1
P V3 =
+
+ ...
1 + 3 f1
(1 + 3f2)2
where
(1 + r4)4
(1 + 3%)4
1 + 3 f1 =
=
3
(1 + r3)
(1 + 2%)3
(1 + r5)5
(1 + 3%)5
(1 + 3f2) =
=
3
(1 + r3)
(1 + 2%)3
2
Therefore,
3
(1 + 2%)
P V3 =
(1 + 3%)3
!
$1
$1
(1 + 2%)3 $1
+
+ ... =
(1 + 3%)
(1 + 3%)2
(1 + 3%)3 3%
Page 38 of Lecture 1
The PV at time 0 then is
P V0 =
P V3
1
$1
=
(1 + r3)3
(1 + 3%)3 3%
Remark: We can solve the problem directly using the present value formula:
P V0 =
$1
$1
$1
+
+
+ ...
(1 + 3%)4
(1 + 3%)5
(1 + 3%)6
To compute the above sum we rewrite it as
$1
$1
$1
$1
+
+
...
3
2
3
(1 + 3%) (1 + 3%)
(1 + 3%)
(1 + 3%)
1
$1
=
(1 + 3%)3 3%
P V0 =
!
Page 39 of Lecture 1
Exercise: You get a car loan of $20,000. The rate on the loan is 10% monthly
APR, and payment extends over 5 years. What is the monthly payment?
Solution: We first solve for the monthly rate
10%
= 0.83%
12
Let C be the monthly payment. Then the PV of the loan repayments is
C
C
C
+
+
...
+
=
2
60
1 + 0.83%
(1 + 0.83%)
(1 + 0.83%)
=C
"
1
0.83%
#
1
= C ⇥ 47.07
60
0.83%(1 + 0.83%)
The PV must equal to $20,000. Therefore, C = $424.9
Page 40 of Lecture 1
Summary
=
The PV (perpetuity)
C
r
The PV (growing perpetuity)
The PV (annuity)
=
"
C
1
r
=
C
r
g
1
(1 + r)T
#
• Points to remember:
? The first cash flow comes one period from today
? If C is an annual (monthly, quarterly, ...)
(monthly, quarterly, ...) rate for r
cash flow, use the annual
Page 41 of Lecture 1
1.5
Net Present Value Rule
Example: You have invented a mobile app. One company o↵ers you $22,000
for the idea, but first you must develop software that implements the idea.
A programmer requires $20,000 payable immediately to program your idea.
Programming will take one year. A second company o↵ers $500 but does not
require you to develop the software. What should you do?
) We need to evaluate a stream of cash flows
• Cash flows are:
-$20,000
u
Years 0
• Need to know the PV of $22,000
$22,000
u
1
-
Page 42 of Lecture 1
• Suppose you can borrow and lend at the annual interest rate of 5%. Then
the net present value is
NP V =
$22, 000
$20, 000 +
= $952 > $500
1.05
) choose the first company
• What if you can borrow and lend at the annual interest rate of 10%?
• Then the net present value is
NP V =
$20, 000 +
$22, 000
= 0 < $500
1.1
) choose the second company
Page 43 of Lecture 1
• Net present value (NPV):
P V (benef its)
P V (costs)
• NPV rule: when making an investment decision, take the alternative with
the highest NPV. In particular,
? Accept positive-NPV projects
? Reject negative-NPV projects
Page 44 of Lecture 1
Main Takeaways
• Financial markets perform a number of important functions. They help
? allocate resources to their most productive use
? move funds from those who have a surplus to those who have a shortage
? move wealth across time
? share and manage risk
• In the absence of financial frictions, we expect to see no arbitrage opportunities and the Law of One Price to hold
• The present value framework and NPV rule are the main tools to value
financial securities and to decide which projects to start
• Next time: how to find discount rates from market prices of financial securities
Lecture Notes
FM 423: Asset Markets
Lecture 2
Fixed-Income Securities I
Igor Makarov
London School of Economics
Page 2 of Lecture 2
Lecture 2: Main Points
1. Overview of Fixed Income Markets
2. Valuation of Bonds
3. Yield to Maturity
Page 3 of Lecture 2
2.1
Bonds
• A fixed-income security is a security that promises cash flows of fixed
amounts at fixed dates
• A prominent example of a fixed-income security is a bond
• A zero-coupon bond (or zero) promises a single cash flow, face value (or
par value), at some future date, maturity
• A coupon bond promises a periodic cash flow, coupon, and the face value
at maturity. The coupon rate is the ratio of the coupon to the face value.
Coupon payments are typically semiannual for US bonds and annual for
European bonds
• The time to maturity is the length of time until maturity
• For notational simplicity, we assume from now on that bonds have a face
value of $100. This is equivalent to expressing bond prices as a percentage
of face value
Page 4 of Lecture 2
Example
• Cash flows of a zero-coupon bond with 3 years to maturity
0
0
100
Time 1
2
3
u
u
u
-
• Cash flows of a bond with coupon rate 10%, annual coupon payments, and
3 years to maturity
10
u
Time 1
10
u
2
110
u
-
3
• Cash flows of a bond with coupon rate 10%, semiannual coupon payments,
and 2 years to maturity
5
u
Time 0.5
5
5
1
1.5
u
u
105
u
2
-
Page 5 of Lecture 2
Where does the name coupon bond come from?
Page 6 of Lecture 2
Who issues bonds?
Source: ICMA analysis using Bloomberg Data (2020)
Page 7 of Lecture 2
Who issues bonds? (cont.)
Source: ICMA analysis using Bloomberg Data (2020)
Page 8 of Lecture 2
Why bonds are so popular?
• Debt contracts are the oldest financial contracts. They help
? transfer funds from those who have a surplus to those who have a shortage
? transfer wealth over time
• Debt contracts require less developed institutions and legal environment than
equity contracts
? To successfully share profits, we need to have a well-developed accounting
system and laws that prevent diverting income streams from firms
? When we issue debt we only need to able to verify whether it has been
repaid or not
Page 9 of Lecture 2
• The main problem when we issue debt is to ensure that it will be repaid in
due time. Four main mechanisms work towards this goal
?
?
?
?
Penalties if the debt is not repaid in time
Collateral
Reputation concerns
Restrictive clauses
• Early on, attaching severe penalties was the most common way to ensure
that a borrower had incentives to repay his debt and not to engage in moral
hazard
• Moral hazard is the risk that a party has incentives to take advantage of a
financial deal or situation, knowing that consequences of bad-decision making
will be born by another party
• Modern debt contracts are very complex. They do not rely on severe penalties
but have many clauses attached, which aim to restrict moral hazard (more
on that in the CF class)
Page 10 of Lecture 2
Examples of moral hazard
• When a firm raises debt firm owners may have incentives to abscond with
the money (or pay themselves a dividend) instead of investing the funds in a
project
• Once a firm issues debt it may have incentives to issue even more debt.
Unless the previously issued debt has a priority over the subsequently issued
debt (in finance parlance, such debt is termed senior) the original lenders will
be worse o↵ if more debt is issued
• When a bank gives a loan it has incentives to run a check on a borrower to
make sure that the borrower will be able to repay the loan. However, if the
bank knows that it will be able to unload this loan from its books (by selling
it to another party) the bank may not run a thorough check
Page 11 of Lecture 2
Default Risk
• Fixed-income securities generally involve default risk, the risk that the issuer
will not meet the cash flow obligations
• Default risk matters for corporate bonds and sovereign bonds of emerging
economies. It has also started to matter for sovereign bonds of advanced
economies (e.g., U.S. and Eurozone)
• This lecture focuses on the default-free bonds
Page 12 of Lecture 2
2.2
Valuation of default-free bonds
Our goal in this lecture will be to show that
• Prices of bonds of di↵erent maturities are linked to each other
• The prices of all bonds can be obtained from the prices of zero-coupon bonds
• Also, the prices of zero-coupon bonds, and hence the discount factors, can
be obtained from the prices of coupon bonds using bootstraping procedure
Page 13 of Lecture 2
• Recall from the previous lecture that the present value (PV) of $1 received t
years from now is dt, where dt is the discount factor
• The price of a zero-coupon bond with t years to maturity is 100 ⇥ dt
• Therefore, we can obtain the discount factor dt, by dividing the price by 100
• Example: Suppose that the prices of zero-coupon bonds with maturities 1,
2, and 3 years, are 95, 88, and 80, respectively. What are the corresponding
discount factors?
Answer: 0.95, 0.88, and 0.8
Page 14 of Lecture 2
2.2.1 Valuation via Zero-Coupon Bonds
• Consider a coupon bond with annual coupon rate 10% and 3 years to
maturity
• Cash flows are:
10
u
Time 1
10
110
2
3
u
u
-
• Suppose that the prices of zero-coupon bonds with maturities 1, 2, and 3
years, are 95, 88, and 80, respectively
• The price of the bond is the PV of the bond’s cash flows:
P = 10 ⇥ d1 + 10 ⇥ d2 + 110 ⇥ d3
= 10 ⇥ 0.95 + 10 ⇥ 0.88 + 110 ⇥ 0.8 = 106.3
• What ensures that this “theoretical” price is the actual market price?
Page 15 of Lecture 2
2.2.2 Synthetic Replication
• We can construct a portfolio of the three zero-coupon bonds, that has the
same cash flow as the coupon bond
Year
Cash Flow
of Coupon Bond
Portfolio of Zeros
1
10
0.1 one-year zeros
2
10
0.1 two-year zeros
3
110
1.1 three-year zeros
• The portfolio of the zero-coupon bonds synthetically replicates the coupon
bond. It is a replicating portfolio for the coupon bond
• By the law of one price the price of the replicating portfolio and coupon bond
must be the same
Page 16 of Lecture 2
• The market value of the replicating portfolio is:
0.1 ⇥ 95 + 0.1 ⇥ 88 + 1.1 ⇥ 80 = 106.3
• Therefore, the price of the coupon bond must be 106.3. Otherwise, there
there exists an arbitrage
• Suppose that a trader o↵ers the coupon bond at 105. Then we can:
? Buy the coupon bond at 105
? Sell the replicating portfolio at 106.3
? The cash flows in years 1, 2, and 3 cancel, and we are left with a gain of
106.3 105 = 1.3 today
• Check yourself: What would you do if the coupon bond is traded at 107?
Page 17 of Lecture 2
2.2.3 Using Coupon Bonds Instead of Zero-Coupon
• We used zero-coupon bonds to:
? Obtain spot rates
? Obtain prices of other bonds
? Synthetically replicate other bonds
• Instead of zero-coupon bonds, we can use coupon bonds
• Bootstrapping is a procedure to obtain the prices of zero-coupon bonds
from prices of coupon bonds
Page 18 of Lecture 2
Bootstrapping
• Example 1: Consider three bonds with the following characteristics:
Bond Maturity
Coupon Rate
(Semiannual)
Price
A
0.5
0%
98
B
1.0
0%
95
C
1.5
8%
102
4
eves b winnings
Determine the 6-month, 1-year, and 18-month discount rates
Solution: We can immediately find that d0.5 = 0.98 and d1 = 0.95
Page 19 of Lecture 2
• To find d1.5 we first, write down bond C’ cash flows
4
u
Time 0.5
4
104
1
1.5
u
u
-
• Applying the PV formula we have
4 ⇥ d0.5 + 4 ⇥ d1 + 104 ⇥ d1.5 = 102
• Since we know d0.5 and d1 we can solve for d1.5 :
d1.5 = (102
4 ⇥ 0.98
6
0
Yon t Yon
4 ⇥ 0.95)/104 = 0.906
• Note that by going long 100/104 units of bond C and going short 4/104
units of bonds A and B we can synthetically replicate a zero-coupon bond
with maturity of 18 months.
Teven
on
100
Page 20 of Lecture 2
Example 2: Consider two bonds with the following characteristics available for
trading:
Bond Maturity
Coupon Rate
(Annual)
Price
A
2
4%
96.37
B
2
8%
103.74
Determine d1 and d2
• Solution: Again, write down bonds’ cash flows
A
4
104
B
8
u
Time 1
108
u
2
-
8
<4 ⇥ d + 104 ⇥ d = 96.37
1
2
)
:8 ⇥ d1 + 108 ⇥ d2 = 103.74
)
d1 = 0.9525 and d2 = 0.89
Page 21 of Lecture 2
• How do we synthetically create a one-year zero-coupon bond by trading in
the two-year bonds?
• Suppose that portfolio consists of x1 units of bond A and x2 units of bond B
Cash Flows
Bond
1
2
A
4
104
B
8
108
)
8
<4x
+ 8x2 = 100
:104x1 + 108x2 = 0
1
The solution is
x1 A + x2 B 4x1 + 8x2 104x1 + 108x2
Zero-coupon
100
0
x1 =
27
and
x2 = 26
Page 22 of Lecture 2
• We need to sell 27 units of the first bond and buy 26 units of the second
• Having synthetically replicated the zero-coupon bond, we can compute its
price
• The market value of the replicating portfolio is:
27 ⇥ 96.37 + 26 ⇥ 103.74 = 95.25
• The price of the zero-coupon bond has to be 95.25. Otherwise, there would
exist an arbitrage
• Having found the price of the zero-coupon bond we can verify the value of
d1 we computed earlier
• Check yourself: How does the analysis change if we want to synthetically
replicate a two-year zero-coupon bond?
Page 23 of Lecture 2
2.2.4 Obtaining Spot Rates
• Annual coupon rate:
1
dt =
(1 + rt)t
)
✓
1
rt =
dt
◆1
t
1
• The term structure of spot rates (or yield curve) is a collection of the
spot rates for di↵erent maturities
Page 24 of Lecture 2
• Semiannual coupon rate:
? It is customary to quote spot rates as semiannual APRs
? Therefore, the price of a bond with semiannual coupon rate c% (i.e.
semiannual coupon payments of c/2), and T years to maturity is
P =
dt =
c
2
1 + r0.5
2
+⇣
1
(1 + r2t )2t
c
2
1+
)
⌘
r1 2
2
+ ··· + ⇣
100 + 2c
1+
0
✓ ◆1
1 2t
@
rt = 2
dt
⌘
rT 2T
2
1
1A
Page 25 of Lecture 2
• Example: Suppose that r0.5 = 8%, r1 = 8.2%, r1.5 = 8.6%, and
r2 = 9%. Compute the price of a bond with semiannual coupon rate 8%
and 2 years to maturity
P =
4
4
4
104
+
+
+
= 98.27
2
3
4
1 + 4% (1 + 4.1%)
(1 + 4.3%)
(1 + 4.5%)
• Example: Find the 6-month, 1-year, and 18-month spot rates expressed as
semiannual APRs if d0.5 = 0.98, d1 = 0.95, and d1.5 = 0.906
Solution:
✓
◆
1
r0.5 = 2
1 = 4.08%
0.98
!
1
r1 = 2
1 = 5.2%
1
2
0.95
!
1
r1.5 = 2
1 = 6.65%
1
0.906 3
Page 26 of Lecture 2
What did we learn?
• Prices of bonds of di↵erent maturities are linked to each other
• Any bond can be synthetically replicated by a portfolio of zero-coupon bonds
• Absence of arbitrage implies that a bond must have the same price as its
replicating portfolio
• Therefore, the prices of all bonds can be obtained from the prices of zerocoupon bonds
• Also, the prices of zero-coupon bonds and therefore, the term structure
of interest rates can be obtained from the prices of coupon bonds using
bootstraping
• Thus, the term structure of interest rates contains information about the
prices of all bonds
Page 27 of Lecture 2
2.2.5 Floating-rate Bonds (Floaters)
• A floating-rate bond is a bond with coupons that adjust periodically based
upon a reference rate
• Many investment products, e.g., student loans or mortgages are examples of
floating-rate bonds
• Example: Consider a floater with 3 years to maturity and annual coupons
based on the one-year interest rate
? Suppose that the current one-year spot rate at inception is 5%
? Denote future one-year spot rates in year one and two by 1r1 and 2r1
? Cash flows:
5
u
Time 1
100 ⇥ 1 r1
u
2
100 ⇥ (1 + 2 r1 )
u
-
3
? How to value this floater? Problem: the future coupons are not known
today
Page 28 of Lecture 2
• The main idea is to find an “equivalent good” that has the same cash flows
as the floating-rate bond and that we know how to price. Then use the Law
of one price to argue that the price of both products should be the same
• The “equivalent good” in this case will be a trading strategy that requires
an initial investment at time 0 and has the same payo↵s at time 1, 2, and 3
as the floater
• We now describe this trading strategy:
? At time 0, invest 100 in a one-year bond. Since the current one-year
interest rate is 5% this bond will return 105 in year 1
? At time 1, collect 100 + 5, use 5 to match the coupon payment of the
floater in year 1, and invest 100 in a one-year bond. This bond will return
100(1 + 1r1) in year 2. We do not know 1r1 today but we will know it in
year 1 when we buy the one-year bond
Page 29 of Lecture 2
? At time 2, as before, collect 100(1 + 1r1), use 100 ⇥ 1r1 to match the
coupon payment of the floater in year 2, and invest 100 again in a one-year
bond. This bond will deliver 100(1 + 2r1) in year 3
? At time 3, collect 100(1 + 2r1) to match the final cash flow of the floater
? Viewed as a whole, the above trading strategy requires an initial investment
of 100 and pays exactly the same cash flows as the floater
Ast
• Hence, by the Law of one price, the price of the floater should be 100.
Otherwise, there is an arbitrage
• In fact, we proved a slightly more general result: On reset dates (after a
current coupon being just paid), a floater is worth par
• To convince ourselves that the price of the floater must be equal to 100
suppose that it is not. Suppose, for example, it is 95. Can we find an
arbitrage strategy that delivers a riskless profit in this case?
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Page 30 of Lecture 2
• The answer is Yes. The arbitrage strategy is as follows.
? Since the fair price of the floater is 100 we know that if its price is quoted
at 95 then it is too cheap. Hence, to realize an arbitrage we should buy
a unit of this bond at 95. To finance this purchase, we use the reverse
strategy. That is we sell a one-year bond worth of 100. Then, in year 1 we
will have to repay 105. We will use the first-year coupon of the floater to
repay 5, and will sell a one-year bond to repay the remaining 100. Then in
year 2, we will need to repay 100(1 + 1r1). We will use the second-year
coupon of the floater to repay 100 ⇥ 1r1 and will sell a one-year bond to
repay the remaining 100. Finally, in year 3 we will use the final payment
of the floater to repay 100(1 + 2r1)
• Check yourself: Find an arbitrage strategy if the price of the floater is 105.
Page 31 of Lecture 2
2.3
Yield to Maturity
• The yield to maturity (YTM) of a bond is the single discount rate that
equates the PV of the bond’s cash flows to the bond’s price
• For a bond with annual coupon rate c% and T years to maturity, the YTM
(y ) is given by:
Price =
c
c
100 + c
+
+
·
·
·
+
1+y
(1 + y)2
(1 + y)T
• Example: If c = 10, T = 3, and the price is 98, then:
98 =
10
10
110
+
+
1+y
(1 + y)2
(1 + y)3
Solving numerically, we get y = 10.82%
Radar line
weighted avenge
of spot rates
Swaps
reference robe
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100
T
in 100
new
Fred
th
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2
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Swipe
CX 100
I
t
0
100
Fristoe
leg
sell
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floating
Thevar
Id sup
value
0
Page 32 of Lecture 2
• The relation between a bond’s YTM and coupon rate tells us how the bond’s
price compares to the face value
? If the YTM is greater than the coupon rate, then the bond sells at a
discount (below face value)
? If the YTM is equal to the coupon rate, then the bond sells at par (at
face value)
? If the YTM is smaller than the coupon rate, then the bond sells at a
premium (above face value)
Page 33 of Lecture 2
Relation Between YTM and Spot Rates
• Consider a bond with annual coupon rate c% and T years to maturity. Its
price is:
c
c
100 + c
Price =
+
+ ··· +
1 + r1
(1 + r2)2
(1 + rT )T
• The YTM is given by:
Price =
c
c
100 + c
+
+
·
·
·
+
1+y
(1 + y)2
(1 + y)T
• Comparing the two equations:
? YTM is a complicated average of the spot rates corresponding to the years
1, .., T
? YTM is simple only if the bond is zero-coupon (c = 0). It is then equal
to the T -year spot rate
Page 34 of Lecture 2
Semiannual Coupon Payments
• When coupon payments are semiannual, the YTM is generally quoted as a
semiannual APR
• For a bond with semiannual coupon rate c% and T years to maturity, the
YTM (y ) is given by:
Price =
c
2
1+
y
2
+⇣
c
2
1+
⌘
y 2
2
100 + 2c
+ ··· + ⇣
1+
⌘
y 2T
2
Page 35 of Lecture 2
YTM with Default Risk
• The YTM in the presence of default risk is defined by the same formula
• A bond with default risk has a lower price, and hence, a higher YTM than
an otherwise identical bond with no default risk
• The di↵erence in the YTMs of the two bonds is the default spread
Page 36 of Lecture 2
2.3.1 Uses and Abuses of YTM
• The main use of the YTM is an alternative way to quote the price of a bond
• Abuse 1: As a measure of the bond’s return
• Abuse 2: As a tool for choosing between di↵erent bonds
Page 37 of Lecture 2
YTM and Return: Zero-Coupon Bonds
• The YTM of a zero-coupon bond is the spot rate corresponding to the bond’s
time to maturity
• This is the return from investing in the bond and holding it until maturity
• But it is not the return for any other investment horizon
Example: Consider a zero-coupon bond with 3 years to maturity and YTM 6%
• The return from investing in the bond and holding it for 3 years is 6%
• However, the return from investing in the bond and selling it after 1 year is
unknown today
? The return depends on the bond’s price in 1 year
? The price in 1 year depends on the 2-year spot rate that will prevail in 1
year. This spot rate is unknown today
? What is the one-year return if the 2-year spot rate in 1 year is 10%?
Page 38 of Lecture 2
? Solution: The bond price today is
P0 =
100
= 83.96
3
(1 + 6%)
? The bond price in one year is
P1 =
100
= 82.64
(1 + 10%)2
? Therefore, the one-year return is
P1
P0
P0
=
1.57%
Page 39 of Lecture 2
YTM and Return: Coupon Bonds
• The YTM of a coupon bond is the return of investing in the bond, holding
it until maturity, and reinvesting the coupons at a rate equal to the YTM
• Problem 1: The YTM is not the return for any investment horizon other
than maturity. This is for the same reason as for zero-coupon bonds
• Problem 2: The YTM is not the return even for investment horizon equal
to maturity. This is because the future spot rates, at which the coupons will
be reinvested, may be di↵erent than the YTM
• Example: Consider a 2-year bond with annual coupon rate 5% and YTM
4%
? What is the return of investing in the bond, holding it until maturity, and
reinvesting intermediate coupons until maturity, if the 1-year spot rate in
1 year turns out to be 6%?
Page 40 of Lecture 2
? Solution: The bond price today is
P0 =
5
105
+
= 101.86
2
1 + 4%
(1 + 4%)
? At the end of year 2 we have
105 + 5 ⇥ 1.06 = 110.30
? Therefore, the (annualized) return over two years is
s
110.30
101.86
1 = 4.06%
Up higher beane
reinvest coupon
at
inger
inverse
nite
Page 41 of Lecture 2
YTM as a Tool for Comparing Bonds
• Using the YTM for comparing bonds is correct only when the bonds have
the same coupon and time to maturity. In all other cases it can be very
misleading
• Suppose, for instance, that the bonds have di↵erent time to maturity
? Recall that the YTM is at best the return of investing until maturity
? Therefore, by comparing YTMs we compare returns for di↵erent investment
horizons
Page 42 of Lecture 2
Main Takeaways
• The term structure of interest rates contains information about prices of
all bonds. It is typically obtained from prices of coupon bonds by the
bootstrapping method
• The Law of one price applies not only to physical goods but also to investment
strategies. Strategies that have the same cash flows must cost the same
(require the same initial investments)
• YTM is a convenient way to quote the price of a bond, but not a tool to
choose between di↵erent bonds
• Next time: term structure theories and main factors that a↵ect the term
structure of interest rates
Page 43 of Lecture 2
Appendix⇤
• The bootstrapping is particularly simple if we use matrix notation
• Suppose we have n bonds that have cash flows across n periods
• Suppose bond i has a cash flow cij in period j
• We can write bonds’ cash flows and a vector of discount rates and prices in
matrix notation as
0
c11 c12 · · · c1n
B
B c21 c22 · · · c2n
C=B
..
..
B ..
@
cn1 cn2 · · · cnn
1
C
C
C,
C
A
0
d1
B
B d2
D=B
B ..
@
dn
1
C
C
C,
C
A
0
P1
B
B P2
P =B
B ..
@
Pn
1
C
C
C
C
A
• Finding a vector of discount rates then amounts to solving a matrix equation:
C⇥D =P
)
D=C
1
P
Page 44 of Lecture 2
Lecture Notes
FM 423: Asset Markets
Lecture 3
Fixed Income Securities II
Igor Makarov
London School of Economics
Page 2 of Lecture 3
Lecture 3: Main Points
1. Term structure of Interest Rates
2. Expectations Hypothesis
3. Monetary Policy and Inflation
4. Major Fixed Income Markets
Page 3 of Lecture 3
3.1
Term Structure of Interest Rates
• Some past US term structures
2‐Jan‐04
3‐Jan‐05
3‐Jan‐06
7
7
7
6
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
1 mo 3 mo 6 mo 1 yr
0
1 mo 3 mo 6 mo 1 yr
2 yr
3 yr
5 yr
7 yr 10 yr 20 yr 30 yr
2‐Jan‐07
2 yr
3 yr
5 yr
7 yr 10 yr 20 yr 30 yr
0
1 mo 3 mo 6 mo 1 yr
2‐Jan‐08
7
7
6
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
1 mo 3 mo 6 mo 1 yr
0
1 mo 3 mo 6 mo 1 yr
3 yr
5 yr
7 yr 10 yr 20 yr 30 yr
4‐Jan‐10
2 yr
3 yr
5 yr
7 yr 10 yr 20 yr 30 yr
0
1 mo 3 mo 6 mo 1 yr
7
7
6
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
1 mo 3 mo 6 mo 1 yr
0
1 mo 3 mo 6 mo 1 yr
3 yr
5 yr
7 yr 10 yr 20 yr 30 yr
2 yr
3 yr
5 yr
5 yr
7 yr 10 yr 20 yr 30 yr
2 yr
3 yr
5 yr
7 yr 10 yr 20 yr 30 yr
3‐Jan‐12
3‐Jan‐11
7
2 yr
3 yr
2‐Jan‐09
7
2 yr
2 yr
7 yr 10 yr 20 yr 30 yr
0
1 mo 3 mo 6 mo 1 yr
2 yr
3 yr
5 yr
7 yr 10 yr 20 yr 30 yr
• Current term structures: http://www.worldgovernmentbonds.com/
Page 4 of Lecture 3
• The term structure can have many shapes
? It generally slopes up. This means that spot rates for long maturities are
generally higher than for short maturities
? However, it can also be hump shaped, inverted hump shaped, or downward
sloping
Page 5 of Lecture 3
? Upward sloping term structures are associated with periods of economic
expansion
? Downward sloping term structures are associated with periods of economic
slowdown/recession
• Important questions:
? What information is contained in the term structure?
? What are main factors that a↵ect the term structure?
Page 6 of Lecture 3
3.2
Expectations Hypothesis (EH)
• Motivating example: Suppose the current one, two, and three-year spot rates
are 2%, 4% and 5%, respectively. Suppose the term structure does not
change over the next year (that is in one year, the one, two, and three-year
spot rates will be 2%, 4% and 5%). Consider one, two, and three-year
zero-coupon bonds. Which bond will have the highest holding one-year
return between today and one year from today?
Solution:
Maturity
Bond Price
today
Bond Price
in one year
1
100
1+2%
= 98.04
2
100
(1+4%)2
= 92.45
100
1+2%
3
100
(1+5%)3
= 86.38
100
(1+4%)2
100
= 98.04
= 92.45
Return
100 98.04
98.04
= 2%
98.04 92.45
92.45
= 6.04%
92.45 86.38
86.38
= 7.03%
Page 7 of Lecture 3
• The three-year bond will have the highest return, followed by the two-year
and one-year bonds
• Thus, any investor who believes that the term structure will not change
and wants to maximize their expected return is better o↵ investing into the
three-year bond
• But if all investors prefer holding the three-year bond then it is inconsistent
with equilibrium (somebody should hold one-year bonds)
• Therefore, it must be that either the term structure in the future should
change or some investors should be willing to hold the one-year bond despite
the fact that it o↵ers a lower expected return
Page 8 of Lecture 3
• In Lecture 2, we showed that the prices of all bonds are linked to the prices
of zero-coupon bonds
• But what can be said about the prices of zero-coupon bonds of di↵erent
maturities?
? In isolation from each other, not much, except for that they are set by
supply and demand
? Intuitively, prices of bonds with similar maturities should be not too
di↵erent from each other since there are many ways to obtain a t period
return in the bond market. For example, one can invest in a t period
zero-coupon bond until maturity, or in a (t 1) period zero-coupon bond
until maturity and then in a one-period bond, or buy a (t + 1) period
zero-coupon bond and sell it after t periods
? One should expect these di↵erent strategies to deliver similar returns (since
otherwise investors could go long the strategy with the highest return and
short the one with the lowest return to realize an oversized profit without
taking a lot of risk)
? The extreme version of the above argument is the Expectations Hypothesis
Page 9 of Lecture 3
• (Pure) Expectations Hypothesis: Investment strategies for the same
horizon must have the same expected return irrespective of bonds maturities.
For example, the expected return from holding a t period bond until maturity
must equal to the expected return from a rollover investment strategy in oneperiod bonds. Alternatively, the expected return from holding a one-period
bond must be equal to the expected return from buying a long-maturity bond
and selling it after one year (there are several alternative formulations)
? Underlying assumption: Investors care only about the expected returns;
bonds of di↵erent maturities are perfect substitutes
• The expectations hypothesis is one of the most tested theories in bond
markets
• The reason why the EH is so important is because if the EH holds then bond
investing is simple: all bonds deliver the same expected returns. In contrast,
if the EH does not hold some bonds deliver higher returns than others
Page 10 of Lecture 3
• Suppose the EH holds. Then
? Investing in a two-year zero-coupon bond until maturity returns (1 + r2)2
in two years
? Investing in a one-year zero-coupon bond and rolling over to another
one-year zero-coupon bond in year 1 returns (1 + r1)(1 + 1r1), where
1 r1 is one-year spot rate in one year from now. The expected return over
two years is (1 + r1)(1 + E 1r1)
? Therefore, if investors care only about the expected returns both strategies
should have the same expected returns
(1 + r1)(1 + E 1r1) = (1 + r2)2,
(1 + r2)2
(1 + E 1r1) =
= 1 + 1 f1 ,
(1 + r1)
where 1f1 is the one-year forward rate
or
) E 1 r1 = 1 f1
Page 11 of Lecture 3
• More generally, assuming that E(1 + trT )T ⇡ (1 + E(trT ))T the EH implies
that the forward rate equals the market expectation of the future short
interest rate:
E(trT ) = tfT
• If the EH holds, long interest rates reflect expectations of future short interest
rates. In particular,
? Term structure slopes up ) Market expects rates to rise
? Term structure slopes down ) Market expects rates to fall
Page 12 of Lecture 3
• Example:
• Two investment strategies:
? Invest in a two-year bond and hold until maturity
? Invest in a one-year bond and after year one year, again in a one-year bond
• If the EH holds then the expected return from following either of the strategies
should be the same. Hence, we should have 5% ⇥ 2 ⇡ 4% + E 1r1 )
E 1r1 = 6%. Thus, the one-year spot rate is expected to increase
Page 13 of Lecture 3
3.2.1
Empirical Evidence: the EH does not hold exactly in the data
• Violation 1: Long-maturity bonds return more, on average, than shortmaturity bonds. This follows from the fact that the term structure is upward
slopping on average (the EH predicts that the term structure should be flat
on average since according to the EH, long and short-maturity bonds should
have the same returns, on average)
• Violation 2: Excess return of long-maturity bonds over short-maturity bonds
is positively related to slope of term structure (according to the EH, it should
not be predictable and should be zero, on average)
? Positive slope predicts outperformance of long-maturity bonds
? Negative slope predicts underperformance of long-maturity bonds
? According to EH, the expected return on long-maturity and short-maturity
bonds should be equal irrespective of the slope of term structure
Page 14 of Lecture 3
• The direction of movement of short rates (with maturities up to one year),
however, is consistent with the EH:
? Steep positive slope ) Short rates are likely to increase
? Negative slope ) Short rates are likely to decrease
• Violations of the EH imply that bonds of di↵erent maturities are not perfect
substitutes and that investors care not only about the expected returns
• What else can investor care about?
? Risk – we will show in Lecture 4 that long-term bonds are riskier than
short-term bonds
? Occasionally, some investors can have preference for specific bond maturities because of their hedging needs. We will provide an example in Lecture
4 (UK pension reform 2004)
Page 15 of Lecture 3
• Suppose investors want to guarantee a return over one year
? They would invest in the two-year bond and sell it after one year only if
this strategy returns more, on average, than an investment in the one-year
bond:
(1 + r2)2
> 1 + r1 , (1 + r2)2 > (1 + 1r1)(1 + r1)
(1 + 1r1)
) The term structure has a positive slope, on average
Page 16 of Lecture 3
3.3
Main factors that a↵ect the term structure
3.3.1 Inflation
• Inflation is the rate at which the general level of prices for goods and services
is rising. Inflation is calculated as a percentage change in a price index,
which trackes prices of a basket of goods. See, for example, Consumer
Price Index (CPI) (http://www.bls.gov/cpi/home.htm)
• Positive inflation means that the purchasing power of money decreases )
the higher the expected inflation the lower the demand for bonds is
US Inflation 1914 – 2022
Page 17 of Lecture 3
Page 18 of Lecture 3
3.3.2 Monetary Policy: A bird’s-eye view
• What is money? — Not a simple question
? Economists define money as anything that is generally accepted in payment
for goods or services or the repayment of debt
? It might be tempting to think of money as currency (paper money and
coins) but this definition would be too narrow. What about checks, bank
deposits, debit or credit cards?
? There are several measures of money supply, referred to as monetary
aggregates. For example, M1 include currency, checking account deposits,
and traveler’s checks
• Functions of money
? Medium of exchange
? Unit of account
? Store of value
Page 19 of Lecture 3
• Throughout history, money evolved together with the evolution of the payments system, the method of conducting transactions in the economy
? Commodity money, money made of precious metals or another valuable
commodity
? Fiat money, paper currency decreed by governments as legal tender
(meaning that it must be accepted as payment for debts) but not convertible into precious metals
? Checking account deposits together with checks and debit cards
? Nearest future: digital currencies
Page 20 of Lecture 3
• The term monetary policy refers to what a central bank does to influence
the amount of money and credit in an economy
• Three goals of modern monetary policy
? Price stability (inflation)
? Stable real economy (high employment and sustainable long-term growth)
? Financial stability (prevention of financial crises and smooth running of
payment system)
• Monetary policy is often a↵ected by political considerations
• To achieve high-level goals, central banks usually choose tactical target,
which they can directly control – typically a short-term interest rate (from
overnight to two weeks)
• There are three basic types of instruments of monetary policy:
? Open market operations
? Standing facilities
? Reserve requirements
Page 21 of Lecture 3
Open market operations
• Open market operations is a primary tool to influence the supply of bank
reserves. Open market operations involve purchases and sales of government
securities on the open market
• When the central bank buys securities through open market operations it
credits the reserve accounts of the institutions that sell the securities
• Additional funds in these institution reserve accounts puts downward pressure
on interest rates
• Lower interest rates encourage consumer and business spending, thereby
stimulating economic activity
Page 22 of Lecture 3
Standing facilities
• Standing facilities are defined as borrowing or deposit facilities available to
banks, usually at parameters set by the central bank; the credit facility allows
banks to borrow from the central bank, the deposit facility allows to place
excess funds
• Announcing a new set of parameters is often a convenient way to send a
message to the markets
• Standing facilities can also become the primary source of funds under unusual
circumstances, when normal functioning of financial markets is disrupted. In
such a case, the central bank serves as lender of last resort
Reserve requirements (rarely used)
• Requirements that banks should hold at the central banks
Page 23 of Lecture 3
Unorthodox monetary policy
• Because the recent recession was so severe, many central banks used a
number of extraordinary monetary policy tools that are not part of their
traditional toolkit
• Traditionally, open market operations involve the buying or selling short-term
securities. With short-rates close to zero or even negative, many central
banks used Quantitavie Easing (QE) – a policy to conduct large-scale
purchases of longer-term securities
• These purchases helped lower longer-term interest rates, including mortgage
rates
Page 24 of Lecture 3
3.4
Major fixed income markets
• The money market (short-term borrowing, maturity < 1 year)
? Government debt and Commercial Paper (Corporate debt)
? Eurodollars (USD-denominated bonds held in a European-based bank)
? Repurchase Agreements (Repos)
? Interbank loans
• Government and corporate bonds (maturities > 1 year)
• Inflation-linked bonds
• Mortgage and asset-baked securities
• Derivatives
Page 25 of Lecture 3
3.4.1 The repo market
• A repurchase agreement (repo) is an agreement to sell some securities to
another party and buy them back at a fixed date and for a fixed amount
• The best way to understand a repo transaction is to consider it as collateralized borrowing
? A trader entering into a repo transaction with a repo dealer is borrowing
cash (the sale price)
? The repo dealer holds the security as a collateral
? The cash amount is typically less than the value of the security, the
di↵erence being called a haircut
? The trader and the repo dealer agree that the trader will return back the
amount borrowed plus the repo rate
• A reverse repo is the opposite transaction, namely, it is the purchase of the
security for cash with the agreement to sell it back to the original owner at
a predetermined price, determined, once again, by the repo rate
Page 26 of Lecture 3
Schematic
repo
transaction
16
AN INTRODUCTION TO FIXED INCOME MARKETS
Figure 1.4
Schematic Repo Transaction
time t
MARKET
buy bond at Pt
=)
(=
pay Pt
TRADER
deliver bond
=)
(=
get Pt haircut
REPO DEALER
time T = t + n days
MARKET
sell bond at PT
(=
=)
get PT
TRADER
pay (Pt
get the bond
(=
REPO DEALER
=)
haircut)⇥ (1+repo rate ⇥ 3n6 0 )
dates, we have
Repo interest =
n
⇥ Repo rate ⇥ (Pt
360
haircut)
(1.1)
where the denumerator “360” stems from the day count convention in the repo market.
P Repo interest. In percentage terms, the
The profit to the trader is then P
Page 27 of Lecture 3
Page 28 of Lecture 3
The role of the repo market
The repo market is pivotal to the efficient working of almost all financial markets
Repos (and reverse repos)
• provide an efficient source of short-term funding
• provide a secure and flexible home for short-term investment
• facilitate central bank operations
• ensure liquidity in the secondary debt market
• allow traders to take short positions
Page 29 of Lecture 3
3.4.2 Interbank lending market
• The interbank lending market is a market in which banks lend funds to one
another for a specified term
• Most interbank loans are for maturities of one week or less, the majority
being over day
• The interbank rate is the rate of interest charged on short-term loans between
banks
• Important interbank rates: federal funds rate (USA), SONIA (LIBOR) (UK)
and the Euribor and ESTER (Eurozone)
Page 30 of Lecture 3
LIBOR: London Interbank O↵ered Rate
• LIBOR is the rate at which major international banks can borrow unsecured
funds from each other
• For a very long time, LIBOR has been one of the most important benchmark
rates with more than $350 trillion in financial contracts being tied to it
• Prior to February 2014 LIBOR was administered by the British Bankers’
Association
• Following the LIBOR scandal, LIBOR is now administered by the Intercontinental Exchange
• On 5 March 2021, the Financial Conduct Authority (FCA), announced
that all LIBOR settings for all currencies will either cease or no longer be
representative immediately after the following dates:
? 31 December 2021, for Sterling, Euro, Swiss Franc and Japanese Yen
? 30 June 2023, for US Dollar Overnight, 1-12-month settings
Beyond Libor
compliant with standards such as the IOSCO principles. Five of the largest currency
areas have all moved to an O/N benchmark as the backbone of the new regime (Table
1). Graph 2 shows a classification of the old and new O/N reference rates based on
31 (i)
ofisLecture
3
key features, ie whether Page
the rate
transaction-based;
(ii) is based on collateralised
(secured) money market instruments; and (iii) reflects borrowing costs from wholesale
non-bank counterparties. For ease of comparison, existing (or old) RFRs as well as
O/N LIBOR are also shown.
Overview of identified alternative RFRs in selected currency areas
Table 1
United States
United Kingdom
Euro area
Switzerland
Japan
SOFR
SONIA
ESTER
SARON
TONA
(euro short-term
rate)
(Swiss average
overnight rate)
(Tokyo overnight
average rate)
ECB
SIX Swiss Exchange
Bank of Japan
MMSR
CHF interbank repo
Money market
brokers
Alternative rate
(secured overnight (sterling overnight
index average)
financing rate)
Administrator
Federal Reserve
Bank of New York
Bank of England
Data source
Triparty repo, FICC
GCF, FICC bilateral
Form SMMD (BoE
data collection)
Wholesale
non-bank
counterparties
Yes
Yes
Yes
No
Yes
Secured
Yes
No
No
Yes
No
Overnight rate
Yes
Yes
Yes
Yes
Yes
Available now?
Yes
Yes
Yes
Yes
Oct 2019
FICC = Fixed Income Clearing Corporation; GCF = general collateral financing; MMSR = money market statistical reporting; SMMD = sterling
money market data collection reporting.
Sources: ECB; Bank of Japan; Bank of England; Federal Reserve Bank of New York; Financial Stability Board; Bank of America Merrill Lynch;
International Swaps and Derivatives Association.
8
LIBOR incorporates both term liquidity and credit premia, although the relative contribution of the
two can differ over time and by maturity (Michaud and Upper (2008), Gefang et al (2010)).
Page 32 of Lecture 3
3.4.3 Inflation-linked bonds
• Inflation-linked bonds are designed to help protect investors from the negative
impact of inflation by contractually linking the bonds’ principal and interest
payments to a nationally recognized inflation measure such as the Retail
Price Index (RPI) in the UK, the European Harmonised Index of Consumer
Prices (HICP) ex-tobacco in Europe, and the Consumer Price Index (CPI) in
the U.S
Source: World Bank Group
Page 33 of Lecture 3
Nominal and Inflation-linked Bonds
• Inflation-linked bonds have their payments scaled with inflation. For example,
a one-year inflation-linked bond pays 100(1 + ⇡) dollars or 100 real (adjusted
for inflation) dollars next year, where ⇡ is the realized inflation over the next
year
• The interest rates derived from nominal (conventional) and inflation-protected
bonds are often called nominal and real interest rates, respectively
? Example: Suppose a nominal bond pays 100 nominal dollars next year and
trades at $95 today. The nominal rate is
100
95 =
) i = 5.26%
1+i
Suppose an inflation-linked bond pays 100 real dollars next year (100(1+⇡)
nominal dollars) and trades at $98 today. The real rate is
98 =
100
) r = 2.04%
1+r
Page 34 of Lecture 3
Break-Even Inflation
• How do the nominal and real rates relate to the inflation rate?
• Note that the nominal return on the inflation-linked bond depends on the
inflation rate:
(1 + r)(1 + ⇡)
• The break-even inflation rate ⇡BR is defined by
1 + i = (1 + r)(1 + ⇡BR)
• When inflation is equal to the break-even inflation rate the returns on nominal
and real bonds are the same
Page 35 of Lecture 3
Page 36 of Lecture 3
• Since investors can invest in both type of bonds we should expect the
expected returns on nominal and inflation-linked bonds to be similar
• If the expected returns from investing in nominal and real bonds are the same
then E(⇡) = ⇡BR
• In general, inflation-linked and nominal bonds can have di↵erent expected
returns because investor might prefer one type of bonds over the other
? For example, suppose investors do not like inflation risk and would like to
have a guaranteed real return. In this case, they would invest in nominal
bonds only if their expected returns are higher than that of real bonds,
that is
(1 + r)(1 + ⇡BR) = 1 + i > (1 + r)(1 + E(⇡)) ) E(⇡) < ⇡BR
• Thus, ⇡BR does not have to be equal to E(⇡) but it is an indication of E(⇡)
Page 37 of Lecture 3
Main Takeaways
• Fixed income markets are central to efficient working of all other markets
• The expectations hypothesis is one of the well known theories of the term
structure
? The consensus is that the expectations hypothesis fails in the U.S. data.
Its failure though, is less strong or mixed in non-U.S. data
• Monetary policy and inflation are the main factors that a↵ect the term
structure of interest rates
• Next time: how to manage interest rate risk
Page 38 of Lecture 3
Lecture Notes
FM 423: Asset Markets
Lecture 4
Risk Management
Igor Makarov
London School of Economics
Page 2 of Lecture 4
Lecture 4: Main Points
1. Interest Rate Risk
2. Duration and Convexity
3. Duration and Convexity Hedging
Page 3 of Lecture 4
4.1
Interest rate risk
• Interest rates move substantially over time
• Bond prices are sensitive to interest rate movements. They go down when
interest rates go up, and vice-versa
• Therefore, the net worth of many market participants is sensitive to interest
rate movements
? Bond mutual funds
? Pension funds
? Banks
? Insurance companies
Page 4 of Lecture 4
Asset–liability Mismatch
• Banks
? Assets: Loans. Long term
? Liabilities: Demand deposits. Short term
? Net worth decreases when interest rates go up
• Pension funds:
? Assets: Fixed-income securities, stocks, etc
? Liabilities: Future pensions to be paid to employees. Long term
? If assets consist of short maturity bonds, net worth decreases when interest
rates go down
• Insurance companies:
? Similar asset and liability structure to pension funds
Page 5 of Lecture 4
• Case study: The Equitable Life Assurance Society (Equitable Life)
? 1762: Equitable Life, the world’s oldest mutual insurer, is founded
? 1913: Equitable Life starts selling pensions
? 1955 – 1988: Equitable Life sells policies which guarantee investors a
minimum annuity rate when they retire
? 1993: The “unthinkable” happens: inflation is barely there. The sold
policies become expensive to honour due to falls in interest rates
Page 6 of Lecture 4
• 2000: Near-collapse of Equitable Life in 2000. Following a July 2000 House
of Lords ruling, and the failure of attempts to find a buyer for the business,
it closed to new business in December 2000 and reduced payouts to existing
members. The estimated investors’ absolute loss was between £2.9B and
£3.7B
Page 7 of Lecture 4
• Example: You are the CFO of a pension fund. Your liabilities consist of
$100M payments in perpetuity starting from year one. Your assets consist of
$2.1B in cash. The term structure is flat at 5%
• The PV of your liabilities is L =
A
L = $2.1B
$100M
5%
= $2B ) your net worth is
$2B = $100M
• Suppose you keep your assets in cash and suppose the term structure shifts
to 4%
• The PV of your liabilities becomes L =
becomes
A
L = $2.1B
$2.5B =
$100M
4%
$400M
= $2.5B ) your net worth
)
Page 8 of Lecture 4
PV
3.0
2.5
A
2.0
L
1.5
3
4
5
r (%)
6
7
Page 9 of Lecture 4
• Suppose you invest all cash in 30-year bonds
• Now if the term structure shifts to 4% the value of your assets becomes
$2.1B ⇥
1 + 5%
1 + 4%
!30
= $2.79B > $2.5B
,
• But suppose the term structure shifts to 6%. Then the value of your assets
becomes
$2.1B ⇥
1 + 5%
1 + 6%
!30
= $1.58B
• The new value of your liabilities is L =
$100M
6%
= $1.66B > $1.58B
/
Page 10 of Lecture 4
• In the previous example, the problem arises because assets and liabilities have
di↵erent sensitivity to interest rate movements
• Therefore, it is important to develop tools for measuring interest rate
sensitivity
• One complication is that unlike in the example, the term-structure is usually
not flat so prices of fixed-income securities depend on a multitude of interest
rates
• To develop ideas, we start with the simplest model for interest-rate risk
management known as duration hedging, which is based on a single risk
variable, the yield-to-maturity
• Duration hedging is still very often used in practice
Page 11 of Lecture 4
4.2
Duration
• The first step consists in writing the price of a portfolio as a function of a
single variable, its yield-to-maturity, y :
P (y) =
C1
C2
CT
+
+
·
·
·
+
1+y
(1 + y)2
(1 + y)T
• Recall that in general, YTM is a complicated average of the spot rates
corresponding to the years 1, .., T . If the term-structure is flat then y = r
• A second step involves the derivation of a Taylor expansion of the value of
the portfolio as an attempt to quantify the magnitude of value changes that
are triggered by small changes y in yield:
P (y) = P (y +
y)
P (y) ⇡ P 0(y)
y,
where P 0(y) is the derivative of the price with respect to y
Page 12 of Lecture 4
3.0
2.5
2.0
DP»aDy
a
Dy
P(y)
1.5
3
4
5
y
y+Dy
6
7
Page 13 of Lecture 4
• In risk management, P 0(y) is known as the dollar duration of the
portfolio (often denoted as DV01 or BPV)
• The percentage change in value of the portfolio is
P (y)
P 0(y)
⇡
P (y)
P (y)
D ⇤(y) =
P 0 (y)
P (y)
y,
is the modified duration of the portfolio
• The modified duration tells us what happens to the value of a dollar invested
in the portfolio for a unit change in yield
• The Macaulay duration, D(y), is defined as
D(y) = (1 + y)D ⇤(y)
Page 14 of Lecture 4
• The main reason to use the Macaulay duration is that it is easy to compute
• Fact: for any T -year zero-coupon bond, the Macaulay duration is T :
100
P (y) =
) P 0 (y) =
T
(1 + y)
T 100
) D(y) =
(1 + y)T +1
P 0 (y)
(1 + y)
=T
P (y)
• If we know the Macaulay duration we can easily compute the modified
duration
D ⇤(y) = D(y)/(1 + y)
Page 15 of Lecture 4
• Suppose we have a portfolio consisting of two securities A and B
Security Value
A
B
Modified
duration
VA
VB
⇤
DA
⇤
DB
• What is the modified duration of the portfolio?
• Answer:
⇤
⇤
D ⇤ = w1 ⇥ D A
+ w2 ⇥ D B
where
w1 =
VA
VA + VB
and w2 =
VB
VA + VB
are the portfolio weights. They represent the share of each security in the
portfolio and sum to 1
Page 16 of Lecture 4
• Proof: Suppose the interest rate changes by
y
• The approximate change in the value of the portfolio is
⇤
VA ⇥ DA
⇥
y
⇤
VB ⇥ DB
⇥
y=
⇤
⇤
+ VB ⇥ DB
(V A ⇥ D A
)⇥
y
• Hence, the percentage change in the value of the portfolio is
⇤
⇤
(w1 ⇥ DA
+ w2 ⇥ D B
)⇥
y
• Therefore, the modified duration of the portfolio is
⇤
⇤
D ⇤ = w1 ⇥ D A
+ w2 ⇥ D B
• Since the Macaulay duration is a multiple of the modified duration
D = w1 ⇥ D A + w2 ⇥ D B
• We can use the last observation to compute the Macaulay duration of a
coupon bond
Page 17 of Lecture 4
• Consider a bond with annual coupon rate c% and T years to maturity
• The Macaulay duration of the bond is
D(y) =
T
X
wtt,
t=1
where
c
1
wt =
(1 + y)t P
for
t = 1, .., T
1,
100 + c 1
wT =
(1 + y)T P
and P is the bond price.
• The Macaulay duration is a weighted average of the years in which the bond
pays cash flows
• The weight of a given year is the PV of that year’s cash flow divided by the
PV of all cash flows. The latter PV is the price of the bond
Page 18 of Lecture 4
• Example: Bond with annual coupon rate 6% and 20 years to maturity and
YTM y = 5%
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
disc. factor cash flow
0.9524
6
0.9070
6
0.8638
6
0.8227
6
0.7835
6
0.7462
6
0.7107
6
0.6768
6
0.6446
6
0.6139
6
0.5847
6
0.5568
6
0.5303
6
0.5051
6
0.4810
6
0.4581
6
0.4363
6
0.4155
6
0.3957
6
0.3769
106
DF*CF
5.7143
5.4422
5.1830
4.9362
4.7012
4.4773
4.2641
4.0610
3.8677
3.6835
3.5081
3.3410
3.1819
3.0304
2.8861
2.7487
2.6178
2.4931
2.3744
39.9503
112.46
Price
weight
0.0508
0.0484
0.0461
0.0439
0.0418
0.0398
0.0379
0.0361
0.0344
0.0328
0.0312
0.0297
0.0283
0.0269
0.0257
0.0244
0.0233
0.0222
0.0211
0.3552
weight*t
0.0508
0.0968
0.1383
0.1756
0.2090
0.2389
0.2654
0.2889
0.3095
0.3275
0.3431
0.3565
0.3678
0.3772
0.3849
0.3911
0.3957
0.3990
0.4011
7.1047
12.62
D in years
Page 19 of Lecture 4
• The Macaulay duration is, in a sense, the e↵ective maturity of the bond
• Properties:
? D always decreases with coupon rate
? D generally increases with time to maturity (but not always)
• For a small change in YTM
approximately
P (y) ⇡
P (y)D ⇤(y)
y , the change in the portfolio’s value is
y
Page 20 of Lecture 4
• Example: Bond with annual coupon rate 6% and 20 years to maturity and
term-structure is flat at r = 5% ) Y T M = y = r = 5%
• Price of the bond is P (y) = 112.46
• Macaulay duration is 12.62, and modified duration is 12.62/(1 + 5%) =
12.02
• Suppose that the term-structure goes up to r +
? Price of the bond becomes P (r +
price is
P = 111.12
112.46 =
r = 5.1%
r) = 111.12. Exact change in bond
1.34
? Approximate change in bond price is
P ⇡
P D⇤
r=
112.46 ⇥ 12.02 ⇥ 0.1% =
1.35
Page 21 of Lecture 4
• Now suppose the term-structure goes down to r +
Exact change:
Approximate change:
r = 4.9%
P = 1.36
P ⇡ 1.35
• For small shifts in the term-structure, duration provides a good approximation
to the actual change
• Now suppose the term-structure change is
down to 4.5%).
Exact change:
P =
r = 0.5% (up to 5.5% or
6.49 for an upward shift
P = 7.05 for a downward shift
Approximate change:
P ⇡
6.76 for an upward shift
P ⇡ 6.76 for a downward shift
• For large shifts in the term-structure, duration provides imprecise approximation to the actual change
Page 22 of Lecture 4
250
Exact
Approximation
200
Price
150
100
50
0
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
Interest Rate
• Duration model:
? Understates the capital gain if interest rates go down
? Overstates the capital loss if interest rates go up
• Approximation error depends on the curvature of the price function
Page 23 of Lecture 4
4.3
Convexity
• To account for large shifts in the term-structure we need to consider a
second-order term in a Taylor expansion:
P (y) ⇡ P 0(y)
1
y + P 00(y)(
2
y)2
or
P (y)
P 0(y)
⇡
P (y)
P (y)
1 P 00(y)
y+
(
2 P (y)
y)2
00
(y)
• The term
= PP (y)
is known as convexity of portfolio P and the term
P 00(y) as dollar convexity
• Convexity is related to the curvature of the price function
• Convexity of a zero-coupon bond with maturity T is
T (T +1)
(1+y)2
Page 24 of Lecture 4
• Convexity of a bond with annual coupon rate c% and T years to maturity is
T
X
1
=
wtt(t + 1),
(1 + y)2 t=1
where the weights wt are as in the definition of duration
• We can plot the exact price, the approximate price obtained using duration,
and the approximate price obtained using duration and convexity:
250
Exact
Approx - D
Approx - D and Gamma
200
Price
150
100
50
0
0%
1%
2%
3%
4%
5%
6%
Interest Rate
7%
8%
9%
10%
Page 25 of Lecture 4
4.4
Duration hedging (immunization)
• In finance, hedging is known as the process that reduces the risk of an
investment
• Duration hedging refers to the process of creating a global portfolio that
has zero duration
• Example: You are the CFO of a pension fund. Your liabilities consist of
$100M payments in perpetuity starting from year one. Your assets consist of
$2.1B in cash. The term structure is flat at 5%
• What is the modified duration of liabilities, DL⇤ ?
C
L(r) =
) L0(r) =
r
C
⇤
)
D
(r) =
L
r2
L0(r)
1
1
= =
= 20
L(r)
r
5%
• Note that even though payments extend to infinity, their duration is close to
that of a twenty-year zero-coupon bond
Page 26 of Lecture 4
• Suppose there are two zero-coupon bonds with maturities 20 and 30 years,
and suppose we want to allocate $2B between these two bonds to create
a portfolio that hedges liabilities exposure to interest rates. How can we
achieve this?
) The Macaulay durations are 20 and 30, and the modified durations are
D1⇤ = 20/(1 + 5%) = 19.05
D2⇤ = 30/(1 + 5%) = 28.56
• Suppose that we invest $x1 in the first bond and $x2 in the second bond
• Then, the value of the portfolio is
x1 + x2
• The dollar duration of the portfolio is
x1D1⇤ + x2D2⇤
Page 27 of Lecture 4
• We need to match the dollar duration of the portfolio with that of liabilities.
Therefore, it must be that
x1D1⇤ + x2D2⇤ = LDL⇤
• Since we want to allocate A = $2B to the hedging portfolio we have:
x 1 + x2 = A
• Therefore, x1 and x2 must satisfy
x1 + x2 = 2B
19.05x1 + 28.56x2 = 40B
• Two equations and two unknowns ) solving this system, we get x1 = $1.8B
and x2 = $0.2B
Page 28 of Lecture 4
• Exercise: How to construct a hedging portfolio if we decide to allocate to
the hedging portfolio $1.5B instead of $2B?
Solution:
? Now x1 and x2 must satisfy
x1 + x2 = 1.5B
19.05x1 + 28.56x2 = 40B
? ) x1 = $0.3B and x2 = $1.2B
? Comparing to the previous case, we now need to invest a higher amount
in the 30-year bond. Why?
Page 29 of Lecture 4
The UK pension reform of 2004
• The pension reform gave the power to pension regulators to take over funds
that were perceived to be at risk of not meeting their obligations
• One of the criteria used by the regulator to determine whether intervention
was necessary was a plan’s “accounting deficit,” the di↵erence between the
market values of a plan’s assets and its liabilities
• The reform created strong incentives for funds to manage interest rate risk
and led to an increased demand for long-term government bonds, which in
turn, led to a significant decline in yields of long-term bonds
PENSION REFORM AND THE TERM
STRUCTURE
2.50
2055 Bond
2035 Bond
2009 Bond
2016 Bond
Yield (% Real)
2.00
1.50
1.00
0.50
0.00
Dec-02
Dec-03
Source: Greenwood-Vayanos (AER 2010).
Dec-04
Dec-05
Dec-06
Page 30 of Lecture 4
4.5
Duration-convexity hedging⇤
• Matching duration of assets and liabilities ensures us against small changes
in the term-structure. If we want to create a hedge against large shifts we
need to match both the dollar duration and dollar convexity of assets and
liabilities
• Note that with only two bonds we cannot match the investment amount,
the duration and convexity because we would have two unknowns and three
equations
• Hence, for the duration-convexity hedging we need at least three securities
• Therefore, suppose that in addition to 20 and 30-year bonds we can also
invest in the 10-year zero-coupon bond
• The duration of the 10-year bond is D3⇤ = 10/1.05 = 9.52
Page 31 of Lecture 4
• Suppose as before, we would like to invest $2B in the three bonds. Suppose
we invest $x3 in the 10-year bond
• The convexity of liabilities,
C
L(r) =
)
r
L
L
is
L00(r)
2
2
=
= 2=
= 800
2
L(r)
r
(5%)
• The bond convexities are
1
2
3
= 20 ⇥ 21/(1 + 5%)2 = 380.95
= 30 ⇥ 31/(1 + 5%)2 = 843.54
= 10 ⇥ 11/(1 + 5%)2 = 99.77
Page 32 of Lecture 4
• Therefore, x1, x2, and x3 must satisfy
x1 + x2 + x3 = 2B
19.05x1 + 28.56x2 + 9.52x3 = 40B
380.95x1 + 843.54x2 + 99.77x3 = 800 ⇥ 2B = 1.6T
• Three equations and three unknowns ) solving this system, we get x1 =
$6.42B , x2 = $4.31B , x3 = $4.11B
Page 33 of Lecture 4
Main Takeaways
• Duration and convexity are the main measures of sensitivity of bond prices
to changes in the interest rate
? Duration provides a good approximation to the actual change for small
shifts in the term-structure
? Approximation error depends on the convexity of the bond
• Duration hedging (immunization) is the process of matching the duration of
assets and liabilities
• Next time: Stocks
Page 34 of Lecture 4
Appendix: Semiannual Coupon Payments
• Consider a bond with semiannual coupon rate c% and T years to maturity
• Assume that r is quoted as a semiannual APR
• The Macaulay duration of the bond is
2T
X
t
D=
wt
2
t=1
where
c
1
2
wt =
for t = 1, .., 2T
tP
r
1+ 2
1
c
100 + 2
1
w2T =
r 2T P
1+ 2
and P is the bond price
• The modified duration is
D
D⇤ =
r
1+ 2
• The convexity of the bond is
2T
X
1
=
wt t(t + 1),
2
r
4 1+ 2
t=1
where the weights wt are as for the Macaulay duration
Lecture Notes
FM 423: Asset Markets
Lecture 5
Stocks
Igor Makarov
London School of Economics
Page 2 of Lecture 5
General Overview
• Lecture 5 (Statistical Facts on Stock Returns):
? Introduction, and basic facts on stock returns
• Lecture 6 (Portfolio Theory):
? How to choose a stock portfolio
• Lectures 7 and 8 (The Capital Asset Pricing Model (CAPM)):
? How are expected returns determined? How are they related to risk?
• Lecture 9 (Valuation of Stocks):
? The constant growth formula
Page 3 of Lecture 5
Lecture 5: Main Points
1. Basic Facts about Stocks
2. Basic Statistics
3. Return on a Portfolio
4. Expected Return and Variance of a Portfolio
5. Benefits of Diversification
Page 4 of Lecture 5
5.1
Introduction
• A stock is a claim to share in the net income and the assets of a business
• Main features of common stock:
? Voting rights
? Rights to dividends
? Limited liability
• Dividends are the cash flows of common stock. They are:
? discretionary. By contrast, debt payments are mandatory
? not tax-deductible (because they are not business expenses). By contrast,
debt payments are tax-deductible
? generally paid quarterly
Page 5 of Lecture 5
Computing stock returns between two dates, 0 and 1
• Invest X dollars in the stock at date 0
• If there are any dividend payments between dates 0 and 1, reinvest them
immediately in the stock
• Suppose that value of the stock investment at date 1 is P dollars
• Return on the stock r is defined as
r=
P
X
X
Page 6 of Lecture 5
A Simple Formula
We will frequently assume that the only dividend payment between dates 0 and
1 is at date 1
• With X dollars, buy x = X/P0 shares of the stock at date 0, where P0 is
the date 0 price
• At date 1, these x shares are worth xP1, where P1 is the date 1 price.
Moreover, they pay a dividend xD1, where D1 is the date 1 dividend
• Value of the stock investment in period 1 is x(P1 + D1)
• Return on the stock is
r=
x(P1 + D1)
X
X
=
D1 + (P1
P0
P0 )
.
• Return is due to dividend payments and capital gains
INVESTING ESSENTIALS
Page 7 of Lecture 5
otson SBBI
®
®
Stocks,
Bonds, 1926–2021
Bills, and Inflation 1926–2021
s, Bonds, Bills,
and Inflation
t?
e financial goals,
secure retirement
for a college
, investing makes
you can see here
wth of $1 over the
a
a a
a to
nt bonds, and
ills should all
ce in a properly
on t
t strategy.
$100k
$56,034
Compound annual return
$14,086
Small stocks
12.1%
Large stocks
10.5
Government bonds 5.5
Treasury bills
3.3
Inflation
2.9
10k
1k
$177
100
$22
$16
10
1926
1936
1946
1956
1966
1976
1986
1996
2006
Trump impeached
by the House
Brexit Referendum
Emergency Economic
Stabili ation Act
Wall St. Reform Act
U.S. Credit Downgrade
Detroit Bankruptcy
Gramm-Leach Bliley Act
September 11
Start of Gulf War
Stock Market Crash
Start of low in ationary period
Gold window closed
Arab oil embargo
May Day, the deregulation
of brokerage fees
Tet offensive in Vietnam
JFK assassinated
Sputnik launched
U.S. Treasury-Federal
Reserve Accord
General Agreement
on Tariffs and Trade
Pearl Harbor
Glass-Steagall Act
Securities Exchange Act
0
Stock Market Crash
1
2016
Past performance is no guarantee of future results.
PastIbbotson
performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 1926. Assumes reinvestment of income and no transaction costs or taxes. This is
Source:
for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar 2022 and Precision Information, dba Financial
Fitness Group 2022. All Rights Reserved.
Not FDIC/NCUA Insured
No Bank Guarantee
Not a Deposit
May Lose Value
Not Insured by Any Government Agency
Page 8 of Lecture 5
Source: Ibbotson
Page 9 of Lecture 5
Monthly Bond Returns
0.2
0.1
0
-0.1
-0.2
1942
1952
1962
1972
1982
1992
2002
2012
2002
2012
Monthly Stock Returns
0.2
0.1
0
-0.1
-0.2
1942
1952
1962
1972
1982
1992
Page 10 of Lecture 5
• Histogram: sort all obs into bins; plot the number of obs in each bin
Stock returns
Bond returns
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Page 11 of Lecture 5
Basic facts
• Historically, stocks delivered higher returns than bonds over the long run
• Stock returns are more volitile than bond returns. Volatility risk refers to
how much the value of an asset fluctuates up and down
) Riskier investments have higher returns on average
Page 12 of Lecture 5
5.2
Basic Statistics
• Suppose a random variable Z takes values Z1, ..., ZK , with probabilities
p1, ..., pK
• The expectation (mean) of Z is
E(Z) = p1Z1 + · · · + pK ZK
• The variance of Z is
V ar(Z) = p1(Z1
E(Z))2 + · · · + pK (ZK
and the standard deviation of Z is
(Z) =
q
V ar(Z)
E(Z))2
Page 13 of Lecture 5
• Example: Suppose we have the following information about two stocks
State of economy Probability Stock A return Stock B return
Boom
0.2
0%
30%
Normal Growth
0.5
5%
10%
Recession
0.3
5%
-10%
• Calculate the expected return on stocks A and B
Solution:
ERA = 0.5 ⇥ 5% + 0.3 ⇥ 5% = 4%
ERB = 0.2 ⇥ 30% + 0.5 ⇥ 10%
0.3 ⇥ 10% = 8%
Page 14 of Lecture 5
• Calculate the standard deviation of stocks A and B
Solution:
A
=
B
=
q
0.2 ⇥ (4%)2 + 0.5 ⇥ (1%)2 + 0.3 ⇥ (1%)2 = 2%
q
0.2 ⇥ (22%)2 + 0.5 ⇥ (2%)2
0.3 ⇥ (18%)2 = 14%
Page 15 of Lecture 5
5.2.1 Empirical estimates
• In practice, we often do not know the exact distribution of variables of interest
but given their realization we can estimate their distribution and statistics
• Suppose we have N realizations of a random variable Z : X1, ..., XN . Then
we can estimate E(Z) using the sample mean
X1 + · · · + XN
X=
N
• To estimate V ar(Z) and (Z) we can use the sample variance
2
s(X) =
(X1
X)2 + · · · + (XN
N 1
and the sample standard deviation
s(X) =
q
s(X)2
X)2
Page 16 of Lecture 5
• The return next year can be viewed as a random variable. If the next year
return generating process is the same as in the past then we can estimate its
? mean (the expected return) by the sample average
? variance by the sample variance
? standard deviation by the sample standard deviation
• Example (Excel)
Period
IBM
GE
sample average sample std sample average sample std
1960-1989
8.9%
27.6%
11.2%
26.7%
1990-2020
11.3%
22.4%
10.8%
20.4%
• Note that empirical estimates of the mean, variance, standard deviation take
di↵erent values for di↵erent realizations, and therefore themselves are random
variables
Page 17 of Lecture 5
• What happens to the sample statistics (e.g., the mean or standard deviation)
when the sample size increases?
? As the sample size increases the sample statistics converge to their probability counterparts (Law of Large Numbers)
? The speed of convergence is given by the Central Limit Theorem. In
particular,
X
!
s(X)
E(Z) ⇠ N ormal 0, p
,
N
where N is the number of observations and Normal stands for the Normal
distribution
Page 18 of Lecture 5
Probability Density Function (PDF) of Normal Distribution N (µ, )
Location
Scale
Final PDF to printer
136
PART II
Portfolio Theory and Practice
Percentage of all obs
in a given interval
68.26%
95.44%
99.74%
23s
22s
21s
0
11s
12s
13s
250
230
210
10
30
50
70
Figure 5.4 The normal distribution with mean 10% and standard deviation 20%.
Page 19 of Lecture 5
• Example (Excel): The Market Risk Premium
• We want to know the expected return of large stocks relative to T-bills. We
refer to this as the market risk premium (MRP). We can obtain an estimate
of the MRP using sample averages
• The 1926-2020 sample averages for large stocks and T-bills are 11.6% and
3.4%, respectively. Therefore, an estimate of the MRP is
X = 11.6%
3.4% = 8.2%
• How precise is this estimate?
? 95% confidence interval is
s(X)
X ± 1.96 p
N
? We have s(X) = 20.26% and N = 95. Therefore, the 95% confidence
interval is [4.1%, 12.3%]
Page 20 of Lecture 5
Worldwide risk premium (%), 1900–2019
Source: Global Investment Returns Yearbook, Credit Suisse, 2020
Page 21 of Lecture 5
Realized vs. Expected Returns
• The 8.2% is (an estimate of) the expected return of large stocks relative to
T-bills
• We expect that large stocks will, on average, outperform T-bills by 8.2%
• This does not mean that in 2021 large stocks will outperform T-bills by 8.2%
for sure
• The return of large stocks relative to T-bills in 2021, will be the realized
return in 2021
Page 22 of Lecture 5
5.2.2 Individual Stocks vs. Indexes
Using monthly (annualized) returns 1980-2020, we can compute:
Sample Average Sample St. Dev.
Coca Cola
15.5
20.2
Disney
17.8
27.8
GM
2.8
34.6
IBM
10.9
25.4
Xerox
9.5
37.4
S&P500
9.8
15.0
• The sample standard deviation of the S&P500 is much smaller than those of
the individual stocks. By contrast, the sample average returns are comparable
• Hence, there are benefits to diversification. By holding a diversified portfolio,
we can reduce risk without sacrificing expected return
Page 23 of Lecture 5
5.2.3 Covariance and Correlation
• We want to measure the association between two random variables X and Y
• Two related measures of association are the covariance
Cov(X, Y ) = E(X
EX)(Y
EY )
and the correlation
⇢(X, Y ) =
Cov(X, Y )
(X) (Y )
• It can be shown that
Cov(↵X + Y, Z) = ↵Cov(X, Z) + Cov(Y, Z)
Page 24 of Lecture 5
• Example: Suppose we have the following information about two stocks
State of economy Probability Stock A return Stock B return
Boom
0.2
0%
30%
Normal Growth
0.5
5%
10%
Recession
0.3
5%
-10%
• Calculate the covariance between stocks A and B
Solution: Recall that ERA = 4% and ERB = 8%. Therefore,
Cov(RA , RB ) =
=
0.2 ⇥ 4% ⇥ 22% + 0.5 ⇥ 1% ⇥ 2%
0.0022
0.3 ⇥ 1% ⇥ 18%
• Calculate the correlation between stocks A and B
Solution: Recall that
⇢(RA, RB ) =
A
= 2% and
0.0022
=
0.02 ⇥ 0.14
B
= 14%. Therefore,
0.78
Page 25 of Lecture 5
• Example: Suppose (RA) = 10%, (RB ) = 15%, (RC ) = 30%,
⇢(RA, RB ) = 0.5, ⇢(RA, RC ) = 0. Find Cov(RA, 0.3RB + 0.7RC )
Solution:
Cov(RA, 0.3RB + 0.7RC ) = 0.3Cov(RA, RB ) + 0.7Cov(RA, RC )
= 0.3⇢(RA, RB ) (RA) (RB ) + 0.7⇢(RA, RC ) (RA) (RC )
= 0.3 ⇥ 0.5 ⇥ 10% ⇥ 15% = 0.00225
Page 26 of Lecture 5
Properties of Covariance and Correlation
• Covariance and correlation have the same sign. They are:
? positive, if X and Y tend to be high at the same time
? zero, if X and Y are unrelated (uncorrelated)
? negative, if X tends to be high when Y is low or vice versa
• Correlation is always a number between -1 and 1
• Correlation is:
? equal to 1 if there is an exact linear relation with positive slope between
X and Y (perfectly positively correlated)
? equal to -1 if there is an exact linear relation with negative slope between
X and Y (perfectly negatively correlated)
Page 27 of Lecture 5
Sample Covariance and Correlation
• If we have realizations of X and Y : X1, ..., XN , and Y1, ..., YN we can
estimate their covariance using sample covariance
s(X, Y ) =
⌃N
i=1 (Xi
X)(Yi
N 1
Y)
and their correlation using sample correlation
s(X, Y )
s(X)s(Y )
Page 28 of Lecture 5
• Example: Coca-Cola vs. S&P500: Monthly returns 1980-2020
Correlation of Stock Returns: A Scatterplot
0.2
Monthly returns on Coca Cola
0.15
‐0.25
0.1
0.05
0
‐0.2
‐0.15
‐0.1
‐0.05
0
0.05
‐0.05
‐0.1
‐0.15
‐0.2
‐0.25
Monthly returns on S&P 500
• Sample correlation is 0.48
0.1
0.15
0.2
0.25
Page 29 of Lecture 5
Correlation of Stock Returns
Using monthly returns 1980-2020, we can compute the following correlation
matrix:
Coca Cola
Coca Cola
1
GM
0.17
IBM
0.15
Disney
0.36
Xerox
0.19
Note that:
• Stocks are positively correlated
• Correlations can be large
GM IBM Disney Xerox
1
0.30 1
0.35 0.35
0.31 0.50
1
0.41
1
Page 30 of Lecture 5
5.3
Return on a Portfolio
• Consider a portfolio consisting of X1 dollars in Disney and X2 dollars in IBM
• The value of the portfolio at date 0 is X = X1 + X2
• The value of the portfolio at date 1 is
X1(1 + R1) + X2(1 + R2),
where R1 is the return on Disney and R2 the return on IBM between dates
0 and 1
• The return on the portfolio is
R=
X1(1 + R1) + X2(1 + R2)
X
X
=
X1
X2
R1 +
R2
X
X
Page 31 of Lecture 5
5.3.1 Portfolio Weights
• The portfolio weights are
w1 =
X1
X
and w2 =
X2
X
They represent the fraction of portfolio value invested in each stock. They
sum to 1
• The return on the portfolio is
R = w 1 R1 + w 2 R2
It is a weighted average of the returns on the individual stocks
• More generally, the return on a portfolio with N stocks is
R=
N
X
n=1
w n Rn ,
wn =
Xn
X
Page 32 of Lecture 5
• Example: Compute the return on a portfolio consisting of $300 in Disney
and $100 in IBM
• We have
w1 =
300
= 0.75
400
and
w2 =
100
= 0.25
400
• The return on the portfolio is
R = 0.75 ⇥ R1 + 0.25 ⇥ R2
Page 33 of Lecture 5
• Example: Start with the same amount of initial capital $400. Instead of
buying $100 of IBM, we sell short $100 of IBM. Compute the return on the
portfolio.
• What are short sales?
• How to compute the return on a portfolio that involves short sales?
Page 34 of Lecture 5
5.3.2 Portfolio Return with Short Sales
• Date 0: Get $100 from selling IBM. Invest $100 + $400 = $500 in Disney
• Date 1:
? Disney is worth $500(1 + R1)
? Pay $100(1 + R2) to buy IBM
? Portfolio value is
500(1 + R1)
100(1 + R2)
• Return on the portfolio is
R =
500(1 + R1)
= 1.25 ⇥ R1
100(1 + R2)
400
400
=
500
R1
400
100
R2
400
0.25 ⇥ R2
• Conclusion: Portfolio return is still weighted average of returns on the
individual stocks. However, weights of stocks that are sold short are
negative. The weights still sum to 1
Page 35 of Lecture 5
5.4
Expected Return and Variance of a Portfolio
Expected Return
• The expected return of a portfolio with N stocks is
E(R) =
N
X
wnE(Rn)
n=1
• It is a weighted average of the expected returns of the individual stocks
• In practice, we do not know the expected returns of the individual stocks,
but can estimate them using sample averages
Page 36 of Lecture 5
• Example: Compute (an estimate of) the expected return of a portfolio
consisting of $300 in Disney and $100 in IBM
• We have
w1 =
300
= 0.75
400
and
w2 =
100
= 0.25
400
• Estimating the expected returns E(R1) and E(R2) using the 1980-2020
data, we have
R1 = 17.9%
and
R2 = 10.9%
• The estimate of the expected return of the portfolio E(R) is
R = 0.75 ⇥ R1 + 0.25 ⇥ R2 = 16.1%
Page 37 of Lecture 5
• Example: Suppose ERA = 5% and ERB = 10%. Find a portfolio of stocks
A and B that has the expected return of 8%
Solution: Let w be the weight of stock A. Then w must solve
5% ⇥ w + 10% ⇥ (1
w) = 8%
)
w = 2/5
Page 38 of Lecture 5
5.4.1 Variance
• The variance of a portfolio depends not only on the variances of the individual
stocks, but also on their covariances
• It is greater when the covariances are positive rather than negative
• The variance of a portfolio with two stocks is
V ar(R) = w12V ar(R1) + w22V ar(R2) + 2w1w2Cov(R1, R2)
• More generally, the variance of a portfolio with N stocks is
V ar(R) =
N
X
n=1
wn2V
ar(Rn) + 2
X
n<m
wnwmCov(Rn, Rm)
Page 39 of Lecture 5
5.4.2 Standard Deviation
• The equations for portfolio variance can be written in terms of standard
deviation and correlation, rather than variance and covariance. Just replace
? V ar(Rn) by (Rn)2
? Cov(Rn, Rm) by ⇢(Rn, Rm) (Rn) (Rm)
• For instance, the standard deviation of a portfolio of two stocks is given by
(R) =
q
w12 (R1)2 + w22 (R2)2 + 2w1w2⇢(R1, R2) (R1) (R2)
• In practice, we do not know the standard deviations and correlations of the
individual stocks, but can estimate them using sample standard deviations
and correlations
Page 40 of Lecture 5
• Example: Compute (an estimate of) the standard deviation of a portfolio
consisting of $300 in Disney and $100 in IBM
• We have
w1 =
300
= 0.75,
400
w2 =
100
= 0.25.
400
• Estimating the standard deviations (R1) and
⇢(R1, R2) using the 1980-2020 data, we get
(R1) = 34.6%,
(R2) = 25.4%,
(R2), and correlation
⇢(R1, R2) = 0.35
• The estimate of the standard deviation of the portfolio is given by
(R)2 = (0.75)2 (R1)2 + (0.25)2 (R2)2
+2(0.75)(0.25)⇢(R1, R2) (R1) (R2)
) (R) = 28.8%.
Page 41 of Lecture 5
5.5
Benefits of Diversification
The table below represents the estimates of the expected return and standard
deviation of a Disney/IBM portfolio, as we vary the weight, w1, on Disney
w1
0.0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E(R) 10.9% 11.6 12.3 13.0 13.7 14.4 15.1 15.8 16.5 17.2 17.9
(R) 25.4 24.3% 23.7 23.5 23.9 24.8 26.1 27.8 29.8 32.1 34.6
• We can represent the previous information graphically
19.00%
17.90%
Expected Return
17.00%
15.00%
13.00%
11.00%
9.00%
15.00%
10.90%
20.00%
25.00%
30.00%
35.00%
Standard Deviation
40.00%
45.00%
50.00%
Page 42 of Lecture 5
• The key observation from the table and the graph is that diversification can
reduce risk substantially
• A diversified portfolio can have lower risk because the individual stocks do
not always move together
• A second observation from the table and the graph is that diversification
does not necessarily reduce the expected return
? By adding Disney to the IBM portfolio, we can simultaneously raise the
expected return and lower the standard deviation
Page 43 of Lecture 5
Main Takeaways
• Stocks are riskier investments than bonds
• Riskier investments deliver higher returns
• Diversification can reduce risk substantially without sacrificing the expected
return
• Next time: how to choose an optimal portfolio of stocks
Page 44 of Lecture 5
Lecture Notes
FM 423: Asset Markets
Lecture 6
Portfolio Theory
Igor Makarov
London School of Economics
Page 2 of Lecture 6
Lecture 6: Main Points
Big question: How to choose a stock portfolio?
1. Mean-Variance Optimization and the Portfolio Frontier
2. Portfolio Frontier with Risky Assets Only
3. Limits of Diversification: Systematic vs. Idiosyncratic risk
4. Portfolio Frontier with a Riskless Asset
5. An Important Property of the Tangent Portfolio
Page 3 of Lecture 6
6.1
Mean-Variance Optimization and the Portfolio Frontier
• The big question: How to choose a stock portfolio?
• This question can be addressed in two steps
• Step 1: Among all portfolios that have a given expected return, which is the
portfolio with the minimum variance?
? Step 1 will give us a set of portfolios, one for each expected return
? This set is the portfolio frontier (PF). Its elements are the frontier
portfolios
? We only need to consider portfolios on the PF. (Assuming that we care
only about mean and variance)
• Step 2: Which is the best portfolio on the PF?
? The answer to Step 2 depends on how we trade o↵ risk and return
Page 4 of Lecture 6
6.1.1 Choosing a Global Portfolio
• To illustrate mean-variance optimization and the PF, we consider the problem
of choosing a global portfolio using historical data
• The “individual stocks” are indices from the largest seven stock markets as
of June 2008. These indices are constructed by Morgan Stanley Capital
International (Source: MSCI, August 2008)
Percentage of World
Market Capitalization
US
29.9%
Japan
8.2
UK
6.8
China
5.4
France
4.4
Hong Kong
4.3
Canada
3.7
Total
62.7
Page 5 of Lecture 6
6.1.2
Statistical Properties of Returns
• Using monthly returns 1/1970-7/2008, we can compute:
Sample Average Sample St. Dev.
Canada
13.2%
19.0%
China
6.5
38.1
France
14.4
22.0
Hong Kong
23.7
36.2
Japan
13.2
21.8
UK
14.0
22.2
US
11.4
15.1
Page 6 of Lecture 6
Sample Correlations
Can
Canada
1
China
0.49
France
0.48
Hong Kong 0.38
Japan
0.33
UK
0.52
US
0.71
CN
Fra HK Jap UK US
1
0.31
0.65
0.17
0.38
0.43
1
0.31 1
0.40 0.31 1
0.58 0.39 0.37 1
0.50 0.36 0.31 0.54 1
Page 7 of Lecture 6
6.2
Portfolio Frontier with Risky Assets Only
The Optimization Problem
• Among all portfolios that have a given expected return (call it E ), which is
the portfolio with the minimum variance?
• Choose portfolio weights wn, n = 1, .., N , to minimize
V ar(R) =
N
X
wn2V
ar(Rn) + 2
n=1
subject to
N
X
wn = 1
n=1
and
E(R) =
N
X
n=1
wnE(Rn) = E
X
n<m
wnwmCov(Rn, Rm)
Page 8 of Lecture 6
Outline
We will construct three portfolio frontiers (PF)
• PF of two countries, US and Japan
• PF of US and Japan, but with short sales
• PF of all countries
Page 9 of Lecture 6
6.2.1 Portfolio Frontier with Two Assets
• With two assets, the PF can be determined very easily. Any portfolio is
a frontier portfolio, because there is no other portfolio having the same
expected return
Portfolio Frontier: US/Japan
14%
Expected Return
13%
Japan
Minimum
Variance
12%
US
11%
10%
9%
10%
12%
14%
16%
18%
Standard Deviation
20%
22%
24%
Page 10 of Lecture 6
6.2.2 Portfolio Frontier with Short Sales
• Short sales expand the PF. Expected returns above Japan or below US can
be achieved. PF becomes a hyperbola
Portfolio Frontier with Short Sales: US/Japan
14%
Japan
Expected Return
13%
12%
US
11%
10%
9%
10%
12%
14%
16%
18%
Standard Deviation
20%
22%
24%
Page 11 of Lecture 6
6.2.3 Portfolio Frontier with More than Two Assets
• With more than two assets, not all portfolios are frontier portfolios. To
determine a frontier portfolio, we need to solve the optimization problem.
This can be done in Excel
Page 12 of Lecture 6
6.2.4 Portfolio Frontier of All Countries
• Not all portfolios are frontier portfolios. In particular, none of the countries,
nor the equally weighted portfolio, are on the PF
Global Portfolio Frontier
25%
Hong Kong
Expected Return
20%
France
UK
Equally Weighted
Japan
Canada
15%
US
10%
China
5%
0%
0%
5%
10%
15%
20%
25%
Standard Deviation
30%
35%
40%
Page 13 of Lecture 6
• Adding assets shifts the PF to the left. The variance that can be achieved
for a given expected return decreases
Comparing Portfolio Frontiers
25%
Hong Kong
Expected Return
20%
France
15%
UK
Equally Weighted
Canada
Japan
US
10%
China
5%
0%
0%
5%
10%
15%
20%
25%
Standard Deviation
30%
35%
40%
Page 14 of Lecture 6
6.3
Limits of Diversification: Systematic vs. Idiosyncratic
Risk
• Adding assets reduces the variance that can be achieved for a given expected
return
• This is simply another way to say that diversification reduces risk
• Question: How much can risk be reduced? Can it be reduced to zero, by
adding very many assets?
Page 15 of Lecture 6
Theory ..
• Consider an equally weighted portfolio of assets, and assume that: (1) all
assets have the same standard deviation, (2) all assets are equally correlated
with each other (correlation ⇢)
1
sigma(Portfolio)/sigma(Asset)
0.8
0.6
rho=0.2
0.4
0.2
rho=0
0
0
50
100
Number of Assets
150
200
Page 16 of Lecture 6
Supporting calculations*
• General formula
v
u N
uX
X
t
2
2
(R) =
wn (Rn) + 2
wnwm⇢(Rn, Rm) (Rn) (Rm)
n<m
n=1
• Suppose we have N assets. (1) all assets have the same standard deviation
( ), (2) all assets are equally correlated with each other (⇢). Compute the
standard deviation of the equally-weighted portfolio
Solution:
P
v
u N
uX
=t
n=1
2
N2
+2
X ⇢
n<m
2
N2
=
s
1
N 1
+⇢
N
N
Page 17 of Lecture 6
.. and some Evidence
• Consider an equally weighted portfolio of randomly selected NYSE stocks
Number of Stocks
in Portfolio
Standard Deviation
of Portfolio
1
2
4
8
20
50
200
500
1000
49.2%
37.4
29.7
25.0
21.7
20.2
19.4
19.2
19.2
Ratio of Portfolio Std. Dev.
to Std. Dev. of a Single
Stock
1.00
0.76
0.60
0.51
0.44
0.41
0.39
0.39
0.39
Source: Statman, Meir, 1987, How many stocks make a diversified portfolio?
Journal of Financial and Quantitative Analysis 22, 353-364.
• This is consistent with the theory, assuming that the average correlation is
around 0.2
Page 18 of Lecture 6
6.3.1 Systematic vs. Idiosyncratic Risk
Consider a group of assets.
• Systematic Risk: Risk which a↵ects all assets
? If, for instance, the assets are US stocks, systematic risk corresponds to
events a↵ecting the US economy
• Idiosyncratic Risk: Risk which a↵ects only one asset
? For US stocks, idiosyncratic risk corresponds to events a↵ecting only the
particular company or industry
• Diversification within the group of assets reduces, and eventually eliminates,
idiosyncratic risk
• However, it cannot reduce systematic risk
• Diversification outside the group of assets (if it is possible) is more e↵ective
in reducing risk
Page 19 of Lecture 6
6.4
Portfolio Frontier with a Riskless Asset
The Optimization Problem
• We only need to choose the weights of the risky assets, wn, n = 1, .., N .
The weight of the riskless asset is
1
N
X
wn
n=1
• The variance of the riskless asset is 0, as is its covariance with all risky assets
• Optimization problem: Choose weights wn, n = 1, .., N , to minimize
V ar(R) =
N
X
wn2V
ar(Rn) + 2
n=1
X
wnwmCov(Rn, Rm)
n<m
subject to
E(R) =
N
X
n=1
wnE(Rn) + (1
N
X
n=1
wn)Rf = E
Page 20 of Lecture 6
6.4.1 Portfolios of a Riskless and a Risky Asset
• To solve the optimization problem, we first consider portfolios of the riskless
asset with only one risky asset
• We denote by w the weight on the risky asset, and by Rr the risky asset’s
return
• The return on the portfolio is
R = wRr + (1
w)Rf
• The expected return is
E(R) = wE(Rr ) + (1
• The variance is
V ar(R) = w2V ar(Rr )
and the standard deviation is
(R) = w (Rr )
w)Rf
Page 21 of Lecture 6
• To represent the portfolios in the standard deviation / expected return space,
we write the expected return of a portfolio as function of the portfolio’s
standard deviation:
(R)
E(R) = Rf + w(E(Rr ) Rf ) = Rf +
(E(Rr ) Rf )
(Rr )
E(R)
6
E(Rr )
Rf
0
s⌘
⌘
|
⌘
⌘
⌘
⌘
⌘
Risky Asset r⌘⌘
(w = 1)⌘⌘
s⌘
⌘
⌘9
⌘
⌘
>
=
⌘
⌘
⌘
{z
w 2 [0, 1]
>
;
}
⌘
⌘
⌘
⌘
E(Rr )
!
(Rr ) w > 1
Rf
-
(R)
Page 22 of Lecture 6
6.4.2 Portfolio Frontier with a Riskless Asset
• Consider the lines linking the riskless asset with the points on and inside the
hyperbola. The portfolio frontier is the line with the steepest slope
E(R)
Portfolio Frontier⌘
6
Tangent Portfoliot⌘⌘
⌘
⌘
Rf
t
⌘
!
⌘
⌘
⌘
⌘
!
⌘
!!
⌘
!
⌘
!
⌘
!!
⌘ !!
⌘ !
⌘!!
!
⌘
!
⌘
⌘
⌘
!
!!
⌘
⌘
⌘
⌘
⌘
⌘
⌘
t
!
!!
!
!
!!
!!
⌘
⌘
⌘
⌘
⌘
⌘
⌘
Line 1 > Line 2
!!
!!
!!
!!
Line 2
Portfolio 1
t
Portfolio 2
-
0
(R)
Page 23 of Lecture 6
6.4.3 Properties of the Portfolio Frontier
• The PF is the line linking the riskless asset with the tangent portfolio (TP)
• All frontier portfolios are combinations of the riskless asset and the TP
• The portfolios below the TP involve a positive weight on the riskless asset
(lending)
• The portfolios above the TP involve a negative weight (borrowing)
• To construct the PF, we only need to
? determine one frontier portfolio, i.e., solve the optimization problem once
? draw the line linking that portfolio to the riskless asset
Page 24 of Lecture 6
Global Portfolio Frontier with Riskless Asset
25%
Hong Kong
Tangent Portfolio
Expected Return
20%
France
15%
UK
Equally Weighted
Canada
Japan
US
10%
China
5%
0%
0%
5%
10%
15%
20%
25%
Standard Deviation
We use the US T-bills as the riskless asset, and assume a return of 5%
30%
35%
40%
Page 25 of Lecture 6
Which Portfolio to Choose?
• We should choose a portfolio on the PF. (Assuming that we care only about
mean and variance)
• Which one, depends on how we trade o↵ risk and return, i.e., on our risk
aversion
? If we are very risk-averse, we should choose a portfolio closer to the riskless
asset
? If we are not very risk-averse, we should choose a portfolio closer to the
TP, and even above the TP
Page 26 of Lecture 6
6.4.4 A Word of Caution
• We should always keep in mind that mean-variance optimization is only as
precise as the input estimates
• The
estimates
for
expected
returns
are
quite
imprecise
(Remember that even for a 90-year sample, the standard error is around
2%)
• The estimates for standard deviations and correlations are generally quite
precise but both change over time
• It is possible to extend the theory to account for the uncertainty about the
estimates
Page 27 of Lecture 6
6.5
An Important Property of the Tangent Portfolio
• The ratio
E(Rn) Rf
2Cov(Rn, R⇤)
is known as the buck for the bang ratio (the change in expected return
(buck) to the change in variance (bang))
• The important property of the TP is that this ratio is independent of the
particular asset n and equal to
E(R⇤) Rf
2V ar(R⇤)
This follows (see the Appendix) from the fact that the tangent portfolio has
the highest possible Sharpe ratio.
Page 28 of Lecture 6
Main Takeaways
• Mean-Variance Optimization and the Portfolio Frontier
• Limits of Diversification: Systematic vs. Idiosyncratic risk
• The buck for the bang ratio
Page 29 of Lecture 6
Appendix⇤
• Suppose that we hold the TP, and decide to
? increase(decrease) the weight wn of a risky asset n
? decrease(increase) the weight of the riskless asset (by the same amount)
• What happens to the Sharpe ratio?
? Can only decrease since by construction the TP has the highest possible
Sharpe ratio
• Hence,
@ E(R⇤) Rf
q
= 0,
⇤
@wn V ar(R )
i = 1, .., n,
where R⇤ denotes the return on the tangent portfolio
Page 30 of Lecture 6
• Using the chain rule we have
@ E(R⇤) Rf
q
=
@wn V ar(R⇤)
@
⇤
(E(R
)
@wn
q
Rf )
V ar(R⇤)
@
⇤
1 E(R⇤) Rf @wn V ar(R )
q
=0
⇤
2 V ar(R⇤)
V ar(R )
• Therefore,
@
⇤
(E(R
) Rf )
@wn
@
V ar(R⇤)
@wn
1 E(R⇤) Rf
=
,
2 V ar(R⇤)
i = 1, .., n
Page 31 of Lecture 6
• The change in expected return
@
(E(R⇤)
@wn
Rf ) = E(Rn)
• The change in variance
Rf
0
1
X
@
⇤
@
V ar(R ) = 2 wnV ar(Rn) +
wmCov(Rn, Rm)A
@wn
m6=n
= 2Cov(Rn, R⇤)
• Notice that the change in variance involves the covariance of asset n with
the tangent portfolio, and not the variance of asset n
• Thus,
@
(E(R⇤) Rf )
@wn
@
⇤)
V
ar(R
@wn
=
E(Rn) Rf
2Cov(Rn, R⇤)
Page 32 of Lecture 6
Lecture Notes
FM 423: Asset Markets
Lecture 7
The Capital Asset Pricing Model (CAPM)
Igor Makarov
London School of Economics
Page 2 of Lecture 7
Lecture 7: Main Points
In Lecture 6 (Portfolio Theory) we studied how to choose a stock portfolio. A
crucial input was the expected return on each stock. In this lecture we seek
to obtain some insight on stocks’ expected returns. How are expected returns
determined? How are they related to risk?
1. The CAPM
2. The CAPM’s Basic Insight
3. CML and SML
4. Uses of the CAPM
Page 3 of Lecture 7
7.1
The CAPM
• The CAPM is a theoretical model which provides insight on assets’ expected
returns
• Assumptions:
? There are N risky assets and a riskless asset
? Trading of assets is costless (including short sales)
? Investors care only about mean and variance
? Investors have the same information (beliefs)
? Investors have an one-period horizon
Page 4 of Lecture 7
Asset Demand
• We first consider the demand for the assets
• A single investor:
? Cares only about mean and variance
? Chooses a portfolio on the portfolio frontier
? Portfolio is a combination of tangent portfolio and riskless asset
- Very risk-averse: Portfolio closer to riskless asset
- Not very risk-averse: Portfolio closer to tangent portfolio, or even above
tangent portfolio
• Investors as a group:
? Demand is a combination of tangent portfolio and riskless asset
Page 5 of Lecture 7
Asset Supply
• We next consider the supply of the assets
• Supply is
PN
n=1 Pn sn
dollars of market portfolio, and the riskless asset
• The market portfolio is the value-weighted portfolio of the N risky assets
? The market value (market capitalization) of asset n is Pnsn, where Pn is
the price of one share and sn the total number of shares
? The market value of the market portfolio is
N
X
P n sn
n=1
? The weight of asset n in the market portfolio is
P n sn
n=1 Pn sn
PN
? We denote by RM the return on the market portfolio
Page 6 of Lecture 7
Market Equilibrium
• In market equilibrium, demand equals supply
• In particular:
Tangent portfolio coincides with market portfolio
• Therefore, R⇤ = RM
• In the last lecture we showed that
E(Rn) Rf
E(RM ) Rf
=
Cov(Rn, RM )
V ar(RM )
• This equation implies that
E(Rn)
Rf =
• This is the CAPM
Cov(Rn, RM )
(E(RM )
V ar(RM )
Rf )
Page 7 of Lecture 7
7.1.1 CAPM Terminology
• The expected excess return of asset n is the asset’s expected return minus
the return on the riskless asset,
E(Rn)
Rf
• The beta of asset n is
n
=
Cov(Rn, RM )
V ar(RM )
• The market risk premium is the expected excess return of the market
portfolio,
E(RM )
Rf
Page 8 of Lecture 7
7.2
The CAPM’s Basic Insight
• The CAPM is
E(Rn)
Rf =
n (E(RM )
Rf )
• An asset’s expected return depends on the asset’s risk
? through the asset’s beta (systematic risk),
? and not through the asset’s sigma (idiosyncratic risk)
• Key insight:
It is systematic risk and not idiosyncratic risk
that is priced in the market
In other words:
The relevant measure of asset risk
is beta and not the variance
Page 9 of Lecture 7
Intuition
• Suppose that an asset has zero beta. The CAPM implies that it has the
same expected return as the riskless asset
? Intuition: The asset’s risk is only idiosyncratic and can be diversified. The
asset does not contribute to portfolio risk
• Suppose that an asset has positive beta. The CAPM implies that it has
higher expected return than the riskless asset
? Intuition: The asset increases portfolio risk
• Suppose that an asset has negative beta. The CAPM implies that it has
lower expected return than the riskless asset
? Intuition: The asset reduces portfolio risk
Page 10 of Lecture 7
Linearity
• The CAPM is
E(Rn)
Rf =
n (E(RM )
Rf ),
and implies that an asset’s expected return depends on risk only through
beta
• It also implies that the asset’s expected excess return is linear in beta
• For instance, if beta is 2, then the asset’s expected excess return is twice the
market risk premium
Page 11 of Lecture 7
7.3
CML and SML
• Two lines which illustrate the CAPM are:
? the Capital Market Line (CML)
? the Security Market Line (SML)
Page 12 of Lecture 7
The Capital Market Line (CML)
E(R)
6
M
E(RM )
Rf
⌘
t
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
t⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘⌘
⌘
-
0
(RM )
(R)
Page 13 of Lecture 7
7.3.1
The Security Market Line (SML)
E(R)
6
M
E(RM )
r
E(Rr )
Rf
⌘
t
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
t⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
t⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘
⌘⌘
⌘
-
0
(Rr )
(RM ) = 1
(R)
Page 14 of Lecture 7
CML vs. SML
• The CML:
? is in the standard deviation/expected return space
? contains only the frontier portfolios
• The SML:
? is in the beta/expected return space
? contains all portfolios (according to the CAPM)
Page 15 of Lecture 7
7.4
Uses of the CAPM
• Valuation: the CAPM provides a risk-adjusted discount rate for the PV
calculation
? Valuation of stocks (Lecture 9)
? Valuation of firms’ investments (FM422)
• Performance evaluation
• Portfolio selection
Page 16 of Lecture 7
Performance evaluation
• Example: We have the following information about the performance of two
money managers
Manager
Realized return
1
19%
1.2
2
16%
1
? Can you tell who is a better stock selector?
) No, not enough information
? Suppose Rf = 6% and MRP = 8%
) Compute abnormal returns:
↵1 = 19% [6% + 1.5 ⇥ 8%] = 19%
18% = 1%
↵2 = 16% [6% + 1 ⇥ 8%] = 16% 14% = 2%
) The second manager is a better stock selector
Page 17 of Lecture 7
Portfolio Selection in a CAPM World
• Portfolio selection in a CAPM world is straightforward
• Suppose that we
? Estimate the betas of all stocks
? Assume that stocks’ expected returns are given by the CAPM
? Care only about mean and variance
• What is our optimal portfolio?
Page 18 of Lecture 7
Portfolio Selection in a Non-CAPM World
• The CAPM can still be useful even in a non-CAPM world because it can
guide portfolio selection
• Suppose that we
? Are confident that expected returns of a few stocks are not given by the
CAPM
? Assume that expected returns of all other stocks are given by the CAPM
• How would we choose our portfolio?
? Compared to its weight in the market portfolio, a stock should get greater
weight if its expected return is greater than that given by the CAPM, i.e.,
if its alpha is positive
Page 19 of Lecture 7
Main Takeaways
• The CAPM is a celebrated finance model which provides insight on assets’
expected returns
• According to the CAPM, the expected excess return on an asset is equal to
the asset’s beta times the MRP
• The market risk premium is the expected excess return of the market portfolio
• The asset’s beta measures how the asset covaries with the market portfolio
Page 20 of Lecture 7
Lecture Notes
FM 423: Asset Markets
Lecture 8
Statistical Tests of the CAPM
Igor Makarov
London School of Economics
Page 2 of Lecture 8
8.1
Regressions
• Consider two random variables X (input) and Y (output)
40.00%
30.00%
20.00%
Y"
10.00%
!25.00%
!20.00%
!15.00%
!10.00%
!5.00%
0.00%
0.00%
!10.00%
!20.00%
X"
!30.00%
5.00%
10.00%
15.00%
Page 3 of Lecture 8
• Goal: given X , make a good prediction of the output Y
• Linear Model: Y = ↵ + X + ✏
? ↵, : constants,
? ✏: random variable, such that
Cov(X, ✏) = 0 and E(✏) = 0
)
V ar(Y ) = V ar( X) + V ar(✏)
• Variation in Y is decomposed into
?
X : variation that can be “explained” by X
? ✏: “unexplained” variation
Page 4 of Lecture 8
• The linear model implies that
✏=Y
↵
X
• For E(✏) = 0 it must be that
↵ = EY
EX
• Notice that Cov(X, ✏) = Cov(X, Y )
V ar(X)
• Therefore, for Cov(X, ✏) = 0 it must be that
Cov(X, Y )
=
V ar(X)
Page 5 of Lecture 8
Estimation
• In general, we do not know ↵ and . However, we can estimate them:
ˆX
↵
ˆ = Y
s(X, Y )
ˆ =
s(X)2
• The line Yb = ↵
ˆ + ˆX is the “best” line that fits Y vs. X
Page 6 of Lecture 8
Regression and Asset Returns
• When studying asset returns, we assume that
? X is excess return of market portfolio, RM
? Y is excess return of asset n, Rn
Rf
Rf
• Regression equation is
Rn
Rf = ↵ n +
n (RM
Rf ) + ✏ n
• Variation in returns of asset n is decomposed into
?
n (RM
Rf ): Systematic risk, i.e., risk that is perfectly correlated with
the market portfolio
? ✏n: Idiosyncratic risk, i.e., risk that is uncorrelated with the market
portfolio
Page 7 of Lecture 8
Asset Characteristics
Three characteristics of an asset:
• Alpha, ↵n
• Beta,
n
• Sigma, (✏n)
Page 8 of Lecture 8
Beta
• Beta:
Rn
Rf = ↵ n +
n
(RM
Rf ) + ✏ n
• Measures the asset’s sensitivity to market movements
• If the return on the market portfolio is higher by 1%, then the return on
asset n is higher by n (holding all else equal)
• Beta is given by
n
Cov(Rn, RM )
=
V ar(RM )
Page 9 of Lecture 8
Alpha and Sigma
• Alpha:
Rn
Rf = ↵ n +
n (RM
Rf ) + ✏ n
• Measures the asset’s attractiveness
• If the CAPM holds then ↵n = 0
• Sigma:
Rn
Rf = ↵ n +
n (RM
Rf ) + ✏ n
• Sigma is the standard deviation of ✏n. It measures the asset’s idiosyncratic
risk
Page 10 of Lecture 8
Regression analysis
• We can estimate ↵, , and (✏) by using regression analysis, which is
available in many software packages
• We denote estimates for ↵, , and (✏) by ↵
ˆ, ˆ, and s(✏); the standard
error of estimates ↵
ˆ and ˆ by s↵ and s
• R-Square (R2):
Explained Variance
V ar( X)
R =
=
Explained Variance + Unexplained Variance
V ar( X) + V ar(✏)
2
Page 11 of Lecture 8
Regression: Example (Excel)
IBM vs. S&P500 (market portfolio). Monthly returns 1/1982-12/2009
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.552348
R Square
0.305088
Adjusted R Square
0.303008
Standard Error
6.638959
Observations
336
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
1
334
335
SS
6463.122311
14721.308
21184.43032
MS
6463.122
44.07577
F
Significance F
146.6366 3.13962E-28
Coefficients
0.410379
0.986585
Standard Error
0.363217925
0.081472955
t Stat
1.129842
12.10936
P-value
Lower 95%
0.259354 -0.304104232
3.14E-28 0.826320623
Upper 95%
1.12486184
1.1468502
• Estimates: ↵
ˆ = 0.41, s↵ = 0.36, ˆ = 0.98, s = 0.08, s(✏n) = 6.63
• R-Square: 30%
Page 12 of Lecture 8
8.2
Statistical Tests of the CAPM
• The CAPM is
E(Rn)
Rf =
n (E(RM )
Rf )
• Implications:
1. Expected excess returns are linear in beta
2. Slope of the security market line is the market risk premium
3. Expected returns depend only on beta
Page 13 of Lecture 8
What is a Good Test?
• Want to minimize two errors:
? Type I Error: the rejection of the true null hypothesis
? Type II Error: the acceptance of a false null hypothesis
• Want to avoid data mining (snooping)
Page 14 of Lecture 8
Regression: Example (Excel)
Google vs. S&P500 (market portfolio). Monthly returns 9/2004-12/2015
Regression)Statistics
Multiple(R
0.454328032
R(Square
0.206413961
Adjusted(R(Square
0.200491677
Standard(Error 0.088716077
Observations
136
ANOVA
df
Regression
Residual
Total
Intercept
X(Variable(1
SS
MS
F
Significance)F
1 0.274318126 0.274318126 34.85377686 2.76366EF08
134 1.054652672 0.007870542
135 1.328970798
Coefficients Standard)Error
t)Stat
P5value
Lower)95% Upper)95%
0.019007236 0.007648849 2.484980012 0.014187453 0.003879146 0.034135325
1.08240049 0.183342461 5.903708738 2.76366EF08 0.719781048 1.445019932
• Estimates: ↵
ˆ = 1.9%, t↵ = 2.48. Should we reject the CAPM?
Page 15 of Lecture 8
Early Tests: Implications 1 and 2
Average risk premium,
1931-1991, percent
30
Market
line
25
20
15
10
Portfolio 1
"-
5
.4
2
•
.6
� •
4
.8
•
•
•
• 8
Portfolio 10
�7
Market
portfolio
1.0
1.2
1.4
1.6
""'---.1----L---.J..._---'---__J_--I...----'----'--------'
.2
Source: Black, Fischer, 1993, Beta and return, The Journal of Portfolio Management.
) Expected excess returns are approximately linear in beta
) Slope of the line is smaller than market risk premium
Portfolio
beta
Page 16 of Lecture 8
Portfolios
• To reduce the impact of idiosyncratic noise and data snooping financial
economists often work with portfolios of stocks
• Portfolios are usually formed based on some stocks characteristics, e.g., stock
beta, market capitalization, past returns, etc
• Consider, for example, portfolios used in the Black (1993) paper
? At the beginning of each year t = 1931, ..., 1990 all stocks are sorted
into 10 bins according to their betas. Stock betas are estimated using
previous five years of monthly return data. Stocks with the smallest betas
are sorted into the first bin, and stocks with the largest betas into bin 10
? Each stock then in each bin receives a weight wn so that the sum of all
stock weights in each bin is equal to one
? Stocks are usually either equal-weighted or value-weighted. In the latter
case, the weight is proportional to the market capitalization of the stock
? Portfolios are held for one year. After that, new portfolios are formed
Page 17 of Lecture 8
Current State
• Subsequent research uncovered many violations of the CAPM. It showed
that in addition to beta, expected returns depend on other factors and stock
characteristics. The most prominent of which are size, value, and momentum
1.2
25 Fama-French Portfotios Sorted on Size and Book-to-Market (1946-2014)
•
1.1
•
:2 1
•
• •• •
+--'
C
0
E 0.9
!,.,,
•
Q)
� 0.8
E
:::J
·E 0.7
Q)
a..
•
••
�
en 0.6
a:
• •
••
•
•
•
•
•
•
•
•
0> 0.5
ctS
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•
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•
0.3
0.2._____��----'---��----'-��--------''-------��---��----��____,_���
0.8
0.9
1
1.1
1.2
Portfolio Beta
1.3
1.4
1.5
Page 18 of Lecture 8
• Size e↵ect: Expected returns of small stocks exceed those of large stocks
(holding beta equal)
• Value e↵ect: Compare market price to accounting measures, e.g., book
value or earnings
? High book-to-market ratio: Value stocks
? Low book-to-market ratio: Growth stocks
• Expected returns of value stocks exceed those of growth stocks (holding beta
equal)
• Momentum e↵ect: Expected returns of stocks with short history of overperformance exceed those with short history of underperformance (holding
beta equal)
Page 19 of Lecture 8
Possible Explanations
• Omitted risk factors
• Investor irrationality
• Frictions
Page 20 of Lecture 8
Omitted Risk Factors
• Example: Value e↵ect can be explained if
? There is an additional factor to the market portfolio, carrying a positive
risk premium
? Value stocks have higher beta with respect to that factor than growth
stocks
• Empirical multi-factor models: Factors are
? Market portfolio
? Value portfolio (HML)
? Momentum portfolio (WML)
? Size portfolio (SMB)
• What is economic interpretation of these factors?
Page 21 of Lecture 8
Irrationality
• Assume that investors process information incorrectly
• Example: Value e↵ect can be explained if
? Investors are too optimistic about future earnings of some stocks (overpricing them) and too pessimistic about future earnings of other stocks
(underpricing them)
• Example: Momentum e↵ect can be explained if
? Optimism/pessimism builds gradually...
? ... and this is not anticipated by rational investors
• Do biases aggregate?
• Rational investors must have limited capital
Page 22 of Lecture 8
Frictions
• There are frictions arising because of
? Short-sale constraints
? Leverage constraints
? Delegation of portfolio management and agency problems
• Example: Momentum e↵ects can be explained if
? Investors invest through asset managers
? Following a manager’s poor performance, investors update negatively on
manager’s ability and withdraw funds gradually
? Manager sells stocks following the withdrawals
• Investors not subject to the frictions must have limited capital
Page 23 of Lecture 8
Main Takeaways
• The CAPM is a simple and intuitive model, used in practice
• Statistical tests reject the CAPM
• A four-factor model with market portfolio, value, momentum and size does
better than the CAPM but is also rejected by the data
• Berk-DeMarzo (Corporate Finance):
“While the CAPM may not be perfect, it is unlikely that a truly perfect
model will be found in the foreseeable future. Furthermore, the imperfections of the CAPM may not be critical in the context of capital
budgeting and corporate finance, where errors in estimating project
cash flows are likely to be far more important than small discrepancies
in the cost of capital. In that sense, the CAPM may be good enough,
especially relative to the cost of implementing a more sophisticated
model.”
Page 24 of Lecture 8
Lecture Notes
FM 423: Asset Markets
Lecture 9
Valuation of Stocks
Igor Makarov
London School of Economics
Page 2 of Lecture 9
Lecture 9: Main Points
Big question: How to value stocks?
1. Valuation Formulas
2. Valuation in Practice
3. Bubbles
Page 3 of Lecture 9
9.1
Valuation Formulas
• Consider a stock which pays annual dividends
• Dividend in year t = 0, 1, .. is Dt
• Ex-dividend price in year t is Pt
• Value the stock: Determine P0
Page 4 of Lecture 9
Price and Expected Return
• Return on the stock between years 0 and 1 is
R=
D1 + (P1
P0
P0 )
• Expected return is
E(R) =
E(D1) + (E(P1)
P0
P0 )
• So far:
? Take P0 as given, and compute E(R)
• Now:
? Take E(R) as given, and compute P0
Page 5 of Lecture 9
Back to the Present Value Rule
• Expected return is
E(D1) + (E(P1)
E(R) =
P0
P0 )
• Set r = E(R), and solve for P0:
P0 =
E(D1) + E(P1)
1+r
• Price P0 is PV of expected cash flows, discounted at a risk-adjusted rate
• Expected cash flows:
? Expected dividend E(D1)
? Expected price E(P1)
? Risk-adjusted rate: Expected return r
• From now on, denote E(D1) by D1, and E(P1) by P1
Page 6 of Lecture 9
One Iteration
• Equation for P0:
P0 =
D1 + P1
1+r
• P0 depends on P1
) Our valuation analysis is incomplete
• Repeating our analysis for years 1 and 2, and assuming that expected return
is also r, we get
P1 =
D2 + P2
1+r
• Plugging back:
D1
D2
P2
P0 =
+
+
1+r
(1 + r)2
(1 + r)2
Page 7 of Lecture 9
Multiple Iterations
• Iterating, we get
P0 =
D1
D2
DT
PT
+
+
·
·
·
+
+
1+r
(1 + r)2
(1 + r)T
(1 + r)T
• “No-bubble” assumption:
PT
(1 + r)T
! 0
T !1
• Interpretation:
? Price is driven only by cash flows (dividends)
? Price is not a bubble
Page 8 of Lecture 9
A General Valuation Formula
• Under the no-bubble assumption, we get
P0 =
D1
D2
DT
+
+
·
·
·
+
+ ···
2
T
1+r
(1 + r)
(1 + r)
• In words:
Price of a stock is PV of expected dividends
discounted at the stock’s expected return
Page 9 of Lecture 9
Constant Growth Model
• Assume that expected dividends grow at a constant rate g , i.e.,
Dt = Dt
1 (1
+ g)
• General valuation formula becomes
D1
D1(1 + g)
D1(1 + g)T
P0 =
+
+ ··· +
1+r
(1 + r)2
(1 + r)T
• Growing perpetuity formula implies that
D1
P0 =
r g
• This is the constant growth valuation formula
1
+ ···
Page 10 of Lecture 9
9.1.1 Obtaining the Formula Inputs
• We will generally use the constant growth formula
• Inputs:
? Expected dividend in year 1, D1 (can be obtained from financial sources)
? Dividend growth rate g (historical or forecasted growth rates)
? Expected return r
Page 11 of Lecture 9
Expected Return
• An estimate of expected return can be obtained from the CAPM:
r = Rf +
⇥ MRP
where
? Rf is the riskless rate (usualy one-month T-bill rate)
?
is the stock’s beta
? MRP is the market risk premium
• Estimates can also be obtained from multi-factor models
• These estimates are more precise than the sample average of the stock’s
realized returns
• Example: The company XY Z is a fast-growing start-up. The beta of XY Z
is expected to be 1.5 for the next 4 years. Starting from year 4, the beta of
XY Z is expected to be 1. Find the expected return of XYZ for di↵erent
years. Assume that the CAMP holds, the market risk-premium is 6%, and
the term-structure of interest rates is constant and equal to 3%.
Solution: Using the CAPM the expected return for years 1 through 4 is
E ir1 = 3% + 1.5 ⇥ 6% = 12%,
i = 0, 1, 2, 3
Afterwards, it is
E ir1 = 3% + 1 ⇥ 6% = 9%,
i = 4, 5, . . .
• Suppose the company XY Z is expected to pay a dividend of $10 next year.
The dividend growth is expected to be 10 percent a year for 3 years (i.e.,
until year 4) and 3% thereafter. Find the current price of XY Z.
Solution: When we discount stock cash flows we use the expected returns
per period
u
D1
D2
D3
1
2
3
u
0
1
1+Er1
u
1
1+E 1 r1
u
-
Time
1
1+E 2 r1
The ex-dividend price of XY Z in year 4 is
D5
10 ⇥ 1.13 ⇥ 1.03
P4 =
=
= 228.488
9% 3%
9% 3%
Therefore, the current price of XY Z is
10
10 ⇥ 1.1
10 ⇥ 1.12
10 ⇥ 1.13 + P4
P0 =
+
+
+
= 179.98
1.12
1.122
1.123
1.124
Page 12 of Lecture 9
9.2
Valuation in Practice
MERCK & CO. NYSE-MRK
TIMELINESS
SAFETY
TECHNICAL
BETA
.85
4
1
3
High:
Low:
Lowered 3/24/17
Raised 4/15/11
RECENT
PRICE
46.4
31.8
61.6
42.3
61.2
22.8
38.4
20.0
65.54
41.6
30.7
P/E
RATIO
37.9
29.5
17.4
RELATIVE
Median: 12.0) P/E RATIO 0.87
17.0 (Trailing:
48.0
36.9
50.4
40.8
62.2
49.3
63.6
45.7
65.5
48.0
DIV’D
YLD
VALUE
LINE
2.9%
66.8
59.1
Target Price Range
2020
2021
2022
LEGENDS
12.0 x ʺCash Flowʺ p sh
Relative Price Strength
Options: Yes
128
96
80
64
48
40
32
24
. . . .
Lowered 7/7/17
Shaded a r ea indicates r ecession
(1.00 = Market)
2020-22 PROJECTIONS
Price
Gain
High
85
(+30%)
Low
70
(+5%)
Insider Decisions
to Buy
Options
to Sell
A
0
3
2
S
0
1
1
O
0
1
1
N
0
4
4
D
0
0
0
Ann’l Total
Return
9%
4%
J
0
0
0
F M
0 0
6 11
7 0
A
0
0
0
% TOT. RETURN 5/17
Institutional Decisions
3Q2016
4Q2016
1Q2017
799
912
856
to Buy
to Sell
805
798
885
Hld’s(000)201491020231812190042
Percent
shares
traded
18
12
6
1 yr.
3 yr.
5 yr.
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
20.99
3.85
3.14
1.37
1.20
7.06
2272.7
22.7
1.16
1.9%
23.07
3.85
3.14
1.41
1.06
8.11
2245.0
17.3
.94
2.6%
10.12
3.56
2.92
1.45
.86
7.01
2221.8
18.2
1.04
2.7%
10.39
3.29
2.61
1.49
.78
7.83
2208.6
16.2
.86
3.5%
10.09
3.34
2.53
1.52
.64
8.21
2181.9
12.1
.64
5.0%
10.44
3.59
2.52
1.52
.45
8.10
2167.8
15.2
.82
4.0%
11.14
2.42
1.49
1.52
.47
8.37
2172.5
34.1
1.81
3.0%
11.32
4.48
3.64
1.52
.62
8.90
2107.7
10.2
.61
4.1%
8.82
3.21
3.25
1.52
.47
19.00
3108.2
9.1
.61
5.1%
14.92
5.87
3.42
1.52
.54
17.64
3082.1
10.5
.67
4.2%
15.80
6.29
3.77
1.52
.57
17.93
3040.8
9.1
.57
4.4%
15.62
6.19
3.82
1.68
.65
17.52
3026.6
10.8
.69
4.1%
15.04
5.95
3.49
1.72
.53
17.00
2927.5
13.3
.75
3.7%
14.88
5.98
3.49
1.76
.46
17.14
2838.1
16.4
.86
3.1%
14.20
5.96
3.59
1.80
.46
16.06
2781.1
15.8
.80
3.2%
14.48
5.83
3.78
1.84
.59
14.58
2748.7
15.2
.80
3.2%
14.65
5.50
3.85
1.88
.55
14.00
2710.0
15.35
5.70
4.20
1.92
.55
13.85
2670.0
24198
31.4%
1988.0
3275.4
2.8%
13.5%
2787.2
3915.8
18185
15.5%
18.0%
NMF
101%
23850
48.5%
1631.2
7808.4
20.4%
32.7%
4986.2
3943.3
18758
34.8%
41.6%
24.1%
42%
27428
24.1%
2576.0
7409.3
20.0%
27.0%
12678
16075
59058
10.1%
12.5%
7.1%
43%
45987
23.3%
7381.0
10715
20.0%
23.3%
13423
15482
54376
15.8%
19.7%
11.0%
44%
48047
32.4%
7427.0
11697
23.4%
24.3%
16936
15525
54517
17.2%
21.5%
12.9%
40%
47267
32.9%
6978.0
11743
23.8%
24.8%
16509
16254
53020
17.4%
22.1%
12.5%
44%
44033
31.6%
6988.0
10443
21.7%
23.7%
17817
20539
49765
15.4%
21.0%
10.6%
49%
42237
32.4%
6691.0
10271
24.3%
24.3%
14407
18699
48647
15.8%
21.1%
10.5%
50%
39498
33.9%
6375.0
10195
21.7%
25.8%
10561
23929
44676
15.3%
22.8%
11.4%
50%
39807
34.4%
5441.0
10580
22.3%
26.6%
13410
24274
40088
17.0%
26.4%
13.6%
48%
CAPITAL STRUCTURE as of 3/31/17
Total Debt $28474 mill. Due in 5 Yrs $9001 mill.
LT Debt $23437 mill.
LT Interest $683 mill.
(37% of Cap’l)
Pension Assets-12/16 $17.6 bill. Oblig. $19.2 bill.
Pfd Stock None
Common Stock 2,735,164,510 shs.
as of 4/30/17
MARKET CAP: $179 billion (Large Cap)
CURRENT POSITION
2015
2016
($MILL.)
Cash Assets
13427
14341
Receivables
6484
7018
Inventory (LIFO)
4700
4866
Other
5153
4389
Current Assets
29764
30614
Accts Payable
2533
2807
Debt Due
2585
568
Other
14085
13829
Current Liab.
19203
17204
ANNUAL RATES
of change (per sh)
Sales
‘‘Cash Flow’’
Earnings
Dividends
Book Value
Calendar
2014
2015
2016
2017
2018
Calendar
2014
2015
2016
2017
2018
Calendar
2013
2014
2015
2016
2017
Past
10 Yrs.
3.5%
5.5%
3.5%
2.0%
7.0%
3/31/17
15249
7066
5146
4069
31530
2484
5037
12302
19823
Past
Est’d ’14-’16
5 Yrs.
to ’20-’22
2.0%
3.5%
3.0%
1.0%
1.0%
5.5%
3.5%
2.0%
-2.5%
-3.0%
QUARTERLY SALES ($ mill.)
Mar.31 Jun.30 Sep.30 Dec.31
10264
10934
10557
10482
9425
9785
10073
10215
9312
9844
10536
10115
9434
9700
10300
10266
9500
10100
10600
10800
EARNINGS PER SHARE A
Mar.31 Jun.30 Sep.30 Dec.31
.88
.85
.90
.87
.85
.86
.96
.93
.89
.93
1.07
.89
.88
.88
1.09
1.00
.95
1.00
1.15
1.10
QUARTERLY DIVIDENDS PAID B■
Mar.31 Jun.30 Sep.30 Dec.31
.43
.43
.43
.43
.44
.44
.44
.44
.45
.45
.45
.45
.46
.46
.46
.46
.47
.47
.47
Full
Year
42237
39498
39807
39700
41000
(A) Diluted earnings (adjusted). Quarters may
not sum due to rounding. Excludes nonrecurring gains (losses): ’05, (43¢); ’06, (13¢); ’09,
$2.40; ’10, ($3.16); ’11, ($1.75); ’12, ($1.66);
Full
Year
3.49
3.59
3.78
3.85
4.20
Full
Year
1.72
1.76
1.80
1.84
Bold figures are
Value Line
estimates
39700
34.0%
4500
10430
22.0%
26.3%
12000
23000
38000
17.5%
27.5%
14.0%
49%
41000
35.0%
4000
11200
22.0%
27.3%
11000
23000
37000
19.0%
30.5%
16.5%
46%
©
THIS
STOCK
VL ARITH.*
INDEX
19.3
23.7
105.1
16.7
22.7
95.1
16
12
VALUE LINE PUB. LLC 20-22
Sales per sh
‘‘Cash Flow’’ per sh
Earnings per sh A
Div’ds Decl’d per sh B■
Cap’l Spending per sh
Book Value per sh
Common Shs Outst’g C
Avg Ann’l P/E Ratio
Relative P/E Ratio
Avg Ann’l Div’d Yield
17.65
6.20
5.00
2.04
.60
13.35
2550.0
15.0
.95
2.7%
Sales ($mill)
Operating Margin
Depreciation ($mill)
Net Profit ($mill)
Income Tax Rate
Net Profit Margin
Working Cap’l ($mill)
Long-Term Debt ($mill)
Shr. Equity ($mill)
Return on Total Cap’l
Return on Shr. Equity
Retained to Com Eq
All Div’ds to Net Prof
45000
36.0%
3000
12750
22.0%
28.3%
10000
23000
34000
23.0%
37.5%
22.0%
41%
BUSINESS: Merck & Co., Inc. is a global health care company that
delivers innovative health solutions through its prescription medicines, vaccines, biologic therapies, and animal health products,
which it markets directly and through joint ventures. Operations
comprised of four segments: Pharmaceutical, Animal Health, Alliances and Healthcare Services. Top-grossing drugs in 2016:
Januvia (diabetes) and Zetia (cholesterol). Acquired ScheringPlough, 11/09. Has 68,000 employees. Off/dirs. own less than 1%
of common stock; BlackRock, 6.7%; Vanguard, 6.7%; Capital World
Investors, 5.1% (4/17 proxy). Chairman/President/CEO: Kenneth
Frazier. Inc.: NJ. Addr.: 2000 Galloping Hill Road., Kenilworth, NJ
07033. Tel.: 908-740-4000. Internet: www.merck.com.
We have slightly raised our 2017 estimates
for
Merck
&
Co.
The
drugmaker ’s
first-quarter
results
beat
consensus expectations on both lines, thanks
to increased cost cutting and better-thanexpected demand in several key products.
Continued momentum in standout oncology asset Keytruda (sales +137%) was a
key highlight, along with a strong seasonal
bump
in
vaccine
franchise
Gardasil
(+41%).
Double-digit
growth
in
animal
health sales (+13%) and a solid contribution from new Hep-C drug Zepatier ($378
million) further bolstered results, helping
to
offset
generic
pressures
on
several
drugs
including,
Zetia / Vytorin
(-35%),
Remicade
(-34%),
Cubicin
(-67%),
and
Nasonex
(-40%).
Following
the
release
(May 2nd), management raised its fullyear adjusted earnings guidance to $3.76$3.88 a share (previously $3.72-$3.87) and
its sales outlook to $39.1 billion-$40.3 billion (previously $38.6 billion-$40.1 billion).
A
new
type
of
cholesterol
drug
(anacetrapib) has shown promise in
late-stage studies. On June 27th, Merck
announced that anacetrapib significantly
reduced heart attacks and other complica-
tions of heart disease in a 30,000 patient
clinical trial. The drug works by blocking a
protein called CETP, which differs from
older cholesterol-lowering medications like
Lipitor and Crestor and newer injectable
methods.
This
represents
a
substantial
commercial opportunity for Merck given
the sheer size of the market. That said,
the timing of a potential regulatory filing
still remains largely uncertain. Full results of the anacetrapib study will be presented at the European Society of Cardiology meeting on August 29th.
Long-term growth remains centered
around Keytruda. The company’s top asset and now second-highest grossing franchise
(behind
Januvia / Janumet) will be
leaned on heavily to drive growth over the
next several years. The drug has locked up
several
highly-coveted
approvals
since
2014 and development efforts are ongoing.
Current
projections
suggest
Keytruda
sales could top $8 billion by 2021.
The stock is ranked 4 (Below Average)
for
Timeliness.
We
continue
to
view
Merck as a solid core holding in the drug
space, however.
’13, ($2.02); ’14, 58¢; ’15, ($2.03); ’16, ($1.74).
Next egs. report due late July.
(B) Dividends historically paid in early January,
April, July, and October. ■ Dividend reinvest-
Michael Ratty
ment plan available.
(C) In millions.
© 2017 Value Line, Inc. All rights reserved. Factual material is obtained from sources believed to be reliable and is provided without warranties of any kind.
THE PUBLISHER IS NOT RESPONSIBLE FOR ANY ERRORS OR OMISSIONS HEREIN. This publication is strictly for subscriber’s own, non-commercial, internal use. No part
of it may be reproduced, resold, stored or transmitted in any printed, electronic or other form, or used for generating or marketing any printed or electronic publication, service or product.
July 7, 2017
Company’s Financial Strength
Stock’s Price Stability
Price Growth Persistence
Earnings Predictability
A++
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60
100
To subscribe call 1-800-VALUELINE
Page 13 of Lecture 9
Dividends
• Forecasted dividend for 2017: 1.88
Dividend Growth Rate
• Historical growth (5-yr average): 2.25%
Expected Return
• One month T-bill rate: 1%
• Market risk premium: 5.0%
• Beta: 0.85
• ) Expected return: 5.25%
Page 14 of Lecture 9
Valuation
P =
D1
1.88
=
= 62.66
r g
5.25% 2.25%
• Actual price: 65.54
• Our valuation of the company is lower than its current market value
• Does this mean that we should short the stock?
• More generally, how should we interpret and use valuation results?
? When we value (default free) bonds we know cash flows and discount rates
with certainty. Therefore, if the market price of a bond is di↵erent than
our theoretical price, we can construct an arbitrage
? When we value stocks we can only use estimates for cash flows and discount
rates. Therefore, our stock valuation results may be quite imprecise
? If the market price is di↵erent than our theoretical price it might be
because we (and not the market) are wrong
Page 15 of Lecture 9
Using Stock Valuation Results
• Although our results may be quite imprecise, they are still useful
• Uses:
? Value assets which are not traded in the market (IPOs, spino↵s, etc.)
? Understand what assumptions (on growth rates, market risk premium, etc)
the market makes to value stocks
? Trade, but only if we disagree with these assumptions very strongly
Page 16 of Lecture 9
9.3
Bubbles
• A bubble is is said to occur if an asset price exceeds its fundamental value
• However, what is the fundamental value? — A difficult question
? In theory, we can define the fundamental value as the value given by the
PV formula:
C1
C2
CT
PV =
+
+ ··· +
+ ···
1 + r1
(1 + r2)2
(1 + rT )T
? However, we have seen that the PV value formula is sensitive to assumptions about growth rate and discount rates, and therefore is generally
consistent with a wide range of values
Page 17 of Lecture 9
Famous historical examples of bubbles
• Dutch Tulip Mania (1634–1637)
Source: Jan Brueghel the Younger, Satire on Tulip Mania (1940), Frans Hals Museum, Netherlands; the image from Wikimedia
Page 18 of Lecture 9
• The South Sea Bubble (1720)
Source: Hogarthian image of the 1720 ”South Sea Bubble” from the mid-19th century, by Edward Matthew Ward, Tate Gallery
Page 19 of Lecture 9
Recent example: the Nasdaq Bubble
Page 20 of Lecture 9
9.3.1 How to detect bubbles?
• Since a bubble occurs when an asset price exceeds its fundamental value it
is natural to expect that financial ratios where one part is the price and the
other is some measure of fundamentals can provide useful information about
the bubble
• Example: The dividend yield of a stock is
DY =
D0
P0
? If the constant growth formula holds then
D0(1 + g)
P0 =
) DY (1 + g) = r
r g
g
? We can approximate this (in most cases, g is small) by
DY = r
g
Page 21 of Lecture 9
Application: the US Stock Market Valuation
• We can apply the constant growth formula at the market level to estimate
the value of the MRP consistent with the current market valuation
• By definition, the expected return on the market is r = rf + M RP
• Using DY = r
M RP = DY
g we can rewrite the MRP as
rf + g
where DY and g are the dividend yield and the dividend growth rate of the
market portfolio
Page 22 of Lecture 9
• Example (Excel):
Page 23 of Lecture 9
• The implied MRP was the lowest during the Nasdaq bubble
? Interpretation: The MRP declined or the market was overvalued
• Caveat: the results are sensitive to the assumptions about the interest rates
Page 24 of Lecture 9
9.3.2
Why do bubbles exist?
“The stock market can remain irrational longer than you can remain solvent”
— John Maynard Keynes
• To take advantage of a bubble one needs to short it
• Limits of Arbitrage
? Costs of short-selling
? Convergence risk: A profitable arbitrage trade might lose money in the
short run
Short-selling in practice
To short a share of a stock (say Gamestop), arbitrageur A must
• Find an existing owner, B , who is willing and able to lend shares
• Leave collateral (usually cash but can be treasuries) with lender B equal
to 102% (haircut) of the market value (marked and settled daily) of the
borrowed share
• If the lender is a broker-dealer, A needs to post an additional 50% in margin
• Pay a loan fee (the fee is embedded in the level of the “rebate” rate, the
interest that B pays A for use of the cash collateral); high fee stocks are
called “special”
• Pay B any dividends/distributions made by Gamestop
• B has the right to recall the share from A at any time
• A short “squeeze” occurs when increasingly optimistic investors compete
with recalled borrowers to buy shares being sold by lenders
Page 25 of Lecture 9
Case study: Volkswagen short squeeze
• 2005-2007: Porsche acquires a 30% stake in Volkswagen
• October 2007: the European Commission overturns the “Volkswagen Act”
— an esoteric German law that requires 80% controlling stake (normally
75%). Since the state of Lower Saxony owned 20.1% the law essentially
prevented any takeover of Volkswagen
• March 2008: Porsche board clears the decision to increase its Volkswagen
stake to 50%
• September 16: Porsche increases its stake to 35% but at the same time
secretly builds a call option position (a call option gives the right to buy
company shares at a fixed price; dealers who sell options usually hold the
underlying stock as a hedge)
Capital structure arbitrage
• September 2008: Unaware of the option stake, several hedge funds establish
short positions in Volkswagen ordinary shares (with voting rights) and long
positions in preference shares (without voting rights) hoping that the price
of the two share classes will converge in the future; 12% short interest
• September 26: Porsche announces that it has a 42.6% stake in ordinary
shares and in addition 31.5% in options (total 74.1%)
• Given the 20.1% stake by the state of Lower Saxony it implies that the
available shares are under 6% — really bad news for the short-sellers who are
short 12%
• September 28: Volkswagen briefly becomes the most valuable company in
the world
• September 29: Porsche provides 5% shares to the stricken shorts
• October 2008: The crisis starts. Porsche does not have enough cash to
complete the transaction
• 2012: Porsche is acquired by Volkswagen
Page 26 of Lecture 9
Case study: Convergence risk: Long-Term Capital Management (LTCM)
• Founded in 1994 by John Meriwether with $1.25B funds under management
• Two Nobel Prize winners, Myron Scholes and Robert Merton, joined as
partners
• Core strategy: convergence trades
? Convergence among European sovereign bonds bond
? Convergence between on-the-run and o↵-the-run U.S. government bonds
Page 27 of Lecture 9
LTCM Convergence Trade: On-the-run/o↵-the-run bonds
• On-the-run (newly issued) bonds are traded at premium over o↵-the-run
(issued half a year ago) bonds; on-the-run bonds are also more liquid
• Trade: bet that the spread will converge; buy o↵-the-run bonds and short
on-the-run bonds
• The price di↵erence between on-the-run and o↵-the-run bonds is small so the
trade is only profitable if the fund uses high leverage:
? LTCM used o↵-the-run bonds as collateral; because of its iconic status it
could borrow 100% of their value
? With $5B of equity, LTCM had a balance sheet of $125B
Page 28 of Lecture 9
What were the risks involved in these trades?
• If the spreads widen
? Position loses money on a marked–to–market basis ) collateral "
? A general deterioration in credit quality of the fund ) collateral "
? Investors can withdraw their funds
• As a result, the fund may need to unwind the trade, which can lead to even
larger spreads
Page 29 of Lecture 9
Bad Times
• August 17th, 1998: Russian default on sovereign debt
? Flight-to-quality: Credit spreads widened
• The fund’s positions lost money on a marked–to–market basis
• Unwinding positions by other funds and investment banks doing similar trades
made the spreads widen even further
• The knowledge that the fund was making many losses made counterparties
require greater collateral and charge greater “haircuts” on their transactions
Page 30 of Lecture 9
LTCM’s Return
Source: Jorion (2000), Risk Management Lessons from Long-Term Capital Management, European Financial Management
Page 31 of Lecture 9
Eventual Outcome
• September 23, the Fed organized a bail-out by a consortium of 14 banks
? $3.6 billion investment for 90% of LTCM
? Argument: LTCM too big to fail