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## Question: Portfolio Management Exam 66% Bond Questions 34% Single Index Model And Capm

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Lecture 7 (Weeks 8 & 9)
Fixed income securities
Dimitris Chronopoulos
Email: dc45@st-andrews.ac.uk
1
MN-5612 Portfolio Management & Investment
2
Portfolio Theory
Lecture outline:
• Types of Bonds
• Spot Rates
• Pricing Bonds
• Duration
• Convexity
• Immunization
Reading
Reily & Brown Ch17, 18, Cuthbertson & Nitzsche Ch 20, Elton
Gruber et al. Ch 20, Sharpe et al. Ch 5, 15
Bonds
• A bond is a form of debt.
• Typically the issuer of a bond (the debtor) promises to repay
the lender (the investor) the amount borrowed at some pre-
determined date in the future in adition to a fixed interest
payment.
• Because the interest paid by the issuer of the bond is fixed,
bonds are sometimes referred to as fixed income securities.
• In the UK, bond trading is carried out on the LSE by gilt-
edged market makers (GEMMS).
4
Bonds - Terminology
• Indenture – the contract between the issuer and the bondholder.
• Parts of bond identure:
– Coupon: The interest payments made to the bondholder.
– Face Value (Par Value, Principal or Maturity Value): Payment at the maturity of
the bond.
– Coupon Rate: Annual interest payment, as a percentage of face value.
– Maturity: Is the life of the loan. Bonds are often a form of long term debt, but the
can also be offered short term. They can also have no maturity date (perpetuities
or consols)
• Zero-Coupon bond - If there is only one payment made to the security
holder.
5
Types Of Bonds
• Government Bonds
– Treasury Securities (Short Term)
– Notes (Medium Term)
– Bonds (Long Term)
– Free from Default Risk (High Credit Rating)
• Corporate Bonds
– Functions the same as government bounds
– Corporations can use both the bond markets and the equity markets to
raise funds
– Pay a higher rate of interest (riskier; credit risk )
Bonds with Option Features
• Callable Bonds: are bonds that can be repaid before maturity
by the issuer. Early repayment may be restricted to a specified
date (European) or allowed at any time (American).
• Puttable Bonds: redemption date is under the control of the
bond holder.
• Convertible Bonds: corporations may sometimes issue debt
that can be converted (at a specified date and time) into a
share of the firm’s equity. In effect, bond with a call option.
• Exchangeable Bonds: grant the bondholder the right to
convert into a predetermined amount of common stock of a
specified company other than the issuing company.
Foreign Bonds and Eurobonds
• A Foreign Bond is a bond issued by a borrower in a country
different from that the borrower’s country of incorporation
(i.e. the borrower is selling debt abroad). The bond is
denominated in the currency of the country in which is sold.
Sterling denominated foreign bonds are known as bulldog
bonds.
• Eurobonds are bonds denominated in the currency of one
country, but actually sold or traded in another. A Eurosterling
bond will be denominated in sterling but sold outside the UK.
8
Pricing Zero Coupon Bonds
• Like other financial assets, the value of a bond is the present
value of its expected future cash flows:
• Where
– P is the bond price
– N is the number of years to maturity
– M is the (face value) payment at maturity
– r is the “risk-adjusted discount rate”.
P
(1 )NN
M
r
=
+
9
Spot Rates
• In practice, the market prices of zero-coupon bonds P are used to
calculate the IRR of the zero-coupon bond.
• This rate is usually called spot rate or spot yield.
• Here, the spot rate rN is the discount rate for a cash flow in year n that
can be locked in today.
– E.g., r3 (3-year spot rate) is the rate the market uses to value a single
payment three years from today.
• Zero-coupon bonds usually do not exist for maturities with more than
one year.
1
P
N
N
M
r = −
10
Pricing Coupon Bonds
• If we have a coupon bond and market-determined spot
rates exist for all maturities, the market price of the bond
is:
• By the law of one price, the coupons are discounted using
the spot rates.
1 2
2
1 2
..
(1 ) (1 ) (1 )
N N N
t N
N
C MC C
P
r r r
+
= + + +
+ + +
11
Bond Returns
• Current Yield - Annual coupon payments divided by bond price.
• Yield To Maturity (YTM)- Interest rate for which the present value of the bond’s
payments equal the market price of the bond. Yield to maturity is derived under
the assumption that coupon payments are reinvested at the rate y. The YTM is
interpreted as the IRR of a bond, if it is held to maturity.
• Holding period return (HPR) - If one holds a coupon bond between t and t+1,
the return is made up of the capital gains plus any coupon payments. HPR is
based on expectations about Pt+1.
1 2
( )
....
(1 ) (1 ) (1 )N
C C C M
P
y y y
+
= + + +
+ + +
1 1
1
t t t
t
t
P P C
R
P
+ +
+
− +
=
12
Coupon bond example
• Take a 3-year 10% coupon bond with face value M= £1000,
assuming annual coupon payments:
• Spot rates: r1=10%, r2=12%, r3=14%
• Yield-to-Maturity:
Price =
100
1.10( )
+
100
1.12( )2
+
1100
1.14( )3
= 913.1
( ) ( ) ( )
( ) ( ) ( )
%7.13
137.1
1100
137.1
100
137.1
100
y1
1100
y1
100
1
100
913.1
32
32
=
++=
+
+
+
+
+
=
y
y
13
Zero coupon bond example
• Price of 3-year zero coupon bond with face value M= £1000
• Spot rates: r1=10%, r2=12%, r3=14%
• Yield-to-Maturity:
( )
675
14.1
1000
Price
3
==
( )
3
1000
675
1 y
14%y
=
+
=
14
YTM of a Bond Portfolio (i)
• The YTM of a bond portfolio is not the average of the YTM of the
individual bonds in the portfolio.
• It is computed by determining the cashflows of the portfolio and
determining the yield that will make the PV of the portfolio’s
cashflows equal to the market value of the portfolio.
• Example: Compute the duration of the following portfolio.
Bond
Coupon
(%)
Maturity
(years)
Face Value
(millions)
Price
(millions) YTM
A 7 5 10 9.21 9
B 10.5 7 20 20 10.5
C 6 3 30 28.1 8.5
15
YTM of a Bond Portfolio (ii)
Years Bond A Bond B Bond C Portfolio
1 0.7 2.1 1.8 4.6
2 0.7 2.1 1.8 4.6
3 0.7 2.1 31.8 34.6
4 0.7 2.1 2.8
5 10.7 2.1 12.8
6 2.1 2.1
7 22.1 22.1
• The YTM of the portfolio is 9.54%.
16
Term Structure of Interest Rates
• Term Structure of Interest Rates: is the relationship between the spot rates
and their maturity dates.
• Yield Curve - Graph of the term structure.
Source: Bodie, Kane and Marcus, 2005
Forward Rates
• Forward Rate - The interest rate, fixed today, on a loan made in the
future at a fixed time.
• Forward rates of interest are implicit in the term structure of interest
rates.
t = 0 1 2 3 4…
• Note the notation: 3f4 means “the forward rate from period 3 to period
4.”
• When the beginning subscript is omitted, it is understood that the
forward rate is for one period only: 3f4 = f4 .
r1
r2
r3
1f2
2f3
3f4
18
Forward Rates Example
• What one-year forward rates are implied by the following spot
rates?
Maturity Year Spot Rate (rt) Forward Rate (ft)
1 4.0% –
2 5.0%
3 5.5%
2
2 1 1 2
2
1 2
1 2
(1 ) (1 )(1 )
(1.05) (1.04)(1 )
6.01%
r r f
f
f
+ = + +
= +
=
3 2
3 2 2 3
3 2
2 3
2 3
(1 ) (1 ) (1 )
(1.055) (1.05) (1 )
6.507%
r r f
f
f
+ = + +
= +
=
6.507%
6.01%
19
General Formula for Forward Rates
• One-period forward rates:
• N-period forward rates?
1 1
1 1
implying that ...
1 1
1
(1 ) (1 ) (1 )
(1 )
1
(1 )
n n
n n n n
n
n
n n n
n
r r f
r
f
r
−
− −
− −
−
+ + +
+
= −
+
(1 + rk +n )
k+ n (1 + rk)
k(1 +
k
fk+ n)
n
implying that...
k
fk+ n =
(1 + rk+ n )
k+ n
(1 + rk )
k
1
n
− 1
20
Risks Associated with Investing in Bonds
• Bonds are generally less risky than stocks, but they do
suffer from several types of risk::
– Credit risk - Risk of default.
– Interest rate risk - Risk of unexpected changes in rates,
causing a capital loss.
– Liquidity risk - The risk that you will not be able to sell
the bond at a price near its full value.
– Other Risks - Reinvestment risk , Call risk.
21
Bond Ratings
• Credit risk is the most important source of risk for bond
holders. As a result, various agencies (S&P, Moody’s,
Fitch, and Dominion Bond) provide bond ratings
services.
• Bond rating is an evaluation of the possibility of default
by a bond issuer. It is based on an analysis of the
issuer's financial condition and profit potential.
22
Bond Ratings (cont.)
Bond Ratings by Agency
Moody's S&P Fitch DBRS DCR Definitions
Aaa AAA AAA AAA AAA Prime. Maximum Safety
Aa1 AA+ AA+ AA+ AA+ High Grade High Quality
Aa2 AA AA AA AA
Aa3 AA- AA- AA- AA-
A1 A+ A+ A+ A+ Upper Medium Grade
A2 A A A A
A3 A- A- A- A-
Baa1 BBB+ BBB+ BBB+ BBB+ Lower Medium Grade
Baa2 BBB BBB BBB BBB
Baa3 BBB- BBB- BBB- BBB-
Ba1 BB+ BB+ BB+ BB+ Non Investment Grade
Ba2 BB BB BB BB Speculative
Ba3 BB- BB- BB- BB-
B1 B+ B+ B+ B+ Highly Speculative
B2 B B B B
B3 B- B- B- B-
Caa1 CCC+ CCC CCC+ CCC Substantial Risk
Caa2 CCC - CCC - In Poor Standing
Caa3 CCC- - CCC- -
Ca - - - - Extremely Speculative
C - - - - May be in Default
- - DDD D - Default
- - DD - DD
- D D - -
- - - - DP
Source: http://www.bondsonline.com/asp/research/bondratings.asp
23
Determinants of Bond Price Volatility (i)
Four factors determine a bond’s price volatility to changing
interest rates:
1. Par value
2. Coupon
3. Years to maturity
4. Prevailing level of market interest rate
24
Determinants of Bond Price Volatility (ii)
• In 1961, Burton Malkiel proved five important bond pricing
theorems that provide information about how bond prices change
as interest rates change.
• Any good bond portfolio manager knows Malkiel’s theorems:
1. Bond prices move inversely with yields.
• If interest rates rise, the price of an existing bond declines.
• If interest rates decline, the price of an existing bond increases.
2. Long-term bonds fluctuate more if interest rates change.
• Maturity is directly related with bond volatility.
• Long (short)-term bonds have more (less) interest rate risk.
25
Determinants of Bond Price Volatility (iii)
3. Price sensitivity increases with maturity at a decreasing rate.
• A given time difference in maturities is more important with shorter-
term bonds.
4. Lower coupon bonds respond more strongly to a given change in
interest rates, thus bond price volatility is inversely related to
coupon.
• Higher coupon bonds have less interest rate risk.
• Money in hand is a sure thing while the present value of an anticipated
future receipt is risky.
5. Price changes are greater when rates fall than they are when rates
rise (asymmetry in price changes).
• Capital gains from an interest rate decline exceed the capital loss from
an equivalent interest rate increase.
26
Duration as A Measure of Interest Rate Risk
• Portfolio manager (and bondholders in general) require a measure of bond price
volatility to calculate the risk of of their investment due to changes in the interest
rates.
• Duration (D) captures the sensitivity of the bond price to the interest rate change.
• Duration is defined as a weighted average of the maturities of the individual
payments.
• In 1938, Frederick Macaulay discovered duration which combines maturity and
coupon rate to describe a bond’s price volatility.
• There are two types of Duration used in the industry
– Macaulay duration - D (we will refer to it as simply “duration”)
– Modified duration - Dm
Macaulay Duration
1 2
1 2
1
/ (1 ) / (1 ) / (1 )
1* 2* ... *
T
T
C Y C Y C M Y
D T
P P P
+ + + +
= + + +
y = the bond’s internal yield
C = annual coupon payment
M = principal payment
T = number of years to maturity
P= current market value of the bond
Macaulay’s Duration: Example
• Consider a bond with the following characteristics:
– £100 annual coupon.
– 2 years to maturity.
– YTM =10%.
– Market Value £1,000.
– The Macaulay Duration for this bond is 1.909 years.
2£100 / (1.1) £1100 / (1.1)
1.909 1 2
£1000 £1000
= +
• Duration measures the average time taken by the security, on a
discounted basis, to pay back the original investment. The longer
the duration, the greater the risk.
Effects of the coupon
• Duration of a 7 year coupon bond
• Shaded area of each box is the present value of cash flow.
• Distance (x-axis) is a measure of time.
Duration
High C, Lower Duration
Low C, Higher Duration
⚫ Duration is similar to the distance to the fulcrum (5.1 years)
“Modified” Duration
• To calculate the % change in a bond’s price for a given change in yield, bond market
participants use a measure known as “modified duration”.
• Modified Duration is the first derivative of the bond price with respect to the interest rates. It
measures the sensitivity of bond prices with respect to changes in yields.
• It gives a measure of the slope of the price-yield curve.
Modified Duration =
Macaulay Duration
(1 +y)
31
Duration of Coupon-Bonds
• Dm is the first derivative of the bond price with respect to the
interest rates.
1 1
1
1
(1 ) (1 ) (1 )
1
1 (1 )
1 (1 )
1
T T
t t
t T t
t t
T
t
t
t
t
tT
t
C CFM
P
y y y
CFP
t
y y y
CFP
y y
t
P y P
= =
=
=
= + =
+ + +
= −
+ +
+
= −
+
D
Dm
32
Duration of zero coupon bonds
• The duration of zero-coupon bonds is the same as their
maturity.
• So long as there are no cash flows during the life of the
security, the duration of the security will be equal to its
maturity.
33
Duration Example
• Consider a 3-year 10% coupon bond selling at $107.87 to yield
7%. Coupon payments are made annually.
1
2 2
3 3
10
( ) 9.35
(1.07)
10
( ) 8.73
(1.07)
110
( ) 89.79
(1.07)
PV CF
PV CF
PV CF
= =
= =
= =
9.35 8.73 89.79
1* 2* 3* 2.7458
107.87 107.87 107.87
D
= + + =
2.7458
2.5661
1.07
mD = =
34
Properties of Duration (i)
• Duration is a measure of bond’s price volatility, as such the properties
of duration have to be in line with those of bond’s price volatility.
• Consider the following example:
Bond Macaulay Duration Modified Duration
9% / 5 years 4.13 3.96
9%/ 25 years 10.33 9.88
6%/ 5 years 4.35 4.16
6%/ 25 years 11.1 10.62
0%/ 5 years 5 4.78
0%/ 25 years 25 23.92
35
Properties of Duration (ii)
• Duration of a bond with coupons is always less than its term to
maturity.
• A zero-coupon bond’s duration equals its maturity.
• There is an inverse relationship between duration and coupon.
• There is a positive relationship between term to maturity and
duration, but duration increases at a decreasing rate with maturity.
• There is an inverse relationship between YTM and duration.
• When calculating duration for semiannual bonds, the resulting
duration will be in semiannual periods. You must adjust this back to
annual by dividing by 2.
36
• Recall that:
• To an approximation (Δy→0):
• Rearranging:
• So Dm measures the sensitivity of the % change in bond price to
changes in yield.
• In other words, if interest rates increase 1%, then the
approximate percentage change in a bond price is (- Dm).
Estimating Bond Price Changes Using Duration
m
P
y
D
P
= −
m
P
y
D
P
= −
m
P
D y
P
= −
37
Modified Duration Example I
• Compare the price sensitivities of:
– Two-year 8% coupon bond with duration of 1.8853 years
– Zero-coupon bond with maturity AND duration of 1.8853 years
• Semiannual yield = 5%
• Suppose yield increases by 1 basis point to 5.01%
• Upshot: Equal duration assets are equally sensitive to interest rate
movements.
Original Price New Price % Change
Coupon bond 964.54 964.19 -.035
Zero bond 835.53 835.24 -.035
38
Modified Duration Example II
• Compare the price sensitivities of:
– Bond A has a modified duration of 4.08 years
– Bond B has a modified duration of 6.14 years
• Annual yield = 3.4%
• Suppose yield increases by 200 basis points to 5.4%
• Why were the estimated changes so far off?
Estimated % Change Actual % Change
Bond A -8.16 -8.6
Bond B -12.28 -13.42
39
Duration Limitations
• First-order approximation. The relationship between price and yield
is non-linear, but duration measures linear changes.
• Accurate for small changes in yield.
• Depends on parallel shifts in a flat yield curve.
• Strictly applicable only to option-free (e.g., non-convertible) bonds.
• However, modified duration correctly tells us that Bond B will
decrease in value significantly more than Bond A. This is really
the important result and the primary scope of duration.
40
Pictorial look at Modified Duration
P0
y
Bond’s price estimation error based on Dm
Bond’s price estimation error based on Dm
Dm is the slope of the tangent drawn
to the price-to-yield curve
Actual Price
Source: Reilly and Brown, 2006
A
41
Convexity
• Measures how much a bond’s price-yield curve deviates from a
straight line.
• Second derivative of price with respect to yield divided by bond
price
• Allows us to improve the duration approximation for bond price
changes.
( )
( )
2
2
2 2
1
2
2
1
(1 ) 1
1
Convexity
T
t
t
t
CFP
t t
y y y
P
P y
=
= +
+ +
=
42
• Recall approximation using only duration:
• The predicted percentage price change accounting for
convexity is:
m
P
D y
P
= −
( ) 2
1
Convexity ( )
2
m
P
D y y
P
= − +
Estimating Bond Price Changes Using Convexity
43
Source: Elton, Gruber, Brown,
and Goetzman, 2007
Actual price and estimated price.
44
Convexity example (i)
• Consider a 20-year 9% coupon bond with face value of
$100 selling at $134.6722 to yield 6%. Coupon payments
are made semiannually.
• Dm= 10.66
• Convexity = 164.106.
45
• If yields increase instantaneously from 6% to 8%, the percentage
price change of this bond is given by:
– Duration approximation :
–10.66 0.02 = -0.2132 = –21.32%
– Duration-Convexity approximation :
–10.66 0.02 + 0.5 164.106 (0.02)2 = –0.2132 + 0.328 = –18.04%
• Actual price change = –18.40%.
Convexity example (ii)
46
Convexity example (iii)
• What if yields fall by 2%?
• If yields decrease instantaneously from 6% to 4%, the percentage price change
of this bond is given by:
– Duration approximation :
–10.66 –0.02 = 0.2132 = 21.32%
– Duration- Convexity approximation :
–10.66 –0.02 + 0.5 164.106 (–.02)2 = 0.2132 + 0.328 = 24.6%
• Actual price change=25.03%
• Note that predicted change is NOT SYMMETRIC.
47
Applications of Duration-Immunization
• These methods allows portfolio managers to be relatively certain
that they will be able to meet a given promised stream of cash
flows.
• Immunization is accomplished by calculating the duration of the
promised outflows and then investing in a portfolio of bonds with
identical duration.
• If a portfolio has one-third of its funds invested in bonds with
duration of six years and two thirds in bonds having duration of
three years, the portfolio has duration 1/3x6+2/3x3 = 4
48
Applications of Duration-Immunization
• Consider the situation where the portfolio manager has one and
only one cash outflow to make from the portfolio-an amount of
$1,000,000 which is to be paid in two years. Because there is
only one cash flow the duration is two years.
• Assume the available bonds are:
– A bond with maturity three years (par = 1,000 coupon = 80 YTM =10%)
– A set of bonds that mature in one year providing the holder a single
payment of $1,070 (coupon +par). The yield to maturity of the bonds is
10% and the price 972.73.
49
Applications of Duration-Immunization
Time Cashflow
(CF)
Discount
Factor
PV of CF PV of
CFxTime
1 80 .9091 72.73 72.73
2 80 .8264 66.19 132.23
3 1,080 .7513 811.40 2,434
Sum 950.25 2,639.17
Duration =2,639.17/950.25
= 2.78 years
50
Applications of Duration-Immunization
• The manager can buy the one year bonds and reinvest the
proceeds one year from now in another one year issue- if interest
rate decline the funds will be reinvested at a lower rate
• A second alternative is to invest all the money in the three year
bond. If interest rates increase after in two years the price of the
bond will fall.
• A third alternative is to invest part of the portfolio in the one year
bonds and part in the three year bonds.
51
Applications of Duration-Immunization
• The manager can invest the funds to both types of bonds. How
much should be placed in each issue?
W1 + W2 = 1
1*W1 + 2.78*W2 = 2
• W1 and W2 are the weights of the portfolio’s funds invested in
the bonds with maturities of one and three years.
• The solution of the two equations is W1 = 0.4382 and W2 =
0.5618.
52
Applications of Duration-Immunization
• The manager will need $1,000,000/(1.1)2 = $826,446 in order to purchase the
bonds that will create a fully immunized portfolio. With this money $362,149
would be used to buy one year bonds and $464,297 would be used to buy
three year bonds. Given market prices we buy 372 one year bonds and 489
three year bonds.
• Why immunization works? If yields rise then the portfolio’s losses owing to
the selling of the three year bond will be exactly offset by the gains from
reinvesting the maturing one year bonds. If yields fall the losses from
reinvesting the maturing one year bonds at a lower rate will be exactly offset
by being able to sell the three-year bonds at a premium.
53
Summary
1. Types of Bonds
2. Spot Rates
3. Pricing Bonds
4. Duration
5. Convexity
6. Immunization
1
MN-5612 Portfolio Management & Investment
Lecture 6 (Week 7)
The CAPM
Dimitris Chronopoulos
Email: dc45@st-andrews.ac.uk
mailto:dc45@st-andrews.ac.uk
2
Portfolio Theory
Lecture outline:
• Asset Pricing Problem.
• Assumptions of the CAPM.
• Separation Theorem.
• CAPM-SML.
• Over- and under-priced securities.
• Estimating betas.
Reading: Bodie et al., Ch9; Sharpe et al., Ch9 & Ch10; Elton et al., Ch13;
3
1. Asset Pricing Models
▪ These are economic models describing how (expected returns on) equities
are or should be priced in equilibrium.
▪ The focus of these models is on what would happen to stock prices if
everybody invests in a similar fashion and markets are perfect.
▪ As such they provide a benchmark against which we can compare actual
prices.
▪ The CAPM follows from the Tobin extension to the Markowitz model but
requires additional assumptions.
▪ Its systematic risk is given by its beta coefficient. A stock’s beta is a
measure of its co-movement with the market portfolio.
4
2. Assumptions of CAPM
▪ All investors are assumed to follow the mean-variance approach i.e. the
risk-averse investor will ascribe to the methodology of reducing portfolio
risk by combining assets with counterbalancing correlations.
▪ Assets are infinitely divisible.
▪ There is a risk-free rate at which an investor may lend or borrow.
▪ Taxes & transactions costs are irrelevant.
▪ All investors have same holding period.
▪ Risk-free rate is the same for all investors.
▪ Information is freely and instantly available to all investors.
5
▪ Investors have homogeneous expectations, i.e., all investors have the
same expectations w.r.t the inputs that are used to derive the Markowitz
efficient portfolios: asset returns, variances and correlations.
▪ Markets are assumed to be perfectly competitive i.e. the number of buyers
and sellers is sufficiently large, and all investors are small enough relative
to the market, so that no individual investor can influence an asset’s price.
Consequently, all investors are price takers, and the market price is
determined where there is equality of supply and demand.
▪ All investor’s face the same efficient set and the only reason why they will
choose different portfolios is their risk tolerance.
▪ Despite the fact that the chosen portfolios will be different, will choose the
same combination of risky assets.
6
3. The Separation Theorem
▪ The optimal combination of risky assets for an investor can be determined
without knowledge of the investor’s preferences.
▪ With CAPM each person faces the same linear set, each person will be
investing in the same tangency portfolio.
▪ Another important implication of the CAPM is that in equilibrium each
security must have a non zero proportion to the composition of the
tangency portfolio.
▪ That means that in equilibrium no security has a weight of zero.
▪ What if the outstanding shares are not enough to cover the demand for
the tangency portfolio?
▪ After adjustments the market will be brought in equilibrium.
▪ Each investor will hold a certain amount of the risky asset.
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▪ The market price will be at a price where the number of shares
outstanding will equal the amount demanded.
▪ In equilibrium the proportions of the tangency portfolio will correspond to
the proportions of what is known as the market portfolio (M).
▪ Fama (1970) showed that M must consist of all assets available to
investors, and each asset must be held in proportion to its market value
relative to the total market value of all assets.
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( )
( ) M f
p f p
M
E r r
E r r
−
= +
M is the market portfolio (M).
The tangent line is the Capital
Market Line (CML).
Portfolios on the left of (M)
represent combinations of risky
assets and the risk-free asset.
Portfolios to the right of (M)
include purchases of risky
assets made with funds
borrowed at the risk-free rate.
9
▪ The theoretical result that all investors will hold a combination of the risk-
free asset and the market portfolio is known as the two-fund monetary
separation theorem.
▪ One fund consists of the risk-free asset and the other is the market
portfolio.
▪ Thus, the numerator of the slope coefficient is the expected return of the
market beyond the risk-free return (the market risk premium). It is a
measure of the reward for holding the risky market portfolio rather than
the risk-free asset.
▪ The denominator is the risk of the market portfolio. Thus, the slope
measures the reward per unit of market risk.
▪ The slope of the line determines the additional return needed to
compensate for a unit change in risk. The slope of the CML is also called
the market price of risk.
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4. Derivation of the CAPM
▪ The market portfolio is the portfolio of all risky assets traded in the stock
market. The total capitalization of a stock i is given by:
price per share shares outstanding
i i
MCAP = .
▪ If we have N stocks, the total market capitalization is given by:
1
N
M i
i
MCAP MCAP
=
.
▪ The market portfolio is the portfolio with weights
,
1, 2......i
i M
M
MCAP
w i N
MCAP
= .
▪ Assume with have K investors in the economy and every investor j has
wealth
j
W . Let f
j
W be the amount invested in the riskless asset and
f
j j
W W− be the wealth invested in the tangent portfolio. Denote as
T
w the
weights of the tangency portfolio.
11
▪ Stock market equilibrium requires that demand equals supply:
1
( )
K
f
j j T M M
j
W W MCAP
=
− = w w
▪ The safe asset is in zero net supply,
1
0
K
f
j
j
W
=
= , and all wealth is invested in
the stock market,
1
K
j M
j
W MCAP
=
= , so T M=w w .
▪ We showed what happens when investors behave according to normative
models. But what are the implications for the individual security?
▪ The capital market line represents the equilibrium relationship between
the expected return and standard deviation for efficient portfolios.
▪ Individual securities plot below the line because a security when held by
itself is an inefficient portfolio.
▪ The CAPM does not imply any particular relationship between the expected
returns and standard deviation (total risk) of an individual security.
12
▪ Remember that the standard deviation of each portfolio is given by:
1/ 2
,
1 1
N N
P i j i j
i j
ww
= =
=
,
where
i
w and
j
w denote the proportions invested in security i , j
respectively and
,i j
denotes the covariance.
▪ The standard deviation of the market portfolio is given by:
1/ 2
, , ,
1 1
N N
M i M j M i j
i j
w w
= =
=
▪ It can be shown that another way of writing the previous equation is:
1/ 2
1, , 1, 2, , 2, , , ,
1 1 1
...
N N N
M M j M j M j M j N M j M N j
j j j
w w w w w w
= = =
= + + +
13
▪ The covariance of security i with the market portfolio,
,i M
, can be
expressed as the weighted average of every security’s covariance with
security i .
, , ,
1
N
j M i j i M
j
w
=
=
▪ Hence we get the fundamental relationship:
1/ 2
1, 1, 2, 2, , ,
...
M M M M M N M N M
w w w = + + +
▪ Therefore the standard deviation of the market portfolio is equal to the
weighted average of the covariances of all securities.
▪ Each investor under the CAPM holds the market portfolio and is concerned
with its standard deviation because this will alter the CML.
▪ The contribution of each security to the standard deviation of the market’s
portfolio depends on the size of the covariance with the market portfolio.
14
▪ Hence, the relevant measure of risk for security i is its covariance with the
market portfolio.
▪ Securities with larger covariances will be viewed by investors as
contributing more to the risk of the market portfolio.
▪ Securities with larger standard deviation should not be viewed necessarily
as adding more risk to the market portfolio.
▪ Securities with larger values of covariance with the market should have
larger expected returns.
▪ The relationship between covariances and expected returns is known as
security market line (SML).
,2
( )
( ) M f
i f i M
M
E r r
E r r
−
= +
15
▪ Or in the more familiar beta version
( )
,
( ) ( )
i f M f i M
E r r E r r = + − , where ,
, 2
i M
i M
M
=
16
5. Security Market Line
Covariance -Version
Beta -Version
17
▪ Betas are standardized around one.
➢ 1 = ... Average risk investment.
➢ 1 ... Above Average risk investment.
➢ 1 ... Below Average risk investment.
➢ 0 = ... Riskless investment.
▪ The average beta across all investments is one (the market).
▪ The beta of a portfolio is the weighted sum of the individual
betas,
,
1
N
P i i M
i
w
=
= .
▪ You can find assets with negative betas. These assets may have negative
expected returns! Why? Because they provide a positive payoff when most
needed, that is, when the market drops. An example of such an asset is a
put option.
18
6. Over- and underpriced securities.
▪ Figure 3 and the SML have security-price implications.
Figure 3
Under- and Overpriced Securities and SML.
19
▪ Points between the SML and the vertical axis, such as point L represent
securities whose prices are lower than they would be in equilibrium. Since
points such as L represents securities with unusually high returns for the
amount of systematic risk they bear, they will enjoy strong demand, which
will bid their prices up until their equilibrium rate of return is driven back
onto the SML at point L.
▪ Likewise, securities represented by points between the SML and the
horizontal axis represent securities whose prices are too high. Securities
such as point H do not offer sufficient return to induce rational investors
to accept the amount of systematic risk they bear. As a result, their prices
will fall, owing to lack of demand. Their prices will continue to fall until
they reach the SML at a point such as H .
20
7. Estimating betas
▪ If β is constant over time, then the beta for a security could be estimated
by examining the historical relationship between the excess returns on the
security and the excess returns on a market index (i.e. ordinary least
squares (OLS) can be used to estimate ex-post betas).
▪ Apart from many econometric issues, there will be a difference in the
calculated beta depending on the following factors.
➢ The length of time over which the return is calculated (e.g. daily,
weekly, monthly).
➢ The number of observations used.
➢ The specific time period used.
➢ The market index selected.
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▪ Find a proxy for the market portfolio (use the SP 500 for US or FTSE 100
for UK). Run the following regression and check the statistical significance
of the parameter estimates
( )
j f M f j
r r a r r − = + − +
✓ 0 … stock did better than expected during estimation period
✓ 0 = … stock did as well as expected during estimation period
✓ 0 … stock did worse than expected during estimation period
▪ Many hedge funds claim that are “alpha generators”. They deliver returns
above the returns justified for holding market returns.
▪ The R squared (
2R ) of the regression provides an estimate of the
proportion of the risk (variance) of a firm that can be attributed to market
(systematic) risk .
▪ The rest (
21 R− ) can be attributed to firm specific (unsystematic) risk.
22
Example:
▪ The following information is provided for a stock market:
▪ Calculate the SML from the information given.
▪ Calculate the beta for securities A and B.
▪ You are told that the mean rates of return for securities A and B are 7.5%
and 4.6%, respectively. What would you infer from this information in the
context of the CAPM?
23
Answer:
▪ Beta can be calculated using:
2 2
0.5 0.6
0.6 1.5, 0.2 0.6
0.2 0.2
im im i m
i A B
m m
= = = = = − = −
▪ From the SML we may calculate the following:
0.05 (0.03 1.5) 9.5%
0.05 (0.03 0.6) 3.2%
A
B
r
r
= + =
= − =
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7. Conclusion.
▪ The CAPM states, based on an equilibrium argument, that the solution to
the Markowitz problem is that the market portfolio is the one fund (and
only fund) of risky assets that anyone need hold. This fund is
supplemented only by the risk-free asset.
▪ The investment recommendation that follows this argument is that an
investor should simply purchase the market portfolio.
▪ The next question, however, is what makes up the market portfolio. If the
world of equity securities is taken as the set of available assets, then each
person should purchase some shares in every available stock, in
proportion dictated by the market value of each stock relative to the total
market value of stocks outstanding. It is not necessary to go to the trouble
of analyzing individual stocks and computing a Markowitz solution. Just
buy the market portfolio.
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▪ Since it would be cumbersome for an individual to assemble the market
portfolio in this context, mutual funds have been designed to match the
market portfolio closely. Such funds are known as index funds, since they
attempt to duplicate the portfolio of a major stock market index, such as
the S&P 500, the Russell 3000 or CRSP. An investor who truly believes in
CAPM being the right asset pricing model would purchase one of these
index funds as well as some risk-free securities such as U.S. Treasury bills.

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