investments - universidade nova de...

InvestmentsMBA teaching notes

Joao Pedro PereiraNova School of Business and Economics

Universidade Nova de Lisboa

[email protected]

http://docentes.fe.unl.pt/∼jpereira/

April 6, 2016

Contents

1 Portfolio Choice 51.1 Risk and return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Holding period return . . . . . . . . . . . . . . . . . . . . . 51.1.2 Expected return and standard deviation . . . . . . . . . . . 61.1.3 Risk premium and Sharpe Ratio . . . . . . . . . . . . . . . 71.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Risk Aversion and Capital Allocation . . . . . . . . . . . . . . . . . 81.2.1 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Portfolio rate of return . . . . . . . . . . . . . . . . . . . . . 101.2.3 Portfolios of one risky asset and a risk-free asset . . . . . . 131.2.4 Risk tolerance and asset allocation . . . . . . . . . . . . . . 141.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Optimal Risky Portfolios . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1 Diversification and portfolio risk . . . . . . . . . . . . . . . 161.3.2 Portfolios of two risky assets . . . . . . . . . . . . . . . . . 161.3.3 Portfolios of two risky assets and a risk-free asset . . . . . . 191.3.4 The Markowitz portfolio optimization model . . . . . . . . 221.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 The Capital Asset Pricing Model 242.1 Assumptions and derivation . . . . . . . . . . . . . . . . . . . . . . 242.2 Efficient frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 Capital Market Line . . . . . . . . . . . . . . . . . . . . . . 262.2.2 Risk aversion and the market risk premium . . . . . . . . . 27

2.3 Expected returns on individual securities . . . . . . . . . . . . . . . 282.3.1 Security Market Line . . . . . . . . . . . . . . . . . . . . . . 282.3.2 Interpretation of beta . . . . . . . . . . . . . . . . . . . . . 292.3.3 Beta of a portfolio . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Arbitrage Pricing Theory and Factor Models 323.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Example with 1 factor: the Market model . . . . . . . . . . . . . . 35

2

Contents 3

3.3 Example with 3 factors: the Fama-French model . . . . . . . . . . 373.3.1 Pricing equation . . . . . . . . . . . . . . . . . . . . . . . . 373.3.2 Details on the factors . . . . . . . . . . . . . . . . . . . . . 37

3.4 Portfolio performance evaluation . . . . . . . . . . . . . . . . . . . 383.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4.2 Adjusting returns for risk . . . . . . . . . . . . . . . . . . . 38

4 Bond Markets 414.1 Rates of return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Effective Annual Rate . . . . . . . . . . . . . . . . . . . . . 414.1.2 Annual Percentage Rate . . . . . . . . . . . . . . . . . . . . 424.1.3 Continuous compounding . . . . . . . . . . . . . . . . . . . 424.1.4 Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Bond Prices and Yields . . . . . . . . . . . . . . . . . . . . . . . . 434.2.1 Bond Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.2 Yield to Maturity . . . . . . . . . . . . . . . . . . . . . . . 464.2.3 Holding-Period Return . . . . . . . . . . . . . . . . . . . . . 474.2.4 Default risk and ratings . . . . . . . . . . . . . . . . . . . . 484.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3 Term structure of interest rates . . . . . . . . . . . . . . . . . . . . 484.3.1 Spot rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.2 Forward rates . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.3 Expectations of future interest rates . . . . . . . . . . . . . 504.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Bond management . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4.1 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4.2 Active bond management . . . . . . . . . . . . . . . . . . . 544.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 The efficient market hypothesis 565.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Random walks and the EMH . . . . . . . . . . . . . . . . . . . . . 565.3 Implications of the EMH . . . . . . . . . . . . . . . . . . . . . . . . 575.4 Are markets efficient? . . . . . . . . . . . . . . . . . . . . . . . . . 585.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Futures markets 606.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.1.1 Forward contract . . . . . . . . . . . . . . . . . . . . . . . . 606.1.2 Futures contract . . . . . . . . . . . . . . . . . . . . . . . . 626.1.3 Differences between Futures and Forwards . . . . . . . . . . 63

6.2 Trading strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.2.1 Speculation and leverage . . . . . . . . . . . . . . . . . . . . 646.2.2 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Contents 4

6.2.3 Spread trading . . . . . . . . . . . . . . . . . . . . . . . . . 666.3 Futures prices of stock indices . . . . . . . . . . . . . . . . . . . . . 686.4 Investment strategies with stock-index futures . . . . . . . . . . . . 71

6.4.1 Creating synthetic stock positions . . . . . . . . . . . . . . 716.4.2 Hedging an equity portfolio from market risk . . . . . . . . 73

6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7 Solutions to Problems 76

Bibliography 77

Chapter 1

Portfolio Choice

1.1 Risk and return

Based on chapter 5 of Bodie, Kane, and Marcus (2014)

1.1.1 Holding period return

Definition 1.1.1: Holding Period Return

The Holding Period Return (HPR) on an investment from today (time 0)until T years from now is

HPR =P (T )− P (0) + Payouts

P (0)(1.1)

where• P (t) is the price of the asset at time t

• “Payouts” represent cash income from the asset between 0 and T(dividends for stocks, coupons for bonds), assumed to be paid at theend of the holding period.

Instead of HPR, will usually just say “rate of return” or simply “return”, anddenote it by r.

5

1.1. Risk and return 6

1.1.2 Expected return and standard deviation

True population moments

We want to characterize the probability distribution of returns across future statesof nature (“scenarios”).

Definition 1.1.2: Mean and Variance

For the random rate of return r,

µ :=E[r] =

S∑

s=1

p(s)r(s)

σ2 :=Var[r] =S∑

s=1

p(s)[r(s)− µ]2

where• s = 1, . . . , S are the possible states of nature (scenarios)

• p(s) is the probability of state s occuring.

The standard-deviation is σ =√

Var[r].

Example 1.1.1. Stock X is trading at $10. You estimate the follow-ing scenarios for next year:

State of Market Prob Year-end price Dividends

Expansion 0.6 $ 13 $ 1Recession 0.4 $ 8 $ 0

Compute the mean and standard-deviation of returns.

Time-series estimation

The “true” moments are not observable, so we have to estimate the inputs to ourmodels

1.1. Risk and return 7

Estimation of mean and variance

Using a sample of T past observations or realized returns,

µ =1

T

T∑

t=1

rt

σ2 =1

T − 1

T∑

t=1

[rt − µ]2

1.1.3 Risk premium and Sharpe Ratio

Terminology for returns in excess of the risk-free rate:

1. Excess return is the difference in any particular period between the actualrate of return on a risky asset and the actual risk-free rate.

2. Risk premium is the difference between the expected HPR on a risky assetand the risk-free rate

The Sharpe Ratio, or Reward-to-Volatility Ratio, is an important measure ofthe trade-off between reward and risk.

Definition 1.1.3: Sharpe Ratio

Given the forward-looking true population moments and a constant risk-freerate, the Sharpe Ratio (SR) is

SR =Risk premium

Std-dev of return

To estimate the SR from historical returns,

SR =Average excess return

Std-dev of excess return

Example 1.1.2. Continuing the previous example for stock X, sup-pose that the 1-year risk free rate is 3%. Compute the Risk Premiumand Sharpe Ratio.

Answer: SR = 0.4423

1.2. Risk Aversion and Capital Allocation 8

1.1.4 Exercises

Ex. 1 — Bodie, Kane, and Marcus (2014) problems at the end of chapter 5:CFA problems 1 through 5 (pages 165–166)

1.2 Risk Aversion and Capital Allocation


1.2.1 Risk Aversion

Definition

A fair game is a gamble with an expected payoff of zero. Example: toss a coinand double or loose your monthly salary.

Definition 1.2.1: Risk aversion

An investor is risk averse if he wishes to avoid a fair game.

Depending on whether you like a fair game or not, you are either:- Risk Averse- Risk Neutral- Risk Lover

We assume that most investors are risk averse:

• They only take risk if expecting a positive risk premium.

• Market prices show that most investors are risk averse

Modelling risk aversion

Different risky portfolios can be ranked with an expected utility function.


Expected utility function

If the investor has mean-variance preferences, the expected utility functionis:

U(r) = E[r]−1

2×A× Var[r] (1.2)

Remarks:

1. The parameter A characterizes the investor

• A > 0 means risk averse

• A = 0 means risk neutral

• A < 0 means risk lover (these investors are usually at the casino)

2. This utility score U can be interpreted as a certainty equivalent rate ofreturn.

• The investor is indifferent between a given risky investment that gen-erates a utility U and a risk-free investment at rate U

Example 1.2.1. (This is concept check 6.2 in BKM). A portfolio hasan expected rate of return of 20% and standard deviation of 30%.T-bills offer a safe rate of return of 7%.

1. Would an investor with risk-aversion parameter A = 4 prefer toinvest in T-bills or the risky portfolio?

2. What if A = 2?

Mean-Variance Dominance

We say that portfolio x mean-variance dominates portfolio y if all risk-averseinvestors with mean-variance preferences prefer x to y, regardless of their particularvalue of A (their degree of risk aversion).


Proposition 1.2.1: Mean-Variance Dominance

Asset x mean-variance dominates asset y iff:

µx ≥ µy and σx < σy

or µx > µy and σx ≤ σy

where µi = E[ri] and σi =√

Var[ri]

✲σ

✻

E[r]

1.2.2 Portfolio rate of return

Definition

Definition 1.2.2: Rate of return on a portfolio

The return on a portfolio is:

rp =

I∑

i=1

wiri (1.3)

where

• I = number of securities in the portfolio

• wi = proportion of funds invested in security i

• ri = return on security i


Example 1.2.2. Consider the following portfolio:

Stock A Stock B

Initial Investment $40,000 $60,000P0 $20 $10Initial Number of shares ... ...P1 $24 $11

Compute the portfolio’s terminal total value:

This implies a portfolio return of 114 000/100, 000 − 1 = 0.14.

Now check that (1.3) gives the same number:

rp = wara + wbrb = . . . = 0.14

Mean and variance

Proposition 1.2.2: Portfolio mean

For a portfolio with 2 assets (I=2):

µp := E[rp] = E

[2∑

i=1

wiri

]

= w1E[r1] + w2E[r2]

and

Proposition 1.2.3: Portfolio variance

For a portfolio with 2 assets (I=2):

σ2p := V ar[rp] = Var

[2∑

i=1

wiri

]

= w21σ

21 + w2

2σ22 + 2w1w2Cov(r1, r2)


Remarks about covariance:

1. By definition, Cov(r1, r2) = E[(r1 − E[r1])(r2 − E[r2])].

2. We sometimes denote σ12 := Cov(r1, r2).

3. The linear correlation coefficient between r1 and r2 is a more intuitive mea-sure and is defined as

ρ :=Cov(r1, r2)

σ1σ2

Always have −1 ≤ ρ ≤ +1.

Example 1.2.3. Consider two stocks with the following parameters:

Stock µ σ

X 10% 20%Y 20% 40%

The correlation is 0.1. We form the following portfolio: wx = 0.4, wy =0.6.

Check that the portfolio mean is 0.16 and the standard-deviation is0.2605.


1.2.3 Portfolios of one risky asset and a risk-free asset

Mean and variance

Consider a complete or combined portfolio c composed of two assets:

• A risk-free asset, such as a T-Bill (denote it by f).

• A risky asset, which may itself be a portfolio of risky assets (denote it by p).

The risk-free asset is not random and thus has Var[rf ] = 0 and σfp = 0.Therefore,

E[rc] = wfrf + wpE[rp]

V ar[rc] = w2pσ

2p ⇒ σc = wpσp

whererc = return on the complete portfolio.

Capital allocation line

Proposition 1.2.4: Capital allocation line

All possible combinations of the risk-free asset and the risky portfolio p lieon the following straight line:

CAL(p): E[rc] = rf +E[rp]− rf

σpσc

Proof. Using the previous equations,

E[rc] = wfrf + wpE[rp]

⇒E[rc] = (1− wp)rf +wpE[rp]

⇒E[rc] = (1−σcσp

)rf +σcσp

E[rp]

⇒E[rc] = rf +E[rp]− rf

σpσc


Example 1.2.4. Consider rf = 7% and a portfolio with E[rp] = 15%and σp = 22%. All possible combinations of these two assets plotalong the Capital Allocation Line:

✲σ

✻

E[r]

For example, plot the combined portfolio for wp = 0.75.

Plot also a leveraged combined portfolio with wp = 1.5. This requireswf = −0.5 (what does this mean?).

The slope of the CAL isE[rp]−rf

σp. This slope is called “reward-to-variability”

ratio. It is also called the Sharpe ratio. The higher this ratio, the better.

1.2.4 Risk tolerance and asset allocation

Proposition 1.2.5: Best complete portfolio

For an investor with mean-variance preferences, the best combination of therisk-free asset and the risky portfolio p is:

w∗

p =E[rp]− rf

Aσ2p

and w∗

f = 1− w∗

p (1.4)

Proof. The investor gets the following utility from a portfolio on CAL(p):

U(rc) = E[rc]− 0.5×A× Var[rc]

= wfrf + wpE[rp]− 0.5×A× w2pσ

2p

The investor picks the particular portfolio on the CAL that maximizes his utilityfunction.

maximizewp

(1− wp)rf + wpE[rp]− 0.5 ×A× w2pσ

2p


The first-order condition is:

− rf +E[rp]−A× σ2p × wp = 0

⇒w∗

p =E[rp]− rf

Aσ2p

Example 1.2.5. (Example 6.4 in BKM) Continuing the previous ex-ample, check that the optimal portfolio for an investor with A = 4 isw∗

p = 0.41. Plot the optimal portfolio on the CAL.

✲σ

✻

E[r]

1.2.5 Exercises

Ex. 2 — Bodie, Kane, and Marcus (2014) problems at the end of chapter 6 (p.192): 2, 13–19

Ex. 3 — The return on stock x over the next year will depend on the state ofthe market as follows:

State of Market Prob rxExpansion 0.6 0.3Recession 0.4 -0.1

1.3. Optimal Risky Portfolios 16

The risk-free rate is 2%.What is the Sharpe Ratio of a portfolio that invests 60% in stock x and 40% inthe risk-free asset?

1.3 Optimal Risky Portfolios


1.3.1 Diversification and portfolio risk

The main idea is “Don’t put all your eggs in one basket”. Different stocks responddifferently to economic shocks. Diversifying your funds into several assets reducesthe risk of the overall portfolio.

The total risk can be decomposed into two parts:

1. The part of the risk that can be easily eliminated through diversification.This is called:

(a) unique risk or firm-specific risk;

(b) nonsystematic risk; or

(c) diversifiable risk

2. The part of the risk that cannot be eliminated and remains even after di-versifying. This is called:

(a) market risk;

(b) systematic risk; or

(c) nondiversifiable risk

1.3.2 Portfolios of two risky assets

Mean and variance

Suppose there are just two risky assets (stocks). Recall that the mean and varianceof the return on the portfolio formed by these two assets is:

E[rp] = w1E[r1] + w2E[r2]

V ar[rp] = w21σ

21 + w2

2σ22 + 2w1w2σ1σ2ρ

where ρ is the correlation coefficient between r1 and r2 (−1 ≤ ρ ≤ +1).


Figure 1.1: Portfolio diversification (fig 7.2 in BKM)

Correlation effects

The investment opportunity set depends critically on the correlation betweenstocks. The smaller the correlation coefficient, the greater the benefits from diver-sification.

Perfect positive correlation (ρ = 1). There is no gain from diversificationsince the assets are essentially identical (the return on one asset is a linear functionof the other). The portfolio standard-deviation is equal to the weighted averageof the two standard-deviations

σp = w1σ1 + (1− w1)σ2

which means that all possible portfolios lie on the straight line between the twoassets (in σ, µ - space).

Imperfect correlation (−1 < ρ < +1). Now we have the diversificationbenefit. At each level of µp, the corresponding σp is less than in the ρ = 1 case.This is because σ2

p increases in ρ (∂σ2p/∂ρ = 2w1w2σ1σ2 > 0).

Whereas the expected return on the portfolio is always the weighted average ofexpected returns on the individual assets, the standard-deviation of the portfoliois now less than the weighted average of the individual standard-deviations.

Note that only the portfolios on the upper part of the curve are efficient, thatis, they (mean-variance) dominate the ones on the lower part of the curve.


Perfect negative correlation (ρ = −1). For this (theoretical) case wewould be able to construct a risk-free asset.

Plot all these cases:

✲σ

✻

E[r]

Minimum variance portfolio

The Minimum-Variance Portfolio (MVP) has the smallest possible risk.

Proposition 1.3.1: Minimum variance portfolio

Given two risky assets, the MVP is given by

w1 =σ22 − σ12

σ21 + σ2

2 − 2σ12and w2 = 1− w1

Proof. The problem is :

minimizew1,w2

σ2p

s.t. w1 + w2 = 1

orminimize

w1

w21σ

21 + (1− w1)

2σ22 + 2w1(1− w1)σ12

The foc for w1 is:

2w1σ21 − 2(1− w1)σ

22 + 2(1− w1)σ12 − 2w1σ12 = 0

⇒w1 =σ22 − σ12

σ21 + σ2

2 − 2σ12


Example 1.3.1. Consider two stocks:

Security µ σ

1 0.10 0.152 0.14 0.20

with ρ12 = 0.2.

Check that the return on MVP has a mean of 11.31% and a standarddeviation of 13.08%. Note that MVP standard deviation is smallerthan both individual standard deviations.

Example 1.3.2. Continuing the previous example, assume instead anegative correlation of ρ12 = −0.3. Check that the return on MVPhas a mean close to the previous case (11.57%), but a much smallerstandard deviation of σMV P = 10.09%.

1.3.3 Portfolios of two risky assets and a risk-free asset

Tangency portfolio

The optimal risky portfolio to combine with a risk-free asset is the one that pro-duces the steepest Capital Allocation Line. It is called the Tangency portfolio.


✲σ

✻

E[r]

Proposition 1.3.2: Tangency portfolio

Given two risky assets and a risk-free rate rf , the tangency portfolio is

w1 =re1σ

22 − re2σ12

re1σ22 + re2σ

21 − (re1 + re2)σ12

and w2 = 1− w1

with rej = E[rj ]− rf , for j = 1, 2.

Proof. We want to find the portfolio that maximizes the slope of the CAL goingthrough it:

maximizew1,w2

E[rp]− rfσp

s.t. E[rp] = w1 E[r1] + w2 E[r2]

σp = [w21σ

21 + w2

2σ22 + 2w1w2σ12]

1/2

w1 + w2 = 1

Substitute w1 = 1− w2 and solve the foc.

Example 1.3.3. Continuing the previous example with ρ = 0.2, fur-ther assume rf = 6%. Check that the Tangency portfolio is (w1 =0.4179, w2 = 0.5821). This portfolio has E[rp] = 0.1233 and σp =0.1428.


Optimal complete portfolio

Once we have the Tangency portfolio, we can find the optimal complete portfo-lio that combines T and the risk-free asset. For an investor with mean-variancepreferences (equation 1.2), the optimal solution is given by (1.4):

w∗

T =E[rT ]− rf

Aσ2T

Example 1.3.4. Continuing the previous example, assume A = 5.Check that the optimal combined portfolio is

(wT = 0.6208, wf = 0.3792)

or equivalently,

(w1 = 0.2594, w2 = 0.3614, wf = 0.3792)

The combined portfolio has E[rc] = 0.0993 and σc = 0.0886

Plot all these portfolios:


✲σ

✻

E[r]

1.3.4 The Markowitz portfolio optimization model

The previous concepts can be extended to the case of 1 risk-free asset and N riskystocks. This is called the Markowitz portfolio model.

Extension to N risky assets. Intuitively, the analysis can be generalizedto 3 risky assets by taking one of the possible two-asset portfolios and a new 3rdasset. Proceeding with these iterations, we could get to N risky assets.

Extension to N risky assets plus 1 risk-free asset. The investor willpick one particular portfolio on the mean-variance frontier — the tangency portfo-lio — to combine with the risk-free asset. The straight line going through rf andµT is the efficient frontier.

✲σ

✻

E[r]


1.3.5 Exercises

Ex. 4 — Bodie, Kane, and Marcus (2014) problems at the end of chapter 7 (p.235): 4–10

Ex. 5 — There are two funds available for investment:

Fund E[r] σ2

Bond fund 0.04 (0.1)2

Equity fund 0.08 (0.2)2

The correlation between the two funds is 0.3. Additionally, it is possible to investin a risk-free asset with rf = 0.02.An investor is willing to tolerate a maximum standard-deviation of 0.15. What isthe expected return on the best portfolio for this investor?

Ex. 6 — Two stocks have the following means and variances:

Stock E[r] σ2

a 0.08 (0.2)2

b 0.12 (0.4)2

The correlation between the two stocks is zero. There is no risk-free asset.Investors have mean-variance preferences given by U(r) = E[r]− A

2 Var[r].Find the optimal portfolio of the two stocks for an investor with A = 3, that is,indicate the optimal weights wa and wb.

Chapter 2

The Capital Asset PricingModel

(Based on chapter 9 of Bodie, Kane, and Marcus (2014))

The value of any asset is the present value, or discounted value, of its futurecash flows. The CAPM gives us a formula for the discount rate. Hence, it isused everyday by corporations and investors to price investment projects, stocks,mutual funds, etc. The CAPM was developed simultaneously in three papers bySharpe in 1964, Lintner in 1965, and Mossin in 1966.

2.1 Assumptions and derivation

Assumptions:

1. All investors are mean-variance optimizers, i.e., they all use the Markowitzmodel.

2. Investors have homogeneous expectations.

3. Investors can borrow or lend at a common (exogenous) risk-free rate.

4. All assets are publicly traded and short positions are allowed.

5. The investors’ planning horizon is a single period.

6. There are no taxes.

7. There are no transaction costs.

24

2.1. Assumptions and derivation 25

Derivation:

1. Since all investors estimate the same inputs (means and covariances), thetangency portfolio is the same for every investor.

2. The efficient frontier (namely, the straight line through rf and T ) is thesame for every investor.

3. Two fund separation:

(a) Every investor allocates his wealth between two portfolios: the risk-freeasset and the Tangency portfolio.

(b) Note that the weights in the risk-free asset and in the tangency portfoliomay differ across investors due to different degrees of risk aversion, butstill everybody invests in just those two assets and in no other portfolio.

4. In equilibrium, all risky assets must belong to T .

To see this, suppose that IBM is not in T (wTIBM = 0). Then, there would

be no demand for this stock, (wiIBM = wT

IBM = 0, for every investor i). Wewould thus have Demand 6= Supply, which is not equilibrium. Therefore, inequilibrium, wT

j > 0,∀ asset j.

5. Furthermore, for every asset, the weight in T must be the same as in thewhole market:

wTj =

Market Capj∑

nMarket Capn=: wM

j ,∀ asset j

If we all put 2% of our risky money into IBM stock, then IBM will have2% of all money invested in the stock market, meaning that the marketcapitalization of IBM will be worth 2% of the whole market capitalization.Note that different investors may put different amounts of money at risk, ie,in the tangency portfolio. But from these amounts, each investor allocatesthe same 2% to IBM.

6. Hence, the Market portfolio is the Tangency portfolio, M = T . This is theeconomic content of the CAPM.

CAPM

The CAPM states that the Market portfolio is mean-variance efficient, thatis,

Market portfolio = Tangency portfolio

2.2. Efficient frontier 26

2.2 Efficient frontier

2.2.1 Capital Market Line

When we use M instead of T, the efficient frontier is called Capital Market Line:

✲σ

✻

E[r]

Proposition 2.2.1: Capital Market Line

Under the CAPM, the efficient frontier is

CML : E[rp] = rf +E[rM ]− rf

σMσp

where p is an efficient portfolio.

Recall that any p ∈ CML is a combination of the risk-free and the marketportfolio, thus σp = wMσM .

Example 2.2.1. You expect the stock market to go up by 10% overthe next year. The standard deviation of the market return is 20%.You can buy 1 year government bonds yielding 4%. You have $100,000to invest and you are willing to tolerate a risk (standard deviation) of15%.

1. What is the best allocation of your money?

2.2. Efficient frontier 27

2. How much money do you expect to have one year from now?(Answer: $108,500)

2.2.2 Risk aversion and the market risk premium

Proposition 2.2.2: Risk premium of the market portfolio

E[rM ]− rf = Aσ2M

where A is the risk-aversion of the “representative” investor.

Proof. Recall that for an investor with risk-aversion parameter A, the best com-plete portfolio p ∈ CML is given by

wM =E[rM ]− rf

Aσ2M

and wf = 1− wM

In the aggregate, the amount of borrowing and lending among investors has to netout to zero. Hence, the “representative” investor will have wf = 0 ⇒ wM = 1.

Denoting by A the risk aversion of the representative investor, 1 =E[rM ]−rf

Aσ2M

.

Example 2.2.2. (Concept check 9.2 in BKM) Data from the last 8decades for the S&P500 index yield the following stats: average excessreturn, 7.9%; standard-deviation, 23.2%.

To the extent that these averages approximated investor expectationsfor the period, what must have been the average coefficient of riskaversion?

2.3. Expected returns on individual securities 28

2.3 Expected returns on individual securities

2.3.1 Security Market Line

Proposition 2.3.1: Security Market Line

Under the CAPM, the equilibrium expected return for any asset j is givenby

SML : E[rj ] = rf + βj ( E[rM ]− rf )

where

βj :=Cov(rj , rM )

Var[rM ]

Different stocks have different betas and thus different expected returns:

✲β

✻

E[r]

Important remarks about the SML:

• The SML applies to every single asset or portfolio (not necessarily on theCML).

• For any asset or portfolio j, the relevant measure of risk is βj , not its vari-ance.1

1But for an efficient portfolio on the CML, including the market portfolio, we can useeither the SML or CML to get its expected return.


Example 2.3.1. Industry-type application of the CAPM. Suppose E[rM ] =10% and rf = 4%. You estimate stock a will pay a dividend of $2 oneyear from now. After that, you expect dividends to grow at 5% peryear. You also estimate the beta of the stock to be βa = 0.9. What isthe equilibrium price of the stock?

Note: recall that the present value of a stream of dividends growingat rate g is P0 = D1

r−g , where r is the discount rate. Thus, you justneed to use the CAPM to estimate the required discount for stock a.Answer: Pa = $45

2.3.2 Interpretation of beta

Why do betas explain differences in expected returns?

1. What matters in βj is Cov(rj , rM ) (the market variance is the same for allstocks)

2. Hence, high beta means high covariance with the market.

3. The contribution of an individual security j to σ2M is proportional to Cov(rj, rM ).

Intuitively, high Cov(rj , rM ) means that j wins/looses in the same statesas the market wins/looses, which increases the variance of the market. SeeBKM for a formal proof.

4. Since all investors hold the market portfolio (in some combination with therisk-free rate), investors do not like stocks with high betas (why?)

5. Therefore, investors require high expected returns to hold high-beta stocks

Example 2.3.2. Consider the following two companies:

• Duke Energy Corporation (DUK): an energy company based inNorth Carolina, with a significant part of its operations in regu-lated markets.

• Ralph Lauren Corporation (RL): sells lifestyle products (cloth-ing, accessories, fragrances).

Which one is likely to have a higher beta? Why?On 2015-06-24, βDUK = 0.28, βRL = 0.93


2.3.3 Beta of a portfolio

Proposition 2.3.2: Portfolio β

For a portfolio p with N securities,

βp =

N∑

i=1

wiβi

where wi are the portfolio weights.

Proof. From the definition of beta,

βp := Cov(rp, rM )/Var(rM ) = Cov(N∑

i=1

wiri, rM )/Var(rM )

=

N∑

i=1

wiCov(ri, rM )/Var(rM ) =

N∑

i=1

wiβi

Example 2.3.3. (This is concept check 9.3 in BKM). Suppose thatthe risk premium on the market portfolio is estimated at 8% witha std of 22%. What is the risk premium on a portfolio p invested25% in Toyota and 75% in Ford, if they have betas of 1.10 and 1.25,respectively?

Note that p in the previous example is not efficient: it will surely have toomuch σ2

p. However, the CAPM does not reward all of σ2p; it only rewards the part

that is nondiversifiable, which depends on Cov(rp, rM ) or βp.

2.4. Exercises 31

Example 2.3.4. Continuing the previous example:

1. Construct an efficient portfolio q on the CML with βq = 1.2125.

2. Since the SML applies to any portfolio and the betas are the same,the risk premium on q is the same as p:2

E[rq]− rf = . . .

This means that E[rq] = E[rp].

3. Plot the CML and the approximate location of p and q

2.4 Exercises

Ex. 7 — Bodie, Kane, and Marcus (2014) problems at the end of chapter 9 (p.317): 3–7, 17, 21.

2Alternatively, to compute the risk premium on q without using the SML, Note thatrq − rf = wMrM + wfrf − rf = wMrM + wf rf − (wM + wf )rf = wM (rM − rf ).

Chapter 3

Arbitrage Pricing Theory andFactor Models

Based (somewhat) on chapters 10 and 24 of Bodie, Kane, and Marcus (2014), withadditional materials from other sources.

3.1 Theory

The APT was developed by Ross (1976).

1. Factor model. The APT starts by assuming that stock returns are gener-ated by K factors:

rj = aj +

K∑

k=1

βjkFk + εj , j = 1, 2, . . . , N (3.1)

with the following assumptions:

A1: E[εj ] = 0,∀j

A2: Cov(Fk, εj) = 0,∀k, j

A3: Cov(εj , εi) = 0,∀j 6= i

The point is to find a small number of factors (K << N) that satisfy thismodel.1

1Equation (3.1) is sometimes stated as deviations from means. Take expectations on

both sides to get E[rj ] = aj +∑K

k=1 βjk E[Fk]. Plug the resulting value for aj into (3.1)

32

3.1. Theory 33

2. Diversification. Under (3.1), the return on a portfolio p with N stocks is

rp = ap +

K∑

k=1

βpkFk + εp

and its risk is

Var[rp]︸︷︷︸

total risk

= Var

[K∑

k=1

βpkFk

]

︸︷︷︸

systematic risk

+ Var[εp]︸︷︷︸

nonsystematic risk

It can be shown that the second term goes to zero in a large, well-diversifiedportfolio (wj = 1/N):

Var[εp]N→∞

−−−−→ 0

Hence, εp ≡ 0 in a very large portfolio (in the limit), and

rp = ap +

K∑

k=1

βpkFk

3. No arbitrage. If we can trade the factors to hedge the random part of rp,then it can be shown that there are no arbitrage opportunities only if

ap =

(

1−

K∑

k=1

βpk

)

rf

Then, the expected return on a well-diversified portfolio must be:

E[rp] = rf +

K∑

k=1

βpk ( E[Fk]− rf ) (3.2)

when Fk are traded portfolios.

For an individual security (which is not a well-diversified portfolio), we onlyget an approximation.

to get

rj = E[rj ] +

K∑

k=1

βjkFk + εj

with Fk := Fk − E[Fk] and thus E[Fk] = 0. Stock returns deviate from their means asa result of unexpected realizations of risk factors. We can further subtract rf from bothsides to get

rj − rf = E[rj − rf ] +

K∑

k=1

βjkFk + εj

which is the version presented in BKM. Note that this is just a mathematical manipulationof (3.1); it is still not saying anything about what variables explain E[rj ].

3.1. Theory 34

In summary,

Definition 3.1.1: Arbitrage Pricing Theory (APT)

If there are no arbitrage opportunities, the expected return on an individualsecurity j is

E[rj] ≈ rf +K∑

k=1

βjkFRPk (3.3)

where:• FRPk, the Factor Risk Premium, is:

– If factor k is a traded positive-investment portfolio, FRPk =E[Fk]− rf = E[Fk − rf ]

– If factor k is a traded zero-investment long-short portfolio,FRPk = E[rlong − rshort]

– If factor k is not a traded portfolio, FRPk is a free parameter tobe estimated.

• βjk is the loading of asset j on factor k. In practice, the loadings areestimated through a time-series regression of excess returns for assetj on the factors.

Remarks:

• In practical applications we typically ignore the nuances and replace the ≈

with =

• The different alternatives for the FRP is advanced material not covered inBKM, you will not be tested on this, and we will only apply the traded-factors version (Fama-French model below).2

For empirical applications with traded factors, the following modification of(3.2) is more useful:

2In any case, the intuition is the following. The general statement of APT, for any typeof factor, is just that E[rep] =

∑K

k=1 βpkFRPk, with FRPk 6= 0, where rep is the excessreturn on a well-diversified portfolio. But for traded factors, we can be more specific. If,for example, factor 1 is traded, we can put F1 on the LHS of the equations above. Thefactor model would obviously have β11 = 1 and β1k = 0, k = 2, . . . ,K. Then, the pricingequation with F1 on the LHS would give E[F e

1 ] = FRP1, thus pinning down the FRP forfactor 1.

3.2. Example with 1 factor: the Market model 35

Proposition 3.1.1: Alternative “alpha statement” of the APT

The risk premium on a well-diversified portfolio p is

E[rp − rf ] = αp +

K∑

k=1

βpkFRPk (3.4)

If all factors are traded and there are no arbitrage opportunities, we musthave

αp = 0

Advantages and disadvantages of the APT:

• The main advantage of the APT is that it can accommodate several sourcesof systematic risk.

Natural macroeconomic factor candidates are: interest rates, GDP growth,inflation, energy prices, etc.

• The main disadvantage is that the theory does not specify which are the“true” factors.

It has been and empirical quest, with the current number of factors in theliterature at over 300 and counting... The horse that is leading the race atthis point is the FF3.

3.2 Example with 1 factor: the Market model

The Market Model states that there is just one factor: the market. The returngenerating process is:

rj = aj + βjrM + εj , ∀j (3.5)

If there are no arbitrage opportunities, expected returns are given by:

E[rj] = rf + βj ( E[rM ]− rf ) ,∀j

which is the twin brother of the CAPM’s SML.

Example 3.2.1. Market neutral strategy. This investment strat-egy is typical of many hedge funds. It is is from BKM (old 2005 ed,sec 10.4).

3.2. Example with 1 factor: the Market model 36

• A portfolio manager has identified an underpriced portfolio pwith the following characteristics:

(rp − rf ) = 0.04 + 1.4(rSP500 − rf ) + ǫp

• If we take expectations on both sides of this equation, we see thatthe APT in (3.4) says that we should not be seing this constantvalue of αp = 0.04.

• Nevertheless, the manager is very confident about its alpha of4%.

• However, even if the manager is right, he may loose money if thewhole market turns down. He would like to explore the relativemispricing of p, regardless of what happens to the market.

Solution:

1. The solution is to construct a tracking portfolio (T ) that matchesthe systematic component of p.

(a) T must have a beta of 1.4, which requires wSP500 = 1.4 andwf = −0.4.

(b) The return on the tracking portfolio is thus

rT = 1.4rSP500 − 0.4rf ⇒ (rT − rf ) = 1.4(rSP500 − rf )

2. The investment strategy is to go long (buy) on p and go short(sell) on T .

3. The combined portfolio C thus has a return of

rC = rp − rT = (rp − rf )− (rT − rf ) = 0.04 + ǫp

This combined position is thus market neutral. Regardless ofwhat happens to the market, the manager earns 4%.3

Remark. This example also illustrates the arbitrage process that restores equi-librium. Investors will want to follow this strategy as much as possible. However,going long on p bids up its price, until alpha disappears.

3Note that there is still some residual risk, ǫp. This will be small if the single marketfactor explains rp well and p is a large portfolio.

3.3. Example with 3 factors: the Fama-French model 37

3.3 Example with 3 factors: the Fama-French

model

3.3.1 Pricing equation

Fama and French (1993) propose the following 3-factor asset pricing model:

Definition 3.3.1: Fama and French 3-factor model

The expected return on stock j is

E[rj ]− rf = βjM( E[rM ]− rf ) + βjs E[SMB] + βjh E[HML] (3.6)

where the loadings (βjM , βjs, βjh) are the slopes in the time-series regression

(rj − rf )t = aj + βjM (rM − rf )t + βjsSMBt + βjhHMLt + εjt (3.7)

3.3.2 Details on the factors

To form the two new factors, FF divide all firms into six buckets depending on theirsize (market equity, ME) and the ratio of book equity to market equity (BE/ME):4

50th ME prct

Small Value Big Value > 70th BE/ME prct

Small Neutral Big NeutralSmall Growth Big Growth < 30th BE/ME prct

“Small” stocks have ME smaller than the median ME. Typically, small stocksperform better than what the CAPM predicts (this is a so called anomaly).

“Value” stocks have BE/ME higher than the 70th BE/ME percentile; theirbook-to-market ratio is High. “Growth” stocks have BE/ME lower than the 30thBE/ME percentile; their book-to-market ratio is Low. Typically, BE/ME is highwhen the ME (denominator) is low. This happens when the firm has had lowreturns and is now near financial distress. Nonetheless, most of these firms usuallyrebound and thus, if you hold a large portfolio of these firms, you end up makingmore money than their CAPM beta would suggest (another CAPM anomaly).

Each month, the factors are computed in the following way:

4See the details at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french

3.4. Portfolio performance evaluation 38

• SMB (Small Minus Big) is the average return on the three small portfoliosminus the average return on the three big portfolios,

SMB = 1/3 (Small Value + Small Neutral + Small Growth)- 1/3 (Big Value + Big Neutral + Big Growth)

Historically, the SMB portfolio generated an annual return somewhere be-tween 1.5% and 3%. This is the size premium.

• HML (High Minus Low) is the average return on the two value portfoliosminus the average return on the two growth portfolios,

HML = 1/2 (Small Value + Big Value) - 1/2 (Small Growth + Big Growth)

Historically, the HML portfolio generated an annual return somewhere be-tween 3.5% and 5%. This is the value premium.

This model has had considerable empirical success in explaining CAPM anoma-lies (portfolios that don’t plot on the SML) and in capturing the variation in thecross-section of expected returns. Thus, Fama and French (1996) argue that SMBand HML mimic combinations of two underlying risk factors of special concern toinvestors.

3.4 Portfolio performance evaluation

3.4.1 Motivation

1. One important question in finance is: How to assess the performance of afund manager?

2. We cannot just look at raw realized returns because we want to distinguishstock-picking skills from simple risk taking. If we see a big return, was itbecause the manager was able to identify mispriced stocks or was it becausehe took large risks and got lucky?

3. Therefore, we need to compute risk adjusted returns, that is, we need tomeasure the difference between the empirical realized returns and the returns“appropriate” for the risk of the fund.

3.4.2 Adjusting returns for risk

The most widely used performance measure is Jensen’s alpha. It is the empiricalcounterpart to the APT equation (3.4).


Definition 3.4.1: Jensen’s alpha

Jensen’s α for fund p is the intercept in the time-series regression

(rp − rf )t = αp +

K∑

k=1

βpkFekt + εpt, t = 1, . . . , T

where F ekt is the excess return on the traded factor k at time t:

• If factor k is a traded positive-investment portfolio, F ekt = (Fk − rf )t

• If factor k is a traded zero-investment long-short portfolio, F ekt =

(rlong − rshort)t

We now have two models to adjust returns for risk.

CAPM

• To evaluate the performance of fund p, estimate the following time-seriesregression:

(rp − rf )t = αp + βp(rM − rf )t + εpt (3.8)

• According to the CAPM, or the 1-factor Market Model APT, we should findαp = 0.

• If instead we find that αp is (statistically significantly) positive, we canconclude that the fund returns are higher than what its level of risk wouldrequire (according to the CAPM). In other words, the manager has skill.

• Graphically, a positive Jensen’s alpha implies that the portfolio lies abovethe SML:

✲β

✻

E[r]


Remark. Model (3.8) is the standard regression to estimate the CAPM beta.BKM call this the “single-index model”. The market model equation (3.5) issometimes also used in the industry to estimate the CAPM beta, but this is onlyequivalent to (3.8) when the interest rate is constant (which is not the case inreality).

FF3

• If we don’t believe that CAPM is a good model to adjust returns for risk,we can use the Fama-French model.

• Run the regression

(rp − rf )t = αp + βpM (rM − rf )t + βpsSMBt + βphHMLt + εpt

• Again, if αp > 0 (statistically), the manager has skill.

Note that the βpM estimator that comes out of this regression is not the CAPMbeta (due to the presence of other regressors).

Chapter 4

Bond Markets

Based on chapters 5, 14–16 of Bodie, Kane, and Marcus (2014).

4.1 Rates of return

Returns are usually expressed in annual terms. There are 3 alternative representa-tions of the same underlying Holding Period Return (HPR, as defined in equation1.1).

4.1.1 Effective Annual Rate

Definition 4.1.1: Effective Annual Rate

The Effective or Equivalent Annual Rate (EAR) corresponding to an in-vestment with a given HPR over T years is

EAR : 1 + EAR = (1 + HPR)n

where n = 1/T is the number of compounding periods per year

Example 4.1.1. A 1-month T-bill (zero coupon bond) is trading at99.75% of par value. Check that:

HPR(T = 1/12) = . . . = 0.2506%

41

4.1. Rates of return 42

and

EAR = . . . = 3.0493%

4.1.2 Annual Percentage Rate

Definition 4.1.2: Annual Percentage Rate

The Annual Percentage Rate (APR) with compounding frequency n, de-noted rn, corresponding to an investment with a given HPR over T < 1years is

rn = HPR× n

where n = 1/T is the number of compounding periods per year

Example 4.1.2. Continuing the previous example, check that theAPR with monthly compounding is

r12 = . . . = 3.0075%

Remarks:

• A complete specification of an APR always needs to state n, unless it is clearfrom the security specifications.

• When n = 1, the APR is also the EAR: r1 = EAR

4.1.3 Continuous compounding

Definition 4.1.3: Annual rate with continuous compounding

The annual rate with continuous compounding, denoted r∞, correspondingto an investment with a given HPR over T years isa

er∞ = 1 + EAR ⇒ r∞ = ln(1 + EAR) =1

Tln(1 +HPR)

aRecall that limn→∞(1 + yn)n = ey

Example 4.1.3. Continuing the previous example, check that

r∞ = . . . = 3.0038%

4.2. Bond Prices and Yields 43

4.1.4 Discounting

Proposition 4.1.1: Discounting

The value at time t of a cash flow C(T ) to be received in T years is

P (t) =

C(T )[

1+rn(t,T )

n

]n×(T−t) , discrete compnd. (n < ∞);

C(T )

er∞(t,T )×(T−t) = C(T )e−r∞(t,T )×(T−t), cts. compnd. (n = ∞)

where rn(t, T ) is the interest rate between t and T , with compounding fre-quency n.

Example 4.1.4. Do the previous examples in reverse, i.e., start fromthe interest rates and compute the T-bill price.

Example 4.1.5. Continuing the previous examples, assume that inthe future you will be able to reinvest at the same T-bill rate that isavailable now. How much money do you need to invest today to have$ 100 000 in 3 months?

Answer: 99 251.87

4.2 Bond Prices and Yields

4.2.1 Bond Pricing

Price units

Bond prices are typically expressed as a percentage of face value. To understandits meaning, note that:

Total Investment (in $) = Price (in % of FV) × Face Value (in $)

Think of FV as a “quantity”.


Example 4.2.1. A Bond with 5% annual coupons is priced at 102.1234%.This means that:

• If we invest 1 M$, we are able to buy a face value (“quantity”)ofFV = 1M$/1.021234 = . . . $

• Our next coupon will be C = . . . $

• At maturity, we will receive a total of $1,028,167.88

Remarks:

• Interpret the symbol “%” as meaning “×0.01” to avoid messing up the unitsin your calculations.

• BKM still show examples of the old days when bonds had $1000 or $100denominations.

Price of fixed-coupon bonds

Definition 4.2.1: Price of fixed-coupon bond

P =m∑

i=1

c/n

[1 + r(0, ti)]ti+

100%

[1 + r(0, tm)]tm(4.1)

where

• P is price of the bond today

• n is the number of coupon payments per year

• c is the annual coupon rate with compounding frequency n

• t1, t2, . . . , tm = T are the years until the coupon payment dates (withT denoting the maturity)

• r(0, ti) is the discount rate or required return from today (t = 0) untilti years, expressed as an Equivalent Annual Rate (EAR)

Important special case: zero-coupon bonds (aka zeroes, or pure discount bonds).


Proposition 4.2.1: Price of a Zero-Coupon Bond (ZCB)

The price of a ZCB maturing in T years is

P =100%

[1 + r(0, T )]T

Quoting conventions

1. The Quoted (or clean, or flat) price quoted by a trader does not includeinterest that accrued since the last coupon date.

2. The Invoice (or dirty, or full, or cash) price is the price that the buyeractually has to pay the seller:

Invoice price = Quoted price + Accrued interest

This is the price given by (4.1).

3. The accrued interest (AI) is:

AI =N. days since last coupon

N. days between coupons× Interest due in full period

Day-count conventions:

• actual / actual(in period) — for T-Notes and T-Bonds.

• actual/360 — used in T-Bills and money markets.

• 30/360 — used in corporate bonds.

Example 4.2.2. Consider a 15/Jul/2020 4% Bond, paying semi-annualcoupons. The settlement date is 10/April/2012 (note that 2012 is aleap year).

• With actual/actual day count,

AI =86

182× 0.02 = 0.9451% of FV

• With 30/360 day count,

AI =85

180× 0.02 = 0.9444% of FV

In Excel, use the functions COUPDAYBS() and COUPDAYS() to count days.


4.2.2 Yield to Maturity

Definition 4.2.2: Yield to Maturity

Given the price of a fixed-coupon bond, the YTM (y, as an EAR) is theconstant discount rate that solves

y : P =m∑

i=1

c/n

[1 + y]ti+

100%

[1 + y]tm(4.2)

Example 4.2.3. A one-month T-Bill sells for 99.67367% (of par value).Check that the one-month yield is y = 4% (EAR).

Plot Price versus Yield and note the inverse relation.

Remarks:

• In general, we need Excel (YIELD or IRR function, or Solver) or a financialcalculator to compute the YTM of a bond with multiple coupons.

• YTM are sometimes quoted as APR, aka “bond equivalent yields”.

Example 4.2.4. Price = 95%, 10-yr Maturity, Coupon rate = 7%with semiannual coupons.

The IRR() fn in Excel outputs 3.8635%, which is a 6-month rate.

Then, we would express the YTM in one of the following ways:

• Bond Equivalent Yield (APR).

yAPR2 = 3.86% × 2 = 7.72%

• Effective Annual Yield (EAR).

yEAR1 = (1.0386)2 − 1 = 7.88%

• Traders also talk about a Current Yield = Annual interest / market price.For the previous example, yCY = 7/95 = 7.37%.


Interpretation. The YTM will be the realized yield if:

• you hold the bond until maturity; and

• all coupons are reinvested at the ytm.

Example 4.2.5. 2-year 8% bond with semiannual coupons, tradingat a ytm of y2 = 8% and a price of 100%. Check that under the twoconditions specified above, the realized yield at the end of 2 years willbe r2 = 8%.

Remark. From the previous example, note that at the ex-coupon date, Y TM =Coupon rate ⇒ P = 100%. Check that Y TM > Coupon rate ⇒ P < 100% andY TM < Coupon rate ⇒ P > 100%

4.2.3 Holding-Period Return

The Realized Yield, or Holding-Period Return, or Realized Compound Return de-pends on:

• Selling price of the bond;

• Reinvestment rate for the coupons.

Example 4.2.6. 2-year 8% bond with semiannual coupons, tradingat a ytm of y2 = 8% and a price of 100%. In six months the ytm fallsto 7%.

1. Check that P6m = 101.4%

2. Check that the HPR for 6 months is r2 = 10.8016% (APR).

Example 4.2.7. Consider a 2-year, 10% bond with annual coupons,trading at a ytm of 10%. You hold the bond until maturity, but areonly able to reinvest the coupons at 8%. Check that the HPR is 9.9%(EAR).

4.3. Term structure of interest rates 48

4.2.4 Default risk and ratings

Corporate bonds trade at an higher yield than risk-free government bonds becausecorporations may default. Default risk is measured by a credit rating.

Long Term Obligation Ratings

S&P Moody’s Meaning

AAA Aaa Highest quality, minimal credit riskAA AaA ABBB Baa Adequate protection, moderate credit risk

BB Ba Speculative, significant riskB BCCC CaaCC CaC C Nonpayment highly likely; in default (Moody’s)D – in Default

Obligors rated BBB or better are called “investment-grade”; investors ratedBB or worse are called “speculative-grade”. Each category can also be appendedwith a + or - sign (S&P) or with 1,2,3 numbers (Moody’s) to show relative standingwithin the category.

4.2.5 Exercises

Ex. 8 — Bodie, Kane, and Marcus (2014) problems at the end of chapter 14 (p.480): 6, 9.

4.3 Term structure of interest rates

4.3.1 Spot rates

Definition 4.3.1: Spot rate

The spot rate r(0, T ) is the yield to maturity on a zero-coupon bond withmaturity T .


Definition 4.3.2: Term structure of interest rates

The term structure of interest rates, or zero yield curve, or simply yieldcurve, is the set of spot rates r(0, T ) for different maturities T .

Example 4.3.1. The following Zero-Coupon bonds are trading in themarket:

Time to Maturity Price (%) YTM

1 95.238 5%2 88.997 6%3 81.630 7%

Hence, the term structure of interest rates is:r(0,1) = 5%r(0,2) = 6%r(0,3) = 7%.

The spot rates are related to the price of coupon-bearing bonds through (4.1).

Example 4.3.2. Consider a bond with 5% annual coupons and 2years to maturity. The total value must be the sum of the present val-ues of all cash flows. Using the spot rates from the previous example:

P = . . . = 98.212%

Remarks:

• The term “yield curve” is also used to denote the set of YTM.

• In addition to the spot rate curve defined above, we can also have forwardrate curves (using the fwd rates defined below)

4.3.2 Forward rates

A forward rate applies to a time period in the future.

Definition 4.3.3: Forward rate

Given a set of spot rates, the forward rate between t1 and t2 (0 < t1 < t2),expressed as an EAR, is given by

f(t1, t2) : [1 + r(0, t1)]t1 [1 + f(t1, t2)]

(t2−t1) = [1 + r(0, t2)]t2


Interpretation. The fwd(1,2) is the rate of return on a future investment thatwill make me indifferent between the following alternatives:

1. Invest in a 2-year zero coupon bond.

2. Invest in a 1-year zero coupon bond. After 1 year reinvest the proceeds inanother 1-year bond.

Example 4.3.3. Continuing the previous example, check that f(1, 2) =7.0095% and f(2, 3) = 9.0284%.

4.3.3 Expectations of future interest rates

Term structure under certainty

If all investors know for sure the path of future interest rates (i.e, there is no risk),then:

1. The forward rate computed today is the value that the spot rate will takeon in the future.

2. All bonds provide the same return over any given holding period.

Example 4.3.4. Consider the following bonds (all have annual coupons):

Bond A B CMaturity (yrs) 1 2 3Coupon rate – 4% 6%Price (%) 90.909 88.045 85.773

1. Compute the spot rates (this procedure is called Bootstrap)

2. Compute the sequence of 1-yr forward rates.

3. Check that the YTM for bond A, B and C are respectively 10%,10.98%, and 11.91%.

4. Plot the YTM curve, the spot-rate curve, and the forward-ratecurve.

5. Check that if we invest for 1 year, all bonds produce the samereturn (recall that under certainty we know the future value ofinterest rates).


Uncertainty and expectations

In reality we don’t know the future value of interest rates. What do forward ratestell us about future spot rates?

Expectations Hypothesis

The forward rate equals the market consensus expectation of the future spotinterest rate. That is,

forward0(t1, t2) = E0[spott1(t1, t2)]

Liquidity Preference Hypothesis

Most investors are short-term and therefore will only hold long-term bondsif they receive a premium. Thus, the forward rate exceeds the expected spotrate by a liquidity premium:

forward0(t1, t2) = E0[spott1(t1, t2)] + liquidity premium

Implications:

1. Expectations of increases in future interest rates must always result in arising yield curve.

2. However, the converse is not necessarily true. Even if the yield, spot, andforward curves are all rising, the market may be expecting constant or evendecreasing future interest rates (why? liquidity premium).

3. In practice, interest rates are very hard to predict...

Relation to the business cycle

The slope of the yield curve is typically a good leading indicator of the businesscycle:

• An upward sloping curve usually forecasts “good times” ahead

• A flat or downward sloping curve usually forecasts “bad times” ahead

For example, the ECB estimates the yield curve for the euro area every day.Check http://www.ecb.europa.eu/stats/money/yc/html/index.en.html

4.4. Bond management 52

4.3.4 Exercises

Ex. 9 — Bodie, Kane, and Marcus (2014) problems at the end of chapter 15 (p.507): 11, 12.

4.4 Bond management

4.4.1 Duration

How to measure the sensitivity of the bond price to changes in interest rates, i.e.,the risk of a bond?

Definition 4.4.1: Macaulay Duration

The Macaulay Duration is

D :=1

P

m∑

i=1

tiCti

(1 + y)ti(4.3)

where• y is the YTM (EAR)

• ti are the coupon payment dates (in years)

• Cti is the total cash flow at date ti: Cti = c/n for i < m, andCti = c/n + 1 for i = m, where n is the coupon payment frequency.

Example 4.4.1. Consider a 3-year 5% bond paying annual coupons.The bond is trading at y = 4%. Check that P = 1.027751 and

D = . . . = 2.8615

Proposition 4.4.1: Price sensitivity to interest rate changes

For a small change in YTM (∆y), the bond price change is well approxi-mated by

∆P

P∼= −D

1

1 + y∆y (4.4)


Proof. The change in price due to a change in interest rates is ∆P ≡ P (y +∆y)−P (y). If the change in interest rates is small and instantaneous (i.e., time isstanding still), the change in price can be approximated by taking a Taylor seriesexpansion of P around y:

P (y +∆y) = P (y) +dP

dy∆y + 1/2 ∗

d2P

dy2(∆y)2 + . . .

Taking derivatives and dividing through by P, we get

∆P

P= −D

1

1 + y∆y +

1

2C(∆y)2 + . . .

where D is as defined in (4.3) and C, denoted Convexity, is

C :=

∑mi=1 ti(ti + 1)

Cti

(1+y)ti

P

1

(1 + y)2

Ignoring term of order higher than ∆y produces the approximation in (4.4).

Meaning of Duration. Bonds with higher Duration are more sensitive tointerest rate changes. Hence, Duration is a measure of the riskiness of a givenbond.

Example 4.4.2. Continuing the previous example, assume that YTMincreases by 20 basis points.

1. The proportional price change for the previous bond is∆PP

∼= . . . = −0.5503%

2. Now consider a 3-year ZCB also trading at an initial YTM of4%. Since

D = . . .

the proportional price change in this bond will be∆PP

∼= . . . = −0.5769%

Remarks:

• The Modified Duration is D∗ := D/(1 + y). Using D∗ instead of D, (4.4)becomes ∆P

P∼= −D∗∆y.


• From (4.3), we can also think of duration as a weighted average of time, i.e.,

D =m∑

t=i

ti ×wti

where the weights are wti =Cti

(1+y)ti/P . Thus, the units of duration are

“years”. In this sense duration measures how fast a bond generates cashflows. For the same maturity, a bond with higher coupons will have lowerduration. This second interpretation of duration is easier to understand, butit is nearly worthless for practical applications.

4.4.2 Active bond management

If a manager is able to forecast unanticipated interest rate movements better thanthe rest of the market, then equation (4.4) suggests an investment rule:

Active bond management with Duration

• Expect interest rates to decrease ⇒ Increase the portfolio duration

• Expect interest rates to increase ⇒ Decrease the portfolio duration

This rule implies that we readjust the bond portfolio, before interest rateschange, according to the following formula:

Proposition 4.4.2: Macaulay Duration of a portfolio

If the term structure is flat, i.e., all bonds have the same YTM, then theMacaulay duration of a bond portfolio is:

Dp =

N∑

i=1

Diwi (4.5)

where• N is the number of bonds in the portfolio

• wi = Vi/Vp is the weight in bond i, where Vi is the amount ($) invested

in bond i and Vp is the total value of the portfolio, Vp =∑N

i=1 Vi.

Caveat:


1. Strictly speaking, this procedure is only correct under the following assump-tions:

(a) the term structure is flat (all bonds have the same YTM);

(b) the change in yield is small and instantaneous.

2. Nevertheless, (4.4) is still a good guide for bond management in most real-lifecases.

4.4.3 Exercises

Ex. 10 — Bodie, Kane, and Marcus (2014) problems at the end of chapter 16(p. 547): 3, 4.

Chapter 5

The efficient market hypothesis


5.1 Motivation

• Do security prices reflect the available information?

• Important implications for

– Investment decisions

– Efficient resource allocation

5.2 Random walks and the EMH

EMH statement:

1. The Efficient Market Hypothesis (EMH) is that stock prices reflect all theavailable information.

2. If EFM is true, then prices change only in response to new information,which is by definition unpredictable

3. The mathematical model for random price changes is a random walk.1

1To be precise, investors expect to be rewarded for risk, so the expected price changeis positive over time (this is called a submartingale). Hence, the common model for pricesis a positive trend with random fluctuations about the trend.

56

5.3. Implications of the EMH 57

EMH and Competition:

1. Once information becomes available, market participants quickly analyze itto trade on it.

2. Hence, competition ensures prices reflect information

3. However, there is a Catch-22:

• Stock analysis ensures markets are efficient

• Stock analysts must be compensated for their work, which can onlyhappen if markets are not efficient

• Equilibrium? Benefit = Cost

Versions of the EMH

1. Weak-form. Prices already reflect all information in past trading data (stockprices and volumes)

2. Semistrong-form. Prices already reflect all public information (past prices,plus accounting, macro, media, etc, information)

3. Strong-from. Prices already reflect ALL information — public and private— related to the firm.

Example 5.2.1. (This is concept check 11.1 in BKM)

1. Suppose high-level managers make superior returns on invest-ments in their company’s stock. Would this be a violation ofweak-form market efficiency? Would it be a violation of strong-form market efficiency?

2. If the weak form of the EMH is valid, must the strong form alsohold? Conversely, does strong-form efficiency imply weak-formefficiency?

5.3 Implications of the EMH

Some implications of the EMH:

• Weak-form efficiency implies that Technical analysis is worthless

5.4. Are markets efficient? 58

• Semistrong-form efficiency implies that Fundamental analysis is worthless

• Semistrong-form efficiency implies that the best investment strategy is Pas-sive Management (buy and hold an index fund), rather than Active Man-agement (stock picking).

Example 5.3.1. (This is concept check 11.3 in BKM). What wouldhappen to market efficiency if all investors attempted to follow a pas-sive strategy?

5.4 Are markets efficient?

Evidence on Mutual Fund Performance is mixed

• Some evidence of persistent positive (superstar or “hot-hand” managers)and negative performance (due to too much trading). But most managersare not consistent.

• How to correctly adjust returns for risk? Some apparent “anomalies” mayrepresent poorly understood risk premia.

Case in point: rise and fall of LTCM in late 1990s.

Conclusion:

• “Don’t try this at home!”

Individual investors (with other day jobs) should not engage in stock picking.Its better to buy a diversified portfolio.

• Professional analysts are probably making an extra return at least equal tothe cost of gathering information.

Caveat. Two economists are walking down the street. They spot a $20 bill onthe sidewalk. One starts to pick it up, but the other one says, “Don’t bother; ifthe bill were real someone would have picked it up already”.

5.5. Exercises 59

5.5 Exercises

Ex. 11 — Bodie, Kane, and Marcus (2014) problems at the end of chapter 11(p. 381): 1, 2, 5, 9, 12.

Ex. 12 — Bodie, Kane, and Marcus (2014) CFA problems at the end of chapter11 (p. 384): 1, 2, 3, 4.

Chapter 6

Futures markets

Based on chapter 22 of Bodie, Kane, and Marcus (2014).

6.1 Definition

6.1.1 Forward contract

Definition 6.1.1: Forward contract

A Forward contract is an OTC agreement between two counterpartieswhereby:

1. One counterparty (“long forward position”) agrees to buy N units ofa given asset, at a future date T , for a forward price F (0, T ) definedtoday

2. The other counterparty (“short forward position”) agrees to sell theasset under the same conditions.

Payments:

1. At inception there is no payment.

2. At the settlement date T , there are two alternatives according to what thecontract specifies:

60

6.1. Definition 61

• Physical settlement. The short delivers the physical asset and the longpays the price initially agreed upon, F (0, T ).

• Cash settlement. The payoff to the long forward counterparty is

CF to long forward at T = N × [S(T )− F (0, T )] (6.1)

where S(T ) is the spot price of the underlying asset at the settlementdate T . The payoff to the short is the symmetric of (6.1).

In either case, the effective price paid for the asset at time T is F (0, T ),rather than S(T ).

Example 6.1.1. Consider a Treasury Bill that matures in 9 months.A bank quotes a 6-month forward contract on this ZCB at F (0, 0.5) =0.99005, with cash settlement.

1. A firm “buys the forward” or “buys the bond forward”, i.e., takesthe long position in the forward. No payments exchange handstoday.

2. Suppose that 6 months from now, r∞(0, 0.25) = 3%.

(a) The forward payoff to the firm isCF to long forward at T = . . . = N × 0.002478

(b) The effective price paid for the bond isEff Price = N × [S(0.5) − CF from forward] = N × . . .

(c) This guarantees that the firm can invest at an effectiverate of r∞(0, 0.25) = 4% (check this), rather than at ther∞(0, 0.25) = 3% currently available in the market.

Draw the Forward contract profit profile at maturity for each position:

✲S(T )

p

F (0, T )

✻

Profit

6.1. Definition 62

6.1.2 Futures contract

Definition 6.1.2: Futures contract

A Futures contract is an Exchange-traded contract between two counter-parties whereby:

1. One counterparty (“long futures position”) agrees to buy N units ofan underlying asset, at a future date T , for a future price F (0, T )defined today

2. The other counterparty (“short futures position”) agrees to sell theasset under the same conditions.

Figure 6.1 lists the most common futures contracts.

Figure 6.1: List of typical futures contracts (from BKM)

One of the most important Futures exchanges is http://www.cmegroup.com/.

Remarks:

http://www.cmegroup.com/

6.1. Definition 63

• Settlement type. Some contracts are only cash settled (eg, stock index fu-tures), whereas others permit physical delivery (eg, crude oil). However,in practice most contracts are closed out, or reversed, before maturity, sothat actual physical delivery only happens in a small fraction of contracts(1%–3%).

• Convergence. The futures price and the spot price must converge at matu-rity; otherwise, there is an arbitrage opportunity. In other words, the basismust go to zero:1

[F (t, T )− S(t)]t→T−−−→ 0

6.1.3 Differences between Futures and Forwards

Futures and forward contracts are very similar. However, there are some importantdifferences:

1. Futures are exchange-traded. In a Futures, the counterparty to every traderis the exchange clearinghouse.

2. Futures are standardized. We can only trade the contracts for the underlyingsecurities, maturities, and quantities that the exchange specifies. This makesfutures more liquid than forward contracts, but less adaptable to specificneeds of traders.

3. Futures are marked-to-market daily. Profits or losses accrue to traders withdaily frequency. If we buy a futures contract today (i.e., enter into the longposition), our cash flow tomorrow is:

CF to long futures at day 1 = N × [F (1 day, T )− F (0, T )]

This mark-to-market is repeated every day until we close the position. If wehold the futures until maturity,

CF to long position over life of the futures = N×

[F (1 day, T )− F (0, T )

+ F (2 days, T )− F (1 day, T )

+ . . .

+ F (T, T )− F (T − 1 day, T )]

= N × [S(T )− F (0, T )]

1The basis can be defined as either F − S or S − F .

6.2. Trading strategies 64

The last line uses the fact that the futures price must converge to the un-derlying security price at the futures maturity date: F (T, T ) = S(T ).

Hence, the total payoff is similar to a forward contract (equation 6.1), despitethe timing of the cash flows being different.

4. Margins. Futures traders need to post an “initial margin” with the exchange(5%–15% of the total value of the contract) and must replenish the accountwhenever the daily losses drive the balance below a given “maintenancemargin”.

Margins and daily mark-to-market virtually eliminate the counterparty creditrisk in futures contracts.

6.2 Trading strategies

6.2.1 Speculation and leverage

Speculation trading rule

• Expect futures price to increase ⇒ Long futures

• Expect futures price to decrease ⇒ Short futures

Example 6.2.1. (Example 22.3 in BKM) Crude oil futures are trad-ing at 91.86 $/bbl. Each contract is on 1 000 barrels. You believe thatoil prices are going to increase.

1. Trading strategy (today): (long/short) futures

2. Result (some days latter): suppose that the futures price in-creases by $2 and you decide to close the position.

Profit = . . . = 2 000 $/contract

Trading in futures allows for much higher leverage than trading in the under-lying spot asset.

Example 6.2.2. (Example 22.4 in BKM) Continuing the previousexample,


• Suppose the initial margin for the oil contract is 10%.

1. Initial margin = . . . = 9 186 $/contract

2. The $2 increase represents a percentage gain ofReturn on posted margin = ...

9 186 $/ctt = 21.77%

• If instead we had invested in spot oil, the return for the same $2increase would have been only

Return on spot oil = ...91.86 $/bbl = 2.177 %

A 10-to-1 ratio!

6.2.2 Hedging

Hedging trading rule

To hedge an underlying cash exposure, take a position in futures such thatthe gains/losses in the futures offset the losses/gains in the cash position.

• Short spot position (ie, will purchase later) or investment sometimein the future (ie, loose if price increases) ⇒ Long futures

• Long spot position (ie, will sell later) or borrowing sometime in thefuture (ie, loose if price decreases) ⇒ Short futures

Example 6.2.3. (Example 22.5 in BKM) Consider an oil producerplanning to sell 100 000 barrels of oil in February that wishes to hedgeagainst a possible decline in oil prices. February futures are tradingat 91.86 $/barrel. Each contract is on 1 000 barrels.

1. To hedge its production, the company should (buy/sell)(number) contracts.

2. Whatever the spot oil price in February, the profit/loss in thefutures exactly offsets the change in the cash position, such thatthe effective sale price is 91.86 $/barrel. Figure 6.2 illustratesthis.


Figure 6.2: Hedging with futures (Fig 22.4 in BKM)

6.2.3 Spread trading

• Definition: a spread trade consists of a long position in one futures contractand a short position in another futures contract.

• Goal: Speculate or hedge a possible price divergence/convergence

• Typical spreads:

– Different but related commodities

– Different delivery months of the same commodity

• Advantages:

– Only relative price changes matter, i.e., do not have to guess directionof the market

– Margin requirements are lower because spreads are less volatile thanabsolute levels. Hence, can get more leverage.


Spread trading rule

Let H be the high-price and L the low-price commodity. The spread isS = H − L.

• If expect S to widen: buy H, sell L

• If expect S to narrow: buy L, sell H

Example 6.2.4. It is now July. Consider the following NYMEX/CMEfutures contracts for New York Harbor delivery, each on 42 000 gallons:

Futures January ($/gallon)

Heating oil 2.96Gasoline 2.67

You believe that the winter in the U.S. East coast will not be as cold asthe market expects, and thus that January heating oil will get closerto gasoline.

1. What is your trading strategy?

2. Suppose you close the position in December, when the Januaryfutures are trading at: gasoline, 2.7 $/gal; heating oil, 2.8 $/gal.Compute your profit if you had traded 10 contracts long and 10short. (Answer: $ 79 800).

Remarks:

• Spread trading is heavily used for hedging in the Energy industry. Typicalspreads include:

– Crack spread — to hedge the profit of an oil refinery

– Spark spread — to hedge the profit of a gas-fired power plant

– Dark spread — to hedge the profit of a coal-fired power plant

• We can also speculate on basis (F −S) changes with the appropriate trades.

Example 6.2.5. (Example 22.6 from BKM). Suppose that gold to-day sells for 991 $/ounce, and the futures price for delivery in 6 monthsis 996 $/ounce. Therefore, the basis is currently 5 $/ounce.

1. To bet that the basis will narrow, what is your trading strategy?

2. Suppose that a few days from now, the spot price increases to 995$/ounce, while the futures price increases to 999 $/ounce. Checkthat your net gain equals the change in the basis, 1 $/ounce

6.3. Futures prices of stock indices 68

6.3 Futures prices of stock indices

Units:

• Stock indices (eg, S&P 500) are usually quoted as unitless numbers (“indexpoints”)

• Futures on those stock indices are quoted in the same “index points”

• One Futures contract calls for delivery of some dollar multiple of the indexnumber: $k × index. For example, the most important futures on the S&P500 has a multiplier of $250, ie, the contract size is $250.

Proposition 6.3.1: Futures pricing

If there are no arbitrage opportunities, the Futures price of a stock index is

F0 = S0(1 + r1)T −DT (6.2)

where• F0 or F (0, T ) is the current futures price for delivery at time T

• S0 or S(0) is the current spot price of the underlying index

• DT is the future value of the dividends paid on the underlying stocksover the life of the futures (assumed known and compounded to timeT at the risk-free rate).a

• r1 or r1(0, T ) is the spot risk-free rate (with annual compounding) forthe maturity of the future.b

aTo avoid having different units on the LHS and RHS of this equation, we needto interpret F0 and S0 as dollar values, that is, as a dollar multiplier ($k) timesthe quoted index points. As this scaling will not affect pricing, it is convenient towork with $k = $1.

bNote that subscripts under F , S, and D denote time, whereas under r (and dbelow) denote compounding frequency.

Intuition. Recall that:

• Stock indices (Dow Jones, S&P 500, etc) usually do not account for dividends

• If you hold a mutual fund that tracks the index, or if you buy the stocksdirectly, you obviously receive dividends


Hence, if we invest in the stock portfolio and sell the index forward to guaranteethe capital appreciation, the gross return of

F0 +DT

S0

should be risk free. Equation (6.2) immediately follows.

Example 6.3.1. (This is from section 22.4 of Bodie, Kane, and Marcus(2014)). We want to price a 1-year futures contract on the S&P 500.

1. Suppose the S&P 500 is currently at 1 000 (this is a unitlessindex).

2. Suppose the futures contract calls for delivery of $1 times thevalue of the SP500 index.2 Hence, S0 = $1000.

3. Suppose that if we invest $1000 in a mutual fund that tracks theSP500, we will receive $20 of dividends over the course of theyear. Assume that all dividends are paid at the end of the year.

4. Assume the risk-free rate is r1(0, 1) = 3%.

5. Then,

F0 = . . . = $1010 = 1010 × $1

which would be quoted as 1 010 index points.

Proof of proposition 6.3.1. If equality does not hold, there is an opportunity forindex arbitrage.

• If F0 > S0(1 + r1)T −DT , the arbitrage strategy is

Cash Flow atTrade t = 0 t = T

Borrow S0 risk-free +S0 −S0(1 + r1)T

Buy S0 of stocks in index −S0 ST +DT

Short Futures 0 F0 − ST

Total 0 F0 − S0(1 + r1)T +DT

The total CF at T is positive by assumption.

• If F0 < S0(1 + r1)T − DT , do the opposite trades. Recall that when you

short stock you need to pay the dividends.

2The two most common contracts traded on CME are actually for $250 or $50 timesthe SP500 index, but this scaling does not matter for pricing.


Example 6.3.2. Continuing the previous example, suppose that the1-year futures is trading at 990. Complete the arbitrage strategy:


Invest S0 risk-freeShort S0 of stocks in indexLong Futures

Total

This guarantees a $20 profit for each futures contract traded.

Remarks about the spot-futures parity relation (6.2):

1. Alternative with D0. With continuous compounding, (6.2) becomes F0 =S0e

r∞T −DT . Let D0 denote the present value of the dividends paid duringthe life of the contract and note thatDT = D0e

r∞T . Then, (6.2) is equivalentto

F0 = (S0 −D0)er∞T (6.3)

2. Alternative with dividend yield. Equation (6.3) can be further modified to

F0 = S0S0 −D0

S0er∞T = S0 exp

(

ln

(S0 −D0

S0

))

er∞T

= S0 exp

(

−1

Tln

(S0

S0 −D0

)

T

)

er∞T

orF0 = S0e

(r∞−d∞)T (6.4)

where d∞ = 1T ln S0

S0−D0is the annual dividend yield (with continuous com-

pounding) estimated with the dividends paid during the life of the futures.3

3Equation (6.4) is the standard representation in Hull (2012).Bodie, Kane, and Marcus (2014) present a discrete-compounding version of (6.4) asF0 = S0(1 + r1 − d?)

T . When the futures maturity is exactly T = 1 year, d? can be easilyestimated by d? = D1/S0. However, I don’t see how to estimate a d? that works for anyT 6= 1, other than by first getting d∞ from d∞ = 1

Tln S0

S0−D0

, and then forcing d? to

be the value that solves 1 + r1 − d? = e(r∞−d∞). When T 6= 1, BKM seem to estimateeffective rates for a short horizon of 1/n years, r(n) and d(n). For example, if n = 12, d(12)is the monthly (not annual) dividend yield from dividends received over the next month.The futures price is then F (t, T ) = St(1 + r(n) − d(n))

n(T−t).

6.4. Investment strategies with stock-index futures 71

3. Investment assets. Equation (6.2) can be applied directly to other assetsheld purely for investment purposes, like gold or silver, by setting DT = 0.

4. Bonds. Equation (6.2) can be applied to bonds by replacing Dividendswith Coupons. However, strictly speaking, the equality would be exact forForward bond prices, but would only be an approximation for Futures bondprices. The reason is the different cash flow timing induced by mark-to-market in futures, which cannot be ignored in interest-rate derivatives.

5. Consumption commodities. (Advanced optional material) Equation (6.2)can be extended to consumption commodities (like crude oil, corn, etc) byincluding storage costs, but we only get a no-arbitrage upper bound on thefutures price: F0 ≤ (S0 + U0)e

rT , where U0 is the present value of storagecosts. For further details on the pricing of these other contracts see Hull(2012).

6.4 Investment strategies with stock-index fu-

tures

Stock-index futures allow for market-timing strategies with much lower transac-tions costs than trading in the individual stocks of the index.

Assumptions:

• These strategies are typically short-term, so we ignore dividends.

• The horizon of the strategy is the same as the maturity of the futures.

6.4.1 Creating synthetic stock positions

(Based on section 23.2 of Bodie, Kane, and Marcus (2014))


Synthetic stock position

To create a synthetic long stock position equivalent to a total initial equityinvestment of $V0 for a period of T years,

1. Invest V0 = S0 ×N dollars in Treasury Bills maturing at T .

2. Buy N = V0/S0 index futures for delivery at T .

The resulting cash flows are:


Invest S0N in T-bills −S0N +S0N(1 + r1)T = F0N

Long N Futures 0 (ST − F0)N

Total −S0N +STN

where• N is the number of futures contracts

• S0 is the dollar spot price of the underlying index ($k× index)

• F0 = S0(1 + r1)T by prop. 6.3.1 when DT = 0

• r1 is the spot risk-free rate for period T (with annual compounding)

Remark: Note that the rate of return is same as on a real stock portfolio(ignoring dividends):

VT /V0 − 1 = (STN)/(S0N)− 1 = ST /S0 − 1

Example 6.4.1. (Example 23.3 in Bodie, Kane, and Marcus (2014)).

• An institutional investor wants to bet 140M$ that the marketwill go up over the next 1 month. To minimize trading costs, hewants to trade in futures.

• The SP500 index is now at 1400.

• The Futures contract has a multiplier of $250.

Investment strategy:

1. The amount to invest in T-bills is 140M$

2. Check that the number of futures to buy is

N = . . . = 400


6.4.2 Hedging an equity portfolio from market risk

(Based on section 3.5 of Hull (2012))

Market risk hedging

To hedge an existing stock portfolio from market movements over a periodof T years, we should short index futures. The optimal number of contractsto short in order to minimize the volatility of the combined position is

N = βV0

F0

where• V0 is the current market value of the equity portfolio

• F0 or F (0, T ) is the dollar value of 1 futures contract (ie, contract sizetimes futures quote)

• β is the CAPM beta of the portfolio.

Remarks:

• This rule is from Hull (2012, equation 3.5)

• Recall that beta can be estimated from the time series regression

(rp − rf )t = αp + βp(rM − rf )t + εpt (6.5)

If the fund is fairly price according to the CAPM, we have αp = 0. Hence,the hedge will work well if

rp,t+1 = rf + βp(rM,t+1 − rf ) (6.6)

Example 6.4.2. The manager of an equity portfolio worth 50M$ isbullish on the market over the long term, but is afraid that there maybe a big drop over the next month. The manager wants to avoid thisrisk, but it would be too expensive to sell the whole portfolio now andrebuild it 1 month from now due to transaction costs.

Other inputs:

• The portfolio has β = 1.5

• The SP500 is at 2050

6.5. Exercises 74

• The risk-free interest rate with monthly compounding is 3% (ie,the interest received at the end of 1 month is 0.25%).

• The 1-month Futures is quoted at 2055.125. The contract size,or multiplier, is $250.

HEDGING STRATEGY:

Check that the number of futures to short is

N = . . . ≈ 146

VERIFICATION:

The following table shows alternative scenarios for the market overthe next month. Note that the combined return is always very closeto the risk-free rate! (Check the numbers in the table)

ScenarioInputs

rm -5.00% -1.00% 0.25% 1.00% 5.00%SP500 1947.50 2029.50 2055.13 2070.50 2152.50

Ouputs

Futures payoffShort position 3 928 313 935 313 0.00 - 561 188 - 3 554 188

Cash Portfoliorp -7.625% -1.625% 0.250% 1.375% 7.375%Vp 46 187 500 49 187 500 50 125 000 50 687 500 53 687 500

Combined PortfolioTotal value 50 115 813 50 122 813 50 125 000 50 126 313 50 133 313rc 0.232% 0.246% 0.250% 0.253% 0.267%

6.5 Exercises

Ex. 13 — Crude oil futures are trading at 50 $/bbl. Each contract is on 1 000barrels. The initial margin is 10%. You believe that oil prices are going to decreaseand want to get exposure to 20,000 barrels.

1. Your trading strategy is to [buy/sell] [number] fu-tures contracts

2. Some days latter you close the position when the futures is trading at 47$/bbl. Compute the rate of return on your initial margin.

6.5. Exercises 75

Ex. 14 — Bodie, Kane, and Marcus (2014) problems at the end of chapter 22(p. 794): 8, 10, 14

Chapter 7

Solutions to Problems

Answer (Ex. 3) — E[rx] = 0.14. V ar[rx] = 0.6 ∗ (0.3 − 0.14)2 + 0.4 ∗ (−0.1 −

0.14)2 = 0.0384 ⇒ σx = 0.1960. Hence,

SRp = SRx = (0.14 − 0.02)/0.196 = 0.6124

Note that all portfolios along the CAL(x) have the same SR.

Answer (Ex. 5) — The Tangency portfolio is given by (1 means bond fund, 2means stock fund):

w1 =re1σ

22 − re2σ12

re1σ22 + re2σ

21 − (re1 + re2)σ12

= 0.4783

and w2 = 0.5217. T hence has

E[rT ] = w1 ∗ 0.04 + w2 ∗ 0.08 = 0.0609

and

σT = ((w12 ∗ 0.12 + w22 ∗ 0.22 + 2 ∗ w1 ∗ w2 ∗ 0.1 ∗ 0.2 ∗ 0.3))0.5 = 0.1272

The optimal portfolio for the investor is

0.15 = wT ∗ 0.1272 ⇒ wT = 1.1796

and wf = −0.1796The complete best portfolio has

E[rp] = wT ∗ E[rT ] + wf ∗ 0.02 = 0.0682

76

77

Answer (Ex. 6) — Since ρ = 0, the variance is just σ2p = w2

aσ2a + w2

bσ2b . Let

wb = 1− wa.The investor picks the portfolio that maximizes his utility function:

maximizewa

[waE[ra] + (1− wa)E[rb]]−A

2[w2

aσ2a + (1− wa)

2σ2b ]

The first-order condition is:

E[ra]− E[rb]−A

2

[2waσ

2a − 2(1− wa)σ

2b

]= 0

⇒wa =

[E[ra]−E[rb]

A+ σ2

b

]

/(σ2a + σ2

b )

Using the numbers provided, wa = 0.7333 and wb = 0.2667

Answer (Ex. 13) — .1) Sell 20 contracts2) Initial margin = 0.1 ∗ 50 ∗ 20000 = 100, 000. Return = 3 ∗ 20000/100000 = 60%

Bibliography

Bodie, Z., A. Kane, and A. Marcus, 2014, Investments. McGraw-Hill, 10thedn.

Fama, E. F., and K. R. French, 1993, “Common Risk Factors in the Returnson Stocks and Bonds,” Journal of Financial Economics, 33, 3–56.

, 1996, “Multifactor explanations of Asset Pricing Anomalies,” Jour-

nal of Finance, 51(1), 55–84.

Hull, J. C., 2012, Options, Futures, and other Derivatives. Pearson Educa-tion.

Ross, S. A., 1976, “The Arbitrage Theory of Capital Asset Pricing,” Journal

of Economic Theory, 13, 341–360.

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