Nguon Goc Cac Cong Thuc
Nguon Goc Cac Cong Thuc
D1 D2 D3
A.1 Summation of Infinite Geometric Series P0 ¼ þ þ þ (A.6)
1 þ k ð1 þ kÞ2 ð1 þ kÞ3
Summation of geometric series can be defined as:
Where P0 ¼ present value of stock price per share
Dt ¼ dividend per share in period t (t ¼ 1, 2,. . .,n)
S ¼ A þ AR þ AR2 þ þ ARn1 (A.1)
If dividends grow at a constant rate, say g, then,
D2 ¼ D1(1 + g), D3 ¼ D2(1 + g) ¼ D1(1 + g)2, and so on.
Multiplying both sides of Equation A.1 by R, we obtain
Then, Equation A.6 can be rewritten as:
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 911
# Springer Science+Business Media New York 2013
Appendix B: Derivation of DOL, DFL and DCL
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 913
# Springer Science+Business Media New York 2013
914 Appendix B
Suppose there are two projects under consideration. Cash Table A.1 NPV of Project A and B under different discount rates
flows of project A, B and B – A are as follows: Discount rate (%) NPV (Project A) NPV (Project B)
0 1500.00 3500.00
Period 0 1 2 3
5 794.68 1725.46
Project A 10,500 10,000 1,000 1,000
10 168.67 251.31
Project B 10,500 1,000 1,000 12,000
15 390.69 984.10
Cash flows of B – A 0 9,000 0 11,000
20 893.52 2027.78
Based upon the information the table above we can cal- 1; 000 10; 000
0 ¼ ½10; 500 ð10; 500Þ þ
culate the NPV of Project A and Project B under different 1 þ R c 1 þ Rc
discount rates. The results are presented in Table A.1. " # " #
1; 000 1; 000 12; 000 1; 000
NPV(B) is higher with low discount rates and NPV(A) þ þ
is higher with high discount rates. This is because the cash ð1 þ Rc Þ2 ð1 þ Rc Þ2 ð 1 þ Rc Þ 3 ð 1 þ Rc Þ 3
flows of project A occur early and those of project B occur (A.10)
later. If we assume a high discount rate, we would favor
project A; if a low discount rate is expected, project B will be Solving Equation A.10 by trial and error method for Rc, Rc
chosen. In order to make the right choice, we can calculate equals 10.55%.
the crossover rate. If the discount rate is higher than the Using the procedure of calculating internal rate of return
crossover rate, we should choose project A; if otherwise, (IRR) as discussed in Equations A.8, A.9, and A.10, we
we should go for project B. The crossover rate, Rc, is the calculate the IRR for both Project A and Project B. The
rate such that NPV(A) equals to NPV(B). IRR for Project A and B are 11.45% and 10.95% respec-
Suppose the crossover rate is Rc, then tively. From this information, we have concluded that Project
A will perform better than Project B without consideration
NPVðAÞ ¼ 10; 500 þ 10; 000=ð1 þ Rc Þ þ 1; 000= for change of discount rate. Therefore, the IRR decision rule
ð1 þ Rc Þ2 þ 1; 000=ð1 þ Rc Þ3 (A.8) cannot be used for capital budgeting decisions when there
exists an increasing or decreasing net cash inflow. This is so
NPVðBÞ ¼ 10; 500 þ 1; 000=ð1 þ Rc Þ þ 1; 000= called “The Timing Problem” for using the IRR method for
capital budgeting decisions.
ð1 þ Rc Þ2 þ 12; 000=ð1 þ Rc Þ3
NPVðAÞ ¼ NPVðBÞ (A.9)
Therefore,
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 915
# Springer Science+Business Media New York 2013
Appendix D: Capital Budgeting Decisions
with Different Lives
D.1 Mutually Exclusive Investment Projects Subtracting Equation A.12 from Equation A.11 gives:
with Different Lives
NPV ðN; tÞ ðHÞNPV ðN; tÞ ¼ NPVðNÞ 1 Htþ1
The traditional NPV technique may not be the appropriate
NPVðNÞ 1 Htþ1
criterion to select a project from mutually exclusive invest- NPV ðN; tÞ ¼
1H
ment projects, if these projects have different lives. The
underlying reason is that, compared with a long-life project,
Taking the limit as the number of replications, t,
a short-life project can be replicated more quickly in the long
approaches infinity gives:
run. In order to compare projects with different lives, we
compute the NPV of an infinite replication of the investment
lim NPV ðN; tÞ ¼ NPV ðN; 1Þ
project. For example, let Projects A and B be two mutually x!1
2 3
exclusive investment projects with the following cash flows.
1
¼ NPV 4 h i5
Year Project A Project B 1 1=ð1 þ K ÞN
0 100 100 " #
1 70 50 ð1 þ K ÞN
¼ NPVðNÞ (A.13)
2 70 50 ð1 þ K ÞN 1
3 50
Equation A.13 is the NPV of an N-year project replicated
By assuming a discount rate of 12%, the traditional NPV
at constant scale an infinite number of times. We can use it to
of Project A is 18.30 and the NPV of Project B is 20.09. This
compare projects with different lives because when their
shows that Project B is a better choice than Project A.
cash-flow streams are replicated forever, it is as if they had
However, the NPV with infinite replications for Project A
the same (infinite) life.
and B should be adjusted into a comparable basis.
Based upon Equation A.13, we can calculate the NPV of
In order to compare Projects A and B, we compute the
Projects A and B as follows:
NPV of an infinite stream of constant scale replications. Let
NPV (N, 1) be the NPV of an N-year project with NPV (N),
replicated forever. This is exactly the same as an annuity For Project A For Project B
paid at the beginning of the first period and at the end of NPV ð2; 1Þ NPV ð3; 1Þ
every N years from that time on. The NPV of the annuity is: " # " #
ð1 þ 0:12Þ2 ð1 þ 0:12Þ3
¼ NPVð2Þ ¼ NPVð3Þ
NPV ðN; 1Þ ¼ NPVðNÞ þ
NPVðNÞ
þ
NPVðNÞ
þ
ð1 þ 0:12Þ2 1 ð1 þ 0:12Þ3 1
ð1 þ K Þ N
ð1 þ K Þ 2N
1:2544 1:4049
¼ ð18:30Þ ¼ 20:09
In order to obtain a closed-form formula, let 0:2544 0:4049
(1/[(1 + K)N]) ¼ H. Then we have: ¼ 90:23 ¼ 69:71
NPV ðN; tÞ ¼ NPVðNÞ 1 þ H þ H 2 þ þ H t (A.11)
Consequently, we would choose to accept Project A over
Multiplying both sides by H, this becomes Project B, because, when the cash flows are adjusted for
different lives, A provides the greater cash flow.
Alternatively, Equation A.13 can be rewritten as an equiv-
H½NPV ðN; tÞ ¼ NPVðNÞ H þ H 2
alent annual NPV version as:
þ þ H t þ H tþ1 (A.12)
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 917
# Springer Science+Business Media New York 2013
918 Appendix D
Equation A.14 can be written as: In this example, following Equation A.13, we can find
NPVðNÞ ¼ K NPV ðN; 1Þ Annuity Factor (A.15) NPV ðN; 1Þ ¼ 1749:47 ð1 þ 0:1Þ4 =½ð1 þ 0:1Þ4 1
¼ 5654:71
Corporate Finance by Ross, Westerfield, and Jaffe (2005,
7th edn, p. 193) has discussed about Equivalent Annual Then following the Equation A.17, we obtain
Cost. The Equivalent Annual Cost (C) can be calculated as
follows: C ¼ K NPV ðN; 1Þ ¼ 0:1 5654:71 ¼ 565:47
NPVðNÞ ¼ C Annuity Factor (A.16) Therefore, the equivalent annual cost C is identical to the
equivalent annual NPV as defined in Equation A.14.
From Equations A.15 and A.16, we obtain
If there is a two security portfolio, its variance can be defined as: s2p ¼ w2D s2D þ w2E s2E þ 2wD wE Covðr D ; r E Þ
We can solve the minimization problem by wD s2D þ wD s2E 2wD Covðr D ; r E Þ ¼ s2E Covðr D ; r E Þ
2
differentiating the s2p with respect to wD and setting the sD þ s2E 2Covðr D ; r E Þ wD ¼ s2E Covðr D ; r E Þ
derivative equal to 0 i.e., we want to solve
Finally, we have
@s2p
¼0 (A.21)
@wD s2E Covðr D ; r E Þ
wD ¼
s2D þ s2E 2Covðr D ; r E Þ
Since, wD + wE ¼ 1 or, wE ¼ 1 wD therefore, the var-
iance, s2P , can be rewritten as
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 919
# Springer Science+Business Media New York 2013
Appendix F: Derivation of an Optimal Weight
Portfolio Using the Sharpe Performance Measure
Solution for the weights of the optimal risky portfolio can be Equation A.22 becomes
found by solving the following maximization problem:
@Sp @ ½f ðwD Þ=gðwD Þ
¼
E rp rf @wD @wD
w D Sp ¼
Max
sp f 0 ðwD ÞgðwD Þ f ðwD Þg0 ðwD Þ
¼ ¼0 (A.28)
½gðwD Þ2
where E(rp) ¼ expected rates of return for portfolio P
@f ðwD Þ
rf ¼ risk free rates of return where f 0 ðwD Þ ¼ ¼ Eðr D Þ Eðr E Þ (A.29)
@wD
Sp ¼ sharpe performance measure, and
sp as defined in Equation A.19 of Appendix 5 @gðwD Þ
g0 ðwD Þ ¼
We can solve the maximization problem by @wD
differentiating the Sp with respect to wD, and setting the 1 h 2 2
derivative equal to 0 i.e., we want to solve ¼ wD sD þ ð1 wD Þ2 s2E þ 2wD ð1 wD Þ
2
Covðr D ; r E Þ1=21
@Sp
¼0 (A.22) 2wD s2D þ 2wD s2E 2s2E þ 2Covðr D ; r E Þ
@wD
4wD CovðrD ; r E Þ
In the case of two securities, we know that ¼ wD s2D þ wD s2E s2E þ Covðr D ,rE Þ
2wD Covðr D ; r E Þ
h
E r p ¼ wD Eðr D Þ þ wE Eðr E Þ (A.23) w2D s2D þ ð1 wD Þ2 s2E þ 2wD ð1 wD Þ
1=2 Covðr D ; r E Þ1=2
sp ¼ w2D s2D þ w2E s2E þ 2wD wE Covðr D ; r E Þ (A.24)
(A.30)
wD þ wE ¼ 1 (A.25)
From Equation A.28,
From above Equations A.23, A.24, and A.25, we can f 0 ðwD ÞgðwD Þ f ðwD Þg0 ðwD Þ ¼ 0; or
rewrite E(rp) rf and sp as: f 0 ðwD ÞgðwD Þ ¼ f ðwD Þg0 ðwD Þ (A.31)
E r p r f ¼ wD Eðr D Þ þ wE Eðr E Þ r f Now, plugging f(wD), g(wD), f0 (wD), and g0 (wD) [Equa-
¼ wD Eðr D Þ þ ð1 wD Þ Eðr E Þ r f tions A.26, A.27, A.29, and A.30] into Equation A.31, we have
f ðw D Þ (A.26)
½Eðr D Þ Eðr E Þ
h i1=2
1=2 w2D s2D þ ð1 wD Þ2 s2E þ 2W D ð1 wD ÞCovðrD ; r E Þ
sp ¼ w2D s2D þ w2E s2E þ 2wD wE Covðr D ; r E Þ
h ¼ wD Eðr D Þ þ ð1 wD ÞEðr E Þ r f
¼ w2D s2D þ ð1 wD Þ2 s2E þ 2wD ð1 wD Þ
wD s2D þ wD s2E s2E þ Covðr D ; r E Þ
Covðr D ; r E Þ1=2 2wD Covðr D ; r E Þ
h
gðwD Þ (A.27) w2D s2D þ ð1 wD Þ2 s2E þ 2wD ð1 wD Þ
Covðr D ; rE Þ1=2
ðA31Þ
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 921
# Springer Science+Business Media New York 2013
922 Appendix F
Rearrange all terms on both hand sides of Equation A.33, i.e., Moving all the terms with wD on one side and leaving the
Left hand side of Equation A.33 rest terms on the other side from Equation A.34, we have
½ Eð r D Þ Eð r E Þ
h i ½Eðr D Þ Eðr E Þ s2E Eðr E Þ r f
w2D s2D þ ð1 wD Þ2 s2E þ 2wD ð1 wD ÞCovðr D ; r E Þ Covðr D ; r E Þ s2E
¼ ½Eðr D Þ Eðr E Þ ¼ Eðr E Þ r f s2D þ s2E 2Covðr D ; r E Þ wD
w2D s2D þ s2E 2wD s2E þ w2D s2E þ 2wD Covðr D ; r E Þ ½Eðr D Þ Eðr E Þ Covðr D ; r E Þ s2E wD (A.35)
2w2D Covðr D ; r E Þ
Rearrange Equation A.35 in order to solve for wD, i.e.,
¼ ½Eðr D Þ Eðr E Þ w2D s2D þ s2E 2Covðr D ; r E Þ
þ2wD Covðr D ; r E Þ s2E þ s2E Eðr D Þ Eðr E Þ þ Eðr E Þ r f s2E
¼ ½Eðr D Þ Eðr E Þ w2D s2D þ s2E 2Covðr D ; r E Þ Eðr E Þ r f Covðr D ; r E Þ
þ ½Eðr D Þ Eðr E Þ 2wD Covðr D ; r E Þ s2E þ ½Eðr D Þ ¼ Eðr E Þ r f s2D þ Eðr E Þ r f s2E
Eðr E Þ s2E ¼ ½Eðr D Þ Eðr E Þ s2D þ s2E Eðr E Þ r f ½2Covðr D ; r E Þ ½Eðr D Þ
2Covðr D ; r E Þw2D þ 2½Eðr D Þ Eðr E Þ Eðr E ÞCovðr D ; r E Þ þ ½Eðr D Þ Eðr E Þs2E wD
Covðr D ; r E Þ s2E wD þ ½Eðr D Þ Eðr E Þ s2E ¼ Eðr D Þ r f s2E þ Eðr E Þ r f s2D ½Eðr D Þ
r f þ Eðr E Þ r f Covðr D ; r E Þ wD
Right hand side of Equation A.33
Finally, we have the optimum weight of security D as
wD Eðr D Þ þ ð1 wD ÞEðr E Þ r f wD s2D þ wD s2E
s2E þ Covðr D ; r E Þ 2wD Covðr D ; r E Þ Eðr D Þ r f s2E Eðr E Þ r f Covðr D ; r E Þ
wD ¼
¼ wD Eðr D Þ þ Eðr E Þ wD Eðr E Þ r f wD s2D Eðr D Þ r f s2E þ Eðr E Þ r f s2D
þwD s2E 2wD Covðr D ; r E Þ s2E þ Covðr D ; r E Þ Eðr D Þ r f þ Eðr E Þ r f Covðr D ; r E Þ
¼ wD ½Eðr D Þ Eðr E Þ þ Eðr E Þ r f wD s2D
þs2E 2Covðr D ; r E Þ þ Covðr D ; r E Þ s2E
¼ wD ½Eðr D Þ Eðr E Þ wD s2D þ s2E 2Covðr D ; r E Þ
þ wD ½Eðr D Þ Eðr E Þ Covðr D ; r E Þ s2E
þ Eðr E Þ r f wD s2D þ s2E 2Covðr D ; r E Þ
þ Eðr E Þ r f Covðr D ; r E Þ s2E
¼ ½Eðr D Þ Eðr E Þ s2D þ s2E 2Covðr D ; r E Þ w2D
þ ½Eðr D Þ Eðr E Þ Covðr D ; r E Þ s2E wD
þ Eðr E Þ r f s2D þ s2E 2Covðr D ; r E Þ wD
þ Eðr E Þ r f Covðr D ; r E Þ s2E
Appendix G: Applications of the Binomial
Distribution to Evaluate Call Options
In this appendix, we show how the binomial distribution is period left to maturity. This option’s value at expiration is
combined with some basic finance concepts to generate a determined by the price of its underlying stock and the
model for determining the price of stock options. exercise price X. The value is either
Cu ¼ Maxð0; uS XÞ (A.36)
G.1 What is an Option?
or
In the most basic sense, an option is a contract conveying the
Cd ¼ Maxð0; dS XÞ (A.37)
right to buy or sell a designated security at a stipulated price.
The contract normally expires at a predetermined date. The Why is the call worth Max (0, uS X) if the stock price
most important aspect of an option contract is that the us uS? The option holder is not obliged to purchase the stock
purchaser is under no obligation to buy; it is, indeed, an at the exercise price of X, so she or he will exercise the
“option.” This attribute of an option contract distinguishes option only when it is beneficial to do so. This means the
it from other financial contracts. For instance, whereas the option can never have a negative value. When is it beneficial
holder of an option may let his or her claim expire unused if for the option holder to exercise the option? When the price
he or she so desires, other financial contracts (such as futures per share of the stock is greater than the price per share at
and forward contracts) obligate their parties to fulfill certain which he or she can purchase the stock by using the option,
conditions. which is the exercise price, X. Thus if the stock price uS
A call option gives its owner the right to buy the underly- exceeds the exercise price X, the investor can exercise the
ing security, a put option the right to sell. The price at which option and buy the stock. Then he or she can immediately
the stock can be bought (for a call option) or sold (for a put sell it for uS, making a profit of uS X (ignoring commis-
option) is known as the exercise price. sion). Likewise, if the stock price declines to dS, the call is
worth Max (0, dS X).
Also for the moment, we will assume that the risk-free
G.2 The Simple Binomial Option Pricing interest rate for both borrowing and lending is equal to r
Model percent over the one time period and that the exercise price
of the option is equal to X.
Before discussing the binomial option model, we must rec- To intuitively grasp the underlying concept of option
ognize its two major underlying assumptions. First, the pricing, we must set up a risk-free portfolio – a combination
binomial approach assumes that trading takes place in dis- of assets that produces the same return in every state of the
crete time, that is, on a period-by-period basis. Second, it is world over our chosen investment horizon. The investment
assumed that the stock price (the price of the underlying horizon is assumed to be one period (the duration of this
asset) can take on only two possible values each period; it period can be any length of time, such as an hour, a day, a
can go up or go down. week, etc.). To do this, we buy h share of the stock and sell
Say we have a stock whose current price per share S can the call option at its current price of C. Moreover, we choose
advance or decline during the next period by a factor of the value of h such that our portfolio will yield the same
either u (up) or d (down). This price either will increase by payoff whether the stock goes up or down.
the proportion u 1 0 or will decrease by the proportion
1 d, 0 < d < 1. Therefore, the value S in the next period hðuSÞ Cu ¼ hðdSÞ Cd (A.38)
will be either uS or dS. Next, suppose that a call option exists
on this stock with a current price per share of C and an By solving for h, we can obtain the number of shares of
exercise price per share of X and that the option has one stock we should buy for each call option we sell.
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 923
# Springer Science+Business Media New York 2013
924 Appendix G
where d < r < u. To simplify this equation, we set Solving the binomial valuation equation as indicated in
Equation A.43, we get
Rd uR
p¼ so 1 p ¼ (A.42) 0:85ð10Þ þ 0:15ð0Þ
ud ud C¼
1:07
Thus we get the option’s value with one period to ¼ $7:94
expiration
The correct value for this particular call option today,
pCu þ ð1 pÞCd under the specified conditions, is $7.94. If the call option
C¼ (A.43) does not sell for $7.94, it will be possible to earn arbitrage
R
profits. That is, it will be possible for the investor to earn a
This is the binomial call option valuation formula in its risk-free profit while using none of his or her own money.
most basic form. In other words, this is the binomial valua- Clearly, this type of opportunity cannot continue to exist
tion formula with one period to expiration of the option. indefinitely.
To illustrate the model’s qualities, let’s plug in the
following values, while assuming the option has one period
to expiration. Let
G.3 The Generalized Binomial Option Pricing
X ¼ $100 Model
S ¼ $100
U ¼ ð1:10Þ; so uS ¼ $110 Suppose we are interested in the case where there is more than
one period until the option expires. We can extend the one-
D ¼ ð0:90Þ; so dS ¼ $90
period binomial model to consideration of two or more periods.
R ¼ 1 þ r ¼ 1 þ 0:07 ¼ 1:07 Because we are assuming that the stock follows a bino-
mial process, from one period to the next it can only go up by
a factor of u or go down by a factor of d. After one period the
First we need to determine the two possible option values stock’s price is either uS or dS. Between the first and second
at maturity, as indicated in Table A.2. periods, the stock’s price can once again go up by u or down
Next we calculate the value of p as indicated in by d, so the possible prices for the stock two periods from
Equation A.42. now are uuS, udS, and ddS. This process is demonstrated in
Appendix G 925
1 Xn
n!
Fig. A.1 Price path of underlying stock (Source: Rendelman and C¼ pk ð1 pÞnk
Bartter, 1979, 1906) R k¼0 k!ðn kÞ!
n
Max 0; uk d nk S X (A.48)
tree diagram (Figure A.1) given in Example A.1 later in this
appendix. To actually get this form of the binomial model, we could
Note that the option’s price at expiration, two periods extend the two-period model to three periods, then from
from now, is a function of the same relationship that deter- three periods to four periods, and so on. Equation A.48
mined its expiration price in the one-period model, more would be the result of these efforts. To show how Equa-
specifically, the call option’s maturity value is always tion A.48 can be used to assess a call option’s value, we
modify the example as follows: S ¼ $100, X ¼ $100,
CT ¼ ½0; ST X (A.44) R ¼ 1.07, n ¼ 3, u ¼ 1.1 and d ¼ 0.90.
First we calculate the value of p from Equation A.42 as
where T designated the maturity date of the option. 0.85, so 1 p is 0.15. Next we calculate the four possible
To derive the option’s price with two periods to go ending values for the call option after three periods in terms
(T ¼ 2), it is helpful as an intermediate step to derive the of Max[0, ukdnkS X].
value of Cu and Cd with one period to expiration when the h i
stock price is either uS or dS, respectively. C1 ¼ 0; ð1:1Þ3 ð0:90Þ0 ð100Þ 100 ¼ 33:10
h i
pCuu þ ð1 pÞCud C2 ¼ 0; ð1:1Þ2 ð0:90Þ ð100Þ 100 ¼ 8:90
Cu ¼ (A.45) h i
R
C3 ¼ 0; ð1:1Þ ð0:90Þ2 ð100Þ 100 ¼ 0
pCdu þ ð1 pÞCdd h i
Cd ¼ (A.46) C4 ¼ 0; ð1:1Þ0 ð0:90Þ3 ð100Þ 100 ¼ 0
R
926 Appendix G
Now we insert these numbers (C1, C2, C3, and C4) into the Equation A.45 in the text, we can write the binomial call
model and sum the terms. option model as
1 3! X
C¼ 3
ð0:85Þ0 ð0:15Þ3 0 C ¼ SB1 ðn; p0 ; mÞ B2 ðn; p; mÞ (A.50)
ð1:07Þ 0!3! Rn
3!
þ ð0:85Þ1 ð0:15Þ2 0 where
1!2!
3!
þ ð0:85Þ2 ð0:15Þ2 8:90 X
n
B1 ðn; p0 ; mÞ ¼ Cnk p0 ð1 p0 Þ
2!1! k nk
3!
þ ð0:85Þ3 ð0:15Þ0 33:10 k¼m
3!0! Xn
B2 ðn; p; mÞ ¼ Cnk pk ð1 pÞnk
1 321
¼ 0þ0þ ð0:7225Þð0:15Þð8:90Þ k¼m
1:225 211
321 and m is the minimum amount of time the stock has to go up
þ ð0:61413Þð1Þð33:10Þ
3211 for the investor to finish in the money (that is, for the stock
1 price to become larger than the exercise price).
¼ ½ð0:32513 8:90Þ þ ð0:61413 33:10Þ
1:225 In this appendix, we showed that by employing the defi-
¼ $18:96 nition of a call option and by making some simplifying
assumptions, we could use the binomial distribution to find
As this example suggests, working out a multiple-period the value of a call option. In the next chapter, we will show
problem by hand with this formula can become laborious as how the binomial distribution is related to the normal distri-
the number of periods increases. Fortunately, programming bution and how this relationship can be used to derive one of
this model into a computer is not too difficult. the most famous valuation equations in finance, the Black-
Now let’s derive a binomial option pricing model in terms Scholes option pricing model.
of the cumulative binomial density function. As a first step,
we can rewrite Equation A.48 as Example A.1
" # A Decision Tree Approach to Analyzing Future Stock Price
X
n
n! uk dnk By making some simplifying assumptions about how a
C¼S pk ð1 pÞnk stock’s price can change from one period to the next, it is
k¼m
k!ðn K Þ! Rn
" # possible to forecast the future price of the stock by means of
X X n
n! nk a decision tree. To illustrate this point, let’s consider the
n p ð1 pÞ
k
(A.49)
R k¼m k!ðn kÞ! following example.
Suppose the price of Company A’s stock is currently
This formula is identical to Equation A.48 except that we $100. Now let’s assume that from one period to the next,
have removed the Max operator. In order to remove the Max the stock can go up by 17.5% or go down by 15%. In
operator, we need to make ukdnkS X positive, which we addition, let us assume that there is a 50% chance that the
can do by changing the counter in the summation from k ¼ 0 stock will go up and a 50% chance that the stock will go
to k ¼ m. What is m? It is the minimum number of upward down. It is also assumed that the price movement of a stock
stock movements necessary for the option to terminate “in (or of the stock market) today is completely independent of
the money” (that is, ukdnkS X > 0). How can we inter- its movement in the past; in other words, the price will rise or
pret Equation A.49? Consider the second term in brackets; it fall today by a random amount. A sequence of these random
is just a cumulative binomial distribution with parameters of increases and decreases is known as a random walk.
n and p. Likewise, via a small algebraic manipulation we can Given this information, we can lay out the paths that the
show that the first term in the brackets is also a cumulative stock’s price may take. Figure A.1 shows the possible stock
binomial distribution. This can be done by defining P0 prices for company A for four periods.
(u/R)p and 1 P0 (d/R)(1 p). Thus Note that in period 1 there are two possible outcomes: the
stock can go up in value by 17.5% to $117.50 or down by
uk dnk 15% to $85.00. In period 2 there are four possible outcomes.
¼ p0 ð 1 p0 Þ
k nk
pk ð1 pÞnk If the stock went up in the first period, it can go up again to
Rn
$138.06 or down in the second period to $99.88. Likewise, if
Therefore the first term in brackets is also a cumulative the stock went down in the first period, it can go down again
binomial distribution with parameters of n and p0 . Using to $72.25 or up in the second period to $99.88. Using the
Appendix G 927
same argument, we can trace the path of the stock’s price for We can also find the standard deviation for the stock’s
all four periods. return.
If we are interested in forecasting the stock’s price at the
end of period 4, we can find the average price of the stock for " #1=2
ð190:61 105:09Þ2 þ þ ð52:20 105:09Þ2
the 16 possible outcomes that can occur in period 4. sP ¼
16
P
16 ¼ $34:39
Pi
190:61 þ 137:89 þ þ 52:20
P ¼ i¼1 ¼ ¼ $105:09 P and sP can be used to predict the future price of stock A.
16 16
Appendix H: Derivation of Modigliani and Miller
(M&M) Proposition I and II with Taxes
H.1 M&M Proposition I with Taxes H.2 M&M Proposition II with Taxes
Assume that the firms are non-growth companies; the market Since market value of levered firm, VL , is equal to total
value of levered firm is equal to the market value of equity, E, plus total debt, D, and based on M&M proposition
unlevered firm plus the present value of the cost of perpetu- I with taxes, the market value of unlevered firm can be
ally total debt. derived as
X1
TDkd TDkd VL ¼ E þ D ¼ VU þ TD ¼> VU ¼ E þ ð1 TÞD (A.54)
VL ¼ VU þ t ¼ VU þ
t¼1 ð1 þ kd Þ
kd
The cash flow from each side of balance sheet must equal,
¼ VU þ TD (A.51) therefore
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 929
# Springer Science+Business Media New York 2013
930
By geometric theory, triangles RfPE and RfMpD in the where Rf ¼ the risk-free rate; RMp ¼ return on market port-
Fig. A.3 above are similar and are, therefore, directly folio Mp; Rp ¼ return on the portfolio consisting of
proportional, combinations of the risk-free asset and portfolio Mp; sp and
sMp ¼ standard deviations of the portfolio and the market;
DPRf E ffi DMp Rf D (A.58) and the operator E denotes expectations.
Therefore,
EðRp Þ Rf sp
¼
EðRMp Þ Rf sMp
sp
¼> EðRp Þ Rf ¼ EðRMp Þ Rf
sMp
sP
¼> EðRP Þ ¼ Rf þ EðRMp Þ Rf (A.59)
sMp
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 931
# Springer Science+Business Media New York 2013
Appendix J: Derivation of Capital Market Line (SML)
Sharpe (1964) used a general risky asset that did not lie on When wi ¼ 0, the security is held in proportion to its total
the CML and dubbed it I in Fig. A.4. The combinations of market value and there is no excess demand for security I.
risk and return possible by combining security I with the This is the key insight to Sharpe’s paper, for when wi ¼ 0, it
market portfolio, M, are shown by figure above. The average is possible to equate the slope of the curve IMI’ with the
return and standard deviation for any I-M combination can capital market line and thus obtain an expression for the
be approached in the same way as for a two-asset case: return on any risky security I. At equilibrium when wi ¼ 0,
the slope along the IMI’ curve will equal
EðRP Þ ¼ wi EðRi Þ þ ð1 wi ÞEðRm Þ (A.60)
@EðRP Þ EðRi Þ EðRm Þ
1 @EðRP Þ @wi sim s2m
sP ¼ w2i s2i þ ð1 wi Þs2m þ 2ð1 wi Þwi sim 2 (A.61) ¼ ¼ (A.64)
@ðsP Þ @ðsP Þ sm
where w1 represents excess demand for I or demand greater @wi
than its equilibrium weight in portfolio M, and sim is the
The slope of the capital market line at point M is
covariance of i and m.
The change in mean and standard deviation as the pro-
EðRm Þ Rf
portion w1 changes are the partial derivatives (A.65)
sm
@EðRP Þ
¼ EðRi Þ EðRm Þ (A.62) Let Eq. A.64 equal to Eq. A.65, then rearranging the
@wi
terms to solve for EðRi Þ gives the equation for the security
h i 1 market line or CAPM
@ðsP Þ 2
¼ 1=2 w2i s2i þ ð1 wi Þ2 s2m þ 2wi ð1 wi Þ
@wi sim
EðRi Þ ¼ Rf þ EðRm Þ Rf 2 (A.66)
2wi s2i 2s2m þ 2wi s2m þ 2sim 4wi sim sm
(A.63)
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 933
# Springer Science+Business Media New York 2013
934 Appendix J: Derivation of Capital Market Line (SML)
Assume the stock prices follow a lognormal distribution and C ¼ exp½rT E½MaxðST X; 0Þ
Z 1
denote the current stock price by S and the stock price at the X
¼ exp½rT S y gðyÞdy
St X S
end of t-th period by St then ¼ expðKt Þ is a random S
Z 1 Z
X 1
St1
variable with a lognormal distribution, where Kt is the rate of ¼ S exp½rT ygðyÞdy exp½rT S gðyÞdy
return in t-thperiod and follows normal distribution with the
X S X
S S
constant mean m and variance s2 , therefore, (A.71)
ST S1 S2 ST Let x ¼ lnðyÞ, then x follows normal distribution with
E ¼E ...
S S S1 ST1 mean mT ¼ r 12 s2 T and variance s2 T, and
Ts2
¼ E expðK1 þ K2 þ ... þ KT Þ ¼ Tm þ (A.68) 1 f ðxÞ
2 dx ¼ dy; ¼ gðyÞ
y y
ð1 ð1
Under the assumption of a risk-neutral investor, the f ðxÞ
gðyÞdy ¼ ðydxÞ
ST X X
ln S y
expected return E is assumed to be exp[rT] (where r is S
ð1
S
¼ f ðxÞdx
the riskless rate of interest). In other words, m ¼ r s2 =2. ln
X
The call option price C can be determined by discounting ð1 S
the expected value of the terminal option price by the risk- ¼ lnðXSÞðr12s2 ÞT hðzÞdz (A.72)
pffiffiffi
less rate of interest (r): s T
C ¼ exp½rTE½MaxðST X; 0Þ (A.69) Where g(y) is the probability density function of y, f(x) is
the probability density function of x, z is standard normal
where T is the time of expiration and X is the striking price, distribution, and h(z) is the probability density function of z.
r is the riskless interest rate, ST is the stock price at time T. The first term of call option can be derived as
Note that
ST X ST X
MaxðST X; 0Þ ¼ S for > (A.70)
S S S S
ST
Let y ¼ has a lognormal distribution with mean
S
1
mT ¼ r s2 T and variance s2 T, then
2
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 935
# Springer Science+Business Media New York 2013
936
ð1 Z 1
f ðxÞ where
erT ygðyÞdy ¼ X erT ex ðydxÞ
X y lnðXSÞðrþ12s2 ÞT lnð S Þþðrþ1s2 ÞT
ln
d1 ¼ pffiffiffi ¼ X spffiffiTffi 2 ;
S
ð1 S s T
¼ erT ex f ðxÞdx
lnðXÞðr1s2 ÞT lnð S Þþðr1s2 ÞT pffiffiffi
d2 ¼ S spffiffiTffi 2 ¼ X spffiffiTffi 2
X
ln S ¼ d1 s T
ðxðr12s2 ÞT Þ
2
ð1 rT þ x
1 2s2 T Based on put-call parity, it can be shown that the relationship
¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi e dx
ln S
X 2ps T 2 between a call option (C) and a put option (P) can be defined
ð1 as
ðxðrþ12s2 ÞT Þ
2
1
¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi e 2s2 T dx
C þ XerT ¼ P þ S
X 2ps2 T
ln S (A.75)
ð1
¼ lnðXSÞðrþ12s2 ÞT hðzÞdz
pffiffiffi Substituting Eqs. A.74 into A.75, we obtain the put option
s T
(A.73) formula as
Where z is standard normal distribution, and h(z) is the P ¼ XerT Nðd2 Þ SNðd1 Þ (A.76)
probability density function of z. Therefore, the call option
pricing model can be rewritten as where S, C, r, T, d1 and d2 are identical to those defined in the
call option model.
ð1
C¼S lnðXSÞðrþ12s2 ÞT hðzÞdz
pffiffiffi
s T
ð1
rT
Xe lnðXSÞðr12s2 ÞT hðzÞdz
pffiffiffi
s T
¼ SNðd1 Þ XerT Nðd2 Þ (A.74)
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Subject Index
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 977
# Springer Science+Business Media New York 2013
978 Subject Index
Interest rate option, 105 IPO. See Initial public offering (IPO)
Interest rate parity, 105, 246 IRA. See Individual retirement account (IRA)
Interest rate risk, 32, 106, 221, 369 IRB. See Industrial revenue bond (IRB)
Interest rates, 45, 72, 77, 106, 173, 377–380 IRR. See Internal rate of return (IRR)
Interest rate structure, 106 Irrelevance result, 109
Interest rate swap, 6, 11, 106, 135, 184 Irrevocable letter of credit, 109
Interest subsidy, 106 ISD. See Implied standard deviation (ISD)
Intermarket spread swap, 106 ISDA. See Institutional Swap Dealers Association (ISDA)
Intermarket trading system (ITS), 537, 540 IS-LM curves, 554
Intermediaries, 106, 510, 607 Iso-expected return line, 109
Internal audit, 106 Iso-variance ellipse, 109
Internal Control Mechanism, 587 Issuer exposure, 110
Internal financing, 106 Itô Process, 110
Internal growth rate, 107 ITS. See Intermarket trading system (ITS)
Internal rate of return (IRR), 22, 66, 107, 426
International, 89, 94, 107, 228, 231–235, 243–249, 581–585,
607–619 J
International asset pricing, 108, 227, 230, 233–235 January effect, 110
International banking, 607–619 Jarrow-Turnbull, 893, 897
International Banking Act, 107, 608 Jensen alpha, 351–354
International banking facilities (IBFs), 100, 107, 608 Jensen, M.C., 34, 110, 279, 297
International capital asset pricing model, 107 Jensen’s inequality, 110
International debt crisis, 581–585 Jensen’s measure, 110
International Fisher effect, 108 Johnson hedge model, 110
International Lending and Supervision Act (ILSA), 108, 582 Joint probabilities, 110
International momentum, 545 Joint venture, 110, 840
International monetary market (IMM), 108 Judgment, 110, 604
International Rating Agencies, 510 Judgmental credit analysis, 110
International system risk, 108 Jumbos, 110
Internet, 347–349 Jump-diffusion model, 525–534
Intertemporal asset pricing, 230 Jump diffusion process, 525, 528, 533
Intertemporal capital asset pricing model (ICAPM), 108, 344 Jump diffusion with conditional heteroscedasticity, 531–533
Intertemporal marginal rate of substitution, 676 Junior liens, 110
Intertemporal risk, 227–235 Junk bonds, 111, 183, 220
Intertemporal substitution, 228, 679
In the money, 108, 198, 422, 717
In-the-money-option, 108 K
Intra-day price volatility, 481 Kappa, 111, 197
Intrinsic value, 108 Keogh plan, 111
Intrinsic value of an option, 108 Key-person insurance, 111
Inventory, 7, 16, 108, 114, 136, 193, 291 Kite, 111
Inventory conversion period, 108 Knock-in option, 111, 195
Inventory loan, 82, 108 Knock-out option, 111, 195
Inventory turnover ratio, 16, 108 Kolmogorov backward equation, 111
Inverted market, 108 Kurtosis, 111, 128, 372
Inverted yield curve, 108
Investable balances, 70, 108
Investment asset, 109, 264, 393 L
Investment bankers, 109, 193, 854 Ladder option, 111
Investment banking, 109, 185 Ladder strategy, 111
Investment company, 109, 259 Lagged reserve accounting, 111
Investment Company Act, 565, 627 Lagging indicators, 30, 111
Investment grade bond, 109, 310 Lambda, 111, 197
Investment Institutions, 109 Latent variables, 276
Investment of different life, 109 Lattice, 111, 383, 430
Investment opportunity schedule (IOS), 109 Lattice method, 429, 431–432
Investment performance, 279, 634 Law of one price (LOP), 111, 244
Investment portfolio, 21, 109, 471 LBO. See Leveraged buyout (LBO)
Investment quality bonds, 109 LDC loans, 582
Investments, 109 Leading economic indicators, 111
Investment trigger price, 109 Leakage, 111
Investor arbitrage, 630–661, 668 LEAPS, 111
Invoice, 24, 109 Learning, 503
Invoicing float, 88, 108 Lease, 112, 165, 844
IO. See Interest only (IO) Lease rate, 112
IOS. See Investment opportunity schedule (IOS) Leasing companies, 112
990 Subject Index
Yield curve, 99, 100, 108, 120, 163, 189, 202, 309 Zero-cost collar, 203
Yield curve swap, 202 Zero coupon bonds, 203, 219–225
Yield-giveup swap, 202 Zero-coupon interest rate, 203
Yield-pickup swap, 202 Zero-coupon swap, 203
Yield rate, 202 Zero-coupon yield curve, 203
Yield to maturity, 202 Zero gap, 203
Zero-investment portfolio, 204
Zero-plus tick, 204
Z Zero rate, 204
Zero-balance accounts, 203 Z-score model, 203, 360
Zero-beta portfolio, 203, 343 Z-tranche, 203
Author Index
C.-F. Lee and A.C. Lee (eds.), Encyclopedia of Finance, DOI 10.1007/978-1-4614-5360-4, 1003
# Springer Science+Business Media New York 2013
1004 Author Index
Koskela, E., 318 Lee, C.-F., 264, 281, 323, 351, 352, 357, 387, 388, 411, 421,
Kosowksi, R., 303 473, 706, 714, 755, 795, 800, 816, 851, 853, 857, 867,
Kothari, S.P., 836 871, 881, 885
KrÄahmer, D., 804 Lee, C.M.C., 547
Kraus, A., 264 Lee, D., 857, 858
Krebs, T., 676 Lee, H.T., 872, 876, 879
Kreps, D., 228, 231, 264, 298, 453 Lee, J.C., 755
Krigman, L., 853, 858 Lee, K.W., 800
Kritzman, M., 368, 735 Lee, M.C., 872, 876, 885, 888
Kroll, Y., 400, 401 Lee, S.B., 367, 377
Kroner, K., 714 Lee, S.W., 316, 317
Kroner, K.F., 872, 880, 887 Lee, T.C., 362
Kroszner, R., 852, 858 Lee, W., 714
Krugman, P., 247 Leftwich, R., 334
Kruschwitz, L., 829 Lehmann, B.N., 268, 279, 299
Kshirsagar, A.M., 361 Lehn, K., 414
Kulatilaka, N.H., 422, 423 Leibowitz, M.L., 312
Kunda, Z., 804 Leigh, R., 413, 415
Kuntz, P., 829 Leland, H., 728, 729, 740–742, 852, 891, 894, 896
Kuo, H.C., 357, 361, 364, 411 Leland, H.E., 185, 646, 647, 654–656, 799
Kuo, J., 714, 872, 874, 876, 884, 887, 888 Lemmon, M., 853, 858
Kupiec, P., 778 Lence, S.H., 872, 873, 877, 883
Kurz, M., 684 Leshno, M., 404
Kwan, C.C.Y., 872, 875, 884, 885 Lesne, J.-P., 454, 457
Kwok, C., 322 Lettau, M., 275, 276, 300, 301, 680
Kwok, Y.K., 796 Leung, K.-W., 582
Kyle, A.S., 323, 659–661 Lev, B., 70, 360, 738
Levhari, D., 351
Levhari, P., 351
L Levine, R., 247, 507
Labys, W.C., 565 Levy, H., 351, 395, 398, 401–407, 409, 811
Lacey, N., 638 Levy, M., 398, 401–407, 409
Lai, K.S., 247, 250 Lewarne, S., 550
Lai, T.Y., 765–776 Lewellen, W.G., 283, 333–339, 448, 828
Laibson, D., 676 Li, D., 666, 781
Laitenberger, J., 829 Li, Q., 781
Lakonishok, J., 362, 867 Li, S., 666, 765, 766, 770
Lalancette, S., 283 Liao, S.L., 800
Lalitha, L., 857, 858 Lichtenstein, S., 804
Lambert, R.A., 601 Lie, E., 854, 858, 866
Lamont, O.A., 660 Liebeskind, J.P., 602
Lamper, D., 502 Lien, D., 872, 875, 876, 878, 879, 881, 882, 884, 886–888
Lamy, R.E., 582, 583 Liew, J.M., 545
Landier, A., 804 Lim, T., 547
Lando, D., 891 Lin, C., 856, 858
Landsman, W.R., 836 Lin, F., 725
Lang, L.H., 519 Lin, L., 357, 361, 363, 365, 411, 418
Lang, W., 326 Lin, W.T., 421, 425, 428, 441
Langer, E.J., 804 Lin, Y., 889
Langetieg, T.C., 418 Linn, S.C., 502
Laplante, M., 284 Lintner, J., 115, 116, 177, 230, 264, 265, 343, 351, 405,
Larcker, D.F., 601 448, 473, 634, 705, 750
Larson, H., 78 Liow, K.H., 705
Latane, H.A., 101, 765 Lippman, S.A., 483
Latta, C., 309 Lipson, M., 326, 409
Lau, K.W., 796 Lipton, A.F., 705
Lau, S., 352 Lipton, M., 593
Laughhunn, D.L., 872, 875 Lischka, F., 796
Lauterbach, B., 542 Litterman, B., 341
Laux, P., 326 Litzenberger, R.H., 354
Lawler, P., 562 Liu, J., 830
LeBaron, B., 502–504 Liu, S., 546
Ledoit, O., 748 Liu, Y.C., 833
Lee, A.C., 756 Livnat, J., 843
Lee, C., 207, 585, 714, 760 Lloyd, W., 869, 957
1012 Author Index
Lo, A.W., 268, 277, 355, 362, 447 Maremont, M., 842
Loderer, C., 855, 858 Mark, N.C., 254
Löffler, A., 833 Markham, J.W., 216
Long, J., 267, 301, 302 Markowitz, H.M., 341, 351, 398, 399, 406, 407,
Long, R.W., 756 409, 566, 705
Longo, J.M., 621–632 Marks, R., 804
Longstaff, F.A., 381, 646, 650–651, 656, 657, Marks, S., 361
799, 891, 894, 896 Mar-Molinero, C., 361
Loomis, C., 623 Marquardt, D., 867
Loria, S., 741 Marschak, J., 341
Lorsch, J.W., 606 Marshall, D., 676
Lothian, J.R., 243, 247, 250 Marston, R.C., 245, 247–249
Loviscek, A.L., 565, 568 Martin, D., 362
Low, V., 563 Martin, K., 851, 855, 856, 858–860
Lowenstein, R., 631 Mason, S.P., 423
Lu, C., 387 Massa, M., 638
Lu, Q., 800 Matheson, T., 777, 780
Lucas, D.J., 675–677 Mathiesen, H., 588
Lucas, R.E. Jr., 250, 265 Mauer, D., 854, 855, 858
Ludvigson, S., 275, 276, 300, 301 Maurer, R., 731, 742
Luehrman, T.A., 423, 424, 442, 825, 829 Mayers, D., 282
Luo, X., 872, 875, 876, 879, 881, 884, 886 Mays, M., 512
Lutje, T., 637 Mazumder, M.I., 705
Lutkepohl, H., 362 Mazuy, K.K., 281, 297, 634
Luttmer, E.G.F., 454–456, 676 McConnell, J.J., 415, 570, 571, 851–853, 858–860, 868
Lynch, A.W., 297, 301, 302, 666, 676 McCormick, T., 326
Lyness, J.N., 432 McDonald, B., 361, 425
Lyons, R.K., 449–451 McDonald, G., 855, 858
McDonald, R., 422, 425
McElroy, M.B., 268
M McEnally, R.W., 309
MacArthur, A.T., 250, 253 McFadden, D.L., 363
Macaulay, F.R., 221, 305, 306, 313, 683 McGee, V., 867
MacBeth, J.D., 168, 351, 752 McGrattan, E.R., 676, 681–683
Machina, M.J., 397 McIntosh, W.R., 407
MacKinlay, A.C., 268, 275, 447 McKenzie, M., 706
Madan, D.B., 891 McKinnon, R.I., 507
Madariaga, J.A., 708 McLean, B., 348
Maddala, G.S., 364 McLeay, S., 361
Madhavan, A., 539 McNamee, M., 350
Madura, J., 515, 582 McNichols, M.F., 604
Maggioni, P., 323, 779 McWilliams, V., 856
Maheu, J.M., 531 Meckling, W.H., 333, 413, 460, 852
Maier, S., 484, 492, 536 Meeks, G., 415
Maillet, B., 733, 742 Meese, R., 450
Mais, E., 851, 853, 858, 860 Megginson, W., 317
Majd, S., 426 Mehra, R., 675–684
Maksimovic, V., 318 Melino, A., 679
Malatesta, P.H., 414, 418 Melnik, A., 315–317
Malkiel, B.G., 267 Mendelson, H., 483, 484, 488, 542, 778
Mallaby, S., 622 Mendes-de-Oliveira, M., 811
Malliaris, A.G., 872, 881, 886, 887 Mensah, Y., 362
Malmendier, U., 803, 804, 806, 811 Merrick, J., 265
Malpass, D., 581 Merton, R.C., 108, 138, 227, 267, 274, 276, 281, 297,
Mamaysky, H., 283 310, 318, 344, 424, 428, 455, 525, 526, 528, 532,
Manaster, S., 765, 766 622, 645, 647, 648, 659, 660, 683, 750, 765, 767,
Mandelbaum, A., 660 796, 798, 799, 894
Mandelbrot, B., 685 Meschke, F., 853, 858
Mandelker, G.N., 418 Meulbroek, L., 660
Maness, T.S., 288 Meulenberg, M., 714
Mankiw, N.G., 676 Miao, J., 832
Manne, H.G., 414 Michaud, R.O., 345
Mansi, S., 852, 855, 858 Mikkelson, W., 853, 858, 866
Manski, C.F., 363 Miles, J.A., 826, 827
March, J.G., 806 Milgrom, P., 412, 414, 416, 488
Marcus, A.J., 358, 365, 412, 413, 417, 418 Miller, E.M., 705
Author Index 1013
Miller, M.H., 41, 65, 99, 109, 111, 127, 177, Nieuwland, F.G.M.C., 533
405, 454, 638 Nijman, T.E., 532, 533
Miller, P.B.W., 844 Nissim, D., 825
Miller, R.E., 635 Noe, T.H., 318
Miller, T.W., 765, 766, 768–771, 775, 776 North, D.J., 413, 415
Milonas, N., 854, 855, 858, 859, 866 Norton, E.A.., 472
Mintel International Group Ltd., 348 NRMLA-Consumer site administered by the National
Mirrlees, J.A., 565 Reverse Mortgage Lenders Association, 691
Mishkin, F.S., 246–249 Nurnberg, H., 846
Mitchell, M., 660, 666 Nyborg, K.G., 796, 799, 828, 829
Mitchell, M.L., 414
Mittnik, S., 685
Mizon, G.E., 882 O
Modest, D.M., 268, 279, 299, 676 O´Brien, T.J., 729, 740–741
Modigliani, F., 41, 65, 109, 111, 113, 127, 142, 177, Obstfeld, M., 555
405, 453, 473, 638, 825, 826, 829 O’Connor, P.F., 460, 461, 463
Modigliani, L., 473, 641, 706, 707 Odean, T., 400, 804
Moffett, M.H., 611 Ofek, E., 660
Mondino, G., 555 Office of Economic and Corporate Development
Montiel, P.J., 555 (OECD), 588
Mooradian, R.M., 418 Ogawa, O., 511
Moore, A., 447 Ogden, J.P., 459–461, 463
Moore, W.T., 815 Oh, G., 508, 511
Morck, R., 603, 851–853, 856, 858–860, 863, 865, 866 O’Hara, M., 262, 457
Morellec, E., 832 Ohlson, J.A., 362, 455
Morgan, J.P., 56, 312 Oksendal, B., 527
Morgenstern, O., 341 Okunev, J., 872, 875, 880, 884, 886
Morris, M.H., 361 Oldfield, G.S., 530
Moskowitz, T.J., 545, 546 Omberg, E., 527, 528
Mossin, J., 265, 351, 473 Ongena, S., 319
Muellbauer, J., 555 Opiela, N., 350
Mullaney, T., 350 Opler, T., 318, 855, 858, 866
Muller III, W.J., 391, 570–572, 574 Orpurt, S., 844
Mullineaux, D.J., 316, 317 Ortiz-Molina, H., 855, 858
Mundell, R.A., 555 Ortobelli, S., 685
Mungthin, N., 508 Osborne, M., 448, 530
Murinde, V., 323, 779 Osterwald-Lenum, M., 879
Murphy, A., 539 Oswald, A.J., 563
Murphy, K.M., 266 Otani, I., 558
Murugesan, B., 638 Otrok, Ch., 680
Musumeci, J.J., 582 Oyer, P., 804
Muthuswamy, J., 715 Ozenbas, D., 484
Myers, D., 283 Oztekin, O., 651
Myers, R.J., 714, 872, 878, 880, 885
Myers, S.C., 220, 318, 365, 412, 418, 423,
426, 461, 825, 829 P
Padmaraj, R., 855
Pagano, M., 493, 494
N Pages, H., 454, 457
Nagao, H., 511 Pain, D., 733, 735, 738, 740
Nakagawa, N., 796 Palazzolo, C., 259
Nandy, D., 316–318 Palepu, K.G., 361, 363–365
Narayanan, P., 242, 361, 364 Palia, D., 803
National Association of Securities Dealers Palmer, R., 502, 504
Automated Quotation (NASDAQ), 131, 149, 325, Palmon, O., 803
326, 329–331, 348, 350, 388, 477, 478, 481, 484, Pamepinto, S., 337
535–539, 542 Pan, J., 685
Nawalkha, S.K., 310 Panchapagesan, V., 536, 537
Neale, M.A., 804 Pantzalis, C., 851, 861, 863
Nelson, C.R., 275 Panyagometh, K., 316
Nelson, K.K., 604 Paolella, M., 685
Neuberger, A., 488, 492 Papazoglou, C., 562
Newey, W.K., 246, 747 Paperman, J., 323
Nguyen, T.H., 529 Paradis, G.E., 400
Nickell, S.J., 563 Park, D., 511
Niederhoffer, V., 448, 530 Park, J., 247, 623
1014 Author Index
Shleifer, A., 263, 274, 659–661, 666 Standard and Poor’s (S&P), 4, 26, 56, 78, 102, 123, 134,
Shores, M.R., 309, 310 149, 157, 165, 178, 181, 186, 192, 246, 259, 280,
Short, H., 858, 860, 866 349, 471, 539, 596, 619, 623–625, 635, 638, 685,
Shoven, J., 358, 365 713–719, 722, 723, 763, 855, 885–888
Shrestha, K., 851–868 Stanton, R., 571, 573, 574
Shreve, S., 891, 894 Stapledon, G.P., 602
Shrieves, R., 416 Starks, L., 282
Shultz, G., 581 Startz, R., 275
Shumway, T., 365, 647 Statman, M., 640, 804
Sick, G.S., 829 Steen, N.M., 430, 431
Sidana, G., 637 Stein, J.C., 546, 547, 555
Siegel, A.F., 271 Sterken, E., 593
Siegel, D., 422, 425 Stevens, D., 416
Siegel, M., 842 Stevens, R.L., 545
Silberberg, E., 565 Stiglitz, J.E., 416, 488, 778, 779
Sim, A.B., 741 Stock, J.H., 882
Simin, T., 268, 275 Stohs, M., 854, 855, 858
Simms, J.M., 582 Stoll, H., 331, 488, 492, 646
Simon, H., 806 Stonehill, A.I., 611
Simon, W., 581 Storesletten, K., 676
Sims, C.A., 438 Storey, D.J., 362
Sing, G.P., 638 Stover, R., 856, 858, 861
Singh, A., 411, 412 Strahan, P.E., 316
Singh, H., 714 Strand, N., 698
Singh, J., 636 Strauss, J., 247
Singh, R., 325 Strebulaev, I.A., 656
Singleton, K.J., 242, 265, 266, 529, 530, 577, Strömberg, P., 825
891, 892, 903 Stulz, R.M., 227, 230, 250, 852, 853
Sinkey, J.F. Jr., 582 Su, D., 322, 323, 856, 858
Sinquefield, R.A., 439 Subha, M.V., 637
Siow, A., 483 Subrahmanyam, A., 483, 765, 766
Sisodiya, A.S., 636 Subrahmanyam, M.G., 766, 767, 775
Skelton, J., 309 Subrahmanyam, V., 855
Skinner, F., 310 Sudhakar, A., 636
Skoulakis, G., 303, 748 Sullivan, M.J., 416, 418
Skully, M., 854, 858 Sultan, J., 714, 872, 880, 887
Slovic, P., 789 Summers, L.H., 581, 660, 778
Slovin, M., 851, 853, 858, 860 Summers, V.P., 778
Smirlock, M., 582 Sumon, C.M., 216
Smith, A.J., 604 Sun, Q., 322
Smith, C. Jr., 854, 858 Sundaresan, S.M., 266, 891
Smith, D.M., 638 Sunder, S., 360, 361, 503
Smith, G., 347, 348 Sundgren, S., 593
Smith, J.L., 429 Sunner, M.W., 259
Smith, R., 853, 858, 859 Surz, R., 483
Smith, T., 275 Sushka, M., 851, 853, 858, 860
Smith, V.L., 400 Svenson, O., 804
Sobaci, T., 209, 211 Svensson, L.E.O., 235
Soderlind, P., 281, 297, 299–301 Swalm, R.O., 399
Sofianos, G., 484 Swaminathan, B., 547
Solnik, B., 230, 235, 244, 246, 247, 250, 275 Swason, P.T., 635
Solomon, D., 842 Sweeney, R.J., 247, 250
Solomon, S., 409 Switzer, L., 714
Sonakul, M.R.C.M., 508 Szafarz, A., 828
Sondhi, H.J., 637
Song, M.H.., 419
Sopranzetti, B.J., 760, 761 T
Sorensen, S.E., 531, 796, 800 Taggart, R.A., 828
Soto, G.G., 310 Takahashi, A., 796
Spatt, C.S., 570 Tanewski, G., 854, 858
Spiegel, M., 283 Tang, R., 786, 789
Staël von Holstein, C.A.S., 804 Taniguchi, T., 508
Stafford, E., 660, 666 Taqqu, M., 685
Stallworthy, E.A., 412, 413, 415 Tate, G., 804, 806, 811
Stambaugh, R.F., 483 Tay, N.S.P., 502
Stambaugh, R.S., 275 Taylor, M.P., 243, 247, 250, 450
Author Index 1017
Zhu, N., 660 Zin, S.E., 228–230, 232, 266, 676, 678, 811
Zhu, Y., 740 Zion, D., 312
Zhuravskaya, E., 660 Zmijewski, M.E., 361
Ziemba, W.T., 685, 731 Zombanakis, G., 562
Zietlow, J.T., 288 Zumpano, L.V., 705