Financial Accounting and Reporting Fundamentals I
Fall 2015
Session 2
George Plesko
Objectives and Game Plan
Understand some key concepts of Financial Accounting
Appreciate the differences between cash basis and accrual
accounting
Develop a mental mode
given by w1. Exercise 1.11. Analyze in detail when only two assets are
considered. Assume the expected return satisfies 1 < 2 and 1 > 0, 2
> 0. Consider first the case (ij )22 is positive definite and then the
case when it degenerate. 14 CHAPTER 1. MEAN-V
simplicity, we use continuously compounded rate. 1.6. MODELS AND
DATA 33 Suppose the security under investigation time period [0, T],
with unit time being one year, has time invariant expected return rate
and variance 2 ; that is, denote by S t the unit
example, a financial security can be the ownership of a stock, the credit
from a bond, or the right to ownership from an option. An option is the
right, but not the obligation, to buy (or sell) an asset under specified
terms. A derivative security, or con
securitys payoff could be as follows: gain $30 if it rains tomorrow and
lost $10 if it is sunny. It is not necessary to specify probabilities. This
security is represented as (30, 10). The central issue here to price
securities. For this, we assume that p
value = 0. We shall manage it appropriately to make money. Namely, at
time t = T /2, we monitor the market to make the following adjustment
to the portfolio: (1) If 1 happens, the value of the portfolio n0 is V1 =
38($) + 4 option. We adjust the portfolio
asset, one expects larger expected return, and hence, it is meaningful
only when 0 > . 1.3. CAPITAL ASSET PRICING MODEL 17 to the right
of the market portfolio (the intersection of the line and the frontier
curve) requires selling the risk free asset shor
investment plan A, B, C. 1.3 Capital Asset Pricing Model Now we take a
look at the Capital Asset Pricing Model, developed by the Nobel Prize
winner William Sharpe and also independently by John Lintner and J.
Mossin, thus called SLM CAPM model. The major
the market portfolio that gives information needed for three stocks A, B,
C. Assume that riskfree rate is 5%, market portfolios expected return is
12% with standard deviation 18%. Information on the portfolio is as
follows: Stock Beta ei weight A 1.10 7.0
priori one does not exactly know the future behavior of stock price,
finding optimal strategy is one of the key here. Example 2.3. Consider
the problem to price a European call option of duration T = 1 (month)
with strike price E = 180($) for a particular
0.167). In the above example, the first asset a1 has expect return 8%
with risk 1 = 0.2, whereas the second asset a1 has expect return 8%
with risk 1 = 0.04. Just comparing these two assets, one can say that
a1 is more preferable than a2. However, asset a
investment, the investment on riskless asset has a known return. As we
shall see, the inclusion of a risk-free asset can improve the risk-return
balance by investing in a portfolio partially in risky assets and partially in
a risk-free asset. Let us denot
distributed with mean and variance 2 . Then is normally
distributed with mean and variance 2/n; that is, n()/ is N(0,
1) (normal with mean zero and unit variance) distributed. Hence, P(z) :=
Z z z e s 2/2ds 2 = Prob n( ) [z, z] = Prob [
z/ n, + z/ n] . T
and the variance of those returns, and covariance between the returns
of different securities. Where do we obtain these parameter values?
One obvious source is historical data of security returns. This method of
extracting the basic parameters from histor
1.64 25/ 20, i.e. [103, 121]. (2) Suppose in a poll the 95 %
confidence interval of percentage of population supporting a candidate
is 39% 3%. We report as follows, the pool indicates that 39%
population supports the candidate, where sample error is 3%. (
(w1(0), w2(0), w3(0) as function of risk-free rate 0. 1.3. CAPITAL
ASSET PRICING MODEL 19 1.3.1 Derivation of the Market Portfolio Here
we derive the formula for the market portfolio. The mathematical
problem is following minimization problem: Given 0, R,
the market is not complete in the sense that one cannot short assets
valued more than the portfolios total worth; (i.e. the sum of all
negative wi is no smaller than 1.) Find the maximum expected return,
regardless how high the risk may be, but still want
multiplied by we obtain 0 = 1(wu 0) 1uw 21w =
10 2 since wu = uw and 1w = w1 = 1. Hence, 2 =
10. Consequently, multiplying the second equation by ( 2C) 1
from the right we obtain w = (1u + 21)C1 2 = 1 1 (u
01)C1 = (u 01)C1 (u 01)C11 where the last equati
return, we need only find the maximum of the slope | 0|/. As (,
) is in the Markowitz bullet, we see that the maximum can only be
attained 16 CHAPTER 1. MEAN-VARIANCE PORTFOLIO THEORY at the
Markowitz curve, i.e., when = p 2 + 2( ) 2. Therefor, the
maximu
expiration date. Notice that the values of options depend mostly on
prices of the underlying stocks. Example 2.1. Consider the a European
call option on a stock with strike price E and expiration T. Let St be the
stock price at t. Then the option has a ca
(u 01)C1 (u 0)C1, 1) . (1.10) (ii) 0 > . 2 In this case, the
capital market line is the extension of the line passing (0, 0) and
tangent to the reflection of Markowitz curve about the axis; see the
thin line in Figure 1.2. One can show that M is obtained
risky assets does not depend on any particular choice of efficient
portfolio. This observation is indeed the key to the CAPM. Theorem 1.4
(The One-Fund Theorem) There is a single fund F of risky assets such
that any efficient portfolio can be constructed
with k freedom has density function (1/2)k/2 (k/2) x k/21 e x/2 , x >
0. Fisher showed that if X 2 k , then 2X 2k 1 is approximately
N(0, 1) distributed when k 1. Thus, when n is larger, p 2(n 1)n r
2n 3 2n 2 o N(0, 1). (1.19) Hence, when n 1, the p-confi
i=1 wiRi . Here R0 = 0 (a.s) is a constant, so that Cov(R0, Ri) = 0 for all i
= 0, , m. With the inclusion of a risk-free asset, the portfolio with
weight w = (1 , w) has the expected return and risk 2 given
by = E(R) = (1 )0 + , = E(R) = (w, u), 2 = Var(
stock as its underlying. The central problem here is to find the current
value of option; namely, determine the present value of a future
payment, that depend on the stock price. Example 2.2. Consider an
American call option on a stock with strike price E
2F1 is hypergeometric function. It is rather complicated to obtain the
confidence interval from the above density function. Quite often, we
use approximations. First let define = tanh1 (12 = 1 2 log 1 + 12 1
12 , := tanh1 (). Then n 1( ) N(0, 1) as n . T
0.15, 0.30). The return M and risk M of this market portfolio are
respectively M = wMu = 0.092, M = p wMCw M = 0.0877. Also,
the market price of risk is M = |M 0| M = 0.25. The Capital
Market line is the line with the equation = 0.07 + 0.25. See Figure
1.
premium, is the additional return beyond the risk-free return 0 that
one may expect for assuming the risk . Of course, it is the presence of
risk that the investor may not actually see this additional return. Hence,
M is also called the market price of ri
of exchanges by financial institutions and their corporate clients in what
are termed as the over-the-counter markets. Other more specialized
derivative securities often form part of a bond or stock issue. In this
chapter, we shall study a derivative pric
securities can be priced. The basic strategy is to find an appropriate
portfolio and manage it optimally according to dynamics of the financial
market so that at the end of day the values of the portfolio equals
exactly that of the derivative security, re
approximations of and 2 : := 1 n Xn i=1 ri , 2 = 1 n 1 Xn i=1 (ri
) 2 . 30 CHAPTER 1. MEAN-VARIANCE PORTFOLIO THEORY Now our
question is how accurate is the estimation , ? To answer
such a question, lets suppose that the n observations are independent;
m
VT /2(1) of the option is zero. (2) Suppose t = T /2 and 2 happens.
Form a portfolio of x share stock and y cash. We want to make it match
the payment of the put option if it is exercised at T. For this, we need
xST + 1.05y = maxcfw_180 ST , 0, i.e. 170x
R. The above argument assumes an ideal functioning of the market:
securities can be arbitrarily divided into two pieces and that there are
no transaction cost. In practice these requirements are not met
perfectly, but when dealing with large numbers of s