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Questions and Answers of Finance

We say that Z is exponentially distributed with parameter A > 0 in the distribution function of Z is given by:P(Z < z) = 1 – e–kz(a) Determine and plot the density function of Z.(b)
A random variable Z ahs Poisson distribution if p(k) = P(Z = ?ke?k / k! For k = 0, 1, 2 . . . (a) Use the expansion To show that (b) Calculate the mean E[Z] and the varianceVar(Z)
Let Y be a random variable with E[Y] < ∞(a) Show that the Mt defined byMt = E[Y|It]Is a martingale.(b) Does this mean that every conditional expection is a martingale given the increasing sequence
Consider the random variable: Where each Bi is obtained as a result of the toss of a fair coin: We let n = 4 and consider X4. (a) Calculate the E [X4| I1], E [X4| I2], E [X4| I4] (b) Let Zi = E
Let Wf be a Wiener process and t denote the time. Are the following stochastic processesmartingales?
You are given the representation: Where the equality holds given the sequence of information sets {It. The underlying process Xt is known to follow the SDE: dXt = ?dt + ?dWt Determine the g(-) in
Given the representation: Can you determine the g(-) if the MT(XT) is the payoff of an plain vanilla European call option at expiration? That is, if MT(XT) is given by: MT(XT) = max [XT?? K,
We consider the random process St, which plays a fundamental role in BIack-Scholes analyses:St = S0e[μ1+σWt]Where Wt is a Wiener process with W0 = 0, μ is ‘a “trend” factor, and(Wt – Ws)
Show that as n ? ?
Let the random variable Xn have a binomial distribution: Where each Bi is independent and i distributed according to We can look at Xn as the cumulated sum of a series of events that occur over
Let Wt be a Wiener process defined over [0, T] and consider the integral: Use the subdivision of [0, t] t0, t1,?, tn?1, tn in the following: (a) Write the approximation of the above integral as
Show that given t0, t1,?, tn?1, tn and Wt0, Wt1,?,Wtn?1, Wtn, We can always write: How is this different from the standard formula for the differentiation of products: d(uv) = (du)v + u(dv)
Now use this information to show that:
Can we say that this is an application of integration by parts?
Differentiate the following functions with respect to the Wiener process Wt, and if applicable, with respect tot.
Suppose the Wti, i =1, 2 are two Wiener processes. Use Ito’s Lemma in obtaining appropriate stochastic differential equations for the following transformations.(a) Xt = (Wt1)4(b) Xt = (Wt1 +
Let H be a Wiener process. Consider the geometric process S again: (a) Calculate dSt. (b) What is the ?expected rate of change? of St? (c) If the exponential term in the definition of St, did not
Consider the following SDE: (a) Write the above SDE in the integral form. (b) What is the value of theintegral.
Consider the geometric SDE:dSt = μStdt + σStdWt,where St is assumed to represent an equity index. The current value of the index isS0 = 940It is known that the annual percentage volatility is 0.15.
Consider the linear SDE that represents the dynamics of a security price:dSt = .01 Stdt + 0.5 StdWtWith S0 = 1 given.Suppose a European call option with expiration T = 1 and strike K 1.5 is written
Consider the SDE: dSt = .05dt + .1dWt. Suppose dWt, is approximated by the following process: (a) Consider Intervals of size ? = 1. Calculate the values of St, beginning from t = 0 to t = 3. Note
You are given a function f(x, z, y) of three variables, x, z, y. The following PDE is called Laplace’s equation:According to this, in Laplace’s equation, the sum of second partials with respect
A function f(x, z, y, t) of four variables, x, z, y, t, that satisfy the following PDE is called the heat equation:Where a is a constant. According to the heat equation, firs partial with respect to
Consider the PDE:fx + .2fy = 0,with X ε[0, 1] and y ε [0,1].(a) What is the unknown in this equation?(b) Explain this equation using plain English. (c) How many functions f(x, y) can you find that
Consider the PDE:fxx + .2ft = 0,With the boundary condition f(x, 1) = max[x – 6, 0].Let0 ≤ x ≤ 12And0 ≤ t ≤ 1.(a) Is the single boundary condition sufficient for calculating a numerical
The exercises in this section prepare the reader for the next three chapters instead of dealing with the PDEs. An interested reader will find several useful problems in Betounes (1998). Let Xt be a
This exercise deals with obtaining martingales. Suppose Xt is a geometric process with drift ? and diffusion parameter ?. (a) When would the e-rt Xt be a martingale? That is, when would the following
Consider Where Xt is an exponential Wiener process:(a) Calculate the expected value of the increment dZ(t).(b) Is Zt a martingale?(c) Calculate E[Zt]. How would you change the definition of Xt to
Consider a random variable ∆x with the following values and the corresponding probabilities:{Δx = 1, p(Δx = 1) = .3},{Δx = –0.5, p(Δx = –0.5) = .2},{Δx = .2, p(Δx = .2) = .5}.(a)
Assume that the return Rt of a stock has the following log-normal distribution for fixed t:log (Rt) ~ N(μ, σ2).Suppose we let the density of log(Rt) be denoted by f(Rt) and hypothesize that μ =
The long rate R and the short rate r are known to have a jointly normal distribution with variance-covariance matrix ? and mean ?. These moments are given by Let the corresponding joint density be
In this exercise we use the Girsanov theorem to price the chooser option. The chooser option is an exotic option that gives the holder the right to choose, at some future date, between a call and a
In this exercise we work with the Black-Scholes setting applied to foreign currency denominated assets. We will see a different use of Girsanov theorem. [for more detail see Musiela and Rutkowski
Plot the payoff diagrams fur the following instruments:(a) A caplet with cap rate Rcap = 6.75% written on 3-wonth Libor Lt, that is about to expire.(b) A forward contract written on a default-free
Which one(s) of the following are assets traded in financial markets:(a) 6-month Libor(b) A 5-year Treasury bond(c) A FRA contract(d) A caplet(e) Returns on 30-year German Bonds(f) Volatility of
Suppose you are given the following information on the spot rate rt:The rt follows:dt =μrt + σrt dWr.The annual drift isμ= .01.15This is the wise because the forward price, Ft], belt belong to the
Suppose at time t = 0, you are given four default-free zero-coupon bond prices P(t, T) with maturities from 1 to 4 years: P(0, 1) = .94, P(0, 2) = .92, P(0, 3) = .87, P(0, 4) .80(a) How can you
Select ten standard, normal random numbers using Mathematica, Maple or Matlab. Suppose interest rates follow the SDE: drt = .02rtdt + .06rtdWt. Assume that the current spot rate is 6%. (a) Discretize
Consider the SDE for the spot rate rt Suppose the parameters a, ?, ? are known, and that as usual, Wt, is a Wiener process. (a) Show that (b) What do these two equations imply for the conditional
Consider a world with two time periods and two possible states at cacti time t = 0, 1. 2. There are only to assets to invest. One is risk-free burrowing and lending at the risk-free rate ri, i = 0,
Consider the equation below that gives interest rate dynamics in a setting where the time axis [0, T] is subdivided into it equal intervals, each of length ?: rt+? = rt + ?rt + ?t(Wt+? ? Wt) + ?2(Wt
Suppose the (vector) Markov process Xt, has the following dynamics, where the error term is jointly normal and serially uncorrelated Suppose rt, is a short rate, while Rt, is a long rate.(a)
Suppose at time t = 0, we are given four zero-coupon bond prices {B1, B2, B3, B4} that mature at times t = 1, 2, 3, 4. This forms the term structure of interest rates. We also have one-period forward
Consider again the setup of Question 1. Suppose we want to price three European style call options written on one period (spot) Libor rates Li with i = 0, 1, 2, 3, as in the above case. Let these
Suppose you are given the following SDE for the instantaneous spot rate:drt = σr1dWt,where the Wt is a Wiener process under the real-world probability and the σ is a constant volatility. The
You are given the spot-rate model; dr1 = ? (k ? rt) dt + bdW, where the Wt, is a Wiener process under the real-world probability. Under this spot rate model, the solution to the PDE that corresponds
Suppose the bond price B(t, T) satisfies the following PDE: ?rtB + Bt + Br(? ? ??B) + ?Brr?2 = 0 B(T, T) = 1. Define the variable V(u) us (a) Let B(t, T) be the bond price, Calculate the d(BV). (b)
A player confronts the following situation. A coin will be tossed at every time t, t = 1, 2, 3, . ., T and the player will get a total reward W1,. He or she can either decide to step or to continue
Consider the problem above again. Suppose we tossed a coin T times and the resulting zt were all +1. The reward will be: (a) Show that the conditional expected reward as we just play one more time
Suppose you are given the following data:• Risk-free interest rate is 6%• The stock price follows:dSt = μSt + σStdWtVolatility is 12 % a year• The stock pays no dividends anti the current
Suppose the stock discussed above pays dividends. Assume all parameters are the same. Consider these three forms of dividends paid by the firm.(a) The stock pays a continuous, known stream of
Consider a policy maker who uses and instrument kt, to control the path followed by some target variable Y,. The policy maker has the following Objective function The environment imposes the
How does a cost-efficient capital market help reduce the prices of goods and services?
Describe the different ways in which capital can be transferred from suppliers of capital to those who are demanding capital. Discuss.
Is an initial public offering an example of a primary or a secondary market transaction? Explain.
Indicate whether the following instruments are examples of money market or capital market securities.a. U.S. Treasury billsb. Long-term corporate bondsc. Common stocksd. Preferred stockse. Dealer
What types of changes have financial markets experienced during the last two decades? Have they been perceived as positive or negative changes? Explain.
Differentiate between dealer markets and stock markets that have a physical location. Discuss.
Describe the three different forms of market efficiency.
Investors expect a company to announce a 10% increase in earnings; instead, the company announces a 1% increase. If the market is semi-strong form efficient, which of the following would you expect
Briefly explain what is meant by the term efficiency continuum.
Explain whether the following statements are true or false.a. Derivative transactions are designed to increase risk and are used almost exclusively by speculators who are looking to capture high
What are the three primary ways in which capital is transferred between savers and borrowers? Describe each one.
What is a market? Differentiate between the following types of markets: physical asset markets versus financial asset markets, spot markets versus futures markets, money markets versus capital
Why are financial markets essential for a healthy economy and economic growth?
What are derivatives? How can derivatives be used to reduce risk? Can derivatives be used to increase risk? Explain.
Briefly describe each of the following financial institutions: commercial banks, investment banks, mutual funds, hedge funds, and private equity companies.
What are the two leading stock markets? Describe the two basic types of stock markets.
If Apple Computer decided to issue additional common stock and Varga purchased 100 shares of this stock from Smyth Barry, the underwriter, would this transaction be a primary or a secondary market
What is an initial public offering (IPO)?
What does it mean for a market to be efficient? Explain why some stock prices may be more efficient than others.
She has read a number of newspaper articles about a huge IPO being carried out by a leading technology company. She wants to get as many shares in the IPO as possible and would even be willing to
What is the effect of multinational operations on capital budgeting decisions.
Who are some of the basic users of financial statements, and how do they use them?
If a “typical” firm reports $20 million of retained earnings on its balance sheet, could its directors declare a $20 million cash dividend without having any qualms about what they were doing?
Explain the following statement: While the balance sheet can be thought of as a snapshot of a firm’s financial position at a point in time, the income statement reports on operations over a period
Financial statements are based on generally accepted accounting principles (GAAP) and are audited by CPA firms. Therefore, do investors need to worry about the validity of those statements? Explain
What is free cash flow? If you were an investor, why might you be more interested in free cash flow than net income?
Would it be possible for a company to report negative free cash flow and still be highly valued by investors; that is, could a negative free cash flow ever be a good thing in the eyes of
What does double taxation of corporate income mean? Could income ever be subject to triple taxation? Explain your answer.
How does the deductibility of interest and dividends by the paying corporation affect the choice of financing (that is, the use of debt versus equity)?
Little Books Inc. recently reported $3 million of net income. Its EBIT was $6 million, and its tax rate was 40%. What was its interest expense?
Pearson Brothers recently reported an EBITDA of $7.5 million and net income of $1.8 million. It had $2.0 million of interest expense, and its corporate tax rate was 40%. What was its charge for
In its most recent financial statements, Newhouse Inc. reported $50 million of net income and $810 million of retained earnings. The previous retained earnings were $780 million. How much in
Which of the following actions are most likely to directly increase cash as shown on a firm’s balance sheet? Explain and state the assumptions that underlie your answer.a. It issues $2 million of
Computer World Inc. paid out $22.5 million in total common dividends and reported $278.9 million of retained earnings at year-end. The prior year’s retained earnings were $212.3 million. What was
W.C. Cycling had $55,000 in cash at year-end 2007 and $25,000 in cash at year-end 2008. Cash flow from long-term investing activities totaled –$250,000, and cash flow from financing activities
Bailey Corporation?s financial statements (dollars and shares are in millions) are provided here. a. What was net working capital for 2007 and 2008? b. What was Bailey?s 2008 free cash flow? c.
Hermann Industries is forecasting the following income statement:Sales $8,000,000Operating costs excluding depr. & amort. 4,400,000EBITDA $3,600,000Depreciation & amortization 800,000EBIT
The Davidson Corporation's balance sheet and income statement are provided here. a. Construct the statement of stockholders' equity for December 31, 2008.b. How much money has been reinvested in
Financial information for Powell Panther Corporation is shown here. a. What was net working capital for 2007 and 2008?b. What was the 2008 free cash flow?c. How would you explain the large increase
Laiho Industries’ 2007 and 2008 balance sheets (in thousands of dollars) are shown.a. Sales for 2008 were $455,150,000, and EBITDA was 15% of sales. Furthermore, depreciation and amortization were
What effect did the expansion have on sales, after-tax operating income, net working capital (NWC), and net income?
What effect did the company’s expansion have on its free cash flow?
D’Leon purchases materials on 30-day terms, meaning that it is supposed to pay for purchases within 30 days of receipt. Judging from its 2008 balance sheet, do you think that D’Leon pays
Suppose D’Leon’s sales manager told the sales staff to start offering 60-day credit terms rather than the 30-day terms now being offered. D’Leon’s competitors react by offering similar terms,
If D’Leon starts depreciating fixed assets over 7 years rather than 10 years, would that affect (1) the physical stock of assets, (2) the balance sheet account for fixed assets, (3) the
Explain how earnings per share, dividends per share, and book value per share are calculated, and what they mean. Why does the market price per share not equal the book value per share?
Explain briefly the tax treatment of (1) interest and dividends paid, (2) interest earned and dividends received, (3) capital gains, and (4) tax loss carry-back and carry-forward. How might each of
Financial ratio analysis is conducted by three main groups of analysts: credit analysts, stock analysts, and managers. What is the primary emphasis of each group, and how would that emphasis affect

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