Questions and Answers of Investment Risk Management

What are some problems that occur in supply chain risk analysis that a simulation model can address?
Describe how the models presented in this chapter can be beneficial to a decision maker involved with a business continuity program.
How important is it to capture the decision maker’s disposition towards risk? How is subjectivity included in the decision modeling process?
What could go wrong with the personnel aspects of executing the strategy?
What could go wrong with the logistical aspects of executing the strategy?
How could weather forecasting inaccuracies interfere with executing the strategy?
The position $X$ has a 1-year probability density function that is normal with a mean of $\$ 100$ million and a standard deviation of $\$ 50$ million. What is the value at risk at the $99 %$
Suppose $X$ is a normal with zero mean and standard deviation of $\$ 10$ million.(a) Find the value at risk for $X$ for the risk tolerances $h=0.01,0.02,0.05,0.10,0.50,0.60$, and 0.95 .(b) Is there
Consider the position $X$ that has a uniform probability density between -40 and 60.Find $\operatorname{VaR}_{h}(X)$ for all $h, 0 \leq h \leq 1$.
General equivalent formulas for VaR are shown here. Assuming the first one, argue that the next two are equivalent to it. [Hint: Consider a distribution $X$ that at a certain point jumps up. Then
Suppose $X_{1}$ and $X_{2}$ are jointly normal positions with parameters $\mu_{1}, \mu_{2}, \sigma_{1}, \sigma_{2}, \sigma_{12}$. Show that\[\operatorname{VaR}_{h}\left(X_{1}+X_{2}\right) \leq
Find $\mathrm{AVaR}_{h}(X)$ for the $X$ of Exercise 3.Data from Exercises 3Consider the position $X$ that has a uniform probability density between -40 and 60.Find $\operatorname{VaR}_{h}(X)$ for all
Let $a>0$ and consider the risk measure $ho(X)=\mathrm{E}[-X]+a \sigma(X)$. Is this a coherent risk measure? If not, which axioms are violated?
Let $X$ be a position with a probability distribution $F$ that is strictly increasing and smooth. Let $f(x)=F^{\prime}(x)$ be the associated probability density.(a) Verify
Find $\mathrm{CVaR}_{h}(X)$ for the linear case of Exercise 3.Data from Exercises 3Consider the position $X$ that has a uniform probability density between -40 and 60.Find $\operatorname{VaR}_{h}(X)$
For Example 10.5 with $h=50 %$, show explicitly that CVaR rewards diversification. Example 10.5 (Diversification failure) Consider the position 1 with probability 1/2 X1 -1 with probability 1/2 } .
Modify Example 10.10 by changing the confidence level to $70 %$ (that is, with $h=30 %$ ).(a) Find the conditional value at risk for $X_{1}$ of the example.(b) Describe the family $\mathcal{P}$ of
Suppose that a bank has position $X$ that has a normal probability density. The value at risk is known to be $V$ at the loss tolerance $h$. The bank plans to take some capital out of reserve, which
Suppose $\$ 1$ were invested in 1776 at $3.3 \%$ interest compounded yearly.(a) Approximately how much would that investment be worth today: $\$ 1,000, \$ 10,000$, $\$ 100,000$, or \(\$
The number of years $n$ required for an investment at interest rate $r$ to double in value must satisfy $(1+r)^{n}=2$. Using $\ln 2=.69$ and the approximation $\ln (1+r) \approx r$ valid
Find the corresponding effective rates for:(a) $3 \%$ compounded monthly.(b) $18 \%$ compounded monthly.(c) $18 \%$ compounded quarterly.
The IRR is generally calculated using an iterative procedure. Suppose that we define $f(\lambda)=-a_{0}+a_{1} \lambda+a_{2} \lambda^{2}+\cdots+a_{n} \lambda^{n}$, where all $a_{i}$ 's are
Suppose that you have the opportunity to plant trees that later can be sold for lumber. This project requires an initial outlay of money in order to purchase and plant the seedlings. No other cash
What rate of interest (to five digits) is equivalent to $10 \%$ yearly under(a) monthly compounding?(b) continuous compounding?
A major lottery advertises that it pays the winner $\$ 10$ million. However, this prize money is paid at the rate of $\$ 500,000$ each year (with the first payment being immediate) for a total of
A young couple has made a nonrefundable deposit of the first month's rent (equal to $\$ 1,000$ ) on a 6 -month apartment lease. The next day they find a different apartment that they like just as
Two copy machines are available. Both have useful lives of 5 years. One machine can be either leased or purchased outright; the other must be purchased. Hence there are a total of three options: A,
You are considering the purchase of a nice home. It is in every way perfect for you and in excellent condition, except for the roof. The roof has only 5 years of life remaining. A new roof would last
A wealthy investor spends $\$ 1$ million to drill and develop an oil well that has estimated reserves of 200,000 barrels. The well is to be operated over 5 years, producing the estimated quantities
Consider the two projects whose cash flows are shown in Table 2.7. Find the IRRs of the two projects and the NPVs at 5\%. Show that the IRR and NPV figures yield different recommendations. Can you
Suppose two competing projects have cash flows of the form $\left(-A_{1}, B_{1}\right.$, $\left.B_{1}, \ldots, B_{1}\right)$ and $\left(-A_{2}, B_{2}, B_{2}, \ldots, B_{2}\right)$, both with
In general, we say that two projects with cash flows $x_{i}$ and $y_{i}, i=0,1,2, \ldots$, $n$, cross if $x_{0}\sum_{i=0}^{n} y_{i}$. Let $P_{x}(d)$ and $P_{y}(d)$ denote the present
In the United States the accelerated cost recovery system (ACRS) must be used for depreciation of assets placed into service after December 1980. In this system, assets are classified into categories
A division of ABBOX Corporation has developed the concept of a new product. Production of the product would require $\$ 10$ million in initial capital expenditure. It is anticipated that 1 million
A debt of $\$ 25,000$ is to be amortized over 7 years at $7 %$ interest. What value of monthly payments will achieve this?
Given a cash flow stream $X=\left(x_{0}, x_{1}, x_{2}, \ldots, x_{n}\right)$, a new stream $X_{\infty}$ of infinite length is made by successively repeating the corresponding finite stream. The
Gavin's grandfather, Mr. Jones, has just turned 90 years old and is applying for a lifetime annuity that will pay $\$ 10,000$ per year, starting 1 year from now, until he dies. He asks Gavin to
For the mortgage listed second in Table 3.1 what are the total fees? TABLE 3.1 MORTGAGE BROKER ADVERTISEMENT Rate Pts Term Max amt APRI 7.625 1.00 30 yr $203,150 7.883 7.875 .50 30 yr $203,150 8.083
An investor purchased a small apartment building for $\$ 250,000$. She made a down payment of $\$ 50,000$ and financed the balance with a 30 year, fixed-rate mortgage at $12 %$ annual interest,
Assume we are at period 0 . The current interest is $r$. Define $x=\frac{1}{1+r}$.(a) Derive the present value $S_{n}$ of the following cash flow in terms of $x$ and $n$ :(b) Derive the present value
The Z Corporation issues a $10 %, 20$-year bond at a time when yields are $10 %$. The bond has a call provision that allows the corporation to force a bond holder to redeem his or her bond at face
Here is a proposal that has been advanced as a way for homeowners to save thousands of dollars on mortgage payments: pay biweekly instead of monthly. Specifically, if monthly payments are $x$, it is
The Smith family just took out a variable-rate mortgage on their new home. The mortgage value is $\$ 100,000$, the term is 30 years, and initially the interest rate is $8 %$. The interest rate is
There are three bonds in the market as follows:1. A bond with $4 %$ coupon rate (paid annually), 10 years to maturity, and $\$ 1,000$ face value 13.2. A bond with $4 %$ plus current (short) rate
In 1997, the U.S. Treasury issued "Treasury inflationprotected securities" (TIPS). These are fixed-income securities that are inflation indexed toprotect their value against inflation. Like
An $8 %$ bond with 18 years to maturity has a yield of $9 %$. What is the price of this bond?
Find the price and duration of a 10 -year, $8 %$ bond that is trading at a yield of $10 %$.
Find the duration $D$ and the modified duration $D_{M}$ of a perpetual annuity that pays an amount $A$ at the beginning of each year, with the first such payment being 1 year from now. Assume a
Consider the four bonds having annual payments as shown in Table 3.9. They are traded to produce a $15 %$ yield.(a) Determine the price of each bond.(b) Determine the duration of each bond (not the
Under continuous compounding the Macaulay duration becomes\[D=\frac{\sum_{k=0}^{n} t_{k} e^{-\lambda t_{k}} c_{k}}{P}\]where $\lambda$ is the yield and\[P=\sum_{k=0}^{n} e^{-\lambda t_{k}}
Show that the limiting value of duration as maturity is increased to infinity is\[D \rightarrow \frac{1+(\lambda / m)}{\lambda}\]For the bonds in Table 3.6 (where $\lambda=.05$ and $m=2$ ) we obtain
Find the convexity of a zero-coupon bond maturing at time $T$ under continuous compounding (that is, when $m \rightarrow \infty$ ).
Suppose that an obligation occurring at a single time period is immunized against interest rate changes with bonds that have only nonnegative cash flows (as in the $\mathrm{X}$ Corporation example).
If the spot rates for 1 and 2 years are $s_{1}=6.3 %$ and $s_{2}=6.9 %$, what is the forward rate $f_{1,2}$ ?
Given the (yearly) spot rate curve $s=(5.0,5.3,5.6,5.8,6.0,6.1)$, find the spot rate curve for next year.
Consider two 5-year bonds; one has a $9 %$ coupon and sells for 101.00 ; the other has a $7 %$ coupon and sells for 93.20 . Find the price of a 5-year zero-coupon bond.
It is November 5 in the year 2012 . The bond quotations of Table 4.6 are available. Assume that all bonds make semiannual coupon payments on the 15 th of the month. Estimate the (continuous-time)
Let $s(t), 0 \leq t \leq \infty$, denote a spot rate curve; that is, the present value of a dollar to be received at time $t$ is $e^{-s(t) t}$. For $t_{1}(a) Find an expression for $f\left(t_{1},
At time zero the one-period discount rates $d_{0,1}, d_{1,2}, d_{2,3}, \ldots, d_{5,6}$ are known to be $0.950,0.940,0.932,0.925,0.919,0.913$. Find the time zero discount factors $d_{0,1}, d_{0,2},
An investor is considering the purchase of 10-year U.S. Treasury bonds and plans to hold them to maturity. Federal taxes on coupons must be paid during the year they are received, and tax must also
Actual zero-coupon bonds are taxed as if implied coupon payments were made each year (or really every 6 months), so tax payments are made each year, even though no coupon payments are received. The
Show explicitly that if the spot rate curve is flat [with $s(k)=r$ for all $k$ ], then all forward rates also equal $r$.
Orange County managed an investment pool into which several municipalities made short-term investments. A total of $\$ 7.5$ billion was invested in this pool, and this money was used to purchase
A (yearly) cash flow stream is $\mathbf{x}=(-40,10,10,10,10,10,10)$. The spot rates are those of Exercise 2.(a) Find the current discount factors $d_{0, k}$ and use them to determine the (net)
You are given an incomplete specification of the term structure, as specified by the spot rates and forward rates noted next. You also know that the price of a 6-year bond with coupon rate $10 %$ is
A certain bond portfolio has a value of $\$ 1,000$ today at a yield of $10 %$. Yesterday the same portfolio had a value of $\$ 990$ at a yield of $10.5 %$.(a) Estimate what the modified duration was
It is sometimes useful to introduce variations of the spot rates that are different from an additive variation. Let $\mathrm{s}^{0}=\left(s_{1}^{0}, s_{2}^{0}, s_{3}^{0}, \ldots, s_{n}^{0}\right)$ be
A company faces a stream of obligations over the next 8 years as shown: where the numbers denote thousands of dollars. The spot rate curve is that of Example 4.8. Try to find a portfolio, consisting
Often a mortgage payment stream is divided into a principal payment stream and an interest payment stream, and the two streams are sold separately. We shall examine the component values. Consider a
(Short rate sensitivity) Gavin Jones sometimes has flashes of brilliance. He asked his instructor if duration would measure the sensitivity of price to a parallel shift in the short rate curve. (That
A firm is considering funding several proposed projects that have the financial properties shown in Table 5.6. The available budget is $\$ 600,000$. What set of projects would be recommended by the
Refer to the transportation alternatives problem of Example 5.2. The bridge at Cay Road is actually part of the road between Augen and Burger. Therefore it is not reasonable for the bridge to have
A company has identified a number of promising projects, as indicated in Table 5.7. The cash flows for the first 2 years are shown (they are all negative). The cash flows in later years are positive,
The cash matching and other problems can be conveniently represented in matrix form. Suppose there are $m$ bonds. We define for each bond $j$ its associated yearly cash flow stream (column) vector
A trinomial lattice is a special case of a trinomial tree. From each node three moves are possible: up, middle, and down. The special feature of the lattice is that certain pairs of moves lead to
You are the manager of XYZ Pension Fund. On November 5, 2021, XYZ must purchase a portfolio of U.S. Treasury bonds to meet the fund's projected liabilities in the future. Short selling is not
Consider the Complexico mine and assume a $10 %$ constant interest rate; also assume the price of gold is constant at $\$ 400 / \mathrm{oz}$.(a) Find the value of the mine (not a 10 -year lease) if
You have purchased a lease for the Little Bear Oil well. This well has initial reserves of 100 thousand barrels of oil. In any year you have three choices of how to operate the well: (a) you can not
The value of a firm is the maximum present value of its possible cash flow streams. This can be expressed as\[ V_{0}=\max
Show that for g < r,\[ \sum_{k=1}^{\infty} \frac{(1+g)^{k-1}}{(1+r)^{k}}=\frac{1}{r-g} \][Let $S$ be the value of the sum. Note that $S=1 /(1+r)+S(1+g) /(1+r)$.]
It is common practice in security analysis to modify the basic dividend growth model by allowing more than one stage of growth, with the growth factors being different in the different stages. As an
Suppose that to short a stock you are required to deposit an amount equal to the initial price $X_{0}$ of the stock. At the end of 1 year the stock price is $X_{1}$ and you liquidate your position.
Two dice are rolled and the two resulting values are multiplied together to form the quantity 2 . What are the expected value and the variance of the random variable $z$ ?
The correlation $ho$ between assets A and B is .1, and other data are given in Table 6.3. [$ho=\sigma_{\mathrm{AB}} /\left(\sigma_{\mathrm{A}} \sigma_{\mathrm{B}}\right)$.](a) Find the proportions
Two stocks are available. The corresponding expected rates of return are $\bar{r}_{1}$ and $\bar{r}_{2}$; the corresponding variances and covariances are $\sigma_{1}^{2}, \sigma_{2}^{2}$. and
Gavin Jones's friend is planning to invest $\$ 1$ million in a rock concert to be held 1 year from now. The friend figures that he will obtain $\$ 3$ million revenue from his $\$ 1$ million
Suppose there are $n$ assets which are uncorrelated. (They might be $n$ different "wild cat" oil well prospects.) You may invest in any one, or in any combination of them. The mean rate of return
There are just three assets with rates of return $r_{1}, r_{2}$, and $r_{3}$, respectively. The covariance matrix and the expected rates of return are\[\mathbf{V}=\left[\begin{array}{lll}2 & 1 & 0
Suppose that it is impractical to use all the assets that are incorporated into a specified portfolio (such as a given efficient portfolio). One alternative is to find the portfolio, made up of a
Derive equation (6.9).\[\left.\frac{\partial}{\partial w_{i}}\left(\sum_{i j}^{n} \sigma_{i j} w_{i} w_{j}\right)^{1 / 2}=\left(\sum_{i j}^{n} \sigma_{i j} w_{i} w_{j}\right)^{-1 / 2} \sum_{j=1}^{n}
Two assets with expected rates of return $\bar{r}_{1}$ and $\bar{r}_{2}$ have identical variances and a known correlation coefficient $ho$. There is a risk-free asset with rate of return $r_{f}$.(a)
Show that for equation (6.10) all rates of return $r$ can be transformed by the linear relation $R=a r+b, a>0$ and that equation (6.10) will still hold for the $R$ 's (although the $v_{i}$ 's will
Two risky assets are derived by a single flip of a coin. For asset $A$, a "heads" outcome pays $\$ 4.00$, while a "tails" outcome pays $\$ 0.00$. For asset B, the corresponding payments are $\$ 3.00$
Assume that the expected rate of return on the market portfolio is $23 %$ and the rate of return on T-bills (the risk-free rate) is $7 %$. The standard deviation of the market is $32 %$. Assume that
Consider a world in which there are only two risky assets, $A$ and $B$, and a risk-free asset $F$. The two risky assets are in equal supply in the market; that is, $M=\frac{1}{2}(A+B)$. The following
Consider a universe of just three securities. They have expected rates of return of $10 %, 20 %$, and $10 %$, respectively. Two portfolios are known to lie on the minimum-variance set. They are
Derive the CAPM formula for $\bar{r}_{k}-r_{f}$ by using Equation (6.9) in Chapter 6.\[\sum_{i=1}^{n} \sigma_{i k} w_{i}=\operatorname{cov}\left(r_{k}, r_{M}\right)\]Apply equation (6.9) both to
Suppose there are $n$ mutually uncorrelated assets. The return on asset $i$ has variance $\sigma_{i}^{2}$. The expected rates of return are unspecified at this point. The total amount of asset $i$ in
In Simpleland there are only two risky stocks, A and B, whose details are listed in Table 7.4.Furthermore, the correlation coefficient between the returns of stocks $A$ and $B$ is $ho_{A
Let $w_{0}$ be the portfolio (weights) of risky assets corresponding to the minimum-variance point in the feasible region. Let $\mathbf{w}_{1}$ be any other portfolio on the efficient frontier.

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