We have a cash flow stream (x0, x1, x2, . . . , xn) (where the indexing is by period). The price is
P(λ) =
We then find that P0(0) = dP(λ)
Duration
DQ does have the units of time; however, it is not exactly an average of the cash flow times because in the numerator, it is
1
[1 + (sk/m)](k+1) instead of 1 [1 + (sk/m)]k.
Formula (Quasi-modified duration)
Under compounding m times per year, the quasi-modified duration of a cash flow stream (x0, x1, x2, . . . , xn) is
where PV denotes the present value of the stream. If all spot rates change to sk + λ, k = 1, 2, . . . , n, the corresponding present value function P(λ)
Immunization
Outline
14 The Yield Curve
15 The Term Structure
16 Forward Rates
17 Term Structure Explanations (Self-learning)
18 Expectation Dynamics
19 Running Present Value
20 Floating Rate Bonds
21 Duration
22 Immunization
Immunization
Example (A million dollar obligation)
Suppose that we have a $1 million obligation payable at the end of 5 years, and we wish to invest enough money today to meet this future obligation. We wish to do this in a way that provides a measure of protection against interest rate risk. To solve this problem, we first
determine the current spot rate curve. A hypothetical spot rate curve sk is shown as the column labeled spot in the table.
Immunization
Example (contd.)
We use a yearly compounding convention and decide to invest in two bonds described as follows:
B1 is a 12-year 6% bond with price 65.95; and B2 is a 5-year 10% bond with price 101.66.
We decide to immunize against a parallel shift in the spot rate curve.
We calculate dP/dλ, denoted by −PV0, by multiplying each cash flow by t and by (1 + st)−(t+1) and then summing these. The quasi-modified duration is then the quotient of these two numbers. For example, the quasi-modified duration of bond B1 is 466/65.95 = 7.07. We also find the present value of the obligation to be $627, 903.01 and the corresponding quasi-modified duration is
5
(1 + s5) = 4.56.
Immunization
Example (contd.)
To determine the appropriate portfolio, let x1 and x2 denote the number of shares of bonds 1 and 2, respectively, in the portfolio (assuming, for simplicity, face values of $100). We solve the following two equations
( P1x1+ P2x2 = PV , P1D1x1+ P2D2x2 = PV · D, ⇒
(x1 = 2, 208.17, x2 = 4, 744.03.
Outline
23 Capital Budgeting
24 Optimal Portfolios
25 Dynamic Cash Flow Processes
26 Optimal Management
27 The Harmony Theorem
28 Valuation of a Firm
Capital Budgeting
Outline
23 Capital Budgeting
24 Optimal Portfolios
25 Dynamic Cash Flow Processes
26 Optimal Management
27 The Harmony Theorem
28 Valuation of a Firm
Capital Budgeting
Definition
Capital budgeting typically refers to allocation among projects or
investments for which there are not well-established markets and where the projects are lumpy in that they each require discrete lumps of cash.
Capital budgeting problems often arise in a firm where several proposed projects compete for funding.
The projects may differ considerably in their scale, their cash requirements, and their benefits.
Even if all proposed projects offer attractive benefits, they cannot all be funded because of a budget limitation.
Capital Budgeting
Independent Projects
The projects are independent in the sense that it is reasonable to select any combination from the list.
Definition (Zero-one programming problem)
Suppose that there are m potential projects. Let bi be the total benefit (usually the net present value) of the i th project, and let ci denote its initial cost. Finally, let C be the total capital available-the budget. For each i , i = 1, 2, . . . , m, we introduce the zero-one variable xi, which is zero if the project is rejected and one if it is accepted.
max
Capital Budgeting
A good approximate solution to zero-one programming problem is the benefit-cost ratio method.
Definition
The benefit-cost ratio is defined as the ratio of the present worth of the benefits to the magnitude of the initial cost.
Example (A selection problem)
During its annual budget planning meeting, a small computer company has identified several proposals for independent projects that could be initiated in the forthcoming year.
These projects include the purchase of equipment, the design of new products, the lease of new facilities, and so forth. The projects all require an initial capital outlay in the coming year.
The company management believes that it can make available up to
$500,000 for these projects.
Capital Budgeting
Example (contd.)
The projects are already listed in order of decreasing benefit-cost ratio.
According to the approximate method, the company would select project 1, 2, 3, 4, and 5 for a total expenditure of $370,000 and a total net present value of $540,000.
Capital Budgeting
Example (contd.)
However, the solution derived by the approximate method is not optimal.
According to the zero-one programming problem, the solution should be to select projects 1, 3, 4, 5, and 6 for a total expenditure of $500,000 and a total net present value of $610,000.
Capital Budgeting
Interdependent Projects
Sometimes various projects are interdependent, the feasibility or one being dependent on whether others are undertaken.
Assume that there are m goals and that associated with the i th goal, there are ni possible projects. Only one project can be selected for any goal. As before, there is a fixed available budget.
We formulate this problem by introducing the zero-one variables xij for i = 1, 2, . . . , m and j = 1, 2, . . . , ni.
Capital Budgeting
Example (County transportation choices)
Suppose that the goals and specific projects shown in the following table are being considered by the County Transportation Authority.
Capital Budgeting
Example (contd.)
To formulate this problem we introduce a zero-one variable for each project (However, for simplicity we index these variables consecutively from 1 through 10, rather than using the double indexing procedure of the general formulation presented earlier).
Optimal Portfolios
Outline
23 Capital Budgeting
24 Optimal Portfolios
25 Dynamic Cash Flow Processes
26 Optimal Management
27 The Harmony Theorem
28 Valuation of a Firm
Optimal Portfolios
Definition
Optimal portfolio usually refers to the construction of a portfolio of financial securities, and more generally, the construction of any portfolio of financial assets, including a “portfolio” of projects.
Definition (Cash matching problem)
Suppose that we face a known sequence of future monetary obligations y = (y1, y2, . . . , yn) and wish to invest now so that these obligations can be met as they occur. If there are m bonds, we denote the stream associated with one unit of bond j by cj = (c1j, c2j, . . . , cnj). The price of
Optimal Portfolios
Example (A 6-year match)
We wish to match cash obligations over a 6-year period. We select 10 bonds for this purpose (and for simplicity all accounting is done on a yearly basis).
The cash flow structure of each bond is shown in the corresponding column in the table. Below this column is the bond’s current price. For example, the first column represents a 10% bond that matures in 6 years. This bond is selling at 109. The second to last column shows the yearly cash
requirements (or obligations) for cash to be generated by the portfolio.
We formulate the standard cash matching problem as a linear programming problem and solve for the optimal portfolio.
Optimal Portfolios
Example (contd.)
The solution is given in the bottom row.
The actual cash generated by the portfolio is shown in the right-hand column. This column is computed by multiplying each bond column j by its solution value xj and then summing these results.
The minimum total cost of the portfolio is also indicated.
Optimal Portfolios
What about the surpluses to extra cash, which is essentially thrown away since it is not used to meet obligations and is not reinvested?
1 Assume that extra cash can be carried forward at zero interest. Such an artificial bond is “purchased” in the year with the −1 and is
“redeemed” the next year.
(0, . . . , 0, −1, 1, 0, . . . , 0).
2 Allow surplus cash to be invested in actual bonds. However, to incorporate this feature, an assumption about future interest rates (or, equivalently, about future bond prices) must be made.
(0, . . . , 0, −1, 1 + r0, 0, . . . , 0).
Other modifications to the basic cash matching problem.
1 To take into account the integer nature of the required solution if the sums involved are not large.
2 To combine immunization with cash matching.
Dynamic Cash Flow Processes
Outline
23 Capital Budgeting
24 Optimal Portfolios
25 Dynamic Cash Flow Processes
26 Optimal Management
27 The Harmony Theorem
28 Valuation of a Firm
Dynamic Cash Flow Processes
Example
Imagine, for example, that you have purchased an oil well. This is an investment project, and to obtain good results from it, it must be carefully managed. In this case you must decide, each month, whether to pump oil from your well or not.
If you do pump oil, you will incur operational costs and receive revenue from the sale of oil, leading to a profit; but you will also reduce the oil reserves.
If you believe that current oil prices are low, you may wisely choose not to pump now, but rather to save the oil for a time of higher prices.
Your current pumping decision clearly influences the future possibilities of production!
Discussion of this type of problem within the context of deterministic cash flow streams is especially useful-both because it is an important class of problems, and because the solution is dynamic programming.
Dynamic Cash Flow Processes