Applied Interest Rate Analysis

1
Applied Interest Rate Analysis

• Purpose of Investment Modeling to improve the
  investment process.
• Components of investment process
• - identification
• - selection
• - combination
• - ongoing management
• This chapter introduces formal ways of
  structuring investments through investment models.

2

• A good background in interest rate theory
  (earlier chapters) provides the basis for the
  vast majority of actual investment studies,
  including those in this chapter.
• Topics in chapter
• capital budgeting
• bond portfolio construction
• management of dynamic investments
• valuation of firms from accounting data (what is
  the stock worth?)

3

• Optimization plays a key role. Often we want to
  make decisions subject to constraints
• - to minimize cost, or maximize profit
• and
• - minimize risk

4

• Capital Budgeting
• Capital budgeting allocation among projects or
  investments for which there are
• - no well-developed markets and
• - where the projects are lumpy (require discrete
  sums of cash)
• Several projects compete for funding. They
  differ in scale, cash requirements, and benefits.
  Not all can be funded (limited budget).

5

6

7

• Steps
• the choice cells are the changing cells
• the choice cells define binary (0-1)
  constraints
• the target cell is the total NPV,G32
• the total cost, F32, must not exceed the
  budget, F33
• use sumprod function to compute total cost,
  total npv
• changing cells are blue
• data is green
• computations are shown in yellow
• budget amount in orange

8

• General Independent Projects Model
• n potential projects
• xj 0-1 (binary) variable 1 if choose project j,
  0 o/w
• bj total benefit of project j (usually NPV)
• cj initial cost of project j
• C available capital (budget)
• max b1 x1 b2 x2 ... bn xn
• subject to
• c1 x1 c2 x2 ... cn xn C
• xj 0 or 1, j 1, ...., n

9

• The benefit-cost ratio approach is a heuristic.
  It usually gives good solutions very quickly, but
  there is no guarantee it will provide an optimal
  solution. The OR and math. programming approach
  can give an optimal solutions.
• Generalizations on Multiple Goals and Projects is
  in the paragraph Independent Projects (PhD
  students should read, and learn the approach
  using the Excel file Proj-Sel.xls)

10

• Optimal Portfolios
• There are several types of optimal portfolios
• - ones consisting of financial securities,
  freely traded in a market,
• - any collection of financial assets.
• The term portfolio optimization refers to the
  first type. Analysis of portfolios including
  stocks comes in Chapter 6, once we develop better
  ways of modeling risk.
• This section considers only portfolios of F-I
  securities. We visualize such securities by
  using the CFS idea. A portfolio is then a
  combination of such CFSs.

11

• Cash Matching
• Large institutions, such as insurance companies
  and pension funds, face sequences of future
  monetary obligations, such as insurance claims or
  annuity payments. They receive funds on a
  regular basis, and will make payments in the
  future. They may want to invest the funds in
  bonds of various maturities, and then use the
  coupon payments and redemption values to meet
  their future obligations.

12

• Cash Matching (Contd)
• An introductory approach to the cash matching
  problem is to consider a one-time model, which
  chooses a portfolio to meet these obligations
  without future changes.
• Note someone saving for retirement, or to put
  children through college, would have a similar
  problem.

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• Steps
• CFS of each bond is shown in green, with coupon
  payments and final payment.
• Bonds are indexed by decreasing maturities.
• Bond prices shown in next to last column.
• Each year we have required financial obligations
  to meet, shown in bottom row.
• We can meet these obligations by buying bonds and
  using their annual payments.
• We want to find the number of shares of each bond
  to buy to minimize our purchase cost AND meet the
  requirements each year.
• The last column shows an optimal solution
  obtained via LP. The next to last row shows how
  much cash we generate at the end of each year -
  always at least as much as our obligation.
• The target cell is the one with value 2381.14,
  the minimal total purchase price.
• The changing cells hold the number of shares we
  purchase.
• The actual cells (except for the last) are the
  LHS of the LP constraints.

15

• General LP Model
• m of available bonds to choose from (i 1,
  ..., m)
• xi shares we purchase of bond i (i 1, ...,
  m)
• pi price of one share of bond i, (i 1, ..., m)
• n periods with payment obligations (j 1,
  ..., n)
• yj payment obligation for period j (j 1,
  ...,n)
• cij entry j in CFS for bond i amount 1 share
  of bond i pays in period j (possibly 0).
• Ci (ci1, ci2, ..., cin) is the CFS for 1 share
  of bond i

16

• General LP Model (Contd)
• cij xi total income from bond i for period j
• c1j x1 c2j x2 ... cmj xm total income from
  all bonds for period j
• yj payment obligation for period j
• LP minimize p1 x1 p2 x2 ... pm xm
• subject to
• c1j x1 c2j x2 ... cmj xm ? yj , j 1,
  ..., n
• xi ? 0, i 1, ..., m

17

• We choose the number of shares of each bond (each
  xi) to purchase to minimize our total bond
  purchase cost while at least meeting our payment
  obligations in each period. Note
• Y (y1, y2, ..., yn)
• is the CFS of our payment obligations. Since
• Ci (ci1, ci2, ..., cin) is the CFS for 1 unit
  of bond i
• the constraints can be written in vector form
• x1 C1 x2 C2 ... xm Cm ? Y.
• We are trying to find an appropriate (linear)
  combination of the bond CFSs to meet or to
  match our obligation CFS at least cost to us.

18

• Consider again what happens in terms of meeting
  the payment obligation CFS in the example
• In years 1 and 4 we obtain more money than we
  need. Also we need much more money in years 3, 5
  and 6 than in other years, so we must buy large
  numbers of bonds (2, 4 and 8) that mature in
  those years. These bonds generate coupon
  payments in earlier years, and we do not need all
  the payments. A smoother set of cash
  requirements would not lead to such surpluses.
• The surpluses identify a flaw in the model. In
  reality, the surpluses would immediately be
  reinvested. We can generalize the model to
  incorporate this reinvestment idea, provided we
  know what sorts of reinvestments are possible.

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• One Reinvestment Approach extra cash is carried
  forward at 0 interest.
• Constraints in initial example without carryover,
  periods 1,2,3
• 10 x1 7 x2 8 x3 6 x4 7 x5 5 x6 10 x7
  8 x8 7 x9 100 x10 ? 100
• 10 x1 7 x2 8 x3 6 x4 7 x5 5 x6 10 x7
  8 x8 107 x9 ? 200
• 10 x1 7 x2 8 x3 6 x4 7 x5 5 x6 110 x7
  108 x8 ? 800

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Let s
denote the surplus we have for period j. We
j
rewrite the constraints as follows
Surplus
10 x
7 x
8 x
6 x
7 x
5 x
10 x
8 x
7 x

1
2
3
4
5
6
7
8
9
100 x
s
100
10
1
10 x
7 x
8 x
6 x
7 x
5 x
10 x
8 x
107 x
1
2
3
4
5
6
7
8
9
s
s
200
1
2
10 x
7 x
8 x
6 x
7 x
5 x
110 x
108 x
s
1
2
3
4
5
6
7
8
2
s
800
3
Carryover
Carryover
22

• In general, instead of
• c1j x1 c2j x2 ... cmj xm ? yj , j 1, ...,
  n
• for period j we have
• c1j x1 c2j x2 ... cmj xm sj-1 sj yj ,
  j 1, ..., n
• with the convention that s0 ? 0
• (there is no sj-1 for j 1).

23

• Model Generalization with Surpluses Carried Over
  (Contd)
• We now buy only bonds 2, 4 and 8. Surpluses in
  years 1, 2 and 4 are used in years 2, 3, and 5.
  This plan costs us 2,305.69 as compared to the
  earlier solution 2,381.13.

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• A More Conventional Looking LP Formulation

25

• Second Reinvestment Approach If we get interest
  on the surplus cash, say a rate of r for 1 year,
  then we could also build that into the model
• 10 x1 7 x2 8 x3 6 x4 7 x5 5 x6 10 x7
  8 x8 7 x9 100 x10 s1 100
• 10 x1 7 x2 8 x3 6 x4 7 x5 5 x6 10 x7
  8 x8 107 x9 (1r) s1 s2 200
• 10 x1 7 x2 8 x3 6 x4 7 x5 5 x6 110 x7
  108 x8 (1r) s2 s3 800

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• Model with 6 Interest on surplus

Bonds
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
price
shares/sur
plus
1
10
10
10
10
10
110
109
0.00
x1
7
7
7
7
7
107
94.8
11.21
2
x2
3
8
8
8
8
8
108
99.5
0.00
x3
6
6
6
6
106
93.1
6.62
4
x4
5
7
7
7
7
107
97.2
0.00
x5
6
5
5
5
105
92.9
0.00
x6
7
10
10
110
110
0.00
x7
8
8
8
108
104
5.96
x8
7
107
102
0.00
9
x9
10
100
95.2
0.00
x1
0
Spls 1
-1
1.06
65.94
s1
-1
1.06
35.83
2
s2
3
-1
1.06
0.00
s3
4
-1
1.06
18.25
s4
5
-1
1.06
0.00
s5
6
-1
0.00
s6
Actual
100
200
800
100
800
1200
2299.81
100
200
800
100
800
1200
Reqd
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• Other Possible Modifications
• Require the number of shares to be integers if
  rounding is unacceptable. (Easy in Excel)
• Immunize the cash matching to interest shifts.
• Recognize that cash-matching may be an on-going
  process instead of a one-shot process

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Dynamic Cash Flow Processes
30

• We have an oil well, with 10 M barrels of oil.
• Each year we either decide to pump out 10 of the
  oil or not. Moving up in the graph indicates
  pumping, down indicates not pumping.
• In order to pump we must have a crew on hand.
• - Y for Yes indicates we have a crew on hand.
• - N for No means we do not have a crew on
  hand.
• The x-axis represents time points (in periods).
• The dots represent states or conditions of the
  process

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• Begin with 10 M barrels and No crew on hand.
• If we decide to pump 10 then next year we have 9
  M barrels and a crew on hand.
• If not, next year we still have 10 M. barrels and
  No crew on hand.
• The graph continues to grow over time in this
  way.

32

• Dots are called nodes or vertices.
• Lines joining nodes are called arcs, branches,
  edges, or links. Links indicate a logical
  connection between two nodes. For example, we
  can start with 10M barrels and no crew, and then
  if we pump in 1 year we have 9 M. barrels and a
  crew on hand. If a crew is on hand we avoid
  having to hire a new crew.

33

34

• There is an area of optimization in OR, called
  dynamic programming, which addresses multistage
  decision problems. Dynamic refers to
  multistage or multiperiod. Programming
  refers to optimization or planning and NOT to
  computer programming.

35

• Cash Flows in Graphs
• Example for cash flows

36

• We can associate a profit with each arc

37

• It is important to state when in a period the
  cash flows occur. For the above example we
  assume they occur at the end of each period.
  Sometimes cash flows occur at the beginning of
  each period. At other times they occur
  throughout the period. The model may be a
  simplification of when the flows occur. By
  shortening the length of a period and having more
  periods we can increase the model accuracy, but
  at the expense of a larger model.
• Sometimes there is a final reward or salvage
  value a cash flow associated with the
  termination of the process. The value can vary
  depending on which is the final node. This value
  was 0 for the oil drilling example, but might in
  reality be the value for which the well could be
  sold.

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• Each path from the initial node to a terminal
  node determines a possible management policy with
  a specific CFS.
• The PV of the CFS of each path is often of
  interest.
• We seek a path with a best PV.
• The number of such paths grows exponentially in
  the number of periods, say n.
• Enumeration of all the paths is impossible for
  large n.
• Dynamic programming can find a best path much
  more efficiently than enumeration.

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• Fishing Problem. Node values are tonnages of
  fish in the lake branch values are cash flows.
• Currently there are 10 tons of fish. For each
  year without fishing, the tonnage doubles. If we
  fish we take 70, and the remaining fish
  reproduce to the previously available population
  during the year. We make a profit of 1.00 per
  ton on fish
• ( scale up to 1,000 if you want to.)

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• Dynamic Programming Recursions
• V2,1 max(c2,11 d2 V3,1, c2,12 d2 V3,2)
• V2,2 max(c2,21 d2 V3,2, c2,22 d2 V3,3)
• V2,3 max(c2,31 d2 V3,3, c2,32 d2 V3,4)
• V1,1 max(c1,11 d1 V2,1, c1,12 d1 V2,2)
• V1,2 max(c1,21 d1 V2,2, c1,22 d1 V2,3)
• V0,1 max(c0,11 d0 V1,1, c0,12 d0 V1,2)
• General form
• Vk,i max(ck,ia dk Vk1,a)
• a

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• PV of some optimal CFS
• PV c0 c1/(1s1) c2/(1s2)2 ...
  cn-1/(1sn-1)n-1
• Vn/(1sn)n
• Instead of working with the spot rates we can use
  equivalent discount factors.

45

• We have seen binomial lattices, but there are
  trinomial lattices... Other lattices, even
  continuous ones, are possible.
• For example, for the oil well problem, from a
  reserve R you could pump any amount z between 0
  and M, leading to a new reserve of R-z. The
  choice of z is continuous, as is the level of
  reserves. We can visualize such a continuous
  lattice as follows.

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• Each vertical line represents the continuum of
  nodes possible at a particular time.
• At the initial time there is only one node.
• The fan emanating from a node represents the
  fan of possibilities for moving to a subsequent
  node.
• Only one fan is indicated for each time, but
  actually there is such a fan from every point on
  each vertical line.
• The dynamic structure works very much like the
  finite-node case.
• Optimizing such a process by dynamic programming
  works backwards (just as with the finite case),
  but is more difficult. A V-value must now be
  assigned to every point on each node line. Hence
  V is a function defined on each line.

48

• Complexico Mining Example, p. 119-121,
• Data
• - 10 year lease
• - Mine has x0 50 K oz. of gold
• - Gold price/oz is g 400
• - Interest rate is 10
• - Discount rate is d 0.909091

49

• Fundamental Question
• How much should we mine each year to maximize the
  NPV?
• Solution Approach Running DP
• Find V9(x9), V8(x8), ..., V1(x1), V0(x0) in this
  order. The last value answers the question.

50

• p(z9x9) g z9 500 z92/x9 g2/2000 x9
  K9 x9 ? V9(x9) p(z9x9) K9 x9 and
• K9 g2/2000

51

• Year 8 (x9 x8-z8)
• V8(x8) maxp(z8x8) d V9(x8-z8) 0 z8 lt x8
• maxg z8 500 z82/x8 d V9(x8-z8) 0 z8 lt
  x8
• maxg z8 500 z82/x8 d K9 (x8-z8) 0 z8 lt
  x8
• z8 (g d K9)x8/1000 ?
• V8(x8) (g d K9)2/2000 d K9 x8 K8 x8
• K8 (g d K9)2/2000 d K9

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• Final recurrent equation
• K9 g2/2000, Kj (g d Kj1)2/2000 d Kj1, j
  0, ...,8.
• Vj(xj) Kj xj, j 0, 1, ..., 9

53

• Valuation of a Firm
• Dividend Discount Models
• An owner of stock in a company can expect to
  receive periodic dividends. Suppose the dividend
  for year k is Dk. Suppose the interest rate (or
  the discount rate) is fixed at r. One way to
  measure the value of the fund to the stockholder
  is as follows
• V0 D1/(1r) D2/(1r)2 D3/(1r)3 ...

54

• Constant-Growth Dividend Model
• Suppose dividends grow at a constant rate g.
• D2 (1g) D1, D3 (1g)2 D1,D4 (1g)3 D1,
  ....
• Assuming g lt r, it is known that substituting the
  above dividend expressions into the equation for
  V0 gives the following
• Gordon Formula
• V0 D1/(r-g).
• (If g ? r, there is no finite value V0.)

55

• Assuming also that, with D0 denoting the current
  (already paid) dividend,
• D1 (1g) D0,
• the Gordon Formula gives the Discounted Growth
  formula
• V0 (1g)D0/(r-g).