! Asset-Liability Management, Stochastic case;
! Each period we can buy and sell a variety of investments,
and we can borrow or lend for one period at a rate
specific to the period. In the stochastic case, the
borrowing and lending rates are random variables.
How should we invest each period?
! Keywords: Financial planning, Asset-Liability management,
Bonds, Yield curve, Stochastic optimization;
SETS:
PERIOD: NEED, BORRMX;
NODE: LEND, BORR, RLEND, RBORR,
PERD, PREV, NWORTH, SHORTF;
ASSET: HOLDMX, HOLDIN;
PXA( PERIOD, ASSET): PAYOUT, PBUY, PSEL;
NXA( NODE, ASSET): HOLD, BUY, SEL;
ENDSETS
DATA:
! Uncertainty is represented by a tree of possible outcome paths;
PERIOD = 1..7; ! There re 7 periods;
! External cash needs by period due to previous commitments;
NEED = -2000 -500 -400 -380 -360 -340 -300;
! Max we can borrow. Cannot borrow(and not repay) at end;
BORRMX = 2000 2000 2000 2000 2000 2000 0;
ASSET= F M D; ! Three investments available;
HOLDMX= 1 1 1; ! Maximum we can hold of each asset;
HOLDIN= 0 0 0; ! Initially we own nothing of anything;
PAYOUT= ! Payout or throw-off ;
0 0 0 ! by asset by period;
-1000 -500 -2000
-1800 1500 -1800
400 1500 1000
1800 1500 1000
1800 200 1000
5500 -1000 6000;
! Price in period t at which we can buy asset i;
PBUY= 3000 2000 2000! Can buy only;
99999 99999 99999 ! in 1st period;
99999 99999 99999
99999 99999 99999
99999 99999 99999
99999 99999 99999
99999 99999 99999;
! Price in period t at which we can sell asset i;
PSEL = -999999; ! Cannot sell these investments once made;
! Below is a description of the tree of all possible outcomes.
Each node corresponds to a particular outcome in a particular period.
A path in the tree represents one possible set of outcomes.
This is a 7 period problem, so every path in the tree has 7
nodes. For example, one possible path is:
1 -> 3 -> 6 -> 12 -> 25 -> 51 -> 102.
Although not essential, the tree used here is a binary tree, i.e.,
all nodes, except for final period nodes, have two successors.
For simplicity, all paths are considered equally likely.
The one period lending rates are based on a yield curve from
2005, fitted with a Black/Derman/Toy stochastic interest rate model.
The borrowing rate is simply set = lending rate + .005
Each node is described by: node number, period, previous or
predecessor node, and the lending and borrowing rate associated with
this outcome;
NODE, PERD, PREV, RLEND, RBORR =
1 1 0 0.0326 0.0376
2 2 1 0.037064518 0.042064518
3 2 1 0.040962628 0.045962628
4 3 2 0.032253001 0.037253001
5 3 2 0.037134275 0.042134275
6 3 3 0.037134275 0.042134275
7 3 3 0.042754296 0.047754296
8 4 4 0.030361098 0.035361098
9 4 4 0.035275441 0.040275441
10 4 5 0.035275441 0.040275441
11 4 5 0.040985234 0.045985234
12 4 6 0.035275441 0.040275441
13 4 6 0.040985234 0.045985234
14 4 7 0.040985234 0.045985234
15 4 7 0.047619232 0.052619232
16 5 8 0.028620814 0.033620814
17 5 8 0.033376231 0.038376231
18 5 9 0.033376231 0.038376231
19 5 9 0.038921772 0.043921772
20 5 10 0.033376231 0.038376231
21 5 10 0.038921772 0.043921772
22 5 11 0.038921772 0.043921772
23 5 11 0.045388718 0.050388718
24 5 12 0.033376231 0.038376231
25 5 12 0.038921772 0.043921772
26 5 13 0.038921772 0.043921772
27 5 13 0.045388718 0.050388718
28 5 14 0.038921772 0.043921772
29 5 14 0.045388718 0.050388718
30 5 15 0.045388718 0.050388718
31 5 15 0.052930163 0.057930163
32 6 16 0.028247864 0.033247864
33 6 16 0.032984564 0.037984564
34 6 17 0.032984564 0.037984564
35 6 17 0.038515529 0.043515529
36 6 18 0.032984564 0.037984564
37 6 18 0.038515529 0.043515529
38 6 19 0.038515529 0.043515529
39 6 19 0.044973946 0.049973946
40 6 20 0.032984564 0.037984564
41 6 20 0.038515529 0.043515529
42 6 21 0.038515529 0.043515529
43 6 21 0.044973946 0.049973946
44 6 22 0.038515529 0.043515529
45 6 22 0.044973946 0.049973946
46 6 23 0.044973946 0.049973946
47 6 23 0.052515332 0.057515332
48 6 24 0.032984564 0.037984564
49 6 24 0.038515529 0.043515529
50 6 25 0.038515529 0.043515529
51 6 25 0.044973946 0.049973946
52 6 26 0.038515529 0.043515529
53 6 26 0.044973946 0.049973946
54 6 27 0.044973946 0.049973946
55 6 27 0.052515332 0.057515332
56 6 28 0.038515529 0.043515529
57 6 28 0.044973946 0.049973946
58 6 29 0.044973946 0.049973946
59 6 29 0.052515332 0.057515332
60 6 30 0.044973946 0.049973946
61 6 30 0.052515332 0.057515332
62 6 31 0.052515332 0.057515332
63 6 31 0.061321284 0.066321284
64 7 32 0.02739986 0.03239986
65 7 32 0.031882856 0.036882856
66 7 33 0.031882856 0.036882856
67 7 33 0.037099334 0.042099334
68 7 34 0.031882856 0.036882856
69 7 34 0.037099334 0.042099334
70 7 35 0.037099334 0.042099334
71 7 35 0.043169299 0.048169299
72 7 36 0.031882856 0.036882856
73 7 36 0.037099334 0.042099334
74 7 37 0.037099334 0.042099334
75 7 37 0.043169299 0.048169299
76 7 38 0.037099334 0.042099334
77 7 38 0.043169299 0.048169299
78 7 39 0.043169299 0.048169299
79 7 39 0.050232395 0.055232395
80 7 40 0.031882856 0.036882856
81 7 40 0.037099334 0.042099334
82 7 41 0.037099334 0.042099334
83 7 41 0.043169299 0.048169299
84 7 42 0.037099334 0.042099334
85 7 42 0.043169299 0.048169299
86 7 43 0.043169299 0.048169299
87 7 43 0.050232395 0.055232395
88 7 44 0.037099334 0.042099334
89 7 44 0.043169299 0.048169299
90 7 45 0.043169299 0.048169299
91 7 45 0.050232395 0.055232395
92 7 46 0.043169299 0.048169299
93 7 46 0.050232395 0.055232395
94 7 47 0.050232395 0.055232395
95 7 47 0.058451111 0.063451111
96 7 48 0.031882856 0.036882856
97 7 48 0.037099334 0.042099334
98 7 49 0.037099334 0.042099334
99 7 49 0.043169299 0.048169299
100 7 50 0.037099334 0.042099334
101 7 50 0.043169299 0.048169299
102 7 51 0.043169299 0.048169299
103 7 51 0.050232395 0.055232395
104 7 52 0.037099334 0.042099334
105 7 52 0.043169299 0.048169299
106 7 53 0.043169299 0.048169299
107 7 53 0.050232395 0.055232395
108 7 54 0.043169299 0.048169299
109 7 54 0.050232395 0.055232395
110 7 55 0.050232395 0.055232395
111 7 55 0.058451111 0.063451111
112 7 56 0.037099334 0.042099334
113 7 56 0.043169299 0.048169299
114 7 57 0.043169299 0.048169299
115 7 57 0.050232395 0.055232395
116 7 58 0.043169299 0.048169299
117 7 58 0.050232395 0.055232395
118 7 59 0.050232395 0.055232395
119 7 59 0.058451111 0.063451111
120 7 60 0.043169299 0.048169299
121 7 60 0.050232395 0.055232395
122 7 61 0.050232395 0.055232395
123 7 61 0.058451111 0.063451111
124 7 62 0.050232395 0.055232395
125 7 62 0.058451111 0.063451111
126 7 63 0.058451111 0.063451111
127 7 63 0.068014522 0.073014522;
! Measuring risk. We consider any final net worth
less than TARGET as bad. Our measure of risk is
the expected amount by which we fall short of the
target, i.e., ESHORTF = E[max(0, TARGET - NWORTH)];
TARGET = 7630;
! MXSHORT = upper limit we place on ESHORTF;
MXSHORT = 99999;
! Punchline: We buy a fair amount of F and M, but nothing of D;
ENDDATA
! Variables:
BUY(t,i) = units bought at beginning of period t of asset i,
SEL(t,i) = units sold at beginning of t of asset i,
HOLD(t,i) = units held at end of t of asset i,
BORR(t) = amount borrowed at beginning of t,
LEND(t) = amount lent at the beginning of t
NWORTH(t) = net worth at node t in final period
SHORTF(t) = max(0,TARGET-NWORTH(t));
! Get index of final period;
TF = @SIZE( PERIOD);
! Get number final period nodes;
NF = @SUM( NODE(n)| PERD(n) #EQ# TF: 1);
! Maximize expected wealth in final period;
MAX = @SUM( NODE( n)| PERD(n) #EQ# TF: NWORTH(n))/ NF;
@FOR( NODE(n) | PERD(N) #EQ# TF:
NWORTH(n)=
@SUM( ASSET(i):
PAYOUT(PERD(n),i)*HOLD(n,i)
- PBUY(TF,i) *BUY(n,i)
+ PSEL(TF,i) *SEL(n,i))
+ (1 + RLEND(PREV(n)))*LEND(PREV(n))
- (1 + RBORR(PREV(n)))*BORR(PREV(n)) - NEED(TF);
);
! First Period;
! Individual asset balances;
@FOR( ASSET(i):
HOLDIN(i) + BUY(1,i) = SEL(1,i) + HOLD(1,i);
HOLD(1,i) <= HOLDMX(i)
);
! Cash balance constraint;
! Cash inflows from investments + borrowings in 1
+ asset sales in 1 = asset purchases in 1 + loans made in 1
+ external liabilities in 1;
@SUM( ASSET(i): PAYOUT(1,i)*HOLDIN(i) + PSEL(1,i)*SEL(1,i)) + BORR(1)
= @SUM( ASSET(i):PBUY(1,i)*BUY(1,i)) + LEND(1) + NEED(1);
BORR(1) <= BORRMX(1); ! Borrowing limit;
! Same applies for subsequent periods;
@FOR( NODE( t) | t #GT# 1:
! For each asset type i: Ending inventory in t-1 + purchases
= sales + ending inventory in t;
@FOR( ASSET( i):
HOLD(PREV(t),i) + BUY(t,i) = SEL(t,i) + HOLD(t,i);
HOLD(t,i) <= HOLDMX(i)
);
! Cash inflows from investments + repayments from loans in t-1
+ borrowings in t + asset sales in t
= asset purchases in t + loans made in t
+ repayment of borrowings in previous period
+ external liabilities in t;
@SUM( ASSET(i): PAYOUT(PERD(t),i)*HOLD(t,i) + PSEL(PERD(t),i)*SEL(t,i))
+(1+RLEND(PREV(t)))*LEND(PREV(t)) + BORR(t) =
@SUM( ASSET(i): PBUY(PERD(t),i)*BUY(t,i)) + (1 + RBORR(PREV(t)))*BORR(PREV(t)) + LEND(t)
+ NEED(PERD(t));
BORR(t) <= BORRMX(PERD(t)); ! Borrowing limit;
);
! Here is the risk computations.
Compute amount short each final outcome;
@FOR( NODE(t)| PERD(t) #EQ# TF:
SHORTF(t) >= TARGET - NWORTH(t);
);
! Compute average, or expected, amount short;
ESHORTF = @SUM( NODE(t) | PERD(t) #EQ# TF: SHORTF(t))/NF;
! Constrain it;
ESHORTF <= MXSHORT;
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