Interest Rate Risk Management

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Interest Rate Risk Management Elias S. W.
Shiu Department of Statistics Actuarial
Science The University of Iowa Iowa City,
Iowa U.S.A.
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Frank M. Redington, F.I.A. Review of the
Principles of Life-Office Valuations Journal of
the Institute of Actuaries Volume 78 (1952),
286-315
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(No Transcript)
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Last sentence in the first paragraph The reader
will perhaps be less disappointed if he is warned
in advance that he is to be taken on a ramble
through the actuarial countryside
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Last sentence in the first paragraph The reader
will perhaps be less disappointed if he is warned
in advance that he is to be taken on a ramble
through the actuarial countryside and that any
interest lies in the journey rather than the
destination.
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For a block of business and for t gt 0, let At
asset cash flow to occur at time t (
investment income capital maturities)
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For a block of business and for t gt 0, let At
asset cash flow to occur at time t (
investment income capital maturities)For
example, if the companys assets consist of one
2-year bond with face value of 100 and
semi-annual coupons at a nominal annual rate of 6
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For a block of business and for t gt 0, let At
asset cash flow to occur at time t (
investment income capital maturities)For
example, if the companys assets consist of one
2-year bond with face value of 100 and
semi-annual coupons at a nominal annual rate of
6 and one 3-year bond at 8
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For a block of business and for t gt 0, let At
asset cash flow to occur at time t (
investment income capital maturities)For
example, if the companys assets consist of one
2-year bond with face value of 100 and
semi-annual coupons at a nominal annual rate of
6 and one 3-year bond at 8, then A0.5 ½
(68) 7
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For a block of business and for t gt 0, let At
asset cash flow to occur at time t (
investment income capital maturities)For
example, if the companys assets consist of one
2-year bond with face value of 100 and
semi-annual coupons at a nominal annual rate of
6 and one 3-year bond at 8, then A0.5 ½
(68) 7, A1 7, A1.5 7
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For a block of business and for t gt 0, let At
asset cash flow to occur at time t (
investment income capital maturities)For
example, if the companys assets consist of one
2-year bond with face value of 100 and
semi-annual coupons at a nominal annual rate of
6 and one 3-year bond at 8, then A0.5 ½
(68) 7, A1 7, A1.5 7, A2 1034
107
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For a block of business and for t gt 0, let At
asset cash flow to occur at time t (
investment income capital maturities)For
example, if the companys assets consist of one
2-year bond with face value of 100 and
semi-annual coupons at a nominal annual rate of
6 and one 3-year bond at 8, then A0.5 ½
(68) 7, A1 7, A1.5 7, A2 1034
107, A2.5 4, A3 104
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For a block of business and for t gt 0, let At
asset cash flow to occur at time t (
investment income capital maturities)Lt
liability cash flow to occur at time t (
policy claims policy surrenders expenses ?
premium income)
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Let A Asset Value at time 0. Then,
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Let A Asset Value at time 0. Then, But
yield curves are not (necessarily) flat.
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Let A Asset Value at time 0. Then, But
yield curves are not (necessarily)
flat. Generalize Then
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Similarly, let L Liability Value at time 0.
Then,
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Similarly, let L Liability Value at time 0.
Then, Surplus (Net Worth or Equity)
Asset Value - Liability Value A - L
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Similarly, let L Liability Value at time 0.
Then, Surplus (Net Worth or Equity)
Asset Value - Liability Value A -
L Instantaneous interest rate shock How
does the surplus change?
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Instantaneous interest rate shock means
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Instantaneous interest rate shock means Ass
ume that the asset cash flows At and liability
cash flows Lt do not change as interest rates
fluctuate.
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Instantaneous interest rate shock means Ass
ume that the asset cash flows At and liability
cash flows Lt do not change as interest rates
fluctuate. That is, there are no embedded
interest-sensitive options.
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Instantaneous interest rate shock means Ass
ume that the asset cash flows At and liability
cash flows Lt do not change as interest rates
fluctuate. That is, there are no embedded
interest-sensitive options. The more general
case of interest-sensitive cash flows is a much
harder problem.
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Changed asset value is
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Changed asset value is Changed liability
value is
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Changed asset value is Changed liability
value is Changed surplus is S A -
L
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Question How will the surplus not decrease?
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Question How will the surplus not
decrease? A - L ? A - L?
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Question How will the surplus not
decrease? A - L ? A - L? Define two
(discrete) random variables X and Y Pr(X t)

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Question How will the surplus not
decrease? A - L ? A - L? Define two
(discrete) random variables X and Y Pr(X t)
Pr(Y t) (The cash flows are assumed
to be non-negative.)
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Define the function f(t)
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Define the function f(t) Then
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Define the function f(t) Then
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Define the function f(t) Then
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Define the function f(t) Then
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Similarly, L L Ef(Y).
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Similarly, L L Ef(Y). Original Surplus
is S A - L.
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Similarly, L L Ef(Y). Original Surplus
is S A - L. Changed Surplus is S A
- L AEf(X) - LEf(Y).
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Similarly, L L Ef(Y). Original Surplus
is S A - L. Changed Surplus is S A
- L AEf(X) - LEf(Y). Now,
assume A L, i.e., assume S 0.
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Similarly, L L Ef(Y). Original Surplus
is S A - L. Changed Surplus is S A
- L AEf(X) - LEf(Y). Now,
assume A L, i.e., assume S 0. Then S
AEf(X) - Ef(Y)
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Similarly, L L Ef(Y). Original Surplus
is S A - L. Changed Surplus is S A
- L AEf(X) - LEf(Y). Now,
assume A L, i.e., assume S 0. Then S
AEf(X) - Ef(Y), and S ? 0 if and only
if
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Similarly, L L Ef(Y). Original Surplus
is S A - L. Changed Surplus is S A
- L AEf(X) - LEf(Y). Now,
assume A L, i.e., assume S 0. Then S
AEf(X) - Ef(Y), and S ? 0 if and only
if Ef(X) ? Ef(Y).
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f(t)
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f(t) In the Redington (1952) model, it
i and i ?, where ? is a positive or
negative constant.
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f(t) In the Redington (1952) model, it
i and i ?, where ? is a positive or
negative constant. Thus f(t) is an exponential
function, which is a convex function.
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f(t) In the Redington (1952) model, it
i and i ?, where ? is a positive or
negative constant. Thus f(t) is an exponential
function, which is a convex function. In the
Fisher Weil (J. of Business 1971) model,
where c is a positive constant.
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f(t) In the Redington (1952) model, it
i and i ?, where ? is a positive or
negative constant. Thus f(t) is an exponential
function, which is a convex function. In the
Fisher Weil (J. of Business 1971) model,
where c is a positive constant. That is,
f(t) ct
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f(t) In the Redington (1952) model, it
i and i ?, where ? is a positive or
negative constant. Thus f(t) is an exponential
function, which is a convex function. In the
Fisher Weil (J. of Business 1971) model,
where c is a positive constant. That is,
f(t) ct, which is also a convex function.
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Jensens Inequality
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Jensens Inequality Ef(X) ? f(EX)
for all convex functions f.
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Jensens Inequality Ef(X) ? f(EX)
for all convex functions f. If Y ? EX,
then Ef(X) ? f(Y)
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Jensens Inequality Ef(X) ? f(EX)
for all convex functions f. If Y ? EX,
then Ef(X) ? f(Y) Ef(Y) for all
convex functions f.
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Jensens Inequality Ef(X) ? f(EX)
for all convex functions f. If Y ? EX,
then Ef(X) ? f(Y) Ef(Y) for all
convex functions f. In the context of assets and
liabilities, Y ? EX means that there is a
single liability cash flow
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Jensens Inequality Ef(X) ? f(EX)
for all convex functions f. If Y ? EX,
then Ef(X) ? f(Y) Ef(Y) for all
convex functions f. In the context of assets and
liabilities, Y ? EX means that there is a
single liability cash flow, and it is to occur at
time t EX.
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The expectation EX is
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The expectation EX is
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The expectation EX is This is a
weighted average of t
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The expectation EX is This is a
weighted average of t (with weights being
cash-flow present values)
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The expectation EX is This is a
weighted average of t (with weights being
cash-flow present values) and it is called
duration.
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Thus, for the case where there is only one
liability cash flow,
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Thus, for the case where there is only one
liability cash flow, Jensens Inequality implies
that if the asset cash flows have the same
present value as the liabilitys present value
(i.e., A L)
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Thus, for the case where there is only one
liability cash flow, Jensens Inequality implies
that if the asset cash flows have the same
present value as the liabilitys present value
(i.e., A L) and have a duration which is when
the liability cash flow is to occur (i.e., EX ?
Y),
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Thus, for the case where there is only one
liability cash flow, Jensens Inequality implies
that if the asset cash flows have the same
present value as the liabilitys present value
(i.e., A L) and have a duration which is when
the liability cash flow is to occur (i.e., EX ?
Y), then for any change in interest rates such
that f(t) is a convex function
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Thus, for the case where there is only one
liability cash flow, Jensens Inequality implies
that if the asset cash flows have the same
present value as the liabilitys present value
(i.e., A L) and have a duration which is when
the liability cash flow is to occur (i.e., EX ?
Y), then for any change in interest rates such
that f(t) is a convex function, we have
S ? 0 (i.e., the surplus will not decrease).
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Thus, for the case where there is only one
liability cash flow, Jensens Inequality implies
that if the asset cash flows have the same
present value as the liabilitys present value
(i.e., A L) and have a duration which is when
the liability cash flow is to occur (i.e., EX ?
Y), then for any change in interest rates such
that f(t) is a convex function, we have
S ? 0 (i.e., the surplus will not decrease).
But if f is concave, .
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We call EX the asset duration
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We call EX the asset duration, and
EY the liability duration.
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We call EX the asset duration, and
EY the liability duration. Remainder
of this talk The more general case of multiple
liability cash flows
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Theorem (J. Karamata 1932) Ef(X) ?
Ef(Y)
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Theorem (J. Karamata 1932) Ef(X) ?
Ef(Y) for all convex f if and only if
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Theorem (J. Karamata 1932) Ef(X) ?
Ef(Y) for all convex f if and only
if (i) EX EY
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Theorem (J. Karamata 1932) Ef(X) ?
Ef(Y) for all convex f if and only
if (i) EX EY and (ii) EX c ? EY
c for all c ? ?.
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Theorem (J. Karamata 1932) Ef(X) ?
Ef(Y) for all convex f if and only
if (i) EX EY and (ii) EX c ? EY
c for all c ? ?. Condition (i), EX
EY, means matching asset duration with
liability duration.
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Theorem (J. Karamata 1932) Ef(X) ?
Ef(Y) for all convex f if and only
if (i) EX EY and (ii) EX c ? EY
c for all c ? ?. Condition (i), EX
EY, means matching asset duration with
liability duration. Fong and Vasicek (1983)
called condition (ii) the Mean Absolute Deviation
(MAD) constraint. The condition implies that the
asset cash flows have to be more dispersed than
the liability cash flows.
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Theorem (J. Karamata 1932) Ef(X) ?
Ef(Y) for all convex f if and only
if (i) EX EY and (ii) EX c ? EY
c for all c ? ?. Remark Write x
max(x, 0). Because x (x
x)/2, condition (ii) can be replaced by E(X
c) ? E(Y c) for all c ? ?.
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For linear programming implementation, the
condition E(X c) ? E(Y c) for all
c ? ? seems easier to use than the condition
EX c ? EY c for all c ? ?.
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For linear programming implementation, the
condition E(X c) ? E(Y c) for all
c ? ? seems easier to use than the condition
EX c ? EY c for all c ? ?. Also, it
is not necessary to check the condition for all
real numbers c.
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For linear programming implementation, the
condition E(X c) ? E(Y c) for all
c ? ? seems easier to use than the condition
EX c ? EY c for all c ? ?. Also, it
is not necessary to check the condition for all
real numbers c. Just check it at those times t
when there is a cash flow.
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Theorem (David Blackwell 1951) Let G and H be two
probability distribution functions.
Then, for all convex functions f if and
only if
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Theorem (David Blackwell 1951) Let G and H be two
probability distribution functions.
Then, for all convex functions f if and
only if there exist random variables X and Y
defined on the same probability space with
G(t) FX(t) and H(t) FY(t) and
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Theorem (David Blackwell 1951) Let G and H be two
probability distribution functions.
Then, for all convex functions f if and
only if there exist random variables X and Y
defined on the same probability space with
G(t) FX(t) and H(t) FY(t) and
EXY Y.
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Theorem (David Blackwell 1951) Let G and H be two
probability distribution functions.
Then, for all convex functions f if and
only if there exist random variables X and Y
defined on the same probability space with
G(t) FX(t) and H(t) FY(t) and
EXY Y. Note Taking expectation of the
last equation yields EX EY.
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With G(t) FX(t) and H(t) FY(t), the
inequality is the same as Ef(X) ?
Ef(Y).
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With G(t) FX(t) and H(t) FY(t), the
inequality is the same as Ef(X) ?
Ef(Y). Proving the if direction in
Blackwells Theorem is easy Assume EXY Y.
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With G(t) FX(t) and H(t) FY(t), the
inequality is the same as Ef(X) ?
Ef(Y). Proving the if direction in
Blackwells Theorem is easy Assume EXY Y.
Let f be a convex function.
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With G(t) FX(t) and H(t) FY(t), the
inequality is the same as Ef(X) ?
Ef(Y). Proving the if direction in
Blackwells Theorem is easy Assume EXY Y.
Let f be a convex function. By Jensens
Inequality, Ef(X)Y ? f(EXY)
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With G(t) FX(t) and H(t) FY(t), the
inequality is the same as Ef(X) ?
Ef(Y). Proving the if direction in
Blackwells Theorem is easy Assume EXY Y.
Let f be a convex function. By Jensens
Inequality, Ef(X)Y ? f(EXY)
f(Y).
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With G(t) FX(t) and H(t) FY(t), the
inequality is the same as Ef(X) ?
Ef(Y). Proving the if direction in
Blackwells Theorem is easy Assume EXY Y.
Let f be a convex function. By Jensens
Inequality, Ef(X)Y ? f(EXY)
f(Y). Then, taking expectations yields Ef(X) ?
Ef(Y).
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With G(t) FX(t) and H(t) FY(t), the
inequality is the same as Ef(X) ?
Ef(Y). Proving the if direction in
Blackwells Theorem is easy Assume EXY Y.
Let f be a convex function. By Jensens
Inequality, Ef(X)Y ? f(EXY)
f(Y). Then, taking expectations yields Ef(X) ?
Ef(Y). The proof of the only if direction is
hard.
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ALM Application of Blackwells Theorem Assume
that there are m asset cash flows to occur at
time t t1, t2, , tm.
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ALM Application of Blackwells Theorem Assume
that there are m asset cash flows to occur at
time t t1, t2, , tm. For simplicity,
write ai i 1, 2, , m.
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ALM Application of Blackwells Theorem Assume
that there are m asset cash flows to occur at
time t t1, t2, , tm. For simplicity,
write ai i 1, 2, , m. Assume that
there are n liability cash flows to occur at time
t s1, s2, , sn.
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ALM Application of Blackwells Theorem Assume
that there are m asset cash flows to occur at
time t t1, t2, , tm. For simplicity,
write ai i 1, 2, , m. Assume that
there are n liability cash flows to occur at time
t s1, s2, , sn. Define lj j 1,
2, , n.
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ALM Application of Blackwells Theorem Assume
that there are m asset cash flows to occur at
time t t1, t2, , tm. For simplicity,
write ai i 1, 2, , m. Assume that
there are n liability cash flows to occur at time
t s1, s2, , sn. Define lj j 1,
2, , n. Then, A ? L for all convex f if and
only if
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there is a nonnegative m?n matrix B, with the sum
of the entries in each of its m rows equal to 1,

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there is a nonnegative m?n matrix B, with the sum
of the entries in each of its m rows equal to 1,
such that (a1, a2, , am)B (l1, l2, ,
ln) and
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there is a nonnegative m?n matrix B, with the sum
of the entries in each of its m rows equal to 1,
such that (a1, a2, , am)B (l1, l2, ,
ln) and (a1t1, a2t2, , amtm)B (l1s1,
l2s2, , lnsn).
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there is a nonnegative m?n matrix B, with the sum
of the entries in each of its m rows equal to 1,
such that (a1, a2, , am)B (l1, l2, ,
ln) and (a1t1, a2t2, , amtm)B (l1s1,
l2s2, , lnsn). Interpretation The n columns
of the matrix B partition the m asset cash flows
into n streams
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there is a nonnegative m?n matrix B, with the sum
of the entries in each of its m rows equal to 1,
such that (a1, a2, , am)B (l1, l2, ,
ln) and (a1t1, a2t2, , amtm)B (l1s1,
l2s2, , lnsn). Interpretation The n columns
of the matrix B partition the m asset cash flows
into n streams, each with the same
present value and same duration as one of the n
liability cash flows .
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there is a nonnegative m?n matrix B, with the sum
of the entries in each of its m rows equal to 1,
such that (a1, a2, , am)B (l1, l2, ,
ln) and (a1t1, a2t2, , amtm)B (l1s1,
l2s2, , lnsn). Interpretation The n columns
of the matrix B partition the m asset cash flows
into n streams, each with the same
present value and same duration as one of the n
liability cash flows . What is the
matrix B?
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For m 3 (three asset cash flows) and n 2 (two
liability cash flows), B is
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For m 3 (three asset cash flows) and n 2 (two
liability cash flows), B is
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For m 3 (three asset cash flows) and n 2 (two
liability cash flows), B is The equation
(a1, a2, a3)B (l1, l2) follows
immediately from
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For m 3 (three asset cash flows) and n 2 (two
liability cash flows), B is The equation
(a1, a2, a3)B (l1, l2) follows
immediately from (Pr(Xt1), Pr(Xt2), Pr(Xt3))B
(Pr(Ys1), Pr(Ys2))
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For m 3 (three asset cash flows) and n 2 (two
liability cash flows), B is The equation
(a1, a2, a3)B (l1, l2) follows
immediately from (Pr(Xt1), Pr(Xt2), Pr(Xt3))B
(Pr(Ys1), Pr(Ys2)) That (a1t1, a2t2,
a3t3)B (l1s1, l2s2)
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For m 3 (three asset cash flows) and n 2 (two
liability cash flows), B is The equation
(a1, a2, a3)B (l1, l2) follows
immediately from (Pr(Xt1), Pr(Xt2), Pr(Xt3))B
(Pr(Ys1), Pr(Ys2)) That (a1t1, a2t2,
a3t3)B (l1s1, l2s2) requires the
condition EXY Y.
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Surplus S A - L
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Surplus S A - L Changed Surplus S

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Surplus S A - L Changed Surplus S
Is there any explicit formula for S -
S ?
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Surplus S A - L Changed Surplus S
Is there any explicit formula for S -
S ? Answer For some asset liability
cashflows, S - S A f"(??Var(X) Var(Y)/2
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Let f be an arbitrary twice differentiable
function, not necessarily convex, f(t) f(0)
f'(0)t (t s) f"(s)ds
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Let f be an arbitrary twice differentiable
function, not necessarily convex, f(t) f(0)
f'(0)t (t s) f"(s)ds f(0)
f'(0)t (t s) f"(s)ds.
114
Let f be an arbitrary twice differentiable
function, not necessarily convex, f(t) f(0)
f'(0)t (t s) f"(s)ds f(0)
f'(0)t (t s) f"(s)ds. Thus, Ef(X)
- Ef(Y) f'(0)EX - EY
E(X s) - E(Y s)f"(s)ds
115
Let f be an arbitrary twice differentiable
function, not necessarily convex, f(t) f(0)
f'(0)t (t s) f"(s)ds f(0)
f'(0)t (t s) f"(s)ds. Thus, Ef(X)
- Ef(Y) f'(0)EX - EY
E(X s) - E(Y s)f"(s)ds 0
E(X s) - E(Y s)f"(s)ds.
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If E(X s) - E(Y s) has the same sign
for all s
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Theorem (J. Karamata 1932) Ef(X) ?
Ef(Y) for all convex f if and only
if (i) EX EY and (ii) E(X s) ?
E(Y s) for all s ? ?.
118
If E(X s) - E(Y s) has the same sign
for all s
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If E(X s) - E(Y s) has the same sign
for all s, then it follows from the Mean Value
Theorem for Integrals that
120
If E(X s) - E(Y s) has the same sign
for all s, then it follows from the Mean Value
Theorem for Integrals that E(X s) - E(Y
s)f"(s)ds
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If E(X s) - E(Y s) has the same sign
for all s, then it follows from the Mean Value
Theorem for Integrals that E(X s) - E(Y
s)f"(s)ds f"(z) E(X s) - E(Y
s)ds
122
If E(X s) - E(Y s) has the same sign
for all s, then it follows from the Mean Value
Theorem for Integrals that E(X s) - E(Y
s)f"(s)ds f"(z) E(X s) - E(Y
s)ds f"(z)EX2 - EY2/2
123
If E(X s) - E(Y s) has the same sign
for all s, then it follows from the Mean Value
Theorem for Integrals that E(X s) - E(Y
s)f"(s)ds f"(z) E(X s) - E(Y
s)ds f"(z)EX2 - EY2/2 f"(z
)Var(X) - Var(Y)/2 because EX EY
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Summary
125
Summary
126
Summary
127
The changed surplus is S AEf(X) -
LEf(Y).
128
The changed surplus is S AEf(X) -
LEf(Y). If A L, then S 0 and the change
in surplus is S - S S
129
The changed surplus is S AEf(X) -
LEf(Y). If A L, then S 0 and the change
in surplus is S - S S AEf(X) -
Ef(Y).
130
The changed surplus is S AEf(X) -
LEf(Y). If A L, then S 0 and the change
in surplus is S - S S AEf(X) -
Ef(Y). If the asset and liability cash flows
are such that Ef(X) ? Ef(Y) for all
131
The changed surplus is S AEf(X) -
LEf(Y). If A L, then S 0 and the change
in surplus is S - S S AEf(X) -
Ef(Y). If the asset and liability cash flows
are such that Ef(X) ? Ef(Y) for all convex
functions f or for all concave functions f,
132
The changed surplus is S AEf(X) -
LEf(Y). If A L, then S 0 and the change
in surplus is S - S S AEf(X) -
Ef(Y). If the asset and liability cash flows
are such that Ef(X) ? Ef(Y) for all convex
functions f or for all concave functions f, then
for each twice differentiable function
f, Ef(X) - Ef(Y) f"(??Var(X)
Var(Y)/2, where z depends on X, Y and f.
133
The changed surplus is S AEf(X) -
LEf(Y). If A L, then S 0 and the change
in surplus is S - S S AEf(X) -
Ef(Y). If the asset and liability cash flows
are such that Ef(X) ? Ef(Y) for all convex
functions f or for all concave functions f, then
for each twice differentiable function
f, Ef(X) - Ef(Y) f"(??Var(X)
Var(Y)/2, where z depends on X, Y and f. Hence,
the change in surplus is S A f"(??Var(X)
Var(Y)/2.
134
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