Download Report

ACTSC 445: Asset-Liability Management
Department of Statistics and Actuarial Science, University of Waterloo
Unit 6 – Immunization
References (recommended readings): Chap. 3 of Financial Economics (on reserve at the library: call
number HG174 .F496 1998).
What is immunization?
• Redington (1952): Immunization implies the investment of assets in such a way that existing
business is immune to a general change in the rate of interest.
• Fisher-Weil (1971): A portfolio of investment is immunized for a holding period if its value at
the end of the holding period, regardless of the course of rates during the holding period, must be
at least as it would have been had the interest rate function been constant throughout the holding
period.
Implication: If the realized return on an investment in bonds is sure to be at least as large as the
appropriately computed yield to the horizon, then that investment is immunized.
• An immunization strategy is a risk management technique designed to ensure that for any
small change in a specified parameter, a portfolio of debt instruments (e.g., T-bills, bonds, GICs
etc) will cover a liability (or liabilities) coming due at a future date (or over a period in the future).
It is a passive management technique because it takes prices as given and then tries to control
the risk appropriately. (By contrast, active management techniques try to exploit changes in (1)
the level of interest rates, (2) the shape of the yield curve (3) yield spreads, by using interest rate
forecasts and identification of mispriced bonds)
⇒ asset allocation problem (i.e., must choose assets that will produce an immunized portfolio)
Single-liability case
We’ll start with the case where there is only one liability in the portfolio, with corresponding cash flow
of Lt at some time t.
The goal is to choose an asset cash flow sequence {At , t > 0} that will, along with Lt , produce an
immunized portfolio. Let’s start with an example.
Example I: Suppose an insurance company faces a liability obligation of $1 million in 5 years. The
available market instruments are: 3-year, 5-year and 7-year zero-coupon bonds, each yielding 6% annual
effective rate.
• Portfolio A: Invest $747,258.17 in the 5-year zero coupon bond
1
• Portfolio B: Invest the same amount (i.e. $747,258.17) in a 3-year zero coupon bond. The
maturity value at t = 3 is $889,996.44.
• Portfolio C: Invest $747,258.17 in a 7-year zero coupon bond. The maturity value at t = 7 is
$1,123,600.00.
If the yields remain unchanged, then the 3 portfolios have the same value of $1 000 000 at time 5.
To verify if these portfolios are immunized or not, we need to look at what happens if, immediately after
the portfolio is acquired, the yield changes instantaneously to yˆ and remains constant at that level.
First, note that for portfolio A, this change has no impact: its value at time 5 is still $1 000 000. But
this is not true for portfolios B and C, as Tables 1 and 2 show.
Table 1: Value of Portfolio B for different yields
yˆ (%)
4.00
5.00
5.90
6.00
6.10
7.00
8.00
Value of Portfolio B
at time 0
at time 5
791203.5944
962620.1495
768812.3874
981221.0751
749377.0511
998114.0975
747258.1729 1000000.0000
745147.2753
1001887.6824
726502.2044
1018956.9242
706507.8685
1038091.8476
Capital Gain
at time 0
−43945.4215
−21554.2146
−2118.8782
0.0000
2110.8975
20755.9684
40750.3044
Implied
Yield (%)
5.20
5.60
5.96
6.00
6.04
6.40
6.80
So for portfolio B, if the yields go up, then we realize a gain at time 5, because we can reinvest the
proceeds obtained at time 3 at a high yield. But if the rates drop, then we realize a loss at time 5. The
problem here is the reinvestment risk.
Table 2: Value of Portfolio C for different yields
yˆ (%)
4.00
5.00
5.90
6.00
6.10
7.00
8.00
Value of Portfolio C
in year 0
at time 5
853843.6549
1038831.3609
798521.5425
1019138.3220
752211.5711
1001889.4658
747258.1729 1000000.0000
742342.0181
998115.8742
699721.6100
981395.7551
655609.8081
963305.8985
Capital Gain
at time 0
106585.4820
51263.3697
4953.3982
0.0000
−4916.1548
−47536.5629
−91648.3647
Implied
Yield (%)
6.81
6.40
6.04
6.00
5.96
5.60
5.21
The situation here is opposite from what we face with Portfolio B: if the rates drop, then we can sell
the 7-year zero bond at a higher price at time 5, which results in a gain. But a yield increase produces
a loss. The problem here is the interest rate or price risk.
2
Observations from Example I
• With a single liability, the best immunization strategy is the one for which the asset cash flow
coincides with the liability cash flow
• When asset cash flows occur prior to (or after) the liability cash flow, the portfolio is subject
to reinvestment risk (or market/interest rate/price risk).
A valid question is: could we construct a portfolio containing cash flows occuring before and after the
liability due date that could be immunized? Motivation:
• Any initial capital loss may be offset in time by greater returns from reinvestment.
• Similarly, any initial capital gain may be offset in time by lower returns from reinvestment.
• Does there exist an “optimum” trade-off? I.e., a way to construct a portfolio like this that
maximizes (in some sense) the gain?
The following example studies this idea.
Example II: Consider Portfolio D, which consists in an investment of $373 629.0864 in 3-year zerocoupon bonds, and $373 629.0864 in 7-year zero-coupon bonds. Their maturity values are, respectively,
444,998.22 and 561,800.00. Note that the Macaulay duration of this portfolio is 5.
If the yields remain unchanged, then at t = 5 we have 373, 629.0864 × 2 × (1.06)3 = 1 000 000.
If the rates change, then we get the following results:
yˆ (%)
4.0
5.0
5.9
6.0
6.1
7.0
8.0
Value of Portfolio D
at time 0
at time 5
822523.6247
1000725.7552
783666.9650
1000179.6986
750794.3111
1000001.7817
747258.1729 1000000.0000
743744.6467
1000001.7783
713111.9072
1000176.3396
681058.8383
1000698.8731
Capital Gain
at time 0
75265.4518
36408.7921
3536.1382
0.0000
-3513.5262
-34146.2657
-66199.3346
Implied
Yield (%)
6.01538
6.00381
6.00004
6.00000
6.00004
6.00374
6.01481
Hence with this portfolio, a gain is realized at time 5 for all alternative yields yˆ considered...
Note that at time 0, there is a capital loss for portfolio D. More generally, we can look at the value of
this portfolio at time t if the initial yield goes from 6% to yˆ. That is, we can consider the value
Vt = 444 998.22(1 + yˆ)−(3−t) + 561 800.00(1 + yˆ)−(7−t)
for t = 1, . . . , 10 and different yˆ’s.
3
t
0
1
2
3
4
5
6
7
8
9
10
if rate drops
4.00%
5.50%
5.90%
822524 765169 750794
855425 807253 795091
889642 851652 842002
925227 898493 891680
962236 947910 944289
1000726 1000045 1000002
1040755 1055047 1059002
1082385 1113075 1121483
1125680 1174294 1187650
1170708 1238880 1257722
1217536 1307018 1331927
y∗
6.00%
747258
792094
839619
889996
943396
1000000
1060000
1123600
1191016
1262477
1338226
if
6.10%
743745
789113
837249
888321
942509
1000002
1061002
1125723
1194392
1267250
1344552
rate rises
6.50%
8.00%
729913 681059
777358 735544
827886 794387
881698 857938
939009 926573
1000044 1000699
1065047 1080755
1134275 1167215
1208003 1260592
1286523 1361440
1370147 1470355
Equivalently, we can look at the corresponding implied yield for the portfolio, which is the value i such
that
747, 258.17 = Vt (1 + i)−t .
t
1
2
3
4
5
6
7
8
9
10
if rate drops
4.00% 5.50% 5.90%
14.48
8.03
6.40
9.11
6.76
6.15
7.38
6.34
6.07
6.53
6.13
6.03
6.02
6.00
6.00
5.68
5.92
5.98
5.44
5.86
5.97
5.26
5.81
5.96
5.11
5.78
5.96
5.00
5.75
5.95
y∗
6.00%
6.00
6.00
6.00
6.00
6.00
6.00
6.00
6.00
6.00
6.00
if rate rises
6.10% 6.50% 8.00%
5.60
4.03 −1.57
5.85
5.26
3.11
5.93
5.67
4.71
5.98
5.88
5.52
6.00
6.00
6.01
6.02
6.08
6.34
6.03
6.14
6.58
6.04
6.19
6.75
6.04
6.22
6.89
6.05
6.25
7.00
Observations from Examples I and II
• Basic risk exposures: reinvestment risk and price risk
• The existence and extent of either risk depends on the nature of the security as well as on the
relative length of the period over which its return and risk are measured.
• Over a period of time in which both risks are in effect, there are some obvious offsetting tendencies.
• We saw in Unit 5 that the degree of capital loss/gain from a change in yields (i.e. price risk)
depends on the duration of the securities held. Example II suggests that the time required to
offset these capital gain/loss from reinvestment of the cash flows also depends on the duration of
the securities held.
4
Target Date Immunization
Here we generalize the ideas examined in Example II. Let Vk (y) be the value of a portfolio of securities
at time k (measured in years) for a given ytm y (assume annual effective rate). Suppose the current
ytm is y ∗ so that the value of the portfolio initially acquired is V0 (y ∗ ).
• If there is no change in the ytm, the investment grows to V0 (y ∗ )(1 + y ∗ )k = Vk (y ∗ ) after k years.
• If there is an instantaneously shift in yield from y ∗ to yˆ, then the year-k portfolio value becomes:
Vk (ˆ
y ) = V0 (ˆ
y )(1 + yˆ)k
• If the investment horizon is T years, then the realized return exceeds the initial yield y ∗ as long
as VT (ˆ
y ) ≥ VT (y ∗ ).
The following result proves that if the portfolio is constructed so that its duration equals the investment
horizon, then the price risk and reinvestment risk cancel out and the realized value of return is at least
y ∗ . The resulting approach is called target-date immunization.
Theorem: For a portfolio initially constructed at ytm y ∗ and with (Macaulay) duration D, then for
any yˆ
VD (ˆ
y ) ≥ VD (y ∗ ).
Remarks:
• When the investment horizon equals the portfolio duration, the price risk and reinvestment risk
cancel out.
• The realized rate of return can never fall below its initial yield.
Proof: First, we look at the behavior of Vk (y) as a function of y: we have that
X
Vk (y) =
At (1 + y)k−t
t
and therefore
dVk (y) X
=
(k − t)At (1 + y)k−t−1
dy
t
and
d2 Vk (y) X
=
(k − t)(k − t − 1)At (1 + y)k−t−2 .
dy 2
t
Note that if t < k − 1 or t > k, then (k − t)(k − t − 1) > 0, and if t = k − 1 or t = k, then
(k − t)(k − t − 1) = 0. Therefore dVk2 (y)/dy 2 ≥ 0, with a strict inequality as long as the cash flows
are not concentrated at time t = k − 1 or t = k. Assuming this is not the case (i.e., assuming that
dVk2 (y)/dy 2 > 0), we can set dVk (y)/dy to 0 to find at which y does the function Vk (y) is minimized.
Doing this, we get
P
tAt (1 + y)−t
dVk (y) X
k−t−1
=
(k − t)At (1 + y)
= 0 ⇔ k = Pt
.
−t
dy
t At (1 + y)
t
5
Trying to solve for y is hard, but notice that if y = y ∗ , then setting k = D makes the derivative equal
to 0, which means that if k = D, then the minimum value for VD (y) is attained when y = y ∗ , which
proves the result.
Single-liability Immunization
Putting it all together, if we have only one liability Lk at time k, the sufficient and necessary conditions
to construct an immunized portfolio are:
X
At (1 + y ∗ )−t = Lk (1 + y ∗ )−k
t>0
X
tAt (1 + y ∗ )−t = kLk (1 + y ∗ )−k
t>0
Remarks:
• The first condition is necessary, and ensures that the PV of the assets equals the PV of liabilities.
• The second condition (sufficient, given the first holds) equates the dollar duration of the assets
to the dollar duration of the liabilities. (Note that if the first condition holds, then the second
condition is equivalent to having the Macaulay duration of the assets equal to that of the liabilities,
which is k.)
Example II revisited: In Example II, the two conditions for immunization are satisfied:
1. PV of assets = 2 × 373, 629.0864 = 747 258.17 and PV of liabilities = 1 000 000(1.06)−5 =
747 258.17.
2. Dollar duration of assets = 3 × 373, 629.0864 + 7 × 373, 629.0864 = 3 736 290.9 and dollar duration
of liabilities = 5 × 1 000 000 × (1.06)−5 = 3 736 290.9. This is why at time 5 (which is equal to
D), we observe V5 (ˆ
y ) ≥ V5 (y ∗ ) for all yˆ considered.
Multiple-liability case: Redington Theory
We now assume that there is a cash flow {Lt , t > 0} of liabilities, which in the context of an insurance
company, could be arising from policy claims, policy surrenders, policy loan payments, policyholder
dividends, expenses and taxes.
Notation
For a given interest rate y, the PV of assets, liabilities and surplus are given by
X At
X Lt
A(y) =
,
L(y)
=
, and
(1 + y)t
(1 + y)t
t
t
X At
X Lt
−
S(y) = A(y) − L(y) =
(1 + y)t
(1 + y)t
t
t
respectively. At the initial ytm y ∗ ,
A ≡ A(y ∗ ), L ≡ L(y ∗ )
and S ≡ S(y ∗ ) = A − L.
The liability obligations are said to be
6
• fully funded if A ≥ L (or S ≥ 0);
• underfunded if A < L (or S < 0);
• exactly fully funded if A = L or S = 0.
Redington’s Problem: How to structure the asset cash flows {At , t = 1, 2, . . .} so that there will be
sufficient cash when liabilities {Lt , t = 1, 2, . . .} arise ? Equivalently, are there ways to ensure that
S(y ∗ + ∆y) ≥ S(y ∗ )?
S(y)
y
y*
Figure 1: Desired behavior for S(y)
The following conditions propose a way to achieve this, at least for small ∆y’s.
Redington Immunization Conditions
1. S = 0, (PV matching criterion)
2. S 0 (y ∗ ) = 0, (duration matching criterion)
3. S 00 (y ∗ ) ≥ 0, (dispersion criterion)
The first condition ensures the PV of assets and liabilities match.
0
0
The second condition is equivalent to having A (y ∗ ) = L (y ∗ ), which implies
X
X
tAt (1 + y ∗ )−t =
tLt (1 + y ∗ )−t .
t>0
t>0
Together, conditions 1 and 2 imply that DA = DL (duration matching), where
DA =
X tAt (1 + y ∗ )−t
A
t>0
=
X tLt (1 + y ∗ )−t
t>0
L
= DL .
00
00
Dispersion criterion: the third condition is equivalent to having A (y ∗ ) ≥ L (y ∗ ), which implies
X t(t + 1)At (1 + y ∗ )−t
t>0
A
≥
X t(t + 1)Lt (1 + y ∗ )−t
t>0
7
L
.
Together, conditions 2 and 3 imply that
X t2 At (1 + y ∗ )−t
A
t>0
≥
X t2 Lt (1 + y ∗ )−t
t>0
L
.
Together, conditions 1 and 3 imply that CA ≥ CL , i.e.,
convexity of assets ≥ convexity of liabilities.
Mathematically speaking, these conditions come from Taylor’s theorem with a remainder, which tells
us that
0
00
S(y ∗ + ∆y) = S(y ∗ ) + ∆yS (y ∗ ) + (∆y)2 S (y ∗ + δ)/2,
0
00
where 0 < δ < ∆y. Therefore, by requiring S (y ∗ ) = 0 and S (y ∗ ) > 0, we get that if ∆y is not too
large, then S(y ∗ + ∆y) > S(y ∗ ).
Example III Suppose there are two liability outflows: $10,000 and $26,620 at the end of years 5 and
8, respectively. An asset cash flow of $36,300 is scheduled at the end of year 7. Does this immunize
the liabilities given y ∗ = 10%?
Solution: The first and second conditions are met, since
10 000(1.1)−5 + 26 620(1.1)−8 = 36 300(1.1)−7 = 18 627.64
5 × 10 000(1.1)−5 + 8 × 26 620(1.1)−8 = 7 × 36 300(1.1)−7 = 130 393.47.
However, losses occur for various values of yˆ, as seen in the following table.
yˆ
0.08
0.09
0.10
0.11
0.12
A(ˆ
y)
21181
19857
18628
17484
16420
L(ˆ
y)
21188
19859
18628
17486
16426
S(ˆ
y)
−7.09
−1.65
0.00
−1.44
−5.36
The problem is that the dispersion criterion is not satisfied... We have that
7 × 8 × 36 300(1.1)−7 = 1 043 147.8
for the assets, while
5 × 6 × 10 000(1.1)−5 + 8 × 9 × 26 620(1.1)−8 = 1 080 403.1
for the liabilities.
If, instead, we have an asset cash flow of $10,625.74 and $27,608.74 due in years 3 and 10, respectively,
then conditions 1 and 2 are still satisfied since
10 625.74(1.1)−3 + 27 608.74(1.1)−10 = 18 627.64
3 × 10 625.74(1.1)−3 + 10 × 27 608.74(1.1)−10 = 130 393.47
8
Also, since
3 × 4 × 10 625.74(1.1)−3 + 10 × 11 × 27 608.74(1.1)−10 = 1 170 880.1,
the dispersion criterion is now satisfied. Hence we can verify that a gain is realized for different values
of yˆ:
yˆ
0.08
0.09
0.10
0.11
0.12
A(ˆ
y)
21223
19867
18628
17493
16452
L(ˆ
y)
21188
19859
18628
17486
16426
S(ˆ
y)
35.45
8.26
0.00
7.18
26.83
M-squared M 2 : another measure for interest-rate risk
The M-squared of the asset flow {At , t > 0} is defined as
MA2 =
X
wtA (t − DA )2
where wtA =
t
At (1 + y)−t
A.
Similarly, we can defined the M-squared of the liability flow, denoted ML2 .
We have that the dispersion criterion, together with conditions 1 and 2, is equivalent to having MA2 ≥
ML2 .
Example III revisited Let’s compute MA2 and ML2 for the two asset portfolios considered in Example
III.
• With A7 = 36 300, we have w7A = 36 300(1.1)−7 /18 627.64 = 1 (obvious since only one cash flow),
and therefore MA2 = (7 − 7)2 = 0.
• With A3 = 10, 625.74 and A10 = 27, 608.74, we have w3A = 10, 625.74(1.1)−3 /18 627.64 = 0.42857
A = 27, 608.74(1.10−10 /18 627.64 = 1 − w A = 0.571428, and therefore in this case, M 2 =
and w10
3
A
0.42857 × (3 − 7)2 + 0.571428 × (10 − 7)2 = 12.
• For the liability portfolio, we have w5L = 10 000(1.1)−5 /18 627.74 = 0.3333 and
w9L = 26 620(1.1)−8 /18 627.74 = 1−w5L = 0.6667, and therefore ML2 = 0.3333×(5−7)2 +0.6667×
(8 − 7)2 = 2.
Properties of M 2
• M 2 ≥ 0 for all nonnegative cash flows
• M 2 = 0 if there is only one cash flow... Hence for the multiple-liability case, we need more than
one cash flow for the assets in order to satisfy Redington’s conditions.
• Going further, consider the following bracketing strategy: let tL
j , j = 1, . . . , n denote the times
at which there are liability cash flows. Suppose we have an asset portfolio consisting of two
L
zero-coupon bonds at time t− and t+. If t− ≤ tL
1 , t+ ≥ tn , and Conditions 1 and 2 are satisfied,
then the dispersion condition is satisfied.
9
• Can write M 2 = Var(T ), where T is a discrete random variable with probability distribution
given by P (T = t) = wt . Note that in this framework, D = E(T ). Hence we can think of the
dispersion condition as a variance condition.
• Following the previous point, we can think of M 2 as a measure of immunization risk.
Remarks on the Redington Model
• Assumes a flat term structure.
• Assumes a parallel yield shift.
• Immunizes only for small instantaneous shifts in yield
• Requires rebalancing dynamically (so that durations of assets and liabilities continue to be equal).
• Assumes cash flows are not interest-sensitive.
• The same discounting rate applies to both asset and liability cash flow.
• Model inconsistency: based on Redington model, we can find a strategy that produces a “free
lunch” ⇒ arbitrage. More precisely, since
1 00
0
S(y ∗ + ∆y) ≈ S(y ∗ ) + S (y ∗ )∆y + S (y ∗ )(∆y)2 ≥ S(y ∗ )
2
00
for small ∆y under Redington’s conditions. That is, since S (y ∗ ) ≥ 0, we are guaranteed to make
a profit when there is a small change in y ∗ . This is also called “second-derivative profit”.
Rebalancing immunized portfolios
The portfolio must be rebalanced (i.e. buying or selling assets) continuously so that the asset duration
is aligned with the liability duration. Why? (1) The maturities decrease as time goes by; (2) If the
yield changes, then it affects PVs and durations. The following example illustrates these ideas.
Example IV: Assume a bullet liability of $20, 000(1.05)13 is due 13 years from now. An immunization
strategy is adopted by investing in a 5-year zero bond and a perpetuity, both with ytm 5% (annual
effective rate). (1) What should be the composition of the asset portfolio? (2) If the ytm remains at
5%, how should the asset portfolio be rebalanced in 1 year?
Solution: Let A5 be the amount to be invested in the 5-year zero-coupon bond, and let A∞ be the
amount to be invested in the perpetuity. Here the PV of the liabilities is L = 20 000. So we must have
A = A5 + A∞ = L. Alternatively, we can write w5 = A5 /A and then solve for
5w5 + 21(1 − w5 ) = 13
since (i) the duration of the perpetuity can be shown to be (1 + y ∗ )/y ∗ = 21; (ii) A∞ /A = 1 − (A5 /A) =
1 − w5 . Thus we get w5 = 0.5, which means A5 = 10 000 and A∞ = 10 000.
(2) Let A15 be the amount to be invested in the zero-coupon bonds (whose maturity is now 4 years) at
time 1, and A1∞ be the amount to be invested in the perpetuity at time 1. We must have that
A15 + A1∞ = 20 000(1.05) = 21 000.
10
Writing w51 = A15 /21 000, we also need to have
4w51 + 21(1 − w51 ) = 12,
which means w51 = 9/17 and thus A15 = 11 117.647 and A1∞ = 9882.35.
Implementation issues
In general, there is not a unique solution that satisfies Redington’s conditions... Means we have several
immunized portfolios to choose from. How should we choose? Must try to optimize some criterion ⇒
optimal asset allocation problem.
More precisely, suppose there are n assets on the market, and let Pj be the price for 1 unit of asset
j, for j = 1, . . . , n. Let nj and xj denote the total number of units, and total dollar amount invested
in the j-th security, respectively: that is, xj = nj Pj . We can then try to maximize/minimize some
objective function f (x1 , . . . , xn ) subject to constraints arising, among other things, from Redington’s
conditions. For instance, the objective function could be to minimize the M -squared of the assets. In
this case, we would have the following optimization problem:
min MA2
subject to MA2 ≥ ML2
DA = DL
A=L
and possibly other constraints
Possible other constraints could be, e.g., that a certain maximum amount can be invested in a given
security. Possible other objective functions could be, e.g., to maximize the portfolio yield.
Generalized Redington Theory of Immunization
In this section, we’ll remove the assumption that the term structure is flat, and also the assumption
that the interest rate change ∆y is small. We’ll introduce some new notation: Nt = At − Lt is the net
cash flow at time t, and P (0, t) is the price at time 0 of a zero-coupon bond maturing for $1 at time t.
The current surplus S is given by
X
S=
Nt P (0, t).
t>0
Consider an instantaneous shock in the term structure that changes P (0, t) to Pˆ (0, t), for each t > 0.
Then the surplus value changes to
X
Sˆ =
Nt Pˆ (0, t).
t>0
As before, we’re wondering if it’s possible to construct the asset portfolio so that
Sˆ − S ≥ 0
11
(1)
for all shocks in the term structure. As it turns out, removing the assumption of a flat term structure
with parallel shifts has the consequence that it is no longer possible to guarantee (1) for all {Pˆ (0, t), t >
0}, unless Sˆ = S, which can be shown to be equivalent to having Nt = 0 for all t > 0: that is, the
assets are perfectly matched to the liabilities. Note that this means the arbitrage opportunities arising
in the classical Redington model no longer exist in the generalized model.
What we want to do now is to understand the behavior of Sˆ − S. First, we define
nt = Nt P (0, t),
the discounted value of the net cash flow Nt with respect to the original term structure. Note that
X
nt = S
t>0
X
tnt = Fisher-Weil dollar duration of surplus
t>0
X
t2 nt = Fisher-Weil dollar convexity of surplus.
t>0
We then introduce the function
g(t) =
Pˆ (0, t)
− 1,
P (0, t)
which we can view as the relative change in the spot rate for period t. Note that g(0) = 1/1 − 1 = 0.
We can then rewrite the change in surplus as
X
X
X
nt g(t).
(2)
Nt P (0, t)g(t) =
Nt (Pˆ (0, t) − P (0, t)) =
Sˆ − S =
t>0
t>0
t>0
The main result for the generalized Redington model is as follows:
Theorem: If (1) the net cash flows {Nt , t > 0} satisfy either
X
nt (t − w)+ ≥ 0 for all w > 0
(3)
t>0
or
X
nt (t − w)+ ≤ 0 for all w > 0;
(4)
t>0
(where x+ = max(x, 0)); (2)
P
t>0 tnt
= 0, then there exists a value χ > 0 such that
X
1 00
Sˆ − S = g (χ)
t2 n t .
2
t>0
In other words, this result gives us two conditions that enable us to give a more compact expression for
the change in surplus. This compact formulation will help us find ways of constructing portfolios that
try to maximize Sˆ − S using linear programming. But before we do that, let’s first try to see where
this result comes from. We’ll first show that
X
X Z t
0
00
ˆ
S − S = g (0)
tnt +
nt
(t − w)g (w)dw.
(5)
t>0
t>0
12
0
Assuming g is twice differentiable, we can use Taylor’s formula with integral remainder, which says
that
Z t
0
00
g(t) = g(0) + tg (0) +
(t − w)g (w)dw.
0
Substituting (2) and using the fact that g(0) = 0, we have that (5) holds.
The next step to prove the above theorem is to show that the second term in (5) can be written as
X
1 00
g (χ)
t2 n t .
2
t>0
To do that, we first write
X
t
Z
00
(t − w)g (w)dw =
nt
0
t>0
=
X
∞
Z
00
(t − w)+ g (w)dw
nt
0
t>0
Z ∞X
00
nt (t − w)+ g (w)dw.
0
(6)
(7)
t>0
To analyze this expression, we’ll use the following result:
Theorem: (Weighted Mean Value Theorem for Integrals) If f and h are continuous functions on the
interval [a, b] and h does not change sign on that interval, then there exists a number ε in [a, b] such
that
Z
Z
b
b
f (x)h(x)dx = f (ε)
a
h(x)dx
a
From this result, and assuming that the net cash flows {Nt , t > 0} satisfy either
X
nt (t − w)+ ≥ 0
for all w > 0
t>0
or
X
nt (t − w)+ ≤ 0
for all w > 0
t>0
then there exists χ > 0 such that
Z ∞X
Z
00
00
nt (t − w)+ g (w)dw = g (χ)
0
∞X
0
t>0
nt (t − w)+ dw.
t>0
Reversing back the order of summation and integration, we have
Z ∞X
X Z ∞
nt (t − w)+ dw =
nt
(t − w)+ dw
0
t>0
0
t>0
=
X
Z
0
t>0
=
X
t>0
13
t
(t − w)dw
nt
nt
t2
2
(8)
To conclude the proof, we simply need to note that Condition (2) implies the first term in (5) vanishes.
P
Note: from (8), we see that the sum t>0 t2 nt can be either positive or negative, depending on which
of (3) or (4) is satisfied;
Also, the classical Redington model can be recovered as follows:
Special case: parallel yield curve shift
Assume the spot rate curve is given by the continously (annualized) compounded rates {s1 , . . . , sn },
and that P (0, t) = e−tst . If we assume that the shifts Pˆ (0, t) take the special form
Pˆ (0, t) = ect P (0, t) = e−(st −c)t .
for some constant c (positive or negative), then we get that
g(t) = ect − 1.
So in this case, under the conditions of Theorem 1, we get that
X
X
1
1 00
Sˆ − S = g (χ)
t2 nt = c2 ecχ
t2 n t .
2
2
t>0
t>0
P
2
In particular, if (3) is satisfied (which is equivalent to having
t>0 t nt ≥ 0, i.e., the Fisher-Weil
convexity of the surplus is non-negative), then Sˆ − S ≥ 0 for any c.
How to use the generalized Redington model
From Theorem 1, we know that under some conditions
X
1 00
t2 n t .
Sˆ − S = g (χ)
2
t>0
How do
P we use this result? Ideally,00 we’d like to structure the assets and liabilities so as to maximize
00
g (χ) t>0 t2 nt ... But the factor g (χ) depends on the interest rate shock, which one cannot predict...
In addition, we don’t even know whether that
P quantity is positive or negative. Consequently, a more
prudent approach is to try to minimize | t>0 t2 nt |. In what follows, we’ll briefly discuss how to
formulate a linear programming model to solve the asset allocation problem within the generalized
Redington model.
Optimization Framework—Linear Programming
To simplify things, assume the cash flows occur only at the end of each time period, and denote by Aj,t
the cash flow at the end ofP
the t-th period for an initial investment of $1 in the jth security. Hence for
each j, we have that 1 = t≥1 Aj,t P (0, t). As before, let xj be the amount of money
to be invested
P
in the jth security. Hence the aggregated cash flow at time t is given by At = j xj Aj,t . The asset
allocation problem is to determine, for a given stream of liabilities {Lt } and surplus S, the “optimal”
amounts xj .
We can use linear programming to solve this problem, which, in this framework, can be formulated as
follows:
14
˛
˛
˛X
˛
˛
˛
2
min ˛
t nt ˛
xj ˛
˛
t>0
X
subject to
nt = S
t>0
X
tnt = 0
t>0
X
nt (t − w)+ ≥ 0 or
t>0
X
nt (t − w)+ ≤ 0 for all w > 0
t>0
where At =
X
xj Aj,t
j
We’ll consider one by one the two possible cases given by (3) and (4). In other words, in each case we
are guessing that the chosen condition can be made to hold, and try to find xj ’s that will minimize
our
while allowing that condition (and others) to hold. Note that since the sum
P objective function,
+ is a piecewise linear function of w, to verify whether it’s positive or negative, we only
n
(t
−
w)
t>0 t
need to check its value at points w where nw 6= 0. Since we
Passumed before that cash flows only occured
at the end of periods, it means we only need to look at t>k nt (t − k)+ for k = 1, 2, . . . .
P
+
Case 1:
t>0 nt (t − w) ≥ 0 for all w > 0
R∞P
P
P
P
2
In this case, P
| t>0 t2 nt | becomes
− w)+ dw = (1/2) t>0 t2 nP
t . Furthert>0 t nt since 0
t>0 nt (t P
P
P
more, minxj t>0 t2 nt ⇔ minxj t>0 t2 (At −Lt )P (0, t) ⇔ minxj t>0 t2 At P (0, t) ⇔ minxj t>0 t2 j xj Aj,t P (0, t) ⇔
P
P
P
minxj j xj t>0 t2 Aj,t P (0, t) ⇔ minxj j xj Cj , where
Cj =
X
t2 Aj,t P (0, t)
t>0
is the Fisher-Weil dollar convexity of the jth security.
Summing up, the following linear programming problem must be solved:
min xj Cj
xj
subject to
X
nt = S
t>0
X
tnt = 0
t>0
X
nt (t − k)+ ≥ 0 for k = 1, 2, . . .
t>k
X
where nt = (
xj Aj,t − Lt )P (0, t)
j
and Cj =
X
t2 Aj,t P (0, t).
t≥1
Case 2:
P
t>0 nt (t
− w)+ ≤ 0 for all w > 0
Using a similar development, in this case we get the following formulation:
15
max xj Cj
xj
subject to
X
nt = S
t>0
X
tnt = 0
t>0
X
nt (t − k)+ ≤ 0 for k = 1, 2, . . .
t>k
X
where nt = (
xj Aj,t − Lt )P (0, t)
j
and Cj =
X
t2 Aj,t P (0, t).
t≥1
Case where S =
P
t>0 nt
=0
P
In the case where S = t>0 nt = 0, we can say a little bit more about the structure of the nt ’s that is
required to satisfy the constraints of the above linear programming problem. More precisely, we have:
Proposition:
Let ni be the discounted
net cash flow at time ti , for i = 1, . . . , m. In order to satisfy
P
Pn
both (i) m
n
=
0
and
(ii)
t
n
i=1 i
i=1 i i = 0, the sequence {n1 , . . . , nm } must have at least two sign
changes.
Proof:(By contradiction) First, there must be at least one change of sign... Then, assume there is
only onePchange of P
sign of the formP
+, − and that
the sign change occurs.
This
Pmtk+1 is the first time P
Pk
k
m
k
k
implies
n
=
|n
|
and
t
n
=
t
|n
|.
However
t
n
<
t
n
k+1
i=0 i
i=k+1 i
i=0 i i
i=k+1 i i
i=0 i i
i=0 i and
Pm
P
m
i=k+1 ti |ni | ≥ tk+1
i=k+1 |ni |, which gives us a contradiction. A similar reasoning can be used to
show that a sign change of the form −, + also leads to a contradiction.
We can then consider the case where we have exactly two sign changes, and study the two possibilities
for that case: + − + or − + − (i.e., + − + means that there exist two indices 1 ≤ k1 < k2 ≤ m such
that n1 , . . . , nk1 ≥ 0, nk1 +1 , . . . , nk2 < 0, nk2 +1 , . . . , nm ≥ 0). We get the following result, which is
based on the concept of Karamata measures:
P
P
Proposition: If m
ni = 0, m
i=1P
i=1 ti ni = 0, and the sequence {n1 , . . . , nm } undergoes the sign change
m
sequence
+
−
+,
then
n
φ(t
i ) ≥ 0 for any convex function φ(·). If the sequence in instead − + −,
i=1 i
Pm
then i=1 ni φ(ti ) ≤ 0 for any convex function φ(·).
The consequence of this result is that if the interest rate shock is such that the function g(·) is convex
(for instance, the function g(t) = ect − 1 corresponding to a parallel shift is convex), and the sequence
{n1 , . . . , nmP
} undergoes exactly 2 sign changes, then using Proposition 2 and the fact that Sˆ − S =
00
(1/2)g (χ) t>0 nt g(t), we can figure out whether Sˆ − S is ≥ or ≤ than 0 depending on whether the
sign change sequence is + − + or − + −.
Example: (Single Premium Immediate Annuity (Shiu, (IME), 1990))
Consider an insurance company that issues single premium immediate annuity policies. It invests all
the premiums it receives for the annuities in a noncallable
P and default-free
P bond. The company has the
policy of matching asset and liability durations. Hence
nt = 0 and
tnt = 0. Therefore, unless the
asset and liability cash flows are perfectly matched, {nt , t > 0} has at least two sign changes. Moreover,
the (expected) annuity cash flows are non-increasing with time, whereas the cash flows from the bond are
level with the exception of the last one, which is larger because of the principal repayment. Therefore
16
the sign change pattern is actually − + −. Thus from Proposition 2, we can conclude that for any
parallel shift in the term structure, the company will lose money.
Other immunization techniques
To conclude this unit, we discuss a few alternative immunization techniques.
Dedication Strategy: Cash Flow Matching
This strategy uses a dedicated bond portfolio, which is constructed so that its monthly cash flows
match the monthly cash requirements of liabilities. Hence this strategy eliminates interest-rate risk.
Some applications of this strategy are for pension benefit funding, structured settlement funding, and
guaranteed investment contract matching. A detailed example is presented in Unit 7.
Usually, the dedicated portfolio is constructed by finding the cheapest combination of bonds that can
provide, at each period, an asset cash flow that is at least as large as the liability cash flow. In other
words, we need to solve
min
X
At P (0, t)
t>0
subject to At ≥ Lt for all t.
Example: Consider funding a stream of liabilities of $300 000, $200 000, and $100 000 at the end of
the first, second, and third year, respectively. Construct an optimal dedicated portfolio based on the
following bonds.
Bond
Bond 1
Bond 2
Bond 3
Bond 4
Bond 5
Bond 6
Year 0
Price/unit
100.50
95.40
105.60
95.00
85.00
75.00
Cash Flow Per Unit of Investment
at Year 1 at Year 2 at Year 3
10
10
110
8
8
108
12
12
112
100
0
0
0
100
0
0
0
100
Solution: Let xj be the amount of units of bond j in the dedicated portfolio. We need to solve
min 100.5x1 + 95.4x2 + 105.6x3 + 95x4 + 85x5 + 75x6
subject to 10x1 + 8x2 + 12x3 + 100x4 ≥ 300 000
10x1 + 8x2 + 12x3 + 100x5 ≥ 200 000
110x1 + 108x2 + 112x3 + 100x6 ≥ 100 000
x1 , x2 , x3 , x4 , x5 , x6 ≥ 0
Using Excel (or Gnumeric) Solver, we find that the optimal solution is
17
x1
0
x2
0
x3
892.857
x4
2892.857
x5
1892.857
x6
0
and that solution exactly matches the liability cash flows. (Or, if we force the xj ’s to be integers
x1
2
x2
0
x3
890
x4
2893
x5
1893
x6
)
1
Advantages of Dedication
• Easy to understand
• Potential to eliminate interest-rate and reinvestment risk
• Not necessary to rebalance (although can be done).
Disadvantages of Dedication
• Liabilities cash flow are not usually known with certainty.
• Perfect matching is hard. Therefore, there is usually still a reinvestment risk, and an assumption
on the reinvestment rate must be made.
• Lack of flexibility: less interesting bonds (e.g., with lower yield) might be chosen only because of
their maturity.
Combination-Matching or Horizon-Matching
A combination-matched portfolio is one that is duration matched with the added constraint that it be
cash-matched for the first few years (usually 5 years). The main advantage of this method over pure
immunization is that the liquidity needs are provided for in the initial cash-flow matched period: this
eliminates the reinvestment risk for that period. Also, since most of the positive slope or inversion of
a yield curve tends to take place in the first few years, by cash-flow matching that portion, we reduce
the risk of non-parallel shifts. The disadvantage is that it is more expensive than immunization.
Contingent Immunization
• Blend of active management with immunization.
• Requires to set a floor return, or safety net: portfolio is actively managed until the return hits
the floor return; at that point, the portfolio manager must commit to an immunized portfolio to
ensure the floor return for the remainder of the investment horizon.
Example: Suppose the current interest rate is 10%, and that a manager’s portfolio is worth 10 millions.
Assume a 5 year investment horizon. (i) What is the immunizable terminal value in year 5? (ii) The
manager wants to pursue an active bond management strategy, provided the net terminal value is not
below 15 millions. Find the corresponding floor return. (iii) Suppose that at time 2, the market interest
rate is now 9%. What is the minimum value of the fund at time 2 that is required in order to guarantee
that the minimum terminal value can be reached?
Solution: (i) The immunizable terminal value in year 5 is 10×106 ×(1.1)5 = 16 105 100. (ii) (15/10)1/5 −
1 = 8.44718% (iii) 15 × 106 × (1.09)−3 = 11 582 752.
18
min.
term.
value
trigger
level
cushion
spread
t*
T
Figure 2: Contingent Immunization
Remarks on Contingent Immunization:
• Has the potential of achieving higher return than an immunized portfolio, but with added uncertainty.
• Requires an objective procedure to monitor the portfolio.
• Portfolio must remain sufficiently liquid so that if it hits the trigger level, actions can be taken
to immunize it.
• Choice of the minimum return and investment horizon: a longer horizon gives more opportunity
to actively manage the portfolio.
We conclude this section with Figure 3, which shows different ALM strategies, going from safe to risky
from left to right.
combination
matching
contingent
immunization
immunization
cash flow
matching
active
management
Figure 3: Spectrum of ALM strategies
Optional Reading Material
• Chapter 47 of Fabozzi (7th edition).
• Asset-Liability Management, Society of Actuaries Professional Actuarial Specialty Guide, 2003.
(UWD1931—also available from www.soa.org)
• Immunization for Pension Plans, Educational Note, Canadian Institute of Actuaries, 1996 (UWD
1940).
19
Exercises
1. An insurance company faces a liability obligation of 4 millions in 3 years. The available market
instruments are 2-year, 4-year and 5-year zero coupon bonds, each with a yield of 5%. (i) Find
two portfolios that match the present value and duration of the liability. (Use the bond yield to
discount the liability). For each portfolio found in (i), compute its value a time t = 0 and t = 3
years if there is an instantaneous change in the yield of (ii) 1%; (iii) -1%.
2. Consider a liability portfolio with a cash flow of 1 million at t = 2 years, and 4 millions at
t = 5. An asset portfolio based on zero-coupon bonds with maturities of 1 or 6 years is to be
constructed. Assume all securities have a yield of 6%. (i) Construct an asset portfolio that
satisfies Redington’s two first conditions. What can you say about the fulfillment of Redington’s
third condition? (ii) Compute the M-squared M 2 of each of the asset and liability portfolios and
verify that Redington’s third condition holds. (iii) How should the asset portfolio be rebalanced at
time 0.5 if the yield is now at 5% (assuming no rebalancing has been done so far)? (iv) Suppose
now that different maturities carry different yields. More precisely, assume the (continuously
compounded) spot rates for 1, 2, P
5 and 6 years are
P 0.05, 0.055, 0.07 and 0.075. Construct an
asset portfolio that satisfies both t nt = 0 and t tnt = 0. (v) Continuing with the setting
given in (iv), what kind of sign change sequence is experienced by the {nt } sequence? Verify by
looking at a parallel shift of 0.5% and then -0.5% that the surplus behaves as predicted.
3. You want to fund the following stream of liabilities: 150 000, 350 000 and 225 00 at the end of the
first, second and third years, respectively. (i) Construct an optimal dedicated portfolio based on
the following bonds (fractions are allowed):
Bond
Bond 1
Bond 2
Bond 3
Bond 4
Bond 5
Bond 6
Year 0
Price/unit
98.50
96.30
102.70
97.00
91.00
82.00
Cash Flow Per Unit of Investment
at Year 1 at Year 2 at Year 3
8
8
108
6
6
106
10
10
110
100
0
0
0
100
0
0
0
100
(ii) Compare the initial amount that needs to be invested for the portfolio constructed in (i) with one
that can only use zero-coupon bonds.
20