Immunization and dedication
techniques refer to strategies that investors use to shield their net worth from interest rate risk.
Immunization
Many banks and thrift institutions have a natural mismatch between the maturities of assets
and liabilities. For example, bank liabilities are primarily the deposits owed to customers;
these liabilities are short term and consequently have low duration. Assets largely comprise
commercial and consumer loans or mortgages, which have longer duration. When interest
immunization
A strategy to shield net worth
from interest rate movements.
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rates increase unexpectedly, banks can suffer decreases in net worth—their assets fall in value
by more than their liabilities.
Similarly, a pension fund may have a mismatch between the interest rate sensitivity of
the assets held in the fund and the present value of its liabilities—the promise to make
payments to retirees. The nearby box illustrates the dangers that pension funds face when
they neglect the interest rate exposure of both assets and liabilities. It points out that when
interest rates change, the present value of the fund’s liabilities changes. For example, in
some recent years pension funds lost ground despite the fact that they enjoyed excellent
investment returns. As interest rates fell, the value of their liabilities grew even faster than
the value of their assets. The conclusion: Funds should match the interest rate exposure of
assets and liabilities so that the value of assets will track the value of liabilities whether
rates rise or fall. In other words, the financial manager might want to immunize the fund
against interest rate volatility.
Pension funds are not alone in this concern. Any institution with a future fixed obligation
might consider immunization a reasonable risk management policy. Insurance companies,
for example, also pursue immunization strategies. In fact, the tactic of immunization was
introduced by F. M. Redington (1952), an actuary for a life insurance company. The idea is
that duration-matched assets and liabilities let the asset portfolio meet the firm’s obligations
despite interest rate movements.
Consider, for example, an insurance company that issues a guaranteed investment con￾tract, or GIC, for $10,000. (GICs are essentially zero-coupon bonds issued by the insurance
company to its customers. They are popular products for individuals’ retirement-savings
accounts.) If the GIC has a five-year maturity and a guaranteed interest rate of 8%, the insur￾ance company promises to pay $10,000 × (1.08)5
 = $14,693.28 in five years.
Suppose that the insurance company chooses to fund its obligation with $10,000 of 8%
annual coupon bonds, selling at par value, with six years to maturity. As long as the market
interest rate stays at 8%, the company has fully funded the obligation, as the present value of
the obligation exactly equals the value of the bonds.
Table 11.4, Panel A, shows that if interest rates remain at 8%, the accumulated funds
from the bond will grow to exactly the $14,693.28 obligation. Over the five-year period,
each year-end coupon payment of $800 is reinvested at the prevailing 8% market interest rate.
On the MARKET FRONT
PENSION FUNDS LOSE GROUND
DESPITE BROAD MARKET GAINS
With the S&P 500 providing a 16% rate of return, 2012 was a good
year for the stock market, and this performance helped boost the
balance sheets of U.S. pension funds. Yet despite the increase
in the value of their assets, the total estimated pension deficit of
400 large U.S. companies rose by nearly $80 billion, and many
of these firms entered 2013 needing to shore up their pension
funds with billions of dollars of additional cash. Ford Motor Com￾pany alone predicted that it would contribute $5 billion to its fund.*
How did this happen? Blame the decline in interest rates during
the year that were in large part the force behind the stock mar￾ket gains. As rates fell, the present value of pension obligations
to retirees rose even faster than the value of the assets backing
those promises. It turns out that the value of pension liabilities is
more sensitive to interest rate changes than is the value of the
typical assets held in those funds. So even though falling rates
tend to pump up asset returns, they pump up liabilities even more.
In other words, the duration of fund investments is shorter than
the duration of its obligations. This duration mismatch makes funds
vulnerable to interest rate declines.
Why don’t funds better match asset and liability durations?
One reason is that fund managers are often evaluated based
on their performance relative to standard bond market indexes.
Those indexes tend to have far shorter durations than pension
fund liabilities. So to some extent, managers may be keeping their
eyes on the wrong ball, one with the wrong interest rate sensitivity.
*These estimates appear in Mike Ramsey and Vipal Monga, “Low Rates
Force Companies to Pour Cash into Pensions,” The Wall Street Journal,
February 3, 2013.
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TABLE 11.4 Terminal value of a six-year-maturity bond portfolio after five years (all proceeds reinvested)
Note: The sale price of the six-year-maturity bond portfolio equals the portfolio’s final payment ($10,800) divided by 1 + r because
the time to maturity of the bonds will be one year at the time of sale.
Payment Number
Years Remaining
until Obligation
Accumulated Value of
Invested Payment
A. Rates remain at 8%
1 4 800 × (1.08)4 = 1,088.39
2 3 800 × (1.08)3 = 1,007.77
3 2 800 × (1.08)2 = 933.12
4 1 800 × (1.08)1 = 864.00
5 0 800 × (1.08)0 = 800.00
Sale of bond 0 10,⁸⁰⁰⁄₁.08 = 10,000.00
14,693.28
B. Rates fall to 7%
1 4 800 × (1.07)4 = 1,048.64
2 3 800 × (1.07)3 = 980.03
3 2 800 × (1.07)2 = 915.92
4 1 800 × (1.07)1 = 856.00
5 0 800 × (1.07)0 = 800.00
Sale of bond 0 10,⁸⁰⁰⁄₁.07 = 10,093.46
14,694.05
C. Rates increase to 9%
1 4 800 × (1.09)4 = 1,129.27
2 3 800 × (1.09)3 = 1,036.02
3 2 800 × (1.09)2 = 950.48
4 1 800 × (1.09)1 = 872.00
5 0 800 × (1.09)0 = 800.00
Sale of bond 0 10,⁸⁰⁰⁄₁.09 =   9,908.26
14,696.02
At the end of the period, the bonds can be sold for $10,000; they still will sell at par value
because the coupon rate still equals the market interest rate. Total income after five years from
reinvested coupons and the sale of the bond is precisely $14,693.28.
If interest rates change, however, two offsetting influences will affect the ability of the fund
to grow to the targeted value of $14,693.28. If rates rise, the fund will suffer a capital loss,
impairing its ability to satisfy the obligation. However, at that higher interest rate, reinvested
coupons will grow faster, offsetting the capital loss. In other words, fixed-income investors
face two offsetting types of interest rate risk: price risk and reinvestment rate risk. Increases
in interest rates cause capital losses but at the same time increase the rate at which reinvested
income will grow. If the portfolio duration is chosen appropriately, these two effects will
cancel out exactly. When portfolio duration equals the investor’s horizon date, the accu￾mulated value of the investment fund at the horizon date will be unaffected by interest rate
fluctuations. For a horizon equal to the portfolio’s duration, price risk and reinvestment risk
are precisely offsetting. The obligation is immunized.
In our example, the duration of the six-year-maturity bonds used to fund the GIC is almost
exactly five years. You can confirm this using either Spreadsheet 11.1 or 11.2. The duration
of the (zero-coupon) GIC is also five years. Because the assets and liabilities have equal dura￾tion, the insurance company is immunized against interest rate fluctuations. To confirm this,
let’s check that the bond can generate enough income to pay off the obligation in five years
regardless of interest rate movements.
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FIGURE 11.3
Growth of invested funds
Note: The solid curve represents
the growth of portfolio value at the
original interest rate. If interest rates
increase at time t*, the portfolio
value falls but increases thereafter
at the faster rate represented by the
broken curve. At time D (duration),
the curves cross.
Accumulated value of invested funds
0
t D = 5 years
t
Obligation
Panels B and C of Table 11.4 illustrate two possible interest rate scenarios: Rates either fall
to 7% or increase to 9%. In both cases, the annual coupon payments are reinvested at the new
interest rate, which is assumed to change before the first coupon payment, and the bond is sold
in year 5 to help satisfy the obligation of the GIC.
Panel B shows that if interest rates fall to 7%, the total funds will accumulate to $14,694.05,
providing a small surplus of $0.77. If rates increase to 9% as in Panel C, the fund accumulates
to $14,696.02, providing a small surplus of $2.74.
Several points are worth highlighting. First, duration matching balances the difference
between the accumulated value of the coupon payments (reinvestment rate risk) and the sale
value of the bond (price risk). That is, when interest rates fall, the coupons grow less than in
the base case, but the higher value of the bond offsets this. When interest rates rise, the value
of the bond falls, but the coupons more than make up for this loss because they are reinvested
at the higher rate. Figure 11.3 illustrates this case. The solid curve traces the accumulated
value of the bonds if interest rates remain at 8%. The dashed curve shows that value if interest
rates happen to increase. The initial impact is a capital loss, but this loss eventually is offset
by the now-faster growth rate of reinvested funds. At the five-year horizon date, equal to the
bond’s duration, the two effects just cancel, leaving the company able to satisfy its obligation
with the accumulated proceeds from the bond.
We can also analyze immunization in terms of present as opposed to future values.
Table 11.5, Panel A, shows the initial balance sheet for the insurance company’s GIC.
Notes: Value of bonds = 800 × Annuity factor(r, 6) + 10,000 × PV factor(r, 6).
Value of obligation =
_________ 14,693.28
(1 + r)5
= 14,693.28 × PV factor(r, 5).
TABLE 11.5 Market value balance sheets
A. Interest rate = 8%
Assets Liabilities
Bonds $10,000.00 Obligation $10,000.00
B. Interest rate = 7%
Assets Liabilities
Bonds $10,476.65 Obligation $10,476.11
C. Interest rate = 9%
Assets Liabilities
Bonds $ 9,551.41 Obligation $ 9,549.62
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Both assets and the obligation have market values of $10,000, so the plan is just fully funded.
Panels B and C show that whether the interest rate increases or decreases, the value of the
bonds funding the GIC and the present value of the company’s obligation change by virtually
identical amounts. In either case, the plan remains fully funded, with the surplus in Panels B
and C just about zero.
Figure 11.4 is a graph of the present values of the bond and the single-payment obliga￾tion as a function of the interest rate. At the current rate of 8%, the values are equal, and the
obligation is fully funded by the bond. Moreover, the two present value curves are tangent
at y = 8%. As interest rates change, the change in value of both the asset and the obligation
are equal, so the obligation remains fully funded. For greater changes in the interest rate,
however, the present value curves diverge. This reflects the fact that the fund actually shows a
small surplus at market interest rates other than 8%.
Why is there any surplus in the fund? After all, we claimed that a duration-matched asset
and liability mix would make the investor indifferent to interest rate shifts. Actually, such
a claim is valid for only small changes in the interest rate because as bond yields change,
so does duration. (Recall Rule 4 for duration.) In fact, while the duration of the bond equals
five years at a yield to maturity of 8%, it rises to 5.02 years when the bond yield falls to 7%
and drops to 4.97 years at y = 9%. That is, the bond and the obligation were not duration￾matched across the interest rate shift, so the position was not exactly immunized.
This example highlights the importance of rebalancing immunized portfolios. As interest
rates and asset durations continually change, managers must adjust the portfolio to realign its
duration with the duration of the obligation. Moreover, even if interest rates do not change,
asset durations will change solely because of the passage of time. Recall from Figure 11.2 that
duration generally decreases less rapidly than maturity as time passes, so even if an obligation
is immunized at the outset, the durations of the asset and liability will fall at different rates.
Without rebalancing, durations will become unmatched. Therefore, immunization is a passive
strategy only in the sense that it does not involve attempts to identify undervalued securities.
Immunization managers still must proactively update and monitor their positions.
rebalancing
Realigning the proportions
of assets in a portfolio as
needed.
EXAMPLE 11.2
Constructing an
Immunized Portfolio
An insurance company must make a payment of $19,487 in seven years. The market interest rate
is 10%, so the present value of its obligation is $10,000. Its portfolio manager wishes to fund the
obligation using three-year zero-coupon bonds and perpetuities paying annual coupons. (We focus
on zeros and perpetuities to keep the algebra simple.) How can she immunize the obligation?
Immunization requires that the duration of the portfolio of assets equal the duration of the liability.
We can proceed in four steps:
Step 1. Calculate the duration of the liability, which in this case is simple to compute. It is a
single-payment obligation with maturity and duration of seven years.
Step 2. Calculate the duration of the asset portfolio, which is the weighted average of the durations
of each component asset, with weights proportional to the funds placed in each asset.
The duration of the zero-coupon bond is simply its maturity, 3 years. The duration of the
perpetuity is 1.10/.10 = 11 years. Therefore, if the fraction of the portfolio invested in the
zero is called w, and the fraction invested in the perpetuity is (1 − w), the portfolio duration is
Asset duration = w × 3 years + (1 − w)× 11 years
Step 3. Find the asset mix that sets the duration of assets equal to the seven-year duration of
liabilities. This requires us to solve for w in the following equation
w × 3 years + (1 − w)× 11 years = 7 years
This implies that w = ¹⁄₂. The manager should invest half the portfolio in the zero and
half in the perpetuity. This will result in an asset duration of seven years.
Step 4. Fully fund the obligation. Because the obligation has a present value of $10,000, and the
fund will be invested equally in the zero and the perpetuity, the manager must purchase
$5,000 of the zero-coupon bond and $5,000 of the perpetuity. Note that the face value of
the zero will be $5,000 × 1.103 = $6,655.
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Even if a position is immunized, however, the portfolio manager still cannot rest. This
is because of the need to rebalance as interest rates fluctuate. Moreover, even if rates
do not change, the passage of time also will affect duration and require rebalancing.
Let’s continue Example 11.2 and see how the portfolio manager can maintain an immu￾nized position.
EXAMPLE 11.3
Rebalancing
Suppose that a year has passed, and the interest rate remains at 10%. The portfolio manager of
Example 11.2 needs to reexamine her position. Is the position still fully funded? Is it still immunized?
If not, what actions are required?
First, examine funding. The present value of the obligation will have grown to $11,000, as it is
one year closer to maturity. The manager’s funds also have grown to $11,000: The zero-coupon
bonds have increased in value from $5,000 to $5,500 with the passage of time, while the perpetuity
has paid its annual $500 coupons and remains worth $5,000. Therefore, the obligation is still fully
funded.
The portfolio weights must be changed, however. The zero-coupon bond now has a duration
of 2 years, while the perpetuity duration remains at 11 years. The obligation is now due in 6 years.
The weights must now satisfy the equation
w × 2 + (1 – w) × 11 = 6
which implies that w = ⁵⁄₉. To rebalance the portfolio and maintain the duration match, the manager
now must invest a total of $11,000 × ⁵⁄₉ = $6,111.11 in the zero-coupon bond. This requires
that the entire $500 coupon payment be invested in the zero, with an additional $111.11 of the
perpetuity sold and invested in the zero-coupon bond.
Of course, rebalancing entails transaction costs. In practice, managers strike a compromise
between the desire for perfect immunization, which requires continual rebalancing, and the
need to control trading costs, which dictates less frequent rebalancing.
CONCEPT
check 11.5 Look again at Example 11.3. What would have been the immunizing weights in the second year
if the interest rate had fallen to 8%?
FIGURE 11.4
Immunization. The
coupon bond fully
funds the obligation
at an interest rate of
8%. Moreover, the
present value curves
are tangent at 8%,
so the obligation will
remain fully funded
even if rates change
by a small amount
Values ($)
14,000
12,000
10,000
8,000
6,000
Coupon bond
Single￾payment
obligation
Interest rate
0 5% 8% 10% 15% 20%
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概括

免疫化和专用技术是投资者用于保护其净资产免受利率风险影响的策略。银行和储蓄机构的资产与负债期限常存在不匹配问题，如短期存款与长期贷款的组合，当利率意外上升时，资产价值跌幅可能超过负债，导致净资产缩水。养老金基金同样面临资产利率敏感性与负债现值不匹配的风险，利率变动会改变负债现值，若资产价值增速不及负债，基金将出现缺口。

免疫策略通过匹配资产与负债的久期来抵消利率波动的影响。例如，保险公司发行5年期保证投资合约（GIC）时，若用久期相同的债券组合匹配负债，无论利率升降，债券的资本损益与再投资收益将相互抵消，确保到期偿付能力。久期匹配需动态调整，因利率变化和时间推移会改变资产久期，需定期再平衡以维持免疫状态。

案例显示，当利率从8%变为7%或9%时，久期匹配的组合仍能基本覆盖负债，微小盈余源于久期的非线性变化。免疫化虽是被动策略，但需主动监控和再平衡，以应对市场变动和期限缩短带来的久期偏移。例如，养老金基金因资产久期短于负债，在利率下降时负债增速超过资产，导致赤字扩大。因此，精确的久期匹配和持续调整是利率风险管理的关键。