Advances in Economics, Business and Management Research, volume 203

Proceedings of the 2021 3rd International Conference on Economic Management and

Cultural Industry (ICEMCI 2021)

Bond Pricing and Immunization Strategy - Asset

Allocation in Bond Investment
Yuyao Wang1, a, † Bixia Han2, b, †
1
Rutgers Newark College, Northeast Normal University, Changchun, 130000, China
2
School of Management, Xiamen University, Fujian Province, 361005, China
a
wangyy085@nenu.edu.cn, b33120182202082@stu.xmu.edu.cn
†
These authors contributed equally.

ABSTRACT
Asset allocation is a widely used financial term that fits most people. However, even in the most basic concept of asset
allocation, there is still existing risk or uncertainty. This paper aims to discuss the impact of different influencing
factors on bond pricing in the bond market and then further discuss the risk aversion strategy that can be analyzed
through the duration, also known as the immunization strategy. We also conduct sensitivity analysis of four different
situations, changing different yield rates and coupons and changing increasing yield rates and coupons at a certain
rate. This paper found that different yield rates, different coupons, and increasing yield rates have a linear relationship
with the bond price. In contrast, the increasing coupon with a certain ratio has a nonlinear relationship. Through
immunization strategy, it can develop an asset portfolio composed by a certain ratio. Also, using the immunization
strategy can reduce the risks that investors need to take and maximize their interests to a certain extent. The
combination of a certain percentage of zero-coupon government bonds and corporate bonds effectively shields
investors from risk. This paper constructs the issue of bond pricing, which is of practical significance for investors to
evaluate their risk-bearing ability and purchasing ability and to use the immunization strategy of hedging risk.

Keywords: Bond Price, Duration, Immunization strategy, Investment portfolio.

1. INTRODUCTION
is of greatest concern. Therefore, many investors turn to
Bonds are one kind of debt security issued in various bond management strategies to protect their
connection with a borrowing arrangement that obligates bond portfolio from interest rate risk. There are overall
the issuer to make specified payments to the two categories of bond management strategies: passive
bondholders on specified dates [1]. Among all debt strategy and active strategy. This article will mainly
securities, including those in the money market and the focus on immunization strategy, one passive bond
capital market, bonds are among the most common management strategy used by many financial and funds
long-term debt securities traded in the capital market. A institutions.
developed bond market can provide local firms with
Next, we will review the literature on bond pricing
access to direct financing, enhance the liquidity of the
and immunization strategy. For defaultable bonds,
overall market and reduce systematic risk in the market
general pricing methods can be divided into two main
[2]. Therefore, given the importance of the bond market,
categories: structural models originated from the pricing
how to price bonds traded in the bond market precisely
model proposed by Merton and reduced-form models or
is one top priority for the healthy and sustainable
intensity models. Merton presented a simplified model
development of the bond market.
that can value each component of the firm’s liability
Moreover, bonds traded in the capital market are mix. Despite being simple and innovative, Merton’s
exposed to various kinds of risks, including default risk, pricing model was inevitably subject to its simplified
liquidity risk, and interest rate risk, and so on. For most assumptions about its liability structure and costless
bond portfolio managers, the sensitivity of bond prices bankruptcy. Therefore, many scholars, including Black
to changes in market interest rates, that is, interest rate and Cox, Leland and Fan, and Sundaresan, had
risk incorporated more realistic assumptions into the Merton
model [3]. For the default-free bond, Longstaff and

Copyright © 2021 The Authors. Published by Atlantis Press International B.V.

 Advances in Economics, Business and Management Research, volume 203
This is an open access article distributed under the CC BY-NC 4.0 license -http://creativecommons.org/licenses/by-nc/4.0/. 454
 Advances in Economics, Business and Management Research, volume

Schwartz found that the risk of a default-free bond immunization strategy and mentions immunization
stems from two major sources: interest rate shifts and strategy’s limitations; Section 6 presents our conclusions.
changes in bond market volatility, and that changes in
the volatility of interest rates have the greatest impact
2. BACKGROUND DESCRIPTION
on the prices and yields of intermediate-term bonds [4].
Over the past few decades, there are lots of studies Macaulay duration is of great significance to bond
focusing on bond portfolio management. Redington, in pricing. The calculated Macaulay duration can be
1952 first introduced the concept of immunization based combined with an immunization strategy to help
on the assumption of flat term-structure [5]. Later, investors avoid risks [7].
Fisher and Weil in 1971 modified Redington’s approach According to Frederick Robertson Macaulay’s
and expanded immunization strategy by taking into research, bond interest rate, maturity period, and yield to
account the natural shifts of interest rate [6]. After 1971, maturity are closely related to bond price. He proposed
several more advanced theories about immunization five theorems of bond pricing that are still regarded as
strategies were continuously proposed, including classic nowadays.
Bierwag and Khang’s maximin theory, Leibowitz and
Weinberger’s contingent immunization, and Bierwag In this research, we conduct about the relationship
and Khang’s multiple shocks, and so on. between bond price and different factors. Generally
speaking, bond pricing is divided into two dimensions:
In the paper, we first introduced discounted cash internal and external factors. Internal factors include the
flow concepts, duration, and modified duration. face value, the coupon interest; the validity period;
Secondly, we listed the pricing methods, including zero- whether it can be redeemed in advance; whether it is
coupon bonds, fixed-rate coupon bonds, and floating- convertible; liquidity, and the possibility of default.
rate coupon bonds. And then conducted a sensitivity External factors include risk-free interest rates, market
analysis of bond prices when changing bond yield rates interest rates, and inflation rates [8].
and coupon rates, respectively. After that, we studied
one passive bond management strategy, immunization Bond price always depends on his present value. The
strategy, and conducted an empirical test to construct present value in accounting measurement refers to the
one real situation where one investor needs to construct future cash flow value after being discounted at an
his bond portfolio to fund future payment and test the appropriate rate. It is a measurement attribute that
effectiveness of immunization strategy. In the last, we considers the time value of money. To understand how
listed some limitations about immunization strategy. to price a particular bond, the concept of present value
should be introduced first, which is needed to be
As for the sensitivity analysis found that bond prices obtained in the calculation and has some practical
and bond yields generally show an inverse relationship. significance for the understanding duration. Generally
When other conditions remain unchanged, as the yield speaking, there are two big categories of bonds: fixed-
rate increases at the same amount, the bond price falls at rate bonds and floating-rate bonds. A zero-coupon bond
a certain rate; while the yield rate increases at a certain is a kind of fixed-rate bond where the coupon rate
rate, the price of the bond falls more and more sharply. It equals zero. This kind of bond is issued at a discount,
is generally a positive correlation between bond price does not carry coupons, and pays a lump sum of
and coupon. When other conditions remain unchanged, principal and interest at face value on maturity.
as the coupon (coupon rate) increases at the same
amount, the price of the bond increases at a certain rate, The immunization strategy is aimed to manage the
while the coupon (coupon rate) increases at a certain bond portfolio. For any bond, interest rate fluctuations
rate, the price of the bond goes up more and more have opposite effects on long-term and short-term bonds
dramatically. As for the empirical test, we found that one [9]. By selecting a bond portfolio with a Macaulay
constructed bond portfolio can be immunized from the duration equal to the maturity of its liabilities (cash
fluctuation of interest rate when the interest rate changes outflow), the manager can ensure a fixed cash flow after
insignificantly. But as time goes by, the duration of a certain period by taking advantage of the offset
bonds and the duration of liability do not match, which characteristics of price risk and reinvestment risk.
means that the investors need to rebalance their bond This paper aims to propose an immunization strategy
portfolio. by integrating different bond varieties, which can lower
The remainder of the paper is organized as follows: the risk for the investors.
Section 2 describes basic background about bond
pricing, Macaulay duration, and immunization strategy; 3. PRICING THE BOND
Section 3 introduces bond pricing methods for different
kinds of bonds and performs one sensitivity analysis of Bond pricing is for risk management, so as
bond prices; Section 4 describes modified duration important as price is the sensitivity of the bond. For
based on the Macaulay duration; Section 5 introduces interest rate sensitivity, the most important is duration
immunization strategy, conducts an empirical test for and convexity.

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Mathematically, you can use these two to approximate a P0 is the price of the zero-coupon bond, F is the face
bond as a quadratic function. More importantly, the
value of a bond, r is yield to maturity (YTM), and n is
values and symbols of the two reflect important
the number of periods to maturity.
characteristics of bonds. For example, the convexity of a
normal bond is positive, but it's negative for mortgage- When we plug the zero-coupon pricing formula into
backed security (MBS). The reason is that when the the general duration formula above, we get the
interest rate is low, the price of ordinary bonds will rise, following formula:
but the price of MBS tends to level off. When the
𝐷0 = 𝑛 （4）
interest rate is low, the buyers will choose to exercise
the second loan to pay off the original high-interest D0 is the duration of a zero-coupon bond, n is the
mortgage, making it difficult for the price to exceed the number of periods to maturity.
parity.
This paper mainly discusses bond pricing based on 3.2. Fixed-Rate Bond
mathematical methods, which uses interest rate and
coupon to calculate the present value of bonds. In Fixed-rate bonds are bonds that pay the same
addition, this paper will compare the influence of interest rate for the entire term of the bond. Investors
different factors on the present value of bonds and can choose from fixed-rate bonds in the form of treasury
combine the concept of duration to make a tentative bills, corporate bonds, municipal bonds, or certificates
plan of immunization strategy. of deposit if they want a guaranteed interest rate for a
specified period.
The following is the mathematical formula for the
present value of bonds:
𝑃 = ∑𝑛
𝐶 𝐹 (1) 3.3. Floating Rate Bond
𝑛
𝑖=1 (1+𝑟)𝑛
+
(1+𝑟)𝑛
A floating rate fund is a fund that invests in financial
where C is periodic coupon payment, F is the face instruments that pays a variable or floating interest rate
value of a bond, r is yield to maturity (YTM), n is the [11]. A floating rate fund, which can be a mutual fund
number of periods to maturity. or an exchange-traded fund (ETF), invests in bonds and
The following formula is used to calculate the debt instruments whose interest payments fluctuate with
Macaulay duration [10]: an underlying interest rate level. Typically, a fixed-rate
𝐶𝐹𝑛 investment will have a stable, predictable income.
𝑛
(1+𝑟)𝑛
However, as interest rates rise, fixed-rate investments
𝐷
𝑖=1
𝑛 (2)
𝑃 lag behind the market since their returns remain fixed.
Floating rate funds aim to provide investors with a
where, CF is periodic cash flow, P is the present flexible interest income in a rising rate environment. As
value of a bond, r is yield to maturity (YTM), n is the a result, floating-rate funds have gained popularity as
number of periods to maturity. investors look to boost the yield of their portfolios.

3.1.Zero-Coupon Bond Next, according to the pricing formulas given above,

we want to investigate different factors’
Based on the above pricing model, we discuss the
different present values of various bonds such as a zero- 3.4. Sensitive Analysis
coupon bond, fixed-rate bond, bond with an increased
bond rate, and bond with different yield rates. Firstly, This section mainly studied the sensitivity of bonds’
the zero-coupon bond is issued at a discount that does value on different factors, including yield rate and
not carry coupons and pays a lump sum of principal and coupon rate.
interest at face value on maturity. The biggest feature of
zero-coupon bonds is that they avoid the risk of 3.4.1. Different Yield Rates
reinvesting the interest earned by the investor. Investors
get a discount (that is, a discount to their face value) Now assume that there are five different bonds, and
when they buy a zero-coupon bond and receive the face they all have the same face value of 1000$, coupon of
value at maturity. Because of these characteristics, zero- 100$, 3 years maturity. We are going to analyze the
coupon bonds are particularly sensitive to changes in relationship between the yield to maturity and present
interest rates. In general, bond prices and interest rates value by using different yield rates:
change in an inverse relationship.
The following formula is used to calculate the zero-
coupon bond:

𝑃0
= (1+𝑟)𝑛
𝐹 (3)

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Figure 1 The relationship between different yield rate

Figure 3 The relationship between different coupon
and bond price
(coupon rate) and bond price
The figure 1 shows that the bond price will decrease
with the different yield rate growing at the same Figure 3 indicates that the bond price will stably
amount. increase with the different coupons (coupon rate),
growing at the same amount.
3.4.2. Increasing Yield Rates
3.4.4. Increasing Coupon Rates
Now make an assumption very similar to the one
above, and there are five different bonds, and they all Five different bonds have similar conditions as we
have the same face value of 1000$, coupon of 100$, 3 mentioned before, and they all have the same face value
years maturity. We are going to analyze the relationship of $1000, the yield rate of 10%, 3 years maturity. We
between the increasing yield to maturity and present will analyze the relationship between the increasing
value by using various yield rates at the increase rate coupon and present value by using various coupon rates
a=10%: of 10% and at the increase rate m=10%.

Figure 4 The relationship between increasing coupon

(coupon rate) and bond price
Figure 2 The relationship between increasing yield rate
Figure 4 indicates that the bond price will increase
and bond price
more and more sharply with the different coupon
Figure 2 shows that the bond price will decrease (coupon rate), which is growing at a stated rate.
more and more sharply with the increasing yield rate, In conclusion, we discussed the general relationship
which is growing at a stated rate. between bond prices and yield rate and coupon rate.
Bond prices and bond yields generally show an inverse
3.4.3. Different Coupon Rates relationship. When other conditions remain unchanged,
as the yield rate increases at the same amount, the bond
After discussing the relationship between the bond price falls at a certain rate, while the yield rate increases
yield and the price, let’s talk about another five different at a certain rate, the price of the bond falls more and
bonds, and they all have the same face value of $1000, more sharply. There is generally a positive correlation
the yield rate of 10%, 3 years maturity. We will analyze between bond price and coupon. When other conditions
the relationship between the increasing coupon and remain unchanged, as the coupon (coupon rate)
present value by using various coupon rates of 10%. increases at the same amount, the price of the bond
increases at a certain rate, while the coupon (coupon
rate) increases at a certain rate, the price of the bond
goes up more and more dramatically.

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4. MODIFIED DURATION 5. IMMUNIZATION STRATEGY

In this section, we introduced the concept of In this section, we introduced one passive bond
modified duration and gave the formula of it. What is management strategy, that is, the immunization model
more, based on the limitation of duration in estimating clarified its main principles and gave its formula.
the bonds’ price change, we discussed the advantage of Moreover, to test its immunization effect, we conducted
using modified duration to calculate the percentage one empirical test and drew the corresponding
change of bonds’ price when the interest rate changes. conclusion. Finally, we discussed its limitations.
Modified duration is defined as
𝐷 5.1. Immunization model
𝐷∗ =(1+𝑦) (5)
Immunization strategy is one passive bond
y refers to the bond’s yield to maturity; D refers to the management strategy used by investors to shield their
duration of one bond. assets from exposure to interest rate risk. To be specific,
Modified duration derives from Macaulay’s this strategy seeks to establish a virtually zero-risk bond
duration. Macaulay’s duration measures the weighted portfolio, in which interest rate fluctuation has no
average of the times to each coupon or principal impact on the firm's value.
payment, which indirectly implies the sensitivity of one Before explaining the immunization strategy in
bond to interest risk. Macaulay’s duration can also be detail, we need to clarify two components of interest risk
expressed as the following formula, that is, first. In general, if the interest rate changes after one
∆𝑃 investor purchased one bond, he may be exposed to two
specific
𝑃
𝐷= ∆(1+𝑦) (6) risks. The first one is price risk, which means that the
(1+𝑦)
bond’s terminal price will be affected by the interest rate
where ∆𝑃 refers to the monetary change of bond’s price; change. Another one is coupon reinvestment risk
P refers to bond’s price; y refers to the bond’s yield to resulting from reinvesting the coupons at interest rates
maturity. different from the yield to maturity on the bond at the
time of purchase. The total effect of these two risks
However, in reality, most investors care about the
varies in opposite directions when the interest rate
proportional price change when the bond’s yield
changes. For example, when the interest rate increase,
changes. Using duration to calculate the percentage of a
the price of a bond will decrease, while the reinvested
bond’s price change is not straightforward enough.
yield of coupons will increase, and vice versa. [13]
Therefore, modified duration was proposed to gauge the
changes in the value of an asset that occurs in response Therefore, an effective immunization requires that
to a parallel shift in interest rates [12], which greatly these two opposite effects offset each other, thereby
simplifies the calculation of the percentage of bond’s guaranteeing that the asset's future value will not be
price change. By using modified duration, we can get affected by the fluctuation of interest rate.
the following formula, that is,
Actually, the concept of immunization was first
𝐷 ∆𝑃 ∆𝑃 proposed by F.M.Redington, an actuary for a life
𝐷∗ = = ∆(1+𝑦)
𝑃
= 𝑃 (7) insurance company, in an article ‘Review of the Principle
(1+𝑦) ∆𝑦

where, ∆𝑃 refers to the monetary change of bond’s of Life-Office Valuations’ in 1952. He assumed a flat
price; P refers to bond’s price; y refers to the bond’s term structure and then showed that if the duration of an
yield to maturity. asset portfolio matched the duration of liabilities,
changes in interest rates over time would affect the
From the perspective of calculus, the modified
value of liabilities to the same extent as these changes
duration can be expressed as the first derivative of the
affected the value of assets.[14]
bond’s price with respect to changes in the bond’s
yield divided by bond price. Thus, the formula for the There are overall two steps in using the
modified duration can also be expressed as: immunization strategy. The first step is to match the
duration of asset and liability investors hold in their
portfolios. The same
1 𝑑
𝐷∗ = − 𝑃 (8) asset and liability duration implies that assets and
× 𝑑𝑦 liabilities have the same sensitivity to interest rate
𝑃
In this way, the percentage of a bond’s price change fluctuation.
is just the product of modified duration and the change
The formula can be expressed as below:
in bond’s yield, which makes the calculation much
easier
𝑛 𝑚
than before. ∑𝑖=1 𝐷𝑖 × = 𝑗=𝑖 𝐷𝑗 × (9)
𝑤𝑖 𝑤𝑗

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Di refers to the duration of bond i; wi refers to the unit of each bond, we can get the exact units needed to be
portfolio share of bond i; Dj refers to the duration of bought.
liability j; wj refers to the share of liability j.
Through the calculations, the units of government
The second step is to match the present value of bond and corporate bond needed to be bought
assets and liabilities. These two rules can, to some respectively are 90.07 and 74.14.
extent, guarantee that assets’ and liabilities’ value
change can offset against each other when interest rate Therefore, from the above working process, we can
fluctuates. conclude that the student should allocate 50.94% of his
fund to invest zero-coupon bond and 49.06% of his fund
to invest corporate coupon bond to make the one-off
5.2. Empirical test
payment three years later.
5.2.1. Constructing the empirical model
5.2.2. Testing the effect of change of interest rate
In this section, we made an empirical test to examine
After constructing the portfolio, the next thing we
the effectiveness of the immunization strategy.
need to do is to test its immunization effect, that is,
Try to assume that one student will need to make a whether this strategy can shield the portfolio from
one-off tuition payment of $17,000 after three years interest risk.
from now. He wonders how to construct the portfolio to
Therefore, we list several scenarios to simulate the
fund future payments. There are two kinds of bonds in
fluctuation of interest rates in the market.
the financial market:
And in each scenario, we can calculate the value of
The first is one zero-coupon government bond, with
assets after three years. The formula of the future value
a face value of $100, a current price of $85.48, and
of assets is given below:
remaining life of 4 years.
Future value of assets
The second is one corporate bond, with a face value
of $100, the current price of $100, an annual coupon of = value of zero-coupon bond + value of coupon bond
$4, and remaining life of 2 years.
Assume all debt has the same level of default risk.
The benchmark yield curve for fixed-income =
face value of zero−coupon bond × units +
(1+discount rate)
investments of the same riskiness is flat at a level of 4% annual coupon of corporate bond
p.a. [ (1+discount rate)2 +
annual coupon+face value
Using the Macaulay duration formula, we can ] × units
(1+discount rate) (
calculate the duration of the zero-coupon bond and the
corporate bond, respectively 4 years and 1.96 years. Figure 5 gives the sensitivity of the future value of
What is more, the duration of the liability is three assets on changes in the interest rate, in a wide range
years. between 2.5% and 5.5%.

For the duration is additive, we can construct the

following formula to match the duration of assets and
liabilities:
𝑤𝐺𝐵 × 4 + 𝑤𝐶𝐵 × 1.9615 = 3(10)
where, wGB refers the proportion of zero-coupon
government bond in the bond portfolio; wCB refers the
proportion of coupon corporate in the bond portfolio.
wGB = 50.94%
wCB = 49.06% Figure 5 Sensitivity of Future Value on Changes in
What is more, we can get the present value of the the Interest Rate
liability to be:
From the above graph, it can be easily observed that
17000 when the interest rate changes insignificantly, the future
(1+4%)3 = 15,112.94 (11)
value of assets changes little, around $17000, which
Therefore, we can get the present value and exact means that despite the fluctuation of interest rate, the
units of each bond we should hold. By multiplying the asset's future value can still keep unaffected.
proportions of two bonds and the total money the
investor has, we can get the respective total amount of
each bond the investor should hold. And then, we know
the price per

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5.3 Some limitations about immunization This paper discusses three aspects: bond pricing
strategy modified duration and immunization strategy.
According to the above analysis, there still exist Firstly, we use sensitivity analysis to study the effect
some limitations to the immunization strategy. First, the of yield rate and coupon rate on the bond price under
duration of bonds will be affected by various factors, different growth conditions. Secondly, we give some
including bonds’ residual maturity and interest rate, details on the modified duration to simplify the
which can be directly seen through the formula of calculation of bonds’ price change when a percentage
Macaulay duration. The change of interest rate will change of bond’s yield is given. To be more specific, the
influence the expected stream of each future payment percentage of a bond’s price change is just the product
and thus the price of a bond. Thus, when the interest of modified duration and the change in the bond’s yield.
rate changes, the duration of one bond will inevitably Thirdly, from the empirical test, it can be known that
vary. Even if we assume that the interest rate does not once the investor matches the duration of the bond
change, the bonds’ duration will still change with time. portfolio and the duration of liability, the investor can
For coupon bonds, duration generally decreases less shield his portfolio from the fluctuation of interest rate
rapidly than maturities dose. However, for the one-off when it changes insignificantly. However, there are
payment liability, its duration equals its maturity. Thus, some limitations to the immunization strategy, including
as time passes, bond portfolio and liability duration will rebalancing, duration limitations, and the assumption of
fall at different rates [15]. the flat term structure of interest. There is a possible
portfolio of bonds by using the immunization strategy.
Therefore, when these factors change, the duration The first is one zero-coupon government bond, with a
of bonds varies as well. The duration of assets and that face value of $100, a current price of $85.48, and
of liability become unmatched. The immunization remaining life of 4 years. The second is one corporate
strategy necessitates continuous rebalancing so that bond, with a face value of $100, a current price of $100,
investors can guarantee that their portfolio is well an annual coupon of $4, and remaining life of 2 years.
immunized. Therefore, from the above working process, we can
Secondly, because immunization strategy is based conclude that the student should allocate 50.94% of his
on duration, immunization strategy is inevitably subject fund to invest zero-coupon bond and 49.06% of his fund
to the limitation of duration. According to the modified to invest corporate coupon bond to make the one-off
duration formula, which directly derives from Macaulay payment three years later.
duration, we know that it implies that the percentage of Although this paper has a sufficient theoretical
a bond’s price change is directly proportional to the structure and derivation process, it still has some defects
change in the bond’s yield. However, if we draw the in the actual process of data collection and innovative
graph of the percentage change in bond price as a modeling. In addition, according to the actual policy and
function of the change in its yield, it can be obviously the current situation of the world economic
found that the plot is not linear. For small changes in environment, this paper should also make some
bond yield, the duration’s approximation is quite corresponding mention and adjustment. Future research
accurate, while for significant changes in bond yield, the should pay more attention to the macro direction of the
duration’s approximation is inaccurate to some extent. world economy to provide the detailed allocation of
Thirdly, the immunization strategy mentioned before bond portfolio and data simulation results for
is based on the concept proposed by F.M.Redington. He subsequent researchers to study
assumed the term structure of interest to be flat so that
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