Lecture 10 h08

Portfolio Management
3-228-07
Albert Lee Chun
Duration, Convexity and
Bond Portfolio Management
Strategies
Lecture 10
27 Nov 2008
0
Bond Portfolio Management
The major source of risk facing bond portfolio
managers has to do with shifts in interest rates.
In this session:
1. We examine how bond prices respond to changes in
this source of risk.
2. We discuss ways of constructing bond portfolios to
insulate against this risk.
Albert Lee Chun
Portfolio Management
1
Today







Review of Bond Fundamentals
Term Structure of Interest Rates
Interest Rate Risk
Sensitivity of Bond Prices to Interest Rates
- Duration
- Convexity
Portfolio Immunization
Yield Curve Strategies
Some Fun Examples
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2
Review of Bond Fundamentals
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3
Bond Definition
Definition: A bond is a debt security that a
corporation or a government issues to borrow on a
long-term basis.
 Normally an interest-only loan (when issued at par)
- Borrower pays only the interest but none of the
principle is paid until the end of the loan.
 Interest is paid in the form of a periodic coupon.

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4
Components of Bond

Bond prices depend on 4 factors
1.
Par or Face Value
Coupon Rate
Number of years to maturity
Yield to Maturity (YTM) The “interest rate” that
makes the discounted present value of the bond’s
coupons equal to its market price (also called
internal rate of return of a bond).
2.
3.
4.
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5
Yield to Maturity



The yield to maturity (YTM) is an interest rate such
that present value of the bond’s coupons and the face
value equals the current market price.
Often simply called the bond’s yield as in “The yield
on the 10 year bond is 5%.”
Important: YTM is quoted as an Annual percentage
rate (APR).
Bond prices are inversely related to yields.
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Equation for Bonds
Suppose a bond pays 1 coupon per year


1
F
PV 
 1 

t 
t
r 
(1  r )  (1  r )
C
Present Value = PV(coupon payments) + PV(face value)
= PV of an annuity + PV of F




F = face value
C = annual coupon amount
= F × coupon rate
r = yield to maturity
t = number of years until maturity
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Equation for Bonds
For a bond paying a semi-annual coupon (2 times a year)


1
F
PV 
 1 

t2 
t2
r/2 
(1  r / 2 )  (1  r / 2 )
C /2




F = face value
C = yearly coupon amount
= F × coupon rate
r = yield to maturity
t = number of periods until maturity
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Equation for Bonds
More generally, if a bond pays m coupons per year


1
F
PV 
 1 

tm 
tm
r/m 
(1  r / m )
(
1

r
/
m
)

C /m





F = face value
C = yearly coupon amount
= F × coupon rate
r = yield to maturity
m = number coupons per year
t = number of years until maturity
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Bond Details

The coupon rate, the face value (or par value) and the
maturity date are all determined by the bond issuer
(corporation or government).

Treasury Bills – One year and less; no coupons
Treasury Notes – Between 2 and 10 years; coupons.
Treasury Bonds – Longer than 10 years; coupons.


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Bond Example
Eggbert’s Egg Company

Simple Example: Eggbert’s Egg Co. issues a bond with time to
maturity of 7 years; the yield to maturity is 10% APR. The
firm pays an interest rate in the form of $40 every 6 months
for 7 years, and a $1,000 principal at the end of 7 years.
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11
Example: Valuing a Bond
Eggbert’s Egg Co. issues a semiannual coupon paying
bond with a coupon rate of 8% and a face value of
$1,000 that matures in 7 years. If we assume, that the
yield to maturity is 10%, what is the price of this bond?
The bondholder receives a payment of $40 every six
months (a total of $80 per year or 8% per year)

1 

1 - 1.05 14  1, 000


Bond Price  40 

 901 . 01
14
0 . 05
1 . 05
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PAR Bonds


The price of a par bond is equal to F.
A par bond has YTM = coupon rate.
Why? Let’s take YTM = coupon rate = r. F = 1000.
Suppose at time t, P(t) = 1000(1+r). At time t-1, the value of the
bond is:
=1000(1+r)/(1+r)
=1000
By induction, since this is true at time T-1, this is true for all t.
Hence, when YTM = coupon rate, PV = 1000.

Most bonds are issued at par, with the coupon rate set
equal to the prevailing market yield.
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Example: Valuing a “Par Bond”

Suppose you are looking at a bond that has a 10%
coupon rate and a face value of $100. There are 10
years to maturity and the yield to maturity is 10%.
What is the price of this bond?

Using the formula:
P
= PV of annuity + PV of F
 P = 5[1 – 1/(1.05)20] / .05 + 100/ (1.05)20
 P = 100.00
Price = F=100.
It’s a par bond.
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14
Example: Yield to Maturity


The price of Eggbert’s Egg Co. is currently trading at 1,200.
The time to maturity is 4-years with a 14% annual coupon.
What is the yield to maturity?
The equation to solve is:
1,200 

140
r


1
1
 1 
 1,000 
4
4
(
1

r
)
(
1

r
)


Using Goal Seek in Excel we get YTM = 2 x r = 7.96%.
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15
Term Structure of Interest Rates
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The World is Not Flat
In earlier courses, you were presented with a rather
simplified view of interest rates, with a constant
single interest rate used to discount all cash flows.
The yield to maturity was assumed to be the same for all
bonds.
In reality, bond of different maturities will have
different yields to maturity.
The yield curve is not flat.
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Term Structure of Interest Rates


The term structure of interest rates gives the relation
between the time to maturity and the yield to
maturity of a bond.
Yield curve – graphical representation of the term
structure.
 Normal – upward-sloping, long-term yields are
higher than short-term yields
 Inverted – downward-sloping, long-term yields are
lower than short-term yields.
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Canada Yield Curve
November 2002
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Canada Yield Curve
May 2006
Canada Yield Curve as of May 2006
4.60%
4.50%
Interest Rate
4.40%
4.30%
4.20%
4.10%
4.00%
3.90%
0.25
2
3.75
5.5
7.25
9
10.75
12.5
14.25
16
17.75
19.5
21.25
23
24.75
26.5
28.25
30
Maturity
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Interest Rate Risk and the Yield Curve
The U.S. central bank sets the short term interest rate –
the fed funds rate.
Long term interest rates are the functions of expected
short-term interest rates, in addition there is a riskpremia associated with uncertainty about the
underlying economy forces, such as the growth rate
of the economy, inflation, etc.
Fed Chairman:
Ben Bernanke
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Links to Macro-economy




The central bank will respond to macroeconomic
forces in setting the interest rate.
Thus, the yield curve embeds information about
current and future macroeconomic conditions.
Conversely, it can serve as a barometer of the
economy.
Inverted yield curves are a leading indicator of
recessions.
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22
Price Sensitivity to Interest Rates
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Interest Rate Risk
Interest Rate Risk ↑ as Time to Maturity ↑
 Interest Rate Risk ↑ as Coupon Rate ↓
 Interest Rate Risk ↑ as Yield to Maturity ↓

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Interest Rate Risk and Maturity
Longer maturity bond prices are more sensitive to
changes in yields than shorter maturity bonds. Interest
rate risk is larger for longer maturity bonds
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Intuition




We can see that the slope of the price-yield curve as a
function of the interest rate is much steeper for the
30-year bond than for the 1-year bond.
The steeper price-yield curve, the more sensitive the
bond is to interest rate changes.
A large portion of a bond’s value is due to the
payment of the face value.
If the bond has a longer maturity, the cash flow from
the payment of the face value is realized further in the
future, this makes it more sensitive to changes in
interest rates.
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Intuition

Why? Because even a small change in the interest rate can
have a significant effect if it is compounded over a longer time.

Let’s see an example:
20 year bond: Price = 1000x(1+r)-20
10 year bond: Price = 1000x(1+r)-10
10 years
20 years
7%
508.3493
258.419
8%
463.1935
214.5482
% Change
0.097488
0.20448
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Interest Rate Risk and the Coupon Rate



Bonds with higher coupon rates are less sensitive to
changes in yields. Interest rate risk is inversely related
to the coupon rate.
Why? because their value depends less on the
discounted face value.
Higher coupon payments move the average maturity
of the bond’s cash flows forward in time, making the
bond less sensitive to the face value payment.
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Coupon Rates and Sensitivity
Zero coupon bonds are more sensitive to yield changes.
Sensitivity increases with maturity.
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Changes in Bond Prices
Bond Coupon Maturity Initial
YTM
A
12%
5 years
10%
B
12%
30 years
10%
C
3%
30 years
10%
D
3%
30 years
6%
Change in yield to maturity (%)
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A
B
C
D
30
Sensitivity of Prices



As the maturity increases from A to B, the bond
becomes more sensitive.
As the coupon rate decreases from B to C, the bond
becomes more sensitive.
As the initial yield to maturity decreases from C to D,
the bond becomes more sensitive (for coupon paying
bonds). At lower yields, the more distant payments
have relatively greater present values and account for
a greater share of the bond’s total value.
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Trading Strategies

Suppose you expect that the Federal Reserve will
lower interest rates at the next FOMC meeting. You
want to build a portfolio of bonds such that you
maximize the gain in the value of your portfolio.
Would you construct a portfolio with
A. Maximum interest rate sensitivity?
B. Minimum interest rate sensitivity?
Answer: A. You want to build a portfolio that will
maximize the price appreciation for a negative change
in interest rates, that is one with maximum interest
rate sensitivity.
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32
Trading Strategies

So how can we construct a portfolio that is most
sensitive to interest rates movement?
We know that we want to choose:
- Portfolio of long maturity bonds over a portfolio of
short maturity bonds
- Portfolio of low coupon bonds over a portfolio of
high coupon bonds
In other words, ideally you would want to hold a
portfolio of long maturity zero-coupon bonds!
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Trading Strategies



Suppose we want to hold a bond portfolio but suspect
that interest rates are about to rise. What should we
do?
We would want to hold a portfolio with minimum
interest rate sensitivity.
Short-maturity bonds with high coupons.
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34
Duration
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The Duration Measure
n
D
n
Ct
 (1  r )
t
t 1
n
t

Ct
 (1  r )
 PV (C )  t
t
t 1
Price
t
t 1
Developed by Frederick R. Macaulay
t = time at which coupon or principal payment occurs
Ct = interest or principal payment that occurs at time t
r = yield to maturity on the bond
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Characteristics of Macaulay Duration





A zero-coupon bond’s duration equals its
maturity
Duration of a bond with coupons is always less
than its term to maturity because duration gives
weight to these interim payments
There is an inverse relationship between duration
and coupon size.
There is a positive relationship between term to
maturity and duration, but duration increases at a
decreasing rate with maturity.
There is an inverse relationship between YTM and
duration.
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Bond Pricing with Continuous Compounding
Note: This is not done in practice, used mostly in academic models, but
it will be useful to illustrate the idea of duration.
Continuous
Compounding
Semi-annual
Compounding
is the price of the i-th cash flow
at time
,
given the yield to maturity
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Continuous Compounding
• If we compound more and more frequently...
• Every minute, every second, every millisecond...
• Then in the limit, we have continuous compounding...
Letting x = YTM and taking the inverse, that is
1/exp(x), we have our 1-year continuous
compounding discount factor.
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Bond Pricing
Price of a
bond with n
coupon
payments.
In General
Continuous
Compounding
where
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Duration (Continuous Compounding)
Duration is the weighted average maturity of a bond's cash
flows.
Note that for continously compounded yields we have:
Cash flow times t
n
weighted by
C

exp(

rt
)

t
 t
present value of
t 1
D
cash flow.
Price
where Ct is the cash flow
at time t, and r is the yield to maturity
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More Generally
For the case of a fixed income security with continuous
compounding with n cash flows Ci each occurring at time ti
is the price,
the i-th cash flow
at time
and
is the yield to maturity
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Sensitivity of Price to Yield
Differentiate price with respect to the yield to maturity.
So given continuously compounded yields, duration is the
(negative) percentage chance in price to change in the yield.
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Modified Duration
For yields compounded m time per year, an adjusted
measure of duration can be used to approximate
the percentage change in price to a change in
yield:
modified duration  Dmod 
Macaulay duration
1
y
m
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Modified Duration
W e k n o w th a t
P 
C
1 
y

1
C
1 
y
2
 ... 
C
1 
y
T

F
1 
y
T
w h e re C is th e c o u p o n , F is th e fa c e v a l u e ,
y is th e yie ld o f th e b o n d , a n d T th e m a tu rity.
D iffe re n tia tin g w ith re s p e c t to y w e g e t th a t
P
y


  1 C
2
1  y 
1
1 
y

 2  C
3
1  y 
 1* C


1

 1  y 
 .. . 
 T  C
T 1
1  y 
2*C
1 
y
2
 ... 

 T  F
T 1
1  y 
T *C
1 
y
T
T *F

1 
y
T




D iv id in g w ith P o n b o th s id e s w e g e t th a t
P 1
y P


1
1 
1
1 
 
Albert Lee Chun
y
y
 1* C


1
1

y





1
1 
y
t*
2*C
1 
 C ash
* D  D
y
2
 ... 
T *C
1 
y
T
T *F

flo w a t tim e t  / 1  y 
1 
y
T
 1

P


t
B o n d P ric e
*
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Modified Duration and Bond Price Volatility
Bond price movements will vary proportionally with
modified duration for small changes in yields
An estimate of the percentage change in bond prices equals
the change in yield multiplied by modified duration
P
P
  Dmod  y
P = change in price for the bond
P = beginning price for the bond
Dmod = the modified duration of the bond
y = yield change
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46
Test your Intuition Again
Suppose you are hired to manage Eggbert Kapital
Management’s portfolio of bonds, and you expect a
significant decrease in interest rates. Should you
invest in long maturity bonds with low coupons, or
low maturity bonds with high coupons? High
duration bond or low duration bonds?

Answer: You should invest bonds whose price will go
up the most if interest rates fell, so pick bonds with
long maturities and low coupon rates. You would
want to pick high duration bonds!
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47
Interpretation of Duration
8-year, 9% annual coupon bond
1200
Cash flow
1000
800
Bond Duration = 5.97 years
600
400
200
0
1
Blue: present value
of each cash flow.
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2
3
5
4
6
7
8
Year
Fulcrum
Point
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Bond Duration in Years Under Different Terms
YTM=6%
COUPON RATES
Years to
Maturity
8
1
5
10
20
50
100
0.02
0.04
0.06
0.08
0.995
4.756
8.891
14.981
19.452
17.567
17.167
0.990
4.558
8.169
12.980
17.129
17.232
17.167
0.985
4.393
7.662
11.904
16.273
17.120
17.167
0.981
4.254
7.286
11.232
15.829
17.064
17.167
Source: L. Fisher and R. L. Weil, "Coping with the Risk of Interest Rate Fluctuations:
Returns to Bondholders from Naïve and Optimal Strategies," Journal of Business 44, no. 4
(October 1971): 418. Copyright 1971, University of Chicago Press.
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Bond Duration vs Maturity
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Portfolio Management
15-50
50
Modified Duration
 Dmod P 
dP
dy
This is the first derivative of price with respect to the
yield.
For small changes this will give a good estimate, but
this is a linear estimate on the tangent line
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51
Example



Eggbert’s Egg Company issued an 8% bond with 3
years to maturity. The yield to maturity is 8% and the
bond pays semi-annual coupons.
What is the Macaulay duration? What is the Modified
Duration?
(See Spreadsheet)
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52
Example
Year
Payment
PV
Weight
Year*Weight
0.5
4
3.846154
0.038462
0.019231
1
4
3.698225
0.036982
0.036982
1.5
4
3.555985
0.03556
0.05334
2
4
3.419217
0.034192
0.068384
2.5
4
3.287708
0.032877
0.082193
3
104
82.19271
0.821927
2.465781
100
1
2.725911
Price
D = 2.5726 and Dm = 2.621
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Short Cut Formulas
Macaulay Duration “Short Cut” Formulas
D 
1 y
y
1  y   T c  y 

T
c 1  y   1  y
D 
1 y /2
y

Semi-annual coupons
1  y / 2   2T c / 2  y / 2 
2T
c 1  y / 2   1  y
Modified Duration “Short Cut” Formula
D 
*
1

1  T c  y  / 1  y 
y


c 1  y   1  y
T
where c = coupon rate per period
y = yield per period
T = periods remaining
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54
Price-Yield Curve
Price
Yield to Maturity
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First-order Approximation


Duration provides a first-order approximation of the price-yield
curve.
Approximating a curve with a straight line
Price
Approximation
Error
slope   Dmod P
Yield
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Convexity
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Convexity
The convexity is the measure of the curvature and is
the second derivative of price with respect to yield
(d2P/dy2) divided by price
Convexity is the percentage change in dP/dy for a
given change in yield
2
d P
Convexity 
dy
2
P
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Correction for Convexity
We can better approximate a curve using a quadratic function.
P
2
1
  D  y 
 Convexity  (y )
2
P
 CFt

2
Convexity 
(t  t )
2 
t
P  (1  y ) t 1  (1  y )

1
T


2T 
CFt / 2
1
t t
1 
Convexity 
( )(  )


y 2 t 1
y t 2 2 2


P  (1  )
(1  )


2
2
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Semi-annual
coupons
59
Duration plus Convexity

The Taylor Series for a function P around y is given by
P(y+Δy) = P(y) + P’(y) Δy + ½ P’’(y) (Δy)2 + ...

Thus, as a second order approximation
P(y+Δy) - P(y) ≈ P’(y) Δy + ½ P’’(y) (Δy)2

Therefore,
ΔP ≈ - Dm P Δy + ½ P C (Δy)2
2
d P
dy
Albert Lee Chun
Portfolio Management
2
 C P
60
Duration plus Convexity



Thus for a small change in the yield Δy, the modified
duration and the convexity give the second-order
approximation to the price-yield curve.
ΔP ≈ - Dm P Δy + ½ P C (Δy)2
Price change due to duration
- Dm P Δy
Price change due to convexity
½ P C (Δy)2
Albert Lee Chun
Portfolio Management
61
Convexity of Bonds
Portfolio
0
Duration
Duration +
Convexity
Change in yield to maturity (%)
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Portfolio Management
62
Duration-Convexity Effects



Changes in a bond’s price resulting from a change in
yields are due to:
 Bond’s modified duration
 Bond’s convexity
Relative effect of these two factors depends on the
characteristics of the bond (its convexity) and the size
of the yield change
Convexity is desirable
Albert Lee Chun
Portfolio Management
63
Portfolio Immunization
Albert Lee Chun
Portfolio Management
64
Protection from Interest Rate Risk



Pension funds, insurance companies and other
financial institutions hold billions of dollars in fixed
income securities.
One of the most widely used analytical techniques is
known as immunization, as it protects a bond
portfolio from interest rate movements.
This is of major practical value so let’s learn how to
structure a portfolio to protect it against interest rate
risk!
Albert Lee Chun
Portfolio Management
65
2 Scenarios


Scenario 1: You want to save money for a major
expense one year from now.
Scenario 2: You want to save money to pay for your
kid’s college tuition 20 years from now.
Given scenario 1: If you invest in one year treasury
bills, there is very little risk. Investing in 20 year
treasuries exposes you to interest rate risk.
Given scenario 2: Holding 20 year treasury bonds
would provide predictable results, investing in one
year T-bills would result in reinvestment risk.
Albert Lee Chun
Portfolio Management
66
Life Insurance Company



Suppose you work for a life insurance company, and
you expect to make a series of cash payments.
One thing you can do is to purchase zero-coupon
bonds, having different maturities so that the
principal exactly matches each separate obligation.
This may not be the best option as corporate bonds
offer higher yields and zero-coupon corporate bonds
are rare.
Albert Lee Chun
Portfolio Management
67
Matching Durations



Match the duration of your portfolio with the duration
of your obligations.
Intuition: If the duration of your bond portfolio
matches the duration of your obligation stream, then
the present value of both your bond portfolio and
your obligation stream will respond (to a first-order
approximation) to a change in the underlying yield.
Specifically, if yields decrease, the present value of
your obligations will increase, but the value of your
bond portfolio will increase (approximately) by the
same amount – so the value of your portfolio will be
enough to cover the obligation!
Albert Lee Chun
Portfolio Management
68
Duration for a Portfolio

If the portfolio has price PV
PV = V1 + V2 + V3 + ... + Vm
D = w1D1 + w2D2 + ... + wmDm
where wi = Vi/ PV in a portfolio of m bonds. Vi is
the value of Bond i in the portfolio.
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Portfolio Management
69
Example


Suppose Eggbert’s Egg Company has to pay for a
new chocolate factory 10 years from now. The cost of
the factory is 1 million dollars and it wishes to invest
that money now.
Suppose no zero-coupon bonds of that maturity are
available. Suppose there are 3 corporate bonds
(FV=100) to choose from:
1
2
3
Coupon
6%
11%
9%
Albert Lee Chun
Maturity
30
10
20
Price
69.04
113.01
100.00
Portfolio Management
Yield
9%
9%
9%
70
Example
Duration of each Bond:
D1 = 11.44
D2 = 6.54
D3 = 9.61
Since Bond 1 has duration greater than 10, we must use
this bond in our portfolio.
Let’s use bonds 1 and 2. (See Spreadsheet)
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Portfolio Management
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Example



Present Value of the obligation is: $414, 642.86
The duration of the obligation is 10 years.
To immunize the portfolio we need to
1. Equate the present value of the portfolio with the
present value of the obligation.
2. Equate the duration of the portfolio with the
duration of the obligation.
Albert Lee Chun
Portfolio Management
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Example

The value of total invested in the Bonds 1 and Bond 2 has to
equal to PV of the obligation
V1 + V2 = PV = $414, 642.86

And the duration of the bond portfolio has to equal the
duration of the obligation
V1/PV * D1 + V2/PV* D2 = 10

We get:
V1/PV * D1 + (PV – V1) /PV* D2 = 10
=> One equation and 1 unknown.
Albert Lee Chun
Portfolio Management
74
Example
From the spreadsheet:
V1 = 292617.60
V2 = 122025.26
We want to purchase
V1/P1 = 4238.20 shares of Bond 1
V2/P2 = 1079.80 shares of Bond 2
Of, course, in practice, you will have to round these numbers.
Albert Lee Chun
Portfolio Management
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Example




Suppose there is a sudden shift in the yield to either
8% or 10%.
The value of the bond portfolio is:
Portfolio
Obligation
Error
8%
457928.3
456386.9
+1541.33
10% 378075.2
376889.5
+1185.69
Due to convexity, the portfolio is always worth more
than the obligation.
Albert Lee Chun
Portfolio Management
76
Homework




See if you can replicate the previous example at
home, without looking at the spreadsheet.
If you look at the spreadsheet answer first, you will
think that this is very easy.
Try doing it from scratch.
Replicate the immunization problem using only
Bonds 1 and 3.
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Portfolio Management
77
Yield Curve Strategies



Bullet strategy: maturity of the securities are highly
concentrated at one point on the curve.
Barbell strategy: maturity of securities included in the
portfolio are concentrated at two extreme maturities.
Ladder strategy: the portfolio is constructed to have
approximately equal amounts of each maturity.
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Portfolio Management
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Yield Curve Strategies
Bond
Coupon
Maturity
Price
YTM
Duration_
Convexity
A
0.085
5
100
0.085
4.0054435
19.81635
B
0.095
20
100
0.095
8.8815081
124.1702
C
0.0925
10
100
0.0925
6.43409
55.45054
Bullet
Portfolio:
100% bond C
Barbell
Portfolio:
50.2% bond A and 49.8% bond B
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Portfolio Management
79
Yield Curve Strategies
Duration of Bullet:
Duration of
Barbell:
Convexity of
Bullet:
6.43409
6.433724
Yield of Bullet:
0.0925
Yield of Barbell:
0.08998
This is the convexity yield
55.45054
0.00252
Convexity of
Barbell:
Albert Lee Chun
Give up yield to get better convexity.
71.78459
Portfolio Management
80
Barbell vs. Bullet




The choice between these strategies depends on the
magnitude of the shift in yields.
Although convexity is preferred, there is the
disadvantage of a convexity yield, as the market
charges a higher price and offers a lower yield.
Thus the benefits from convexity are only realized for
large shifts in yields.
See the article in the course reader for details.
Albert Lee Chun
Portfolio Management
81