Encyclopedia of Finance Part 3 doc

Chapter 3
COMPARATIVE ANALYSIS OF
ZERO-COUPON AND
COUPON-PRE-FUNDED BONDS
A. LINDA BEYER, Alaska Supply Chain Integrators, USA
KEN HUNG, National Dong Hwa University, Taiwan
SURESH C. SRIVASTAVA, University of Alaska Anchorage, USA
Abstract
Coupon-prefunded bonds have been developed and
sold by investment bankers in place of zero-coupon
bonds to raise funds for companies facing cash flow
problems. Additional bonds are issued and proceeds
are deposited in an escrow account to finance the
coupon payment. Our analysis indicates that a cou-
pon-prefunded bond is equivalent to a zero-coupon
bond only if the return from the escrow account is
the same as the yield to maturity of the prefunded
issue. In reality, the escrow return is lower than the
bond yield. As a result, the firm provides interest
subsidy through issuing additional bonds which
leads to higher leverage, greater risk, and loss of
value compared to a zero-coupon issue.
Keywords: zero-coupon bond; Macaulay dura-
tion; escrow account; Treasury STRIPS; junk
bonds; coupon collateralization; financial engin-
eering; coupon pre-funded bond; cash flows; and
value loss
3.1. Introduction
Coupon-prefunded bonds, new to financial mar-
kets, were first issued in 1994 (Doherty, 1997).
1

They were introduced as a means to raise capital
for firms unable to generate cash flow to make
coupon payments, while still meeting the needs of
investors to receive coupon income. With a pre-
funded bond structure, additional bonds are issued
and an escrow account is established to finance
coupon payments over the life of the bond. In this
manner, the bond is considered prefunded. The firm
is not required to generate cash flow to meet coupon
obligations; it is paid out of the escrow account
usually collateralized by treasury securities. The
risk-free coupon payment allows the firm to set a
lower coupon rate on the bond than the yield on a
comparable zero-coupon bond. In general, the cost
of funding the escrow account is greater than the
return of the escrow account. This leads to an inter-
est rate subsidy and the loss of value. In this paper,
we compare zero-coupon bonds to prefunded bonds
and ascertain conditions under which the two fund-
ing options are equivalent. A prefunded issue sim-
ultaneously creates an asset and a liability. The net
duration of the pre-funded issue is the weighted
average of the asset and liability durations. The
model of net duration developed in this paper in-
corporates increased leverage of the pre-funded
issue, and appropriately assess its increased risk. In
spite of the fact that a prefunded bond is an inter-
esting concept of financial engineering, there is very
little academic research on this topic.
The remainder of this paper is made up of four

sections. Section 3.2 discusses the options available
to a firm interested in issuing debt. In Section 3.3,
we derive a mathematical model for Macaulay
duration of the prefunded issue to determine the
interest rate risk and calculate the loss in value due
to interest rate subsidy. A numerical example and
its analysis are presented in Section 3.4. Section 3.5
concludes the paper.
3.2. Funding Options
A firm wants to raise funds to finance a new pro-
ject. The pecking-order theory of capital structure
suggests that managers prefer internal equity to
external financing (Myers, 1984). In case the in-
ternal equity (retained earnings) is not available
then issuing new debt is preferred over issuing
preferred or additional common stock. Further,
firms would like to reduce the interest payment
burden. Hence, conventional coupon bond or hy-
brid financing such as convertible bonds or bonds
with warrants are ruled out. The available funding
options are (1) zero-coupon bonds, (2) step-up
bonds – initially coupon payment is set at a low
value and later stepped up, (3) deferred interest
bonds – initially there is no interest payment, but
it is resumed in 3–7 years, (4) paid-in-kind bonds –
issuer has right to pay interest in cash or with
similar bonds
2
, and (5) prefunded bonds. The
focus of the study is to compare zero-coupon and

coupon-prefunded bonds.
3.2.1. Zero-Coupon Bonds
Pure discount bonds are often called zero-coupon
bonds. It was first issued by J.C. Penney Company
Inc. in 1982 (Brigham and Daves, 2004). In recent
years, other firms (e.g. IBM, GMAC, Alcoa and
Martin-Marietta) have issued zero-coupon bonds.
Municipalities started issuing zero-coupon bonds in
1983. These bonds are sold at a deep discount and
increase in value as they approach maturity. Zero-
coupon bonds do not provide interest or coupon
payments at regular intervals like other types of
bonds. Implicit coupons are automatically rein-
vested by the issuer at yield to maturity. Interest
accrues over the life of the bond and a return is
earned as the bond appreciates. At maturity its
value equals the face value, and the bond holder
receives the yield to maturity expected at the time of
purchase. If held to maturity, the investor faces no
reinvestment risk but high-interest rate risk, as its
market price fluctuates considerably with move-
ments in market rates.
Corporate and municipal zero-coupon bonds
are usually callable and rated as junk bonds.
3
The
financial condition of the company issuing bonds
predicates the use of junk bonds, i.e. the firm is
unable to generate cash flows to meet coupon pay-
ments. Junk bonds are typically rated BB or lower

by Standard and Poor’s, or BA or lower by Moo-
dy’s. Junk bonds offer a high-expected return but
require investors to take on higher default risk.
Covenants on junk bonds are less restrictive, and
therefore provide alternatives for firms that may
not meet the more restrictive covenants of conven-
tional bonds.
3.2.2. Coupon Pre-Funded Bonds
In raising capital with a prefunded bond issue,
additional bonds are issued and an escrow account
is established. The firm is not required to generate
cash flow to meet coupon obligations over the life
of the bond. Bond interests are paid out of an
escrow account, which is usually collateralized by
treasury securities. In this manner, the bond is
considered prefunded. A prefunded bond issue
simultaneously creates an asset and a liability.
The risk characteristics of prefunded bonds’ inter-
est payments are different from that of traditional
coupon-bearing bonds because prefunded bonds’
coupon payments are asset based. The default free
nature of the coupon payment allows the firm to
set a lower coupon rate than the yield on a com-
parable zero-coupon bond. In general, the cost of
funding the escrow account is greater than the
return from the escrow account. This spread
leads to an interest rate subsidy which necessitates
COMPARATIVE ANALYSIS OF ZERO-COUPON AND COUPON-PRE-FUNDED BONDS 315
issuing more bonds, and hence a loss of value.
Greater the spread between the cost of funding

the escrow account and the return from the
escrow account, the larger the total face value of
the prefunded issue and the value loss. With a
prefunded bond issue, there are additional flota-
tion costs and cost of establishing the escrow ac-
count. However, for this analysis, we consider
the escrow costs and additional flotation costs to
be negligible.
Market price of prefunded bonds fluctuates
with movements in market rates, but it does not
move as dramatically as zero-coupon bond prices.
The reason for this difference is that zero-coupon
bonds do not provide any cash flow until maturity.
Coupon payments reduce the impact of interest
rate changes on prefunded bonds. Market condi-
tions where interest rate movements are frequent
and highly variable make prefunded bonds more
attractive than zero-coupon bonds. The risk pro-
files of zero-coupon and prefunded bonds can be
summarized as follows: A zero-coupon bond has
no reinvestment risk, higher price elasticity to
interest rate changes, and a default risk consistent
with its junk bond rating. The prefunded bond
has reinvestment risk but lower price elasticity to
interest rate changes. For a meaningful analysis
of the interest rate risk, one must examine the
combined interest rate sensitivity of the escrow
asset and the bond liability. The default risk of
the prefunded issue should be decomposed into
two components: the default risk of the coupon

payments and the default risk of the maturity
payment. The coupon payments are default free
but the default risk of the maturity payments is
much higher. This is due to the increased leverage
of the prefunded issue compared to zero-coupon
financing. In spite of the default-free coupon pay-
ments, the prefunded bonds are usually rated as
junk bonds.
In the next section, the combined interest
rate sensitivity of the escrow asset and the bond
liability is examined. A model for the net Macau-
lay duration of the prefunded issue is developed,
and loss of value due to interest rate subsidy is
calculated.
3.3. Macaulay Duration and Value Loss
In this section, we calculate the total face value of
the prefunded bonds issued, initial balance of the
escrow account, interest rate subsidy provided by
the firm, effective cost of the prefunded issue, and
resulting loss of value. Also, we derive an expres-
sion for the net Macaulay duration of the pre-
funded issue, i.e. the weighted average durations
of the coupon bond and the escrow asset
The face values of zero-coupon bonds issued, to
raise an amount B,is
B
z
¼ B(1 þ r
z
)

n
(3:1)
where r
z
is the discount rate for the zero-coupon
bond with maturity n. The Macaulay duration of
zero-coupon bond is its maturity (Fabbozzi,
2000).
Let B
pf
be the face value of the prefunded bonds
issued to raise an amount B. The annual coupon
payment is B
pf
(r
pf
), where r
pf
is the prefunded
bond yield. The initial balance in the escrow annu-
ity account set up to meet the coupon payments is
B
pf
À B. Hence,
B
pf
À B ¼ B
pf
r
pf

ÀÁ
PVIFA
r
es
,n
ÀÁ
B
pf
¼
B
1 À r
pf
PVIFA
r
es
,n
ÀÁ
(3:2)
where PVIFA indicates present value interest fac-
tor of an annuity, n is the maturity, and r
es
is the
rate of return on the escrow account. Substituting
the algebraic expression for PVIFA we get
4
B
pf
¼
r
es

(1 þ r
es
)
n
B
r
pf
À (r
pf
À r
es
)(1 þ r
es
)
n
(3:3)
The initial balance in the escrow account is
B
pf
À B ¼
r
pf
½(1 þ r
es
)
n
À 1B
r
pf
À (r

pf
À r
es
)(1 þ r
es
)
n
(3:4)
Escrow account is funded at a cost of r
pf
and
provides a return of r
es
. Consequently, the firm is
providing a pre-tax interest subsidy of (r
pf
B
pf
)
(r
pf
À r
es
) per year, which increases the cost of
prefunded issue and leads to loss of value.
316 ENCYCLOPEDIA OF FINANCE
The loss of value is:
Value Loss ¼(r
pf
B

pf
)(r
pf
Àr
es
)
(1 þr
pf
)
n
À1
(1 þr
pf
)
n
(3:5)
and the effective cost of the prefunded issue is
given by:
r
eff
¼
r
es
(1 þ r
es
)
n
r
pf
À (r

pf
À r
es
)(1 þ r
es
)
n

1=n
À1(3:6)
The concept of duration was introduced by
Macaulay (1938) as a measure of price sensitivity
of an asset or liability to a change in interest rates.
Working independently, Samuelson (1945) and
Redington (1952) developed the same concept
about the interest rate risk of bonds. Details of
duration computation can be found in any finance
text (Fabbozzi, 2000). A prefunded bond issue cre-
ates an asset, the escrow account annuity with mar-
ket value B
pf
À B; and a liability, coupon bonds
with market value B
pf
. The net market value of the
prefunded issue is B. Let D
es
and D
pf
represent the

duration of escrow annuity and the bond liability
respectively. Duration D
es
is the Macaulay duration
of an n-year annuity with yield r
es
and D
pf
is the
Macaulay duration of an n-year coupon bond with
yield to maturity r
pf
. The net duration of the pre-
funded issue isthe weighted average of thedurations
of the escrow account and the coupon bond. Hence
D
net
¼
B
pf
B
 D
pf
À
B
pf
À B
B
 D
es

(3:7)
where (B
pf
=B) and (À(B
pf
À B)=B) are the weights
of the coupon bond and the escrow annuity re-
spectively. This definition of net duration, D
net
,
captures the increased risk due to additional lever-
age caused by prefunding of coupon payments and
interest subsidy provided by the firm.
3.4. Numerical Example and Analysis
A firm wants to raise $10 million by issuing either
zero-coupon bonds or prefunded bonds with five
or ten year maturity. We assume that transaction
costs are identical for both issues and negligible.
5
Further, we assume that financial market views the
zero-coupon and prefunded bonds to be equivalent
securities, and prices them with identical yields.
Four different yields, 8 percent, 7 percent, 6 per-
cent, and 5 percent,.on zero-coupon and prefunded
bonds are considered for this analysis. Later, we
modify this assumption and consider the situation
where market views prefunded bond to be safer
and erroneously prices them with yields lower
than the comparable zero-coupon yields by 25,
50, and 75 basis points. In doing so, market over-

looks the added default risk associated with in-
creased leverage.
Table 3.1 presents the face value of zero-coupon
bonds issued to meet the $10 million funding need.
For 5-year maturity with discount rates of 8 per-
cent, 7 percent, 6 percent, and 5 percent, the firm
issues zero-coupon bonds with total face values of
Table 3.1. Zero-coupon bond
B
z
¼ B(1 þr
z
)
n
and D
z
¼ n
Discount rate, r
z
8% 7% 6% 5%
Maturity, n 5 years Funds needed, B $10,000,000 $10,000,000 $10,000,000 $10,000,000
Face value of bonds issued, B
z
$14,693,281 $14,025,517 $13,382,256 $12,762,816
Market value of bonds issued $10,000,000 $10,000,000 $10,000,000 $10,000,000
Duration, D
z
5 years 5 years 5 years 5 years
Maturity, n 10 years Funds needed, B $10,000,000 $10,000,000 $10,000,000 $10,000,000
Face value of bonds issued, B

z
$21,589,250 $19,671,514 $17,908,477 $16,288,946
Market value of bonds issued $10,000,000 $10,000,000 $10,000,000 $10,000,000
Duration, D
z
10 years 10 years 10 years 10 years
COMPARATIVE ANALYSIS OF ZERO-COUPON AND COUPON-PRE-FUNDED BONDS 317
$14,693,281, $14,025,517, $13,382,256, and
$12,762,816 respectively. These values are calcu-
lated using Equation (3.1). The Macaulay duration
of the 5-year zero-coupon bond is 5 years. For
10-year zero-coupon bonds, an 8 percent, 7 per-
cent, 6 percent, and 5 percent discount rate leads to
total face values of $21,589,250, $19,671,514,
$17,908,477, and $16,288,946 respectively. The
Macaulay duration of the 10-year zero-coupon
bond is 10 years.
In Table 3.2, we present the total face value of
the prefunded issue, amount of annual coupon
payment disbursed from escrow account, and the
effective cost of prefunded issue. It provides the
following important inferences.
First, when the prefunded bond yield, r
pf
, is the
same as the escrow account return, r
es
, then (i) the
total face value of the pre-funded issued is the same
as the total face value of the zero-coupon bonds

and (ii) the effective cost of prefunded issue, r
eff
,is
Table 3.2. Total face value and effective cost of prefunded issue
B
pf
¼
r
es
(1 þ r
es
)
n
B
r
pf
À (r
pf
À r
es
)(1 þ r
es
)
n
and r
eff
¼
r
es
(1 þ r

es
)
n
r
pf
À (r
pf
À r
es
)(1 þ r
es
)
n

1=n
À1
Prefunded bond yield, r
pf
nr
es
8% 7% 6% 5%
5 Face value, B
pf
$14,693,281
8% Escrow payment $1,175,462
Effective cost, r
eff
8.000%
Face value, B
pf

$14,881,302 $14,025,517
7% Escrow payment $1,190,504 $981,786
Effective cost, r
eff
8.275% 7.000%
Face value, B
pf
$15,082,708 $14,181,691 $13,382,256
6% Escrow payment $1,206,617 $992,718 $802,935
Effective cost, r
eff
8.567% 7.237% 6.000%
Face value, B
pf
$15,298,893 $14,368,507 $13,509,289 $12,762,816
5% Escrow payment $1,223,912 $1,004,395 $810,557 $638,141
Effective cost, r
eff
8.876% 7.518% 6.201% 5.000%
Face value, B
pf
$21,589,250
8% Escrow payment $1,727,140
10 Effective cost, r
eff
8.000%
Face value, B
pf
$22,825,137 $19,671,514
7% Escrow payment $1,826,011 $1,377,006

Effective cost, r
eff
8.603% 7.000%
Face value, B
pf
$24,319,478 $20,627,322 $17,098,477
6% Escrow payment $1,945,558 $1,443,913 $1,074,509
Effective cost, r
eff
9.294% 7.509% 6.000%
Face value, B
pf
$26,160,123 $21,763,801 $18,632,525 $16,288,946
5% Escrow payment $2,092,810 $1,523,466 $1,117,952 $814,447
Effective cost, r
eff
10.094% 8.087% 6.421% 5.000%
r
es
¼ escrow return. Maturity ¼ n years. Empty cell represents the improbable case of r
pf
< r
z
.
318 ENCYCLOPEDIA OF FINANCE
the same as the yield to maturity of the zero-
coupon bond, r
z
. Second, increase in the spread
between r

pf
and r
es
increases the total face value
of the bonds issued and its effective cost. Finally,
for a given spread the total face value of the bonds
issued and its effective cost increases with matur-
ity. For example, consider the case when both r
pf
and r
es
are equal to 8 percent and the firm wants to
issue 5-year maturity bonds to raise $10 million. It
can issue either zero-coupon bonds or prefunded-
coupon bonds with $14,693,281 face value and
8 percent effective costs. For 10-year maturity, it
will have to issue $21,589,250 zero-coupon or pre-
funded bonds. However, with a 3 percent spread,
i.e. r
pf
¼ 8 percent and r
es
¼ 5 percent, the firm will
have to issue $15,298,893 coupon bonds with ma-
turity 5 years or $26,160,132 coupon bonds with
maturity 10 years. The effective cost of 5-year and
10-year prefunded issues will rise to 8.876 percent
and 10.094 percent respectively.
Examples of net duration of pre-funded issue, i.e.
the weighted average durations of the escrow asset

and coupon bond liability are presented in Tables
3.3 and 3.4. In Table 3.3, we present a 5-year bond
issue without spread, i.e. both r
pf
and r
es
are equal to
8 percent. Firm issues $14,693,281 bonds with an-
nual coupon payment of $1,175,462. Coupon pay-
ments are disbursed out of an escrow account with
$4,693,281 initial balance. Panel A of Table 3.3
shows that duration of the coupon bond, D
pf
,is
4.3121 years. Panel B of Table 3.3 shows that the
duration of the escrow annuity, D
es
, is 2.8465 years.
Table 3.3. Net duration of the prefunded issue without spread
D
net
¼
B
pf
B
 D
pf
À
B
pf

À B
B
 D
es
Panel A: Bonds issued
Time, t Cash outflow, CF PVIF
8%,5
CF
Ã
PVIF t
Ã
CF
Ã
PVIF Duration, D
pf
1 $1,175,462 0.9259 $ 1,088,391 $1,088,391
2 1,175,462 0.8573 1,007,769 2,015,538
3 1,175,462 0.7938 933,120 2,799,359
4 1,175,462 0.7350 864,000 3,455,999
5 15,868,743 0.6806 10,800,000 53,999,999
$14,693,280 $63,359,286 4.3121
Panel B: Escrow annuity
Time, t Cash inflow, CF PVIF
8%,5
CF
Ã
PVIF t
Ã
CF
ÃÃ

PVIF Duration, D
es
1 $1,175,462 0.9259 $1,088,391 $1,088,391
2 1,175,462 0.8573 1,007,769 2,015,538
3 1,175,462 0.7938 933,120 2,799,359
4 1,175,462 0.7350 864,000 3,455,999
5 1,175,462 0.6806 800,000 3,999,998
$4,693,280 $13,359,282 2.8465
Panel C: Net durations
Fund raised, B $10,000,000 Escrow amount, B
pf
À B $4,693,281
Face value of bond, B
pf
$14,693,281 Escrow return, r
es
8%
Bond yield, r
pf
8.00% Escrow weight, (B À B
pf
)=B À0.469
Bond weight, B
pf
=B 1.469 Escrow duration, D
es
2.847
Bond duration, D
pf
4.312 Net duration, D

net
5.000
If escrow return equals the bond yield, i.e. r
es
¼ r
pf
, then the net duration equals the maturity.
COMPARATIVE ANALYSIS OF ZERO-COUPON AND COUPON-PRE-FUNDED BONDS 319
Panel C of Table 3.3 shows that the weights of bond
liability and escrow asset are 1.469 and À:0:469
respectively. Hence, the net duration, D
net
, of the
prefunded issue is 5 years, which is identical to the
duration of a zero-coupon bond. The result is
understandable because the firm has no net cash
outflow for years one to four, the only cash outflow
of $14,693,281 is in year five.
In Table 3.4, we present an example of a 5-year
prefunded bond issue with 3 percent spread, i.e.
r
pf
¼ 8 percent and r
es
¼ 5 percent. Firm issues
$15,298,250 bonds with annual coupon payment
of $1,223,912. Coupon payments are disbursed
out of an escrow account with $5,298,250 initial
balance. Firm provides the interest subsidy by issu-
ing additional bonds compared to the example in

Table 3.3. Panel A of Table 3.4 shows that the
duration of the coupon bond, D
pf
, is 4.3121 years,
same as the example in Table 3.3. But the duration
of the escrow annuity, D
es
, increases to 2.9025
years. The weights of bond liability and escrow
asset, reported in Panel C of Table 3.4, are 1.530
and À0:530 respectively. The net duration, D
net
,of
the prefunded issue increases to 5.059 years. The
interest subsidy creates the additional leverage, and
which stretches the duration beyond its maturity.
6
Because interest subsidy is a realistic condition, the
prefunded bond issue has greater interest rate risk
than the comparable zero-coupon bond.
Table 3.5 presents net duration, interest subsidy
and loss of value associated with a prefunded bond
issue for different bond yields and escrow returns.
When r
pf
¼ r
es
, then there is no interest subsidy or
loss of value and the net duration of the pre-funded
Table 3.4. Net duration of the prefunded issue with spread

D
net
¼
B
pf
B
 D
pf
À
B
pf
À B
B
 D
es
Panel A: Bonds issued
Time, t Cash outflow, CF PVIF
8%,5
CF
Ã
PVIF t
Ã
CF
Ã
PVIF Duration, D
pf
1 $1,223,912 0.9259 $1,133,252 $1,133,252
2 1,223,912 0.8573 1,049,307 2,098,615
3 1,223,912 0.7938 971,581 2,914,742
4 1,223,912 0.7350 899,612 3,598,447

5 16,522,162 0.6806 11,244,706 56,223,529
$15,298,458 $65,968,585 4.3121
Panel B: Escrow annuity
Time, t Cash inflow, CF PVIF
5%,5
CF
Ã
PVIF t
Ã
CF
Ã
PVIF Duration, D
es
1 $1,223,912 0.9524 $1,165,630 $1,165,630
2 1,223,912 0.9070 1,110,124 2,220,249
3 1,223,912 0.8638 1,057,261 3,171,784
4 1,223,912 0.8227 1,006,915 4,027,662
5 1,223,912 0.7835 958,967 4,794,835
$5,298,897 $15,380,160 2.9025
Panel C: Net durations
Fund raised, B $10,000,000 Escrow amount, B
pf
À B $5,298,250
Face value of bond, B
pf
$15,298,250 Escrow return, r
es
5%
Bond yield, r
pf

8.00% Escrow weight, (B À B
pf
)=B À0.530
Bond weight, B
pf
=B 1.530 Escrow duration, D
es
2.903
Bond duration, D
pf
4.312 Net duration, D
net
5.059
If escrow return is less than the bond yield, i.e. r
es
< r
pf
, then the net duration exceeds maturity.
320 ENCYCLOPEDIA OF FINANCE
issue is equal to bond maturity. The net duration,
interest subsidy, and loss of value increases with
the increase in the spread, r
pf
¼ r
es
.
Table 3.6 presents the case whenprefunded bonds
are priced to yield lower than the zero-coupons. The
asset-based coupon payments ofthe prefundedissue
are default free, thus market lowers the yield by 25,

50, or 75 basis points from the comparable zero-
coupon yield. We recalculate the total face value,
net duration, interest subsidy, and loss of value
under these conditions. Results in Table 3.6 indicate
that the impact of the spread, r
pf
À r
es
is still dom-
inant. The total face value and net duration of the
prefunded issue is greater than corresponding values
for the zero-coupon bond.
3.5. Conclusion
Coupon-prefunded bonds have been developed
and sold by investment bankers in place of zero-
coupon bonds to raise funds for companies facing
cash flow problems. Additional bonds are issued
and proceeds are deposited in an escrow account to
finance the coupon payment. Our analysis indi-
cates that when the prefunded bond yield is the
same as the escrow return then total face value of
Table 3.5. Net duration, interest subsidy, and value loss of prefunded bonds
Pre-tax Interest Subsidy ¼ (r
pf
B
pf
)(r
pf
À r
es

) per year
Value loss ¼ (r
pf
B
pf
)(r
pf
À r
es
)
(1 þ r
pf
)
n
À 1
(1 þ r
pf
)
n
Escrow return, r
es
Prefunded bond yield, r
pf
8% 7% 7% 7%
Maturity, n 5 years 8% Net duration, yrs 5
Interest subsidy 0
Value loss 0
7% Net duration, yrs 5.019 5
Interest subsidy $11,905 0
Value loss ($47,533) 0

6% Net duration, yrs 5.038 5.016 5
Interest subsidy $24,132 $9,927 0
Value loss ($96,353) ($40,703) 0
5% Net duration, yrs 5.059 5.037 5.013 5
Interest subsidy $36,717 $20,088 $8,106 0
Value loss ($146,602) ($82,364) ($34,144) 0
Maturity, n 10 years 8% Net duration, yrs 10
Interest subsidy 0
Value loss 0
7% Net duration, yrs 10.198 10
Interest subsidy $18,260 0
Value loss ($122,527) 0
6% Net duration, yrs 10.433 10.165 10
Interest subsidy $38,911 $14,439 0
Value loss ($261,097) ($101,414) 0
5% Net duration, yrs 10.715 10.358 10.135 10
Interest subsidy $62,784 $30,469 $11,180 0
Value loss ($421,288) ($214,004) ($82,282) 0
Empty cell represents the improbable case of r
pf
< r
z
.
Zero-coupon and prefunded bonds are priced by market as equivalent securities.
COMPARATIVE ANALYSIS OF ZERO-COUPON AND COUPON-PRE-FUNDED BONDS 321
the prefunded issued is the same as the total face
value of the zero-coupon bonds and the effective
cost of prefunded issue is the same as the yield to
maturity of the zero-coupon bond. Also, increase
in the spread between prefunded bond yield and

zero-coupon yield increases the total face value
of the bonds issued and its effective cost. The
interest subsidy creates additional leverage, which
stretches the net duration of the prefunded issue
beyond its maturity. Further, an increase in the
yield spread between prefunded bonds and zero-
coupon bonds increases net duration, interest
subsidy, and loss of value. Even when prefunded
bonds are priced to yield lower than the zero-
coupons, impact of the spread is dominant – total
face value and net duration of the prefunded
issue is still greater than corresponding values for
the zero-coupon bond.
NOTES
1. For the remainder of this paper we will adopt popu-
lar finance nomenclature and refer it as prefunded
bonds. However, one must keep in mind that only
coupon payments are prefunded.
2. See Goodman and Cohen (1989) for detailed discus-
sion of paid-in-kind bonds.
3. U.S. Treasury sells risk-free zero-coupon bonds in
the form of STRIPs.
4. See Ross, Westerfield, and Jaffe (2005) for algebraic
expression of PVIFA.
5. Alternately, we can assume that all yields are net of
transaction costs.
6. This is analogous to a situation in portfolio construc-
tion. Consider two assets with standard deviations
10 percent and 20 percent. For an investor who is
long on both assets, the portfolio standard deviation

will be between 10 percent and 20 percent. However,
if the investor is short on the first asset and long on
the second asset then portfolio standard deviation
will exceed 20 percent.
REFERENCES
Brigham, E.F. and Phillip, R.D. (2004). Intermediate
Financial Management. Mason, OH: Thomson
Southwestern Publishing.
Doherty, J. (1997). ‘‘For junk borrowers, pre-funded
bonds pick up steam, but they may pose greater
risk than zeros.’’ Barrons, MW15.
Fabbozzi, F.J. (2000). Bond Markets, Analysis and
Strategies. Englewood Cliffs, NJ: Prentice-Hall.
Goodman, L.S. and. Cohen,A.H. ( 1989). ‘‘Payment-in-
kind debentures: an innovation.’’ Journal of Portfolio
Management, 15: 9–19.
Myers, S.C. (1984). ‘‘The capital structure puzzle.’’
Journal of Finance, 39: 575–592.
Table 3.6. Face value, net duration, interest subsidy, and value loss of prefunded bonds
Prefunded bond yield, B
pf
r
z
B
z
r
z
r
z
À :25% r

z
À :50% r
z
À :75%
8% $21,589,250 Face value of pre-funded, B
pf
$26,160,123 $24,902,535 $23,760,313 $22,718,277
Duration, D
net
10.718 yrs 10.611 yrs 10.516 yrs 10.432 yrs
Interest subsidy $62,784 $53,074 $44,551 $37,059
Value loss ($421,288) ($360,179) ($305,799) ($257,307)
7% $19,671,514 Face value of pre-funded, B
pf
$21,763,801 $20,886,293 $20,076,805 $19,327,721
Duration, D
net
10.358 yrs 10.291 yrs 10.266 yrs 10.181 yrs
Interest subsidy $30,469 $24,672 $19,575 $15,100
Value loss ($214,004) ($175,306) ($140,721) ($109,831)
6% $13,382,256 Face value of pre-funded, B
pf
$18,632,525 $17,985,604 $17,382,097 $16,817,777
Duration, D
net
10.135 yrs 10.094 yrs 10.058 yrs 10.027 yrs
Interest subsidy $11,180 $7,756 $4,780 $2,207
Value loss ($82,282) ($57,769) ($36,030) ($16,839)
r
z

¼ discount rate on zero-coupon bonds, B
z
¼ face value of zero-coupon bonds with 10-year maturity. Escrow account yield ¼
5%. Prefunded bonds are priced to yield lower than comparable zero-coupon bonds.
322 ENCYCLOPEDIA OF FINANCE
Macaulay, F. (1938). Some Theoretical Problems Sug-
gested by the Movement of Interest Rates, Bond
Yields, and Stock Prices in the US since 1856. New
York: National Bureau of Economic Research.
Redington, F.M. (1952). ‘‘Review of the principles of
life office valuation.’’ Journal of the Institute of
Actuaries, 78: 286–340.
Ross, S.A., Westerfield, R.W., and Jaffe, J. (2005). Cor-
porate Finance. Homewood, IL: Irwin McGraw-Hill.
Samuelson, P.A. (1945). ‘‘The effect of interest rate
increases on the banking system.’’ American Eco-
nomic Review, 35: 16–27.
COMPARATIVE ANALYSIS OF ZERO-COUPON AND COUPON-PRE-FUNDED BONDS 323
Chapter 4
INTERTEMPORAL RISK
AND CURRENCY RISK
JOW-RAN CHANG, National Tsing Hua University, Taiwan
MAO-WEI HUNG, National Taiwan University, Taiwan
Abstract
Empirical work on portfolio choice and asset pricing
has shown that an investor’s current asset demand is
affected by the possibility of uncertain changes in
future investment opportunities. In addition, differ-
ent countries have different prices for goods when
there is a common numeraire in the international

portfolio choice and asset pricing. In this survey,
we present an intertemporal international asset pri-
cing model (IAPM) that prices market hedging risk
and exchange rate hedging risk in addition to market
risk and exchange rate risk. This model allows us to
explicitly separate hedging against changes in the
investment opportunity set from hedging against ex-
change rate changes as well as separate exchange
rate risk from intertemporal hedging risk.
Keywords: currency risk; exchange rate risk; hedg-
ing risk; inflation risk; international asset pricing;
intertemporal asset pricing; intertemporal risk;
intertemporal substitution; purchasing power par-
ity; recursive preference; risk aversion
4.1. Introduction
In a dynamic economy, it is often believed that if
investors anticipate information shifts, they will
adjust their portfolios to hedge these shifts. To
capture the dynamic hedging effect, Merton
(1973) developed a continuous-time asset pricing
model which explicitly takes into account hedging
demand. In contrast to the Arbitrage Pricing The-
ory (APT) framework, there are two factors, which
are theoretically derived from Merton’s model: a
market factor and a hedging factor. Stulz (1981)
extended the intertemporal model of Merton
(1973) to develop an international asset pricing
model. However, an empirical investigation is not
easy to implement in the continuous-time model.
In a recent paper, Campbell (1993) developed a

discrete-time counterpart of Merton’s model. Mo-
tivated by Campbell’s results, Chang and Hung
(2000) adopted a conditional two-factor asset pri-
cing model to explain the cross-sectional pricing
relationships among international stock markets.
In their setup, assets are priced using their covar-
iance with the market portfolio as well as with the
hedging portfolio, both of which account for
changes in the investment set. Under their pro-
posed international two-factor asset pricing
model framework, the international capital asset
pricing model (CAPM) is misspecified and esti-
mates of the CAPM model are subject to the omit-
ted variable bias.
If purchasing power parity (PPP) is violated,
investors from different countries will have differ-
ent evaluations for real returns for investment in
the same security. This implies that the optimal
portfolio choices are different across investors
residing in different countries, and any investment
in a foreign asset is exposed to currency risk.
Therefore, it is reasonable to assume that investors
from different countries have different estimations
for real returns. This phenomenon clearly shows
the existence of currency risk as well as market
risk.
There are two goals in this survey. First, we
want to know whether hedging demand is import-
ant to an international investor. Second, we want
to separate currency hedging risk from intertem-

poral market hedging risk on an international asset
pricing model.
The approach we describe here was first pro-
posed by Epstein and Zin (1989, 1991). In their
model, the investor is assumed to use a nonex-
pected utility that distinguishes the coefficient of
relative risk aversion and the elasticity of intertem-
poral substitution. Campbell (1993) applied a log-
linear approximation to the budget constraint in
order to replace consumption from a standard
intertemporal asset pricing model. Chang and
Hung (2000) used this model to explain the inter-
temporal behavior in the international financial
markets under no differences in consumption
opportunity set.
An important challenge therefore remains – how
to build a more realistic intertemporal inter-
national asset pricing model (e.g. when the con-
sumption opportunity set is different). This essay
surveys the progress that has been made on this
front, drawing primarily from Chang and Hung
(2000) and Chang et al. (2004).
In Section 4.2, we present a testable intertem-
poral capital asset pricing model proposed by
Campbell. Hence, we can examine whether Camp-
bell’s model explains the intertemporal behavior of
a number of international financial markets. In
Section 4.3, we separate currency hedging risk
from intertemporal market hedging risk. This is
accomplished by extending Campbell’s model to

an international framework in which investor’s
utility depends on real returns rather than on nom-
inal returns and PPP deviation.
4.2. No Differences in Consumption
Opportunity Set
This section describes the international asset pri-
cing model we employ to estimate and test the
pricing relationships among the world’s five main
equity markets. The model we use is a two-factor
model based on Campbell (1993). We first review
the theory of nonexpected utility proposed by Weil
(1989) and Epstein and Zin (1991). Then we apply
a log-linear approximation to the budget con-
straint to derive an international asset pricing
model, which is used in this chapter.
4.2.1. Asset Pricing Model
4.2.1.1. Nonexpected Utility
We consider an economy in which a single, infin-
itely lived representative international agent
chooses consumption and portfolio composition
to maximize utility and uses U.S. dollar as the
numeraire and where there is one good and N
assets in the economy. The international agent in
this economy is assumed to be different to the
timing of the resolution of uncertainty over tem-
poral lotteries. The agent’s preferences are as-
sumed to be represented recursively by
V
t
¼ W (C

t
, E
t
[V
tþ1
jI
t
]), (4:1)
where W(.,.) is the aggregator function, C
t
is the
consumption level at time t, and E
t
is the math-
ematical expectation conditional on the informa-
tion set at time t. As shown by Kreps and Porteus
(1978), the agent prefers early resolution of uncer-
tainty over temporal lotteries if W(.,.) is convex in
its second argument. Alternatively, if W(.,.) is con-
cave in its second argument, the agent will prefer
late resolution of uncertainty over temporal
lotteries.
The aggregator function is further parameter-
ized by
V
t
¼ [(1 Àd)C
1Àr
t
þ d(EV

1Àl
tþ1
)
(1Àr)=(1Àl)
]
1=(1Àr)
¼ [(1 Àd)C
(1Àl)=u
t
þ d(E
t
V
1Àl
tþ1
)
1=u
]
u=(1Àl)
(4:2)
INTERTEMPORAL RISK AND CURRENCY RISK 325
Parameter d is the agent’s subjective time-
discount factor and l is interpreted as the Arrow–
Pratt coefficient of relative risk aversion. In add-
ition, 1=r measures the elasticity of intertemporal
substitution. For instance, if the agent’s coefficient
of relative risk aversion (l) is greater than the re-
ciprocal of the agent’s elasticity of intertemporal
substitution ( r), then the agent would prefer an
early resolution towards uncertainty. Conversely,
if the reciprocal of the agent’s elasticity of inter-

temporal substitution is larger than the agent’s co-
efficient of relative risk aversion, then the agent
would prefer a late resolution of uncertainty. If l
is equal to r, the agent’s utility becomes an isoelas-
tic, von Neumann–Morgenstern utility, and the
agent would be indifferent to the timing of the
resolution of uncertainty.
Furthermore, u is defined as u ¼ (1 À l)=(1 À r)
in accordance with Giovannini and Weil (1989).
Three special cases are worth mentioning. First,
u ! 0 when l ! 1. Second, u !1when r ! 1.
Third, u ¼ 1 when l ¼ r. Under these circumstan-
ces, Equation (4.2) becomes the von Neumann–
Morgenstern expected utility
V
t
¼ (1 À d)E
t
X
1
j¼1
d
j
C
~
1Àg
tþj
"#
1=(1Àl)
: (4:3)

4.2.1.2. Log-Linear Budget Constraint
We now turn to the characterization of the budget
constraint of the representative investor who can
invest wealth in N assets. The gross rate of return
on asset i held throughout period t is given by
R
i, tþ1
. Let
R
m,tþ1

X
N
i¼1
a
i,t
R
i,tþ1
(4:4)
denote the rate of return on the market portfolio,
and a
i,t
be the fraction of the investor’s total
wealth held in the i th asset in period t. There are
only N À 1 independent elements in a
i, t
since the
constraint
X
N

i¼1
a
i,t
¼ 1(4:5)
holds for all t. The representative agent’s dynamic
budget constraint can be given by
W
tþ1
¼ R
m,tþ1
(W
t
À C
t
), (4:6)
where W
tþ1
is the investor’s wealth at time t. The
budget constraint in Equation (4.6) is nonlinear
because of the interaction between subtraction
and multiplication. In addition, the investor is cap-
able of affecting future consumption flows by trad-
ing in risky assets. Campbell linearizes the budget
constraint by dividing Equation (4.6) by W
t
, tak-
ing log, and then using a first-order Taylor expan-
sion around the mean log consumption=wealth
ratio, log (C=W ). If we define the parameter
b ¼ 1 À exp (

c
t
À w
t
), the approximation to the
intertemporal budget constraint is
Dw
tþ1
ffi r
m,tþ1
þ k þ 1 À
1
b

(c
t
À w
t
), (4:7)
where the log form of the variable is indicated by
lowercase letters and k is a constant.
Combining Equation (4.7) with the following
equality,
Dw
tþ1
¼ Dc
tþ1
þ (c
t
À w

t
) À (c
tþ1
À w
tþ1
), (4:8)
we obtain a different equation in the log consump-
tion=wealth ratio, c
t
À w
t
. Campbell (1993) shows
that if the log consumption=wealth ratio is sta-
tionary, i.e. lim
j!1
b
j
(c
tþj
À w
tþj
) ¼ 0, then the
approximation can be written as
c
tþ1
À E
t
c
tþ1
¼ (E

tþ1
À E
t
)
X
1
j¼0
b
j
r
m:tþ1þj
À (E
tþ1
À E
t
)
X
1
j¼1
b
j
Dc
tþ1þj
: (4:9)
Equation (4.9) can be used to express the fact that
an unexpected increase in consumption today is
determined by an unexpected return on wealth
today (the first term in the first sum on the right-
hand side of the equation), or by news that future
326 ENCYCLOPEDIA OF FINANCE

returns will be higher (the remaining terms in the
first sum), or by a downward revision in expected
future consumption growth (the second sum on the
right-hand side).
4.2.1.3. Euler Equations
In this setup, Epstein and Zin (1989) derive the
following Euler equation for each asset:
1 ¼ E
t
d
C
tþ1
C
t

Àr
&'
u
1
R
m,tþ1
&'
1Àu
R
i,tþ1
"#
(4:10)
Assume for the present that asset prices and
consumption are jointly lognormal or apply a sec-
ond-order Taylor expansion to the Euler equation.

Then, the log version of the Euler equation (4.10)
can be represented as
0 ¼ u log d À urE
t
Dc
tþ1
þ (u À 1)E
t
r
m,tþ1
þ E
t
r
i,tþ1
þ
1
2
[(ur)
2
V
cc
þ (u À 1)
2
V
mm
þ V
ii
À 2ur(u À 1)V
cm
À 2urV

ci
þ 2(u À 1)V
im
]
(4:11)
where V
cc
denotes var(c
tþ1
), V
jj
denotes var(r
j,tþ1
)
8j ¼ i,m, V
cj
denotes cov(c
tþ1
, r
j,tþ1
) 8j ¼ i,m, and
V
im
denotes cov(r
i,tþ1
, r
m,tþ1
).
By replacing asset i by the market portfolio and
rearranging Equation (4.11), we obtain a relation-

ship between expected consumption growth and
expected return on the market portfolio
E
t
Dc
tþ1
¼
1
r
log d þ
1
2

urV
cc
þ
u
r

V
mm
À 2uV
cm
!
þ
1
r
E
t
r

m,tþ1
:
(4:12)
When we subtract Equation (4.11) for the risk-free
asset from that for asset i, we obtain
E
t
r
i,tþ1
À r
f,tþ1
¼À
V
ii
2
þ u(rV
ic
) þ (1 À u)V
im
(4:13)
where r
f ,tþ1
is a log riskless interest rate. Equation
(4.13) expresses the expected excess log return on an
asset (adjusted for Jensen’s inequality effect) as a
weighted sum of two terms. The first term, with a
weight u, is the asset covariance with consumption
multiplied by the intertemporal elasticity of substi-
tution, r. The second term, with a weight 1 À u,is
the asset covariance with the return from the mar-

ket portfolio.
4.2.1.4. Substituting Consumption out of the Asset
Pricing Model
Now, we combine the log-linear Euler equation
with the approximated log-linear budget constraint
to get an intertemporal asset pricing model without
consumption. Substituting Equation (4.12) into
Equation (4.9), we obtain
c
tþ1
ÀE
t
c
tþ1
¼ r
m,tþ1
À E
t
r
m,tþ1
þ 1 À
1
r

(E
tþ1
À E
t
)
X

1
j¼1
b
j
r
m,tþ1þj
(4:14)
Equation (4.14) implies that the unexpected con-
sumption comes from an unexpected return on
invested wealth today or expected future returns.
Based on Equation (4.14), the conditional cov-
ariance of any asset return with consumption can
be rewritten in terms of the covariance with the
market return and revisions in expectations of fu-
ture market returns which is given by
cov
t
(r
i,tþ1
, Dc
tþ1
)  V
ic
¼ V
im
þ 1 À
1
r

V

ih
(4:15)
where V
ih
¼ cov
t
À
r
i,tþ1
,(E
tþ1
À E
t
)
P
1
j¼1
b
j
r
m,tþ1þj
Á
:
Substituting Equation (4.15) into Equation
(4.13), we obtain an international asset pricing
model that is not related to consumption:
E
t
r
i,tþ1

À r
f ,tþ1
¼À
V
ii
2
þ lV
im
þ (l À 1)V
ih
:
(4:16)
Equation (4.16) states that the expected excess
log return in an asset, adjusted for Jensen’s in-
equality effect, is a weighted average of two covar-
iances—the covariance with the return from the
market portfolio and the covariance with news
about future returns on invested wealth.
INTERTEMPORAL RISK AND CURRENCY RISK 327
4.2.2. Empirical Evidence
The relationship between risk and return has been
the focus of recent finance research. Numerous pa-
pers have derived various versions of the inter-
national asset pricing model. For example, Solnik
(1974)extendsthe staticCapital AssetPricing Model
of Sharpe (1964) and Lintner (1965) to an inter-
national framework. His empirical findings reveal
that national factors are important in the pricing of
stock markets. Furthermore, Korajczyk and Viallet
(1989) propose that the international CAPM out-

performs its domestic counterpart in explaining the
pricing behavior of equity markets.
In a fruitful attempt to extend the conditional
version of the static CAPM, Harvey (1991) em-
ploys the Generalized Method of Moments
(GMM) to examine an international asset pricing
model that captures some of the dynamic behavior
of the country returns. De Santis and Gerard
(1997) test the conditional CAPM on international
stock markets, but they apply a parsimonious Gen-
eralized Auto-Regressive Conditional Hetero-
scedasticity (GARCH) parameterization as the
specification for second moments. Their results
indicate that a one-factor model cannot fully ex-
plain the dynamics of international expected re-
turns and the price of market risk is not significant.
On the other hand, recent studies have applied
the APTof Ross (1976) to an international setting.
For instance, Cho et al. (1986) employ factor an-
alysis to demonstrate that additional factors other
than covariance risk are able to explain the inter-
national capital market. Ferson and Harvey (1993)
investigate the predictability of national stock mar-
ket returns and its relation to global economic risk.
Their model includes a world market portfolio,
exchange rate fluctuations, world interest rates,
and international default risk. They use multifac-
tor asset pricing models with time-varying risk
premiums to examine the issue of predictability.
But, one of the drawbacks of the APT approach

is that the number and identity of the factors are
determined either ad hoc or statistically from data
rather than from asset pricing models directly.
Several international asset pricing models expli-
citly takeinto account currencyrisk, forexample, see
Solnik (1974), Stulz (1981), and Adler and Dumas
(1983). But investors in these models are assumed to
maximize a time-additive, von Neumann–Morgen-
stern expected utility of lifetime consumption func-
tion. This implies that two distinct concepts of
intertemporal substitution and risk aversion are
characterized by the same parameter. Another ap-
proach examines consumption risk. Cumby (1990)
proposes a consumption-based international asset
pricing model. Difficulty occurs in the usage of ag-
gregate consumption data, which are measured with
error, and are time-aggregated. Chang and Hung
(2000) show that estimations of price of market risk
obtained from the De Santis and Gerard (1997) con-
ditional CAPM model may be biased downward due
to the omission of the hedging risk, which is nega-
tively correlated to the market risk.
4.3. Differences in Consumption Opportunity Set
In this section, we consider the problem of optimal
consumption and portfolio allocation in a unified
world capital market with no taxes and transac-
tions costs. Moreover, investors’ preferences are
assumed to be nationally heterogeneous and asset
selection is the same for investors in different coun-
tries. Consider a world of M þ1 countries and a set

of S equity securities. All returns are measured in
the Mþ1st country’s currency in excess of the risk-
free rate and this currency is referred to as the
numeraire currency. Investors are assumed to
maximize Kreps–Porteus utility for their lifetime
consumption function.
4.3.1. Portfolio Choice in an International Setting
4.3.1.1. Kreps–Porteus Preferences
Define C
t
as the current nominal consumption level
at time t, and P
t
as the price level index at time t,
expressed in the numeraire currency. In the setup of
Kreps and Porteus (1978) nonexpected utility, the
investor’s value function can be represented as:
V
t
¼ U
C
t
P
t
, E
t
V
tþ1
!
,(4:17)

where V
t
is the lifetime utility at time t, E
t
is the
expected value function conditional on the infor-
328 ENCYCLOPEDIA OF FINANCE
mation available to the investor at time t, U [.,.] is
the aggregator function that aggregates current
consumption with expected future value. As
shown by Kreps and Porteus, the agent prefers
early resolution of uncertainty over temporal lot-
teries if U [.,.] is convex in its second argument. On
the other hand, if U [.,.] is concave in its second
argument, the agent will prefer late resolution of
uncertainty over temporal lotteries.
Furthermore, the aggregator function is para-
meterized to be homogenous of degree one in cur-
rent real consumption and in the value of future
state-dependent real consumption:
U
C
t
P
t
, E
t
V
tþ1
!

¼ (1 À d)
C
t
P

1Àr
"
þd(E
t
V
tþ1
)
(1Àr)=(1Àl)
#
(1Àl)=(1Àr)
,
(4:18)
where l is the Arrow–Pratt coefficient of relative
risk aversion, r can be interpreted as the elasticity
of intertemporal substitution, and d 2 (0,1) is the
subjective discount factor.
The Kreps–Porteus preference allows the separ-
ation of risk aversion from intertemporal substitu-
tion. For instance, if the agent’s coefficient of
relative risk aversion, l, is greater than the recip-
rocal of the agent’s elasticity of intertemporal sub-
stitution, r, then the agent prefers early resolution
of uncertainty. Conversely, if the reciprocal of the
agent’s elasticity of intertemporal substitution is
larger than the agent’s coefficient of relative risk

aversion, the agent prefers late resolution of uncer-
tainty. When r ¼ l, the objective function is the
time-separable power utility function with relative
risk aversion l. In addition, when both l and r
equal 1, we have standard time-separable log uti-
lity function. Hence, the standard time- and state-
separable expected utility is a special case under
Kreps–Porteus preferences.
4.3.1.2. Optimal Consumption and Portfolio
Allocation
We now turn to the characterization of the budget
constraint of the representative investor who can
invest his wealth in N(¼MþS) assets that include
M currencies and S equities. Currencies may be
taken as the nominal bank deposits denominated
in the nonnumeraire currencies. The gross rate of
nominal return on asset i held throughout period t
is given by R
i,tþ1
. Let
R
m,tþ1

X
N
i¼1
a
i,t
R
i,tþ1

(4:19)
denote the rate of return on the market portfolio,
and a
i,t
be the fraction of the investor’s total
wealth held in the i th asset in period t. There are
only N À 1 independent elements in a
i,t
, since the
constraint
X
N
i¼1
a
i,t
¼ 1(4:20)
holds for all t. The representative agent’s dynamic
budget constraint in terms of real variables can be
written as:
W
tþ1
P
tþ1
¼ R
m,tþ1
P
t
P
tþ1
W

t
P
t
À
C
t
P
t

(4:21)
where W
tþ1
is the investor’s nominal wealth at time
t. The budget constraint in Equation (4.21) is non-
linear because of the interaction between subtrac-
tion and multiplication.
Define I
t
as the information set available to the
representative agent at time t. Denoting by
V(W=P, I) the maximum value of Equation (4.17)
subject to Equation (4.20), the standard Bellman
equations can then be written as:
V
W
t
P
t
, I
t


¼ max
C
t
,{a
i,t
}
N
i¼1
(1 À d)
C
t
P
t

1Àr
(
þd E
t
V
W
tþ1
P
tþ1
, I
tþ1
 !
(1Àr)=(1Àl)
)
(1Àl)=(1Àr)

:
(4:22)
Due to the homogeneity of the recursive struc-
ture of preferences, the value function can be writ-
ten in the following functional form:
V
W
t
P
t
, I
t

¼F(I
t
)
W
t
P
t

1Àl
F
t
W
t
P
t

1Àl

,
(4:23)
where F(:) is an unknown function. The homogen-
eity of degree zero of the recursive utility function
INTERTEMPORAL RISK AND CURRENCY RISK 329
implies that V (W=P, I ) satisfying Equation (4.22)
must be homogeneous of degree zero in W and P.
Let the derivatives with respect to the decision
variables C
t
equal zero, we then obtain:
C
Àr
t
¼
d
1 À d
c(W
t
À C
t
)
Àr
,(4:24)
where c
t
¼ E
t
F
tþ1

R
m,tþ1
P
t
P
tþ1

1Àl

(1Àr)=(1Àl)
.
Given the structure of the problem, the nominal
consumption function is linear in nominal wealth.
Hence, we can rewrite Equation (4.24) as:
m
Àr
t
¼
d
1 À d
c(1 À m
t
)
Àr
(4:25)
where C(W
t
, I
t
) ¼ m(I

t
)W
t
 m
t
W
t
. Combining the
Envelope condition with respect to W
t
with the
first-order condition in Equation (4.25), we obtain
the following functional form:
F
t
¼ (1 À d)
(1Àl)=(1Àr)
C
t
W
t

Àr

(1Àl)=(1Àr)
(4:26)
Substituting this expression into Equation
(4.25), we obtain the following Euler equation for
optimal consumption decision:
E

t
d
C
tþ1
=P
tþ1
C
t
=P
t

Àr

(1Àl)=(1Àr)
(
R
m,tþ1
P
t
P
tþ1

(1Àl)=(1Àr)
)
¼ 1, i ¼ 1, , N
(4:27)
The maximization with respect to the decision vari-
able a
i
(i ¼ 2, , N), given a

1
¼ 1 À
P
N
i¼2
a
i
,on
the right hand side of Equation (4.22), is equivalent
to the following problem:
max
{a
i,t
}
N
i¼2
E
t
F
tþ1
X
N
i¼1
a
i,t
R
i,tþ1
P
t
P

tþ1
!
1Àl
2
4
3
5
s:t:
X
N
i¼1
a
i,t
¼ 1
(4:28)
Using this optimal problem along with Equa-
tion (4.26), it is straightforward to show that the
necessary conditions can be derived as:
E
t
d
C
tþ1
=P
tþ1
C
t
=P
t


Àr

1Àl=1Àr
(
R
m,tþ1
P
t
P
tþ1

[(1Àl)=(1Àr)]À1
(R
i,tþ1
À R
1,tþ1
)
P
t
P
tþ1
)
¼ 0, i ¼ 1, , N
(4:29)
Taking Equations (4.27) and (4.29) together to
represent the Euler equations of the optimal prob-
lem defined in Equation (4.22), we obtain a set of
N equations that provide a more direct comparison
with the traditional expected utility Euler equa-
tions. Multiplying Equation (4.29) by a

i,t
, sum-
ming up by i, and substituting from Equation
(4.27), we obtain:
E
t
d
C
tþ1
=P
tþ1
C
t
=P
t

Àr

u
R
m,tþ1
P
t
P
tþ1

uÀ1
(
R
i,tþ1

P
t
P
tþ1
)
¼ 1, i ¼ 1, , N
(4:30)
where u ¼ (1 À l)=(1 À r). These are the real form
Euler equations which are similar to the nominal
form Euler equations seen in Epstein and Zin
(1989).
When r ¼ l, the Euler equations of the time
additive expected utility model are also obtained
in terms of real variables:
E
t
d
C
tþ1
=P
tþ1
C
t
=P
t

Àr
R
i,tþ1
P

t
P
tþ1

¼ 1, i ¼ 1, , N
(4:31)
Another special case of this model is the loga-
rithmic risk preferences where r ¼ l ¼ 1. Then,
the real Euler equations are equal to the nominal
Euler equations, and can be written in two alge-
braically identical functional forms:
E
t
d
C
tþ1
C
t

À1
R
i,tþ1
"#
¼ 1, i ¼ 1, , N (4:32)
330 ENCYCLOPEDIA OF FINANCE
or
E
t
[R
i,tþ1

=R
m,tþ1
] ¼ 1, i ¼ 1, , N (4:33)
In this case, the parameter r governing inter-
temporal substitutability cannot be identified
from these equations. Hence, there is no difference
between Euler equations of the nonexpected utility
model with logarithmic risk preferences and those
of the expected utility model with logarithmic risk
preferences.
Assume that asset prices and consumption are
jointly lognormal or use a second-order Taylor
expansion in the Euler equation when we assume
that asset prices and consumption are conditional
homoskedastic, then the log-version of the real
Euler equation (4.30) can be represented as:
0 ¼ u log d ÀurE
t
Dc
tþ1
þ (u À 1)E
t
r
m,tþ1
þ E
t
r
i,tþ1
þ u(r À 1)E
t

Dp
tþ1
þ
1
2
[(ur)
2
V
cc
þ (u À 1)
2
V
mm
þ V
ii
À 2ur ( u À 1)V
cm
À 2ur V
ci
þ 2(u À 1)V
im
]
þ
1
2
([(u(r À 1)]
2
V
pp
À 2u

2
r(r À 1)V
pc
þ 2u(u À 1)(r À 1)V
pm
þ 2u(r À 1)V
pi
]
(4:34)
where V
cc
denotes var
t
(c
tþ1
), V
jj
denotes
var
t
(r
j,tþ1
)8j ¼ i,m, V
cj
denotes cov
t
(c
tþ1
,r
j,tþ1

)8j
¼ i,m, V
im
denotes cov
t
(r
i,tþ1
,r
m,tþ1
), V
ip
¼ cov
t
(r
i,tþ1
, p
tþ1
), and p
tþ1
¼ d ln (P
tþ1
) ¼
dP
tþ1
P
tþ1
.
Replacing asset i by market portfolio and under-
going some rearrangement, we are able to obtain a
relationship between expected consumption growth

and the expected return on the market portfolio:
E
t
Dc
tþ1
¼ m
m
þ
1
r
E
t
r
m,tþ1
þ 1 À
1
r

E
t
p
tþ1
(4:35)
where m
m
¼
1
r
log d þ
1

2
urV
cc
þ u
1
r
V
mm

þ21À
1
r

u(r À 1)V
pp
!
À
1
2
"
2uV
cm
þ 2u(r À1)
V
pc
À 2u 1 À
1
r

V

pm
#
When the second moments are conditional
homoskedastic, Equation (4.35) indicates that the
consumption growth is linearly related to the
expected world market return and expected infla-
tion. In addition, the coefficients of these two vari-
ables are summed up to 1.
When we subtract the risk free version of Equa-
tion (4.34) from the general version, we obtain:
E
t
r
i,tþ1
À r
f ,tþ1
¼À
V
ii
2
þ urV
ic
þ (u À ur)V
ip
þ (1 À u)V
im
(4:36)
where r
f ,tþ1
is a log riskless real interest rate.

This result is similar to that of Campbell (1993)
except for the inflation term. Equation (4.36)
shows that the expected excess log return on an
asset is a linear combination of its own variance,
which is produced by Jensen’s inequality, and by a
weighted average of three covariances. The weights
on the consumption, inflation, and market are
ur,(u À ur), and (1 À u), respectively. Moreover,
the weights are summed up to 1. This is one of the
most important differences between Campbell’s
model and our real model.
If the objective function is a time-separable
power utility function, a real functional form of a
log-linear version of the consumption CAPM pri-
cing formula can thus be obtained:
E
t
r
i,tþ1
À r
f , t þ 1
¼À
V
ii
2
þ r V
ic
þ (1 À r)V
ip
(4:37)

The weights on the consumption and inflation
are r and (1 À r), respectively. These weights are
also summed up to 1. However, when the coeffi-
cient of relative risk aversion l ¼ 1, then u ¼ 0.
The model is reduced to the real functional form
of log-linear static CAPM, which is the same as the
nominal structure of log-linear static CAPM.
4.3.2. International Asset Pricing Model
Without Consumption
In order to get a pricing formula without con-
sumption, we apply the technique of Campbell
(1993). Campbell (1993) suggests to linearize the
INTERTEMPORAL RISK AND CURRENCY RISK 331
budget constraint by dividing the nominal form of
Equation (4.21) by W
t
, taking log, and then using a
first-order Taylor approximation around the mean
log consumption=wealth ratio (log (C=W )). Fol-
lowing his approach, approximation of the nom-
inal budget constraint is:
Dw
tþ1
ffi r
m,tþ1
þ k þ 1 À
1
b

(c

t
À w
t
)(4:38)
where the log form of the variable is indicated by
lowercase letters, b ¼ 1 À exp (c
t
À w
t
), and k is a
constant.
Combining Equation (4.38) with the following
trivial equality
Dw
tþ1
¼ Dc
tþ1
þ (c
t
À w
t
) À (c
tþ1
À w
tþ1
), (4:39)
we obtain a difference equation in the log con-
sumption=wealth ratio, c
t
À w

t
. When the log
consumption=wealth ratio is stationary, i.e.
lim
j!1
b
j
(c
tþj
À w
tþj
) ¼ 0, Equation (4.38) implies
that the innovation in logarithm of consumption
can be represented as the innovation in the dis-
counted present value of the world market return
minus the innovation in the discounted present
value of consumption growth:
c
tþ1
À E
t
c
tþ1
¼(E
tþ1
À E
t
)
X
1

j¼0
b
j
r
m:tþ1þj
À ( E
tþ1
À E
t
)
X
1
j¼1
b
j
Dc
tþ1þj
(4:40)
Now we are ready to derive an international
asset pricing model without consumption in terms
of real variables by connecting the log-linear Euler
equation to the approximation log-linear budget
constraint. Substituting Equation (4.35) into
Equation (4.40), we obtain:
c
tþ1
À E
t
c
tþ1

¼ r
m,tþ1
À E
t
r
m,tþ1
þ 1 À
1
r

(E
tþ1
À E
t
)
X
1
j¼1
b
j
r
m,tþ1þj
À 1 À
1
r

(E
tþ1
À E
t

)
X
1
j¼1
b
j
p
m,tþ1þj
(4:41)
Equation (4.41) implies that an unexpected con-
sumption may come from three sources. The first
one is the unexpected return on invested wealth
today. The second one is the expected future
nominal returns. The direction of influence de-
pends on whether 1=r is less or greater than 1.
When 1=r is less than 1, an increase (or decrease)
in the expected future nominal return increases
(or decreases) the unexpected consumption. Con-
versely, when 1=r is greater than 1, an increase
(or decrease) in the expected future nominal
return decreases (or increases) the unexpected con-
sumption. The third one is the inflation in the
investor’s own country. The direction of influence
also depends on whether 1=r is less or greater than
1. When 1=r is less than 1, an increase (or decrease)
in the inflation decreases (or increases) the unex-
pected consumption. Conversely, when 1=r is
greater than 1, an increase (or decrease) in the
inflation increases (or decreases) the unexpected
inflation.

Based on Equation (4.41), the conditional cov-
ariance of any asset return with consumption can
be rewritten in terms of covariance with market
return and revisions in expectations of future mar-
ket return as:
cov
t
(r
i,tþ1
, Dc
tþ1
)  V
ic
¼ V
im
þ 1 À
1
r

V
ih
À 1 À
1
r

V
ihp
,
(4:42)
where

V
ih
¼ Cov
t
r
i,tþ1
,(E
tþ1
À E
t
)
X
1
j¼1
b
j
r
m,tþ1þj
!
and
V
ihp
¼ Cov
t
r
i,tþ1
,(E
tþ1
À E
t

)
X
1
j¼1
b
j
p
tþ1þj
!
Substituting Equation (4.42) into Equation
(4.36), we thus obtain an international asset pricing
model, which is not related to consumption:
332 ENCYCLOPEDIA OF FINANCE
E
t
r
i,tþ1
À r
f ,tþ1
¼À
V
ii
2
þ lV
im
þ (l À 1)
V
ih
þ (1 À l)V
ip

þ (1 À l)V
ihp
(4:43)
The only preference parameter that enters
Equation (4.43) is the coefficient of relative risk
aversion (l). The elasticity of intertemporal sub-
stitution r is not present under this international
pricing model. Equation (4.43) states that the
expected excess log return in an asset, adjusted
for Jensen’s inequality effect, is a weighted average
of four covariances. These are the covariance with
the market return, the covariance with news about
future returns on invested wealth, the covariance
with return from inflation, and the covariance
with news about future inflation. This result is
different from both the international model of
Adler and Dumas (1983) and the intertemporal
model of Campbell (1993). Adler and Dumas use
von Neumann–Morgenstern utility and assume a
constant investment opportunity set to derive the
international model, and therefore neither V
ih
or
V
ihp
is included in their pricing formula. Since the
intertemporal model of Campbell is a domestic
model, it does not deal with the issues of inflation
and currency that are emphasized in our inter-
national asset pricing model.

4.3.3. International Asset Pricing Model When
PPP Deviate
Let us now turn to the problem of aggregation
across investors. It is true that different investors
use different information set and different methods
to forecast future world market return and infla-
tion. To obtain the aggregation results, we first
superimpose Equation (4.43) by a superscript l to
indicate optimal condition for an investor l:
E
t
r
i,tþ1
À r
f ,tþ1
¼À
V
ii
2
þ l
l
V
l
im
þ (l
l
À 1)
V
l
ih

þ (1 À l
l
)V
l
ip
þ (1 À l
l
)V
l
ihp
(4:43)
Then, Equation (4.44) can be aggregated across
all investors in all countries.
The operation is to multiply Equation (4.44) by
h
l
, which indicates risk tolerance where h
l
¼ 1=l
l
and to take an average of all investors, where
weights are their relative wealth. After aggregating
all investors, we obtain:
E
t
r
i,tþ1
À r
f ,tþ1
¼À

V
ii
2
þ
1
h
m
V
m
im
þ
1
h
m
À 1

X
l
v
l
V
l
ih
þ 1 À
1
h
m

X
l

v
l
V
l
ip
þ 1 À
1
h
m

X
l
v
l
V
l
ihp
(4:45)
where h
m
¼
(
P
l
W
l
h
l
)
(

P
l
W
l
)
and v
l
¼
(1 À l
l
)W
l
P
l
(1 À l
l
)W
l
.
There are several interesting and intuitive results
in this equation. First, Equation (4.45)shows that
an international asset risk premium adjusted for
one-half its own variance is related to its covar-
iance with four variables. These are the world mar-
ket portfolio, aggregate of the innovation in
discounted expected future world market returns
from different investors across countries, aggregate
of the inflation from different countries, and ag-
gregate of the innovation in discounted expected
future inflation from different investors across

countries. The weights are 1=h
m
,1=h
m
À 1,
1 À 1=h
m
, and 1 À 1=h
m
, respectively. The sum of
these weights is equal to 1. Moreover, it is noted
that the market hedging risk is a weighted average
of world market portfolio for investors from dif-
ferent countries. This is different from the domestic
counterpart of Campbell (1993).
Second, an international asset can be priced
without referring to its covariance with consump-
tion growth. Rather, it depends on its covariance
with world market return, the weighted average of
news about future world market return for inves-
tors from different countries, inflation, and the
weighted average of news about future inflation
for investors from different countries.
Third, the coefficient of risk tolerance, h
m
, is the
only preference parameter that enters Equation
(4.45). When consumption is substituted out in
INTERTEMPORAL RISK AND CURRENCY RISK 333
this model, the coefficient of intertemporal substi-

tution r disappears. Similar results have been
documented by Kocherlakota (1990) and Svensson
(1989). They show that when asset returns are
independently and identically distributed over
time, the coefficient of intertemporal substitution
is irrelevant for asset returns.
If we are willing to make some more assump-
tions, we can obtain a more compact result.
Namely, if investors are assumed to have the
same world market portfolio and use the same
method to forecast world market portfolio return,
we can multiply Equation (4.44) by l
l
, and take an
average of all investors, where weights are their
relative wealth, to get a simple version of the inter-
national asset pricing model:
E
t
r
i,tþ1
À r
f ,tþ1
¼À
V
ii
2
þ l
m
V

im
þ (l
m
À 1)V
ih
þ (1 À l
m
)
X
l
v
l
V
l
ip
þ (1 À l
m
)
X
l
v
l
V
l
ihp
(4:46)
where l
m
¼ (
P

l
W
l
l
l
)=(
P
l
W
l
) and v
l
¼
(1Àl
l
)W
l
P
l
(1Àl
l
)W
l
Both hedging risk V
ih
and currency risk V
l
ip
are
related to expected return. In addition, they all

depend on whether l
m
is different from 1 or not.
Furthermore, when we assume that domestic
inflation is nonstochastic, the only random com-
ponent in p would be the relative change in the
exchange rate between the numeraire currency and
the currency of the country, where the investor
resides. Hence, V
l
ip
is a pure measure of the expos-
ure of asset i to the currency risk of the country,
where investor l resides and V
l
ihp
is also a measure of
the exposure of asset i to hedge against the currency
risk of the country, where investor l resides.
Equation (4.46) also states that the currency risk
is different from the hedging risk. However, if V
ih
and V
l
ip
are large enough, then whether V
ih
and V
l
ip

and related to expected return depends on whether
or not l
m
is different from 1: This may be the
reason why Dumas and Solnik (1995) argue that
exchange rate risk premium may be equivalent to
intertemporal risk premium. But, their conjecture
is based on an empirical ‘‘horse race’’ test between
international model and intertemporalmodel rather
than a theoretical derivation.
4.4. Conclusion
The international asset pricing model without con-
sumption developed by Chang and Hung (2004)
argues that the real expected asset return is deter-
mined by market risk, market hedging risk, cur-
rency risk, and currency hedging risk. The weights
are related only to relative risk aversion. More-
over, the weights are summed up to 1. Their results
may be contrasted with the pioneering work of
Adler and Dumas (1983), who assume a constant
investment opportunity set, thus their model lacks
market hedging risk and currency hedging risk.
In the Chang et al. (2004) model, the price of
market hedging risk is equal to the negative price
of the currency risk. This may be the reason why
Dumas and Solnik (1995) argue that currency risk
is equivalent to market hedging risk. But, their
conjecture is based on a ‘‘horse race’’ test between
international model and intertemporal model ra-
ther than on a theoretical derivation.

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