Morningstar - Q1 2023 - 49

T
P(y, c, T) =
Yielding to Inflation
In the current environment, there are
few places to hide.
QUANT U
Paul D. Kaplan
In the previous issue of Quant U, I presented
Milton Friedman's version of the quantity
theory of money with rational expectations as an
explanation for inflation. One of the building
blocks of that theory is the Fisher Identity, which
states that the nominal interest rate is the
real interest rate plus the expected rate of inflation.
In this issue of Quant U, I discuss the historical
relationships between inflation, interest
rates (bond yields), bond prices, and bond returns.
Bond Yields and Prices
The rate of interest on a bond is called the bond's
yield to maturity, or yield for short. While there
are many types of bonds, here, I only consider
coupon bonds. Coupon bonds pay interest at fixed
intervals and pay the bond's face value at
maturity. To further simplify things, I set the
face value to $1, because face value in my analysis
is just an arbitrary scaling factor.
The relationship between bond price and
yield can be expressed using a bond price function
as follows:
T
P(y, c, T) =
P(y, c, T) =
c
Rt
M5
T
Rt
M20
∑
t=1
= periodic yield (3% per year = 0.03,
for example)
c
y
(1 ) T
+y
=
=
= number of periods to maturity (five years,
for example)
+ P
= periodic coupon rate (3% per year =
0.03, for example)
12
yt -1
A5
12
yt -1
A20
The bond price function can also be written:
12
12
12
yt
A20
,
+ P
yt
A5
,
12
yt -1
A20
, 239
- 1
yt -1
A5
, 59
- 1
+y
c
(1 ) t
+y
P() = the bond price function
y
1 -
1
+
+
1
(1 ) T
+y
1
(1 ) T
P(y, c, T) =
Rt
Rt
M5
yt -1
A5
12
12
∑
t=1
c
y
c
(1 ) t
+y
1 -
1
(1 ) T
+y
=
=
+ P
+ P
yt
M20
yt
A5
yt -1
A5
12
12
,
,
, 59
+
+
1
(1 ) T
+y
1
(1 ) T
+y
Note that P(y,y,T) = 1. In other words,
if the yield and the coupon rate are equal, the
price of the bond is the same as its
face value. In this case, we say that the bond
is trading at par. Coupon bonds are often
issued at par.
- 1
12
yt -1
A20
A20
12
EXHIBIT 1 shows the relationship between yield
and price for a five-year bond and a 20-year
bond, both with an annual coupon rate of 3%.
When the annual yield is 3%, both bonds are at
par. At lower yields, both bonds are above
par, and at higher yields, both bonds are below
par. Overall, there is a negative relationship
between yields and prices, but the yield/price
curve is much steeper for the 20-year bond
than the five-year bond. In general, the longer the
term of a bond, the more sensitive its price
is to changes in yield. In other words, the longer
the term of the bond, the more it is subject
to interest-rate risk.
EXHIBIT 1
Yield vs. Bond Price for Five- and 20-Year Bonds With 3% Coupon Rate
$1.50
1.25
Par
1.00
0.75
Source: Morningstar Direct.
1%
Yield
20%
2
3
4
5
5-Year Bond
20-Year Bond
gIntermediate-term government bonds: yields.
gLong-term government bonds: yields.
g30-day U.S. Treasury bills: total returns.
gInflation.
In SBBI, intermediate-term bonds have a term of
about five years and long-term government
bonds have a term of about 20 years. So, from now
on, I refer to them as five-year and 20-year
bonds, respectively.
To measure how five-year bond investors
fared over the nearly 97-year period, I assume
the following:
16.4% 5-Year Bond
Yield Peak (August 1981)
14.8% 20-Year Bond Yield
Peak (September 1981)
morningstar.com/products/magazine
49
The History of Inflation and Bonds
To study the history of inflation and bonds,
I obtained monthly U.S. historical data for the
period January 1926 through October 2022.
The data through December 2021 appear in the
2022 SBBI Yearbook, now published by Kroll,
and are updated monthly in Morningstar Direct.
(SBBI stands for " Stocks, Bonds, Bills, and
Inflation. " ) I exported the following series from
Morningstar Direct:
o
yt -1
A20
, 239
- 1
Price
http://www.morningstar.com/products/magazine
Morningstar - Q1 2023

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