Morningstar - Q1 2023 - 50

Strategies
Par
1.25
Par
EXHIBIT 2
gIn January 1926, the investor invests all of their
portfolio in 60-month bonds at par that pay
a monthly coupon.
gIn each subsequent month, the investor collects
the coupons from the bonds purchased
the previous month, which are added to their
portfolio value.
gThe investor sells the bonds, which now
have a term of 59 months.
gThe investor invests their portfolio in 60-month
bonds at par.
The process is the same for 20-year bonds, except
that the bonds purchased have a 240-month term
and the ones sold have a 239-month term.
In reality, U.S. government bonds pay coupons
every six months. However, assuming that
coupons are paid monthly simplifies the math
with little effect on the results.
We can now give the formulas for the
monthly returns on five-year and 20-year
bonds. Let:
yt
yt
A5
A20
Rt
Rt
M5
= the annual yield on five-year bonds
in month t
= the annual yield on 20-year bonds
in month t
P(y, c, T) =
M20
P(y, c, T) =
= the monthly return on five-year bonds
in month t
T
∑
t=1
I calculate the monthly returns as follows:(1 ) T
c
y
Rt
Rt
M5
yt -1
A5
=
12
M20
yt -1
A20
=
12
+ P
1 -
(1 ) T
+y
+ P
yt
yt
A5
,
12
A20
,
12
12
EXHIBIT 2 shows the history of yields and
cumulative returns over the entire period. The
upper panel shows the five-year and
20-year yields. The lower panel shows the
cumulative values of $1 invested in January 1926
in five-year bonds, 20-year bonds, and 30-day
Treasury bills. It also shows the cumulative
level of inflation.
50
Morningstar Q1 2023
12
yt -1
A20
, 239
- 1
yt -1
A5
, 59
+
+y
Here are a few important points that stand out
in EXHIBIT 2 :
- 1
gIn accordance with the formulas for bond
prices and bond returns, bond returns
were positive when yields were falling and
negative when yields were sharply rising,
the effect being greater for 20-year bonds than
for five-year bonds.
o
gYields rose from the 1950s until reaching their
peak in 1981, and then fell until recently when they
began to rise again. The periods of rising and
falling interest rates roughly correspond to periods
of rising and falling inflation. If bond investors
form their inflation expectations based on recent
c
(1 ) t
+y
1
+
1
(1 ) T
+y
= the monthly return on 20-year bonds
in month t
1
The Long-Term View, January 1926 to October 2022
0.75
1%
Yield
Historical Bond Yields
20%
1%
Yield
10
15
20%
10
15
5
01/1926
5
Cumulative Values of Bonds and Treasury Bills, and Cumulative Inflation (Compound Annual Rates
of Return in Parentheses)
$200.0
$200.0
100.0
10.0
10.0
1.0
0.1
1.0 01/1926
0.1
01/1926
Source: Morningstar Direct.
f = n- qT
n = qT
n
inflation (so-called adaptive expectations),
then the historical record is consistent with the
Fisher Identity.
P =
c
q
∑
j=1
n
gOver the nearly 97-year period, consumer prices
have risen more than 16.5 times their level in
January 1926, for an overall inflation rate of just
under 3% per year.
P(y; c, T) = ∑(1+
j=1
c
q
D(y; c, T) =
∑ n
j=1
j -f
q
y
q
)
+ 1+
(
c
q
y
q
)
P(y; c, T) = P(y; c,
c
q
(
P(y; c, T)
1+
y
q
)
=
c
q
(
P(y; c,
(
f-j
=
c
q
n
q
)(1+
(
y
q
)
f
( ) (1+
y
q
f-j
P(y; c, T)
gThirty-day Treasury bills barely kept up with
inflation; $1 invested in January 1926 rose to just
over $21.84 for an annual rate of return of
3.24%. This means that the real return on Treasury
bills was just under 0.3%. Investors who want
to outperform inflation by any significant amount
need to look elsewhere.
1+
P(y; c,
y
1+ q
)
n
q
-j
)
)
f
n
q
y
q
)
f-j
)(1+
y
q
)
f
)
+ 1+T
(
f-j
- qT
-j
+ 1+
(
y
q
)
-n
(
1+
y
q
)
f
y
q
)
- qT
01/1950
01/1974
01/1998
10/2022
01/1950
01/1974
01/1998
10/2022
Inflation
16.56 (2.94%)
100.0 01/1926
Logarithmic Scale
Logarithmic Scale
01/1950
01/1974
01/1998
10/2022
01/1950
01/1974
01/1998
10/2022
2
16.4% 5-Year Bond
Yield Peak (August 1981)
16.4% 5-Year Bond
Yield Peak (August 1981)
3
4
14.8% 20-Year Bond Yield
Peak (September 1981)
14.8% 20-Year Bond Yield
Peak (September 1981)
20-Year Bond
4.52
5-Year Bond
4.33
5-Year Bond
4.33
20-Year Bond
96.31 (4.83%)
5-Year Bond
62.83 (4.37%)
20-Year Bond
96.31 (4.83%)
30-Day
Treasury Bills
21.84 (3.24%)
Inflation
16.56 (2.94%)
30-Day
Treasury Bills
21.84 (3.24%)
5-Year Bond
62.83 (4.37%)
20-Year Bond
4.52
5
2
3
4
5
1.00
1.00
0.75
5-Year Bond
20-Year Bond
5-Year Bond
20-Year Bond
Cumulative Value
Cumulative Value
Yield
Yield
Price
Price
Morningstar - Q1 2023

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