Fixed Income Predictability
Andrea Buraschi & Paul Whelan
Imperial College Business School
March 2011
Andrea Buraschi & Paul Whelan
The Term Structure of Interest Rates
“The quest for understanding what moves bond yields has produced an enormous literature
with its own journals and graduate courses. Those who want to join the quest are faced
with considerable obstacles. The literature has evolved mostly in continuous time, where
stochastic calculus reigns and partial differential equations (PDEs) spit fire.The knights in
this literature are fighting for different goals, which makes it often difficult to comprehend
why the quest is moving in certain directions. But the quest is moving fast, and dragons
are being defeated.”, Monika Piazzesi.
Andrea Buraschi & Paul Whelan
The expectations hypothesis
I The 1-year spot rate is 3%. Agents have perfect foresight and know the spot rate will
be 5% in one year. What is today’s 2-year yield?
P(1, 2) = e −r (1,2)×1 × 100 = e −5% × 100 = 95.12
Discounting this price we have:
P(0, 2) = e −r (0,1)×1 × P(1, 2) = 0.97 × 95.12 = 92.31
I Then r (0, 2) = − ln(0.92)/2 = 4% = the average between todays spot rate and next
years expected spot rate.
I Writing P(0, 2) = e −r (0,1)−r (1,2) × 100 and from the definition of the 2-year yield
P(0, 2) = e −r (0,2)×2 × 100, we have
r (0, 2) =
1
1
r (0, 1) + r (1, 2)
2
2
The long-term yield is a weighted average of the current short and future expected
short yields.
I The positive relationship between market participants expectations about future rates
and the current shape of the term structure is know as the expectations hypothesis .
Andrea Buraschi & Paul Whelan
The expectations hypothesis
I The expectations hypothesis can generally be written as three equivalent statements:
1. The N-period yield is the average of expected future one-period yields:
ytN =
N−1
1 X
Et [yt+j ] + RP
N
j=0
2. The forward rate equals the expected future spot rate
1 ftN→N+1 = Et yt+N
+ RP
3. The expected holding period returns on bond of all maturity are equal
N Et rett+1
= yt1 + RP
I The expectations hypothesis says that the risk premia are constant adjustments for
maturity specific risk, i.e., something like duration risk, rollover risk, or inflation risk.
Andrea Buraschi & Paul Whelan
Why does the term structure slope upwards?
I The shape of the yield curve is related to the risk of investing in long term bonds
versus short-term bonds.
I Assume that yields are log-normally distributed.
I For a given yield r (t + 1, T ) the value of a zero at time t + 1 with maturity T will be
P(t + 1, T ) = e −r (t+1,T )×(τ −1) × 100
where τ is the time to maturity of the bond at t.
I The value of the bond maturity at T is then
P(t, T ) = e −r (t,t+1)+λ × Et [P(t + 1, T )]
P(t, T ) = e −r (t,t+1)+λ × e
(τ −1)2
−Et [r (t+1,T )×(τ −1)+
2
(1)
Vt (r (t+1,T ))
× 100
where λ denotes the risk premium for investing in long-term bonds for a 1-year
horizon compared to risk-free 1-years zeros.
Andrea Buraschi & Paul Whelan
(2)
Why does the term structure slope upwards?
I From P(t, T ) = e −r (t,T )×τ × 100 we have:
»
r (t, T ) =
–
τ −1
1
× r (t, t + 1) +
× Et [r (t + 1, T )]
τ
τ
λ
+
τ
(τ − 1)2
−
Vt (r (t + 1, T ))
2τ
Expected future yield
Risk Premium
Convexity
I The role of λ is can be understood by rewriting 1:
»
Et
– »
–
P(t + 1, T )
100
=
× eλ
P(t, T )
P(t, t + 1)
I Thus, in an uncertain world the term structure is rising even if investors do not
expect yields to rise.
Andrea Buraschi & Paul Whelan
Campbell and Shiller (1991)
I Setting the risk premium and convexity terms equal we can re-write the above as:
Et [r (t + t, t + τ ) − r (t, t + τ )] =
1
[r (t, t + τ ) − r (t, t + 1)]
τ −1
I The slope of the term structure is positively related to expected yield changes as
postulated by the expectations hypothesis.
I Campbell and Shiller test this relation by running
r (t + t, t + τ ) − r (t, t + τ ) = α + β
1
[r (t, t + τ ) − r (t, t + 1)] + (t + 1)
τ −1
I high yield spreads should forecast increases in long rates.
Andrea Buraschi & Paul Whelan
Campbell and Schiller (1991)
Maturity
2
Long yield changes
se
Short yield changes
se
3
6
12
24
48
120
0.003 −0.145 −0.835 −1.435 −1.448 −2.262 −4.226
(0.191) (0.282) (0.442) (0.599) (1.004) (1.458) (2.076)
0.502
0.467
0.320
0.272
0.363
0.442
1.402
(0.096) (0.148) (0.146) (0.208) (0.223) (0.384) (0.147)
Table: Monthly zero coupon bonds 1952:1 to 1991:2 from McCulloch-Kwon data set.
I According to the expectations hypothesis β = 1.
I The first two rows report results from the regression above:
a high yield spread between a longer-term and a shorter-term interests predicts a
declining yield on the longer-term bond over the life of the short term bond.
I The first two rows report results from a similar specification which says high yield
spreads should forecast rising short rates over the life of the long term bond:
a high yield spread between a longer-term and a shorter-term interests predicts rising
shorter-term interest rates over the life of the long term bond.
Andrea Buraschi & Paul Whelan
Fama and Bliss (1987)
I The relationship between the slope of the term structure is also revealed by the
return on investments of long over short-term bonds.
I Fama and Bliss address x2 questions:
I
I
Do forward rates contain information about expected returns on bonds?
Do forward rates forecast future interest rates?
I The time t price of a zero is the present value of 1-dollar discounted at the expected
values of the future 1-year expected returns:
n
n−1
1
ptn = exp (−Et [rt+1
] − Et [rt+2
] − · · · − Et [yt+n−1
])
which is a tautology.
I Adding the assumption of rational expectations then price contains rational forecasts
of equilibrium expected returns.
Andrea Buraschi & Paul Whelan
Fama and Bliss (1987)
I Summing the last n − 1 terms in the exponential:
n
n−1
ptn = exp (−Et [rt+1
] − Et [yt+1
])
I Subbing into the definition of a forward contract we obtain:
“
” “
”
n
n−1
ftn−1→n − yt1 = Et [rt+1
] − yt1 + Et [yt+1
] − ytn−1
I The forward-spot spread contains:
1. the term premium for a 1-year return on an n-year bond over the spot
rate.
2. and the expected change over the next year of the yield on n − 1 year
bonds.
I The expectations hypothesis says that expected returns are constant thus forward
rates are optimal forecasters of expected future spot rates.
I Sound familiar? dividend growth should be forecastable so that returns are not
forecastable.
Andrea Buraschi & Paul Whelan
Fama and Bliss (1987)
“
(1)
(n)
I Running the regression: rxt,t+1
= α + β ftn−1→n − yt
”
(n)
+ t+1
I Evidence that β is positive implies that term premia are time-varying.
Maturity
2
(se)
3
(se)
4
(se)
5
(se)
α
β
R2
−0.00
0.00
−0.00
0.01
−0.01
0.01
−0.01
0.01
1.10
0.31
1.29
0.41
1.48
0.57
1.20
0.76
0.18
0.19
0.20
0.06
Table: Fama Bliss term premium regression. Sample Period 1964 : 2008
Andrea Buraschi & Paul Whelan
Fama and Bliss (1987)
I All β’s are within one standard error of 1.0
I This implies that on a 1-year horizon yield changes follow a random walk. If
“
”
(1)
(n)
n
rxt+1
= 0 + 1 × ftn−1→n − yt
+ t+1
then, writing out the definition of a holding period return and forward rate
(n)
n−1
pt+1
− ptn + pt1 = 0 + 1 × (ptn−1 − ptn + pt1 ) + t+1
(n)
n−1
pt+1
= 0 + 1 × (ptn−1 ) + t+1
(n)
n−1
yt+1
= 0 + 1 × (ytn−1 ) − t+1 /(n − 1)
I a coefficient of 1.0 implies yields or prices that follow random walks.
I caveat: yields are obviously stationary and thus predictable, however, they are very
persistent and thus difficult to predict at short horizons!
Andrea Buraschi & Paul Whelan
Fama and Bliss (1987)
I Summing the first n − 1 terms in the above exponential we can obtain a different
forecasting regression:
n
1
ptn = exp (−Et [rt+n−1
] − Et [yt+n−1
])
I Again, subbing into the definition of a forward
“
” “
”
1
n
ftn−1→n − yt1 = Et [yt+n−1
] − yt1 + Et [rt+n−1
] − ytn−1
I Written this way the n − 1-year forward-spot spread contains:
1. the term premium for an (n − 1)-year return on an n-year bond over the
spot rate.
2. and the expected change of over the next n − 1 years of the spot rate.
I Again, these are complimentary regressions: mechanically forward spreads cannot
forecast short term changes in spot rates.
Andrea Buraschi & Paul Whelan
Fama and Bliss (1987)
Maturity
1
se
2
se
3
se
4
se
α
β
R2
0.19
0.33
0.36
0.68
0.48
0.73
0.84
0.58
−0.10
0.28
0.56
0.43
1.36
0.25
1.74
0.34
0.00
0.05
0.26
0.47
Table: Fama-Bliss forecast of changes in yields. Sample period - 1964:2008
Andrea Buraschi & Paul Whelan
Summary Statistics
Maturity
n
E [rett+1
]
1
2
3
4
5
5.52
5.92
6.23
6.39
6.39
Andrea Buraschi & Paul Whelan
√σ
N
0.11
0.13
0.17
0.21
0.24
n
σ(rett+1
)
Sharpe
2.86
3.61
4.50
5.45
6.25
0.00
0.23
0.22
0.20
0.16
Fama and Bliss (1987)
I In a nutshell ...
I The expectations hypothesis does badly at short horizons but performs much better
at longer horizons.
I A high forward rate seems to entirely indicate that you will earn that more in holding
long term bonds than short. How do we reconcile this with the summary statistics
table?
I Short rates do eventually rise to meet the forward rate forecast.
I The response is slow as the short rate adjustment is sluggish.
I Fama-Bliss attribute the predictability of the short rate at longer horizons to a slow
mean-reverting tendency the 1-year interest rate.
Andrea Buraschi & Paul Whelan
Cochrane and Piazzesi (2005)
I Forecasts 1 year treasury bond returns over the 1 year rate:
(n)
(n)
rxt+1 = an + bn CPt + t+1
I R 2 up to 44% from Fama-Bliss / Campbell Shiller 15%
I a single factor γ 0 ft = CPt forecasts bonds of all maturities.
I Tent shaped factor is correlated with slope but is definitely not a slope proxy. The
improvement above the slope is because it signals when to bail - when rates will rate
in an upward sloping environment.
Andrea Buraschi & Paul Whelan
A single factor for expected bond returns
“
”
(n)
(n)
rxt,t+1 = bn γ0 + γ1 y (1) + γ2 f (1→2) + γ3 f (2→3) + γ4 f (3→4) + γ5 f (4→5) + εt+1
with the restriction
5
1X
bn = 1
4 n=2
Andrea Buraschi & Paul Whelan
Andrea Buraschi & Paul Whelan
I
Run regression of bond excess returns on all forward rates not just
maturity specific spreads (Fama-Bliss)
I
The same linear combination of forward rates forecasts all bond returns.
Andrea Buraschi & Paul Whelan
Andrea Buraschi & Paul Whelan
More Lags
Andrea Buraschi & Paul Whelan
I
More lags are significant + same pattern. Suggests moving averages
(n)
(n)
rxt+1 = an + bn γ (α0 ft + α1 ft−1 + · · · αk ft−k ) + t+1
(n)
= an + bn α0 (γ 0 ft ) + α1 (γ 0 ft − 1) + · · · αk (γ 0 ft−k ) + t+1
k
R̄ 2 0.35
I
1
2
3
4
5
0.41
0.43
0.44
0.44
0.43
6
Interpretation: yields should (we hope) be Markov, so a small transitory
measurement error ft−1/12 is informative about the true ft so it enters
with the same pattern.
Andrea Buraschi & Paul Whelan
I 5 year bond had b = 1.43. Thus, 1.73 − 2.11 is what you expect for a perpetuity
I Does better than D/P or spread. Drives out spread; survives with D/P.
I a common risk premium embedded in stocks and bonds. Reassuring since stocks can
be viewed as a default free bonds plus cash flow risk.
Andrea Buraschi & Paul Whelan
Interest Rate Forecasts
(2)
(1)
(1)
(2)
Et (rxt+1 ) = −Et (yt+1 − yt ) + (ft
(1)
− yt )
(2)
I Expectations hypothesis: Et (rxt+1
) = constant
(1)
(1)
I Fama-Bliss: Et (yt+1
− yt ) = constant
(2)
(2)
(1)
I Cochrane-Piazzesi: Et (rxt+1
) is now even more variable than (ft − yt ).
I the short rate must then be forecastable precisely because the expectations
hypothesis is even more wrong than Fama and Bliss suspected
I CPt must predict short rate changes in roughly the wrong direction that the EH
predicts. (capital gains on long bonds, not just riding on yields).
Andrea Buraschi & Paul Whelan
I
γ 0 f and slope are correlated: both show a rising yield curve with no rate
rise.
I
γ 0 f tells you when a steep yield curve will NOT be followed by a decline
in rates.
I
The signal is the tent.
Andrea Buraschi & Paul Whelan
Macroeconomic Interpretation
I The slope of the term is correlated with recessions. Inverted yield curves are a
recession predictor.
I CPt is closely linked to the business cycle: expected returns are high in bad times and
low in good times.
I CPt is correlated with the level of unemployment.
I CPt is uncorrelated with inflation.
I CPt is a level not growth factor.
Andrea Buraschi & Paul Whelan
Finance Interpretation
Andrea Buraschi & Paul Whelan
Yield Factors
Andrea Buraschi & Paul Whelan
I Panel A: yields are a linear combination of forwards.
γ 0 f = γ ?0 y ;
γ ? ≈ slope plus 4-5 spread
I Panel B: γ 0 f has nothing to do with slope or curvature.
I Panel C: you cannot approximate γ 0 f well with level, slope and curvature factors.
I Moral of the story: term structure models needs level, slope and curvature factors to
get ∆yt+1 AND γ 0 f to get Et [rxt+1 ]. Adding γ 0 f does very little to improve pricing
errors but does help to get transition dynamics right.
Andrea Buraschi & Paul Whelan
Yield Factors
Andrea Buraschi & Paul Whelan
Measurement Error
Danger: if pt is measured to high then rt+1 = pt+1 − pt will too low, and a high pt will
seem to forecast a low rt+1 . Is this what CP is picking up on?
Probably not.
1. Lags also forecast with no common price.
2. Also forecasts stock returns.
3. Measurement error gives a pattern that the n period yield at t forecasts the n period
bond return. It does not show the pattern that the m period yield forecasts the n
bond returns. Measurement error cannot produce the central finding of a common
forecasting factor.
Andrea Buraschi & Paul Whelan