Deflation Risk
Matthias Fleckenstein, Francis A. Longstaff, Hanno Lustig
Q Group Conference
October 2015
Introduction
Deflation has played a central role during the worst economic
meltdowns in U.S. history.
Panic of 1837
Long Depression of 1874-1896
Great Depression of the 1930s
Growing fears of deflation in the financial press.
”Nightmare scenario”
”Looming disaster”
”Growing threat”
Mitigating risk of deflation is an explicit motivation behind many
recent measures by the Federal Reserve such as the Quantitative
Easing Programs.
Deflation Risk
2 / 32
Motivation
Relatively little is known about the probability of deflation.
Reason may be that the distribution of inflation is difficult to measure.
Ang, Bekaert, and Wei (2007) show that econometric models perform
poorly in estimating first moment of inflation.
Survey data does better, but only looks at first moments, not tail
probabilities.
Deflation Risk
3 / 32
Recent Decline in Global Inflation
Global Inflation
6
US
UK
G7
Ger
5
4
3
2
1
0
−1
−2
−3
08
∗
09
10
11
12
13
14
year-over-year inflation rate in percentages.
Deflation Risk
4 / 32
Deflation in Europe’s Periphery
Deflation in Europe
6
Spain
Ireland
Greece
Rate of year−over−year Inflation (in percent)
4
2
0
−2
−4
−6
−8
08
∗
09
10
11
12
13
14
year-over-year inflation rate in percentages.
Deflation Risk
5 / 32
History of U.S. Deflation
U.S. Deflation
16
14
Percentage rate of deflation
12
10
8
6
4
2
0
∗
1861
1889
1916
1943
1971
1998
year-over-year deflation rate in percentages. shaded areas are NBER recessions
Deflation Risk
6 / 32
This Paper
We use a new market based approach to measure deflation risk.
First we solve for the risk-neutral density of inflation from the market
prices of inflation calls and puts.
Then we solve for the inflation risk premium via MLE estimation of an
affine term structure model using the term structure of inflation swap
rates.
Finally, we solve for the objective density of inflation by adjusting the
pricing measure for the market price of risk.
Deflation Risk
7 / 32
Results
Long run expected inflation is about 3.00 percent.
Average inflation risk premium is close to zero for horizons out to
about 20 years.
Inflation volatility is roughly 1.50 percent.
Probability of deflation sizable.
Bernanke, Aug 27, 2010, ”Falling into deflation is not a significant risk.”
On same date, market-implied probability of deflation was 27.35
percent for two-year horizon, 16.15 percent for five-year horizon, and
6.39 percent for a ten-year horizon.
Tail risk of deflation priced similarly to other types of tail risk such as
catastrophe insurance and corporate bond defaults.
Deflation correlated with other types of financial and economic tail
risk such as systemic credit risk, liquidity risk, and unemployment.
Inflation risk is priced less severely by the market.
Deflation Risk
8 / 32
Dynamics of the Price Level
Objective Measure
dI = XIdt + σIdZI
dX = κ (Y − X ) dt + ηdZX
dY = (µ − ξY ) dt + sdZY
Risk-Neutral Measure
dI = XIdt + σIdZI
dX = λ (Y − X ) dt + ηdZX
dY = (α − ξY ) dt + sdZY
Deflation Risk
9 / 32
Valuing Inflation Swaps
The payoff on a T -year inflation swap paying f on the fixed leg is
(1 + f )10 − I10 .
The value of the inflation swap at time zero is
F (X , Y , T ) = exp (−A(T ) − B(T )X − C (T )Y )
where
2
1 σ2
−λT
−2λT
T
−
1
−
e
+
1
−
e
...
2λ2
λ
2λ
A (T ) =
− 1 − e −λT
B (T ) =
λ λ
1
1
−ξT
−λT
C (T ) =
1−e
−− 1−e
ξ −λ ξ
λ
Deflation Risk
10 / 32
Valuing Inflation Options
The payoff on a European-style
inflation cap at expiration
date T is
T
max 0, IT − (1 + K ) . Similarly, for a floor max 0, (1 + K )T − IT .
The value of the call option at time zero is
C (X , Y , V , T ) = D (T ) E Q
∗
h
i
max (0, IT − (1 + K )T ,
The expectation is taken with respect to the adjusted risk-neutral
measure Q ∗ for inflation defined by the following dynamics
dI = XIdt + σIdZI
h
i
dX = λ (Y − X ) + η 2 B (T − t) dt + ηdZX
h
i
dY = α − ξY + s 2 C (T − t) dt + sdZY
Deflation Risk
11 / 32
Data
Inflation swaps
Maturities from one to 55 years for the period from July 23, 2004 to
July 29, 2014.
Inflation caps and floors
Strikes from −2% to 6 % in increments of 50bps, and maturities from
1 to 30 years for the period from October 5, 2009 to July 29, 2014.
Inflation surveys
University of Michigan Survey of Consumers, the Philadelphia Federal
Reserve Bank Survey of Professional Forecasters, and the Livingston
Survey for the period from July 2004 to July 2014.
Measures of financial tail risk
One-year Refcorp-Treasury yield spread, 10–15 % CDX IG index
tranche prices, one-year Libor-Treasury spread, five-year swap spread,
VIX index, Merrill Lynch MOVE index, Baa spread over the five-year
Treasury rate, spread for a five-year CDS contract on the U.S. Treasury.
Macroeconomic variables
U.S. industrial production, U.S. unemployment rate, University of
Michigan consumer confidence index.
Deflation Risk
12 / 32
Table 1
Summary Statistics for Inflation Swap Rates. This table reports summary statistics for the inflation swap rates for the indicated
maturities. Swap maturity is expressed in years. Inflation swap rates are expressed as percentages. The sample consists of daily observations
for the period from July 23, 2004 to July 29, 2014.
Swap
Maturity
Mean
Standard
Deviation
Minimum
Median
Maximum
N
1
2
3
4
5
6
7
8
9
10
12
15
20
25
30
35
40
45
50
55
1.743
1.919
2.077
2.207
2.314
2.394
2.465
2.522
2.570
2.615
2.671
2.739
2.800
2.841
2.888
2.810
2.769
2.866
2.784
2.878
1.244
0.987
0.775
0.626
0.525
0.450
0.389
0.344
0.304
0.272
0.255
0.256
0.266
0.277
0.276
0.156
0.115
0.169
0.139
0.165
−4.545
−3.605
−2.047
−1.228
−0.570
−0.080
0.402
0.640
0.904
1.146
1.280
1.161
1.070
1.211
1.455
2.353
1.454
2.220
1.465
2.386
1.862
1.992
2.141
2.268
2.377
2.468
2.530
2.580
2.622
2.664
2.713
2.778
2.845
2.887
2.923
2.809
2.828
2.894
2.821
2.861
3.802
3.460
3.351
3.342
3.310
3.310
3.229
3.195
3.135
3.145
3.160
3.330
3.360
3.390
3.500
3.119
3.377
3.305
3.500
3.287
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
1023
1649
1023
1649
1023
Table 2
Summary Statistics for Inflation Caps and Floors. This table reports the average values for inflation caps and floors for the indicated
maturities and strikes. The average values are expressed in terms of basis points per $100 notional. Option Maturity is expressed in years.
Ave. denotes the average number of caps and floors available each day from which the risk-neutral density of inflation is estimated. N denotes
the number of days for which the risk-neutral density of inflation is estimated. The sample consists of daily observations for the period from
October 5, 2009 to July 29, 2014.
Average Floor Value by Strike
Average Cap Value by Strike
Option
Maturity
−2
−1
0
1
2
3
4
5
−1
0
1
2
3
4
5
6
Ave.
N
1
2
3
5
7
10
12
15
20
30
4
12
14
23
23
19
17
16
15
10
8
19
22
33
34
31
28
27
25
20
17
33
37
50
56
59
58
55
54
52
38
67
76
103
115
134
136
136
136
154
85
151
184
241
278
335
354
372
399
533
161
304
399
563
679
832
910
1014
1182
1755
294
544
752
1058
1298
1561
1704
1920
2259
2867
398
749
1053
1565
1999
2536
2863
3346
3959
5896
237
523
832
1456
2031
2752
3117
3532
3975
4511
153
349
571
1044
1504
2111
2419
2788
3266
3883
92
201
346
658
1010
1523
1803
2164
2667
3607
38
85
151
309
516
849
1030
1260
1582
2227
13
34
59
118
208
395
484
597
748
1144
6
16
27
56
105
172
224
277
333
584
3
9
16
31
58
99
113
141
186
328
2
6
10
20
37
58
71
78
104
197
22.5
22.6
23.5
23.1
22.9
22.5
23.0
22.8
22.4
20.4
1210
1103
1222
1075
1124
1214
1218
1198
1121
789
Maximum Likelihood Estimation
We assume that the two-year and 30-year inflation swap rates are
measured without error.
Thus, given a parameter vector Θ, X and Y can be expressed as
explicit linear functions of the two inflation swap prices F (X , Y , 2)
and F (X , Y , 30).
ln F (X , Y , 2) = −A(2) − B(2)X − C (2)Y ,
ln F (X , Y , 30) = −A(30) − B(30)X − C (30)Y ,
The log of the joint likelihood function LLKt of the two-year and
30-year inflation swap prices is
= − ln 2πσX σY
− 2ρXY
p
1 − ρXY
Xt+∆t − µXt
σX
(Xt+∆t − µXt )2
1
−
2 1 − ρ2XY
σX2
Yt+∆t − µYt
σY
"
(Yt+∆t − µYt )2
+
σY2
#
We maximize the log likelihood function over the 22-dimensional
parameter
Deflation Risk vector
15 / 32
Table 3
Maximum Likelihood Estimation of the Inflation Swap Model. This table reports the maximum likelihood estimates of the parameters of the inflation swap model along with their asymptotic
standard errors. The model is estimated using daily inflation swap prices for the period from July
23, 2004 to July 29, 2014.
Parameter
κ
η
μ
ξ
s
λ
α
v1
v3
v4
v5
v6
v7
v8
v9
v10
v12
v15
v20
v25
Value
Standard
Error
0.963868
0.042007
0.006743
0.026974
0.008720
0.866571
0.001680
0.00000665
0.00000086
0.00000180
0.00000246
0.00000263
0.00000287
0.00000298
0.00000331
0.00000365
0.00000295
0.00000199
0.00000101
0.00000047
0.324027
0.000625
0.002709
0.111689
0.000008
0.005804
0.000001
0.00000075
0.00000021
0.00000035
0.00000050
0.00000055
0.00000064
0.00000069
0.00000091
0.00000144
0.00000068
0.00000039
0.00000023
0.00000016
Table 4
Summary Statistics for Inflation Risk Premia. This table reports summary statistics for the estimated inflation risk premia for the
indicated horizons. Horizon is expressed in years. The inflation risk premia are measured in basis points. The inflation risk premia are
estimated using the period from July 23, 2004 to July 29, 2014.
Horizon
1
2
3
4
5
6
7
8
9
10
12
15
20
25
30
Mean
Standard
Deviation
Minimum
Median
Maximum
N
−5.45
−7.79
−7.68
−6.35
−4.53
−2.64
−0.89
0.63
1.86
2.79
3.72
2.91
−3.55
−15.04
−30.20
4.58
5.43
5.08
4.45
3.83
3.32
2.90
2.57
2.30
2.08
1.74
1.40
1.05
0.83
0.67
−30.01
−37.28
−35.50
−30.84
−25.74
−21.06
−17.04
−13.70
−10.99
−8.66
−6.08
−4.99
−9.47
−19.69
−33.94
−4.80
−7.01
−6.96
−5.72
−3.99
−2.17
−0.48
1.00
2.19
3.08
3.99
3.13
−3.39
−14.92
−30.10
4.46
3.81
3.07
3.01
3.49
4.26
5.12
5.92
6.58
7.04
7.26
5.73
−1.45
−13.36
−28.80
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
Basis Points
Inflation Risk Premia
40
40
20
20
0
0
−20
−20
Basis Points
−40
2006
2008
2010
2012
2014
−40
40
40
20
20
0
0
−20
−20
−40
2006
Deflation Risk
2008
2010
2012
2014
−40
2006
2008
2010
2012
2014
2006
2008
2010
2012
2014
19 / 32
Figure 1. Inflation Densities. This figure plots the time series of inflation
densities for horizons of one year (upper left), two years (upper right), five years
(lower left), and ten years (lower right).
40
Table 5
Summary Statistics for Expected Inflation. This table reports summary statistics for the expected inflation rate for the indicated horizons.
Horizon is expressed in years. Expected inflation rates are expressed as percentages. The sample consists of daily observations for the period
from July 23, 2004 to July 29, 2014.
Horizon
Mean
Standard
Deviation
Minimum
Median
Maximum
N
1
2
3
4
5
6
7
8
9
10
12
15
20
25
30
1.798
1.997
2.154
2.270
2.359
2.421
2.474
2.516
2.551
2.587
2.634
2.710
2.836
2.992
3.190
1.201
0.936
0.728
0.586
0.491
0.422
0.365
0.324
0.286
0.256
0.243
0.247
0.259
0.271
0.273
−4.263
−3.232
−1.692
−0.930
−0.322
0.112
0.572
0.776
0.960
1.184
1.299
1.177
1.139
1.387
1.777
1.914
2.065
2.215
2.326
2.422
2.488
2.536
2.570
2.600
2.632
2.672
2.747
2.878
3.034
3.224
3.763
3.421
3.326
3.324
3.288
3.271
3.181
3.163
3.104
3.106
3.113
3.256
3.385
3.532
3.797
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
2613
Percent
Expected Inflation
6
6
4
4
2
2
0
0
−2
−2
−4
−4
Percent
−6
2006
2008
2010
2012
2014
−6
6
6
4
4
2
2
0
0
−2
−2
−4
−4
−6
2006
Deflation Risk
2008
2010
2012
2014
−6
2006
2008
2010
2012
2014
2006
2008
2010
2012
2014
21 / 32
Table 6
Comparison of Survey Forecasts with Market-Implied Forecasts. This table reports the
average values of the survey forecasts for the indicated forecast horizon along with the corresponding average of the market-implied expected inflation for the same horizon. Inflation forecasts are
expressed as percentages. The sample period is July 2004 to July 2014.
Forecast
Horizon
Survey
Forecast
MarketImplied
Forecast
N
Survey
Frequency
1 Year
SPF
Livingston
Michigan
Quarterly
Semiannual
Monthly
2.12
2.13
3.23
1.78
1.60
1.80
40
20
120
5 Years
Michigan
Monthly
2.91
2.31
106
10 Years
SPF
Livingston
Quarterly
Semiannual
2.40
2.42
2.60
2.56
120
20
Table 7
Summary Statistics for Inflation Volatility. This table reports summary statistics for the volatility of the annualized inflation rate for
the indicated horizons. Horizon is expressed in years. Inflation rates are expressed as percentages. The sample consists of daily observations
for the period from July 23, 2004 to July 29, 2014.
Horizon
Mean
Standard
Deviation
Minimum
Median
Maximum
N
1
2
3
5
7
10
12
15
20
30
1.385
1.400
1.342
1.351
1.396
1.455
1.445
1.420
1.420
1.397
0.403
0.422
0.356
0.266
0.237
0.232
0.197
0.201
0.206
0.134
0.818
0.715
0.685
0.688
0.775
0.932
1.019
1.072
0.991
0.796
1.373
1.386
1.305
1.399
1.421
1.432
1.448
1.465
1.474
1.414
2.973
2.365
2.231
2.186
2.073
2.032
1.910
1.874
1.842
1.708
1210
1103
1222
1075
1124
1214
1218
1198
1121
789
Table 8
Summary Statistics for Deflation Probabilities. This table reports summary statistics for the probability of the average inflation rate
being below zero for the indicated horizons. Horizon is expressed in years. Probabilities are expressed as percentages. The sample consists of
daily observations for the period from October 5, 2009 to July 29, 2014.
Horizon
Mean
Standard
Deviation
Minimum
Median
Maximum
N
1
2
3
5
7
10
12
15
20
30
13.73
11.36
8.32
6.34
5.52
5.23
4.83
4.27
3.96
2.77
10.66
9.06
7.01
4.36
3.18
2.82
2.41
2.37
2.30
1.71
0.14
0.16
0.06
0.04
0.10
0.31
0.66
0.48
0.25
0.00
10.61
8.44
5.93
5.81
5.27
4.51
4.34
3.94
3.83
2.34
41.73
34.00
27.70
22.23
18.27
11.93
11.80
11.77
12.94
9.60
1210
1103
1222
1075
1124
1214
1218
1198
1121
789
Percent
Deflation Probabilities
40
40
30
30
20
20
10
10
Percent
0
2010
2011
2012
2013
2014
0
40
40
30
30
20
20
10
10
0
2010
Deflation Risk
2011
2012
2013
2014
0
2010
2011
2012
2013
2014
2010
2011
2012
2013
2014
25 / 32
Table 9
Summary Statistics for the Pricing of Deflation Tail Risk. This table reports the means of the ratio of the probability of inflation
being below the indicated threshhold under the pricing measure divided by the probability of the same event under the actual measure. The
mean is taken over only the observations where the probability of the event is greater than 0.01 percent under the actual measure. Horizon is
expressed in years. The sample consists of daily observations for the period from October 5, 2009 to July 29, 2014.
Probability Inflation < 0.00
Probability Inflation < −1.00
Probability Inflation < −2.00
Horizon
Mean
N
Mean
N
Mean
N
1
2
3
5
7
10
12
15
20
30
1.159
1.287
1.373
1.348
1.251
1.139
1.100
1.086
1.159
1.713
1210
1103
1222
1075
1124
1214
1218
1198
1121
788
1.303
1.518
1.704
1.682
1.495
1.287
1.214
1.178
1.261
2.052
1188
1039
1161
1050
1107
1213
1218
1198
1119
788
1.404
1.677
1.956
1.880
1.748
1.481
1.355
1.269
1.325
2.294
972
835
902
766
883
1084
1063
968
895
603
Table 10
Results from the Regression of Monthly Changes in Deflation Probabilities on Financial and Macroeconomic Variables. This
table reports the t-statistics and adjusted R2 s from from the regression of monthly changes in the deflation probabilities for the indicated
horizon on the monthly changes in the following variables: the spread between three-month Libor and the overnight index swap (OIS) rate, the
five-year swap spread, the VIX volatility index, the CDX North American Investment Grade CDS Index, the return on the CRSP value-weighted
stock index, the five-year U.S. Treasury CDS spread, the five-year German CDS spread, industrial production (IP, percentage change), the
unemployment rate (Unemp), and the Conference Board’s Consumer Confidence Index (Conf). Horizon in measured in years. The t-statistics
are based on the Newey-West estimator of the covariance matrix (three lags). The superscript ∗∗ denotes significance at the five-percent level;
the superscript ∗ denotes significance at the ten-percent level. The sample consists of monthly observations for the period from October 2009
to July 2014.
Horizon
1
2
3
5
7
10
12
15
20
30
Libor
Spread
1.55
2.29∗∗
0.00
0.13
−0.84
−1.60
−0.09
0.22
0.59
1.57
Swap
Spread
2.04∗∗
2.11∗∗
1.56
0.22
0.55
1.46
−0.39
0.84
1.35
−0.11
VIX
CDX
−1.35
0.05
−2.43∗∗ −0.80
−2.91∗∗ −0.76
−2.29∗∗
1.38
−1.51
1.14
0.82
−1.78∗
0.21
1.96∗
0.24
1.56
−1.70∗
1.39
0.31
2.30∗∗
Stock
Return
−2.40∗∗
−3.85∗∗
−5.60∗∗
−2.69∗∗
−1.15
−0.31
0.84
1.19
−0.04
1.53
Trsy
CDS
−0.96
−1.84∗
−1.72∗
0.29
−0.29
−0.82
0.95
0.79
−0.47
−0.41
German
CDS
IP
0.50
1.05
1.81∗
1.23
2.32∗∗ −0.22
1.52
0.72
2.19∗∗
0.14
−0.56
1.15
1.83∗ −0.66
1.42
0.18
0.62
1.68∗
−0.40
2.33∗∗
Unemp
0.24
−0.52
−0.79
−2.27∗∗
−1.94
1.64
−1.99∗
−1.31
−1.54
−2.08∗∗
Conf
−2.34∗∗
−0.62
−0.51
−0.96
−0.13
−0.08
0.45
−0.03
−2.12∗∗
−2.74∗∗
Adj.
R2
N
0.251
0.510
0.455
0.419
0.277
−0.003
0.155
0.053
0.204
0.492
56
56
56
54
56
56
56
56
54
39
Table 12
Summary Statistics for the Probabilities of Inflationary Scenarios. This table reports summary statistics for the probability of the
average inflation rate being above the indicated thresholds for the respective horizons. Horizon is expressed in years. Probabilities are expressed
as percentages. The sample consists of daily observations for the period from October 5, 2009 to July 29, 2014.
Probability Inflation > 4.00
Probability Inflation > 5.00
Probability Inflation > 6.00
Horizon
Mean
Min.
Max
Mean
Min.
Max
Mean
Min.
Max
N
1
2
3
5
7
10
12
15
20
30
5.14
11.36
7.09
9.37
11.91
14.56
15.03
15.53
16.57
19.80
0.23
0.16
0.59
0.89
2.36
5.51
5.50
5.09
4.82
13.53
22.20
34.00
19.56
23.25
23.13
25.67
24.34
24.04
24.10
26.58
1.40
6.33
1.90
2.53
3.48
4.66
4.72
4.82
5.24
6.30
0.00
0.21
0.01
0.01
0.06
0.59
0.83
0.65
0.47
2.67
8.81
19.67
9.88
12.13
10.29
13.04
10.96
10.73
10.64
11.18
0.35
1.89
0.47
0.55
0.81
1.20
1.15
1.15
1.27
1.42
0.00
0.00
0.00
0.00
0.00
0.02
0.06
0.04
0.02
0.09
4.68
8.84
4.31
5.45
3.67
5.58
4.07
3.94
3.79
3.73
1210
1103
1222
1075
1124
1214
1218
1198
1121
789
Percent
Inflation Probabilities
30
30
25
25
20
20
15
15
10
10
5
5
Percent
0
2010
2011
2012
2013
2014
0
30
30
25
25
20
20
15
15
10
10
5
5
0
2010
Deflation Risk
2011
2012
2013
2014
0
2010
2011
2012
2013
2014
2010
2011
2012
2013
2014
30 / 32
Table 13
Summary Statistics for the Pricing of Inflation Tail Risk. This table reports the means of the ratio of the probability of inflation being
above the indicated threshhold under the pricing measure divided by the probability of the same event under the actual measure. The mean is
taken over only the observations where the probability of the event is greater than 0.01 percent under the actual measure. Horizon is expressed
in years. The sample consists of daily observations for the period from October 5, 2009 to July 29, 2014.
Probability Inflation > 4.00
Probability Inflation > 5.00
Probability Inflation > 6.00
Horizon
Mean
N
Mean
N
Mean
N
1
2
3
5
7
10
12
15
20
30
1.023
1.043
1.062
1.089
1.097
1.094
1.088
1.065
0.981
0.737
1210
1103
1222
1075
1124
1214
1218
1198
1121
789
1.099
1.161
1.227
1.262
1.244
1.204
1.182
1.135
1.001
0.665
1160
1055
1217
1071
1124
1214
1218
1198
1121
789
1.151
1.255
1.372
1.461
1.467
1.374
1.322
1.240
1.037
0.602
863
818
988
933
1100
1214
1218
1198
1121
789
Conclusion
We solve for the objective distribution of inflation using the market
prices of inflation swap and option contracts and study the nature of
deflation risk.
Market-implied probabilities of deflation are substantial, even though
the expected inflation rate is roughly 2.50 to 3.00 percent for horizons
of up to 30 years.
Deflation risk is priced by the market in a manner similar to that of
other major types of tail risk such as catastrophic insurance losses or
corporate bond defaults.
Deflation risk is significantly related to measures capturing stress in
the financial system and credit risk in the economy.
Our results support the view that the risk of economic shocks severe
enough to result in deflation is fundamentally related to the risk of
major systemic shocks in the financial market.
Deflation Risk
32 / 32