A Probability-Based Stress Test of
Federal Reserve Assets and Income
Jens H. E. Christensen
Jose A. Lopez
Glenn D. Rudebusch
Federal Reserve Bank of San Francisco
Term Structure Modeling and the Lower Bound Problem
Lecture III.3
European University Institute
Florence, September 9, 2015
The views expressed here are solely the responsibility of the authors and should not be interpreted as reflecting the views
of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System.
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5000
Federal Reserve Assets
3000
2000
1000
QE2
QE1
Other Assets
Non−Treasury
Securities
Treasury
Securities
Dates of
analysis
0
Billions of dollars
4000
QE3
2008 2009 2010 2011 2012 2013 2014 2015
2 / 37
Concerns: Capital Losses and Income Shortfalls
Former Fed Governor Mishkin (2010): “Major holdings of
long-term securities expose the Fed’s balance sheet to potentially
large losses if interest rates rise. Such losses would result in severe
criticism of the Fed and a weakening of its independence.”
Former Fed Vice Chairman Kohn (2014): “As long-term rates rise,
the Federal Reserve will have mark-to-market losses on its balance
sheet. These losses are not a threat to the Federal Reserve’s ability
to tighten nor do they have any economic significance, but losses
could be used as a political weapon by those who seek to curtail
the Federal Reserve’s independence or limit its powers.”
Minutes of March 20, 2013 FOMC meeting: “Some participants
were concerned that a substantial decline in remittances might lead
to an adverse public reaction or potentially undermine Federal
Reserve credibility or effectiveness.”
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Fed’s Interest Rate Risk (Two Types)
Balance sheet risk: Increases in longer-term interest rates
erode the market value of the Fed’s portfolio. (Fed does not
mark-to-market, so these would be unrealized losses—unless
Fed sells securities.)
Income risk: Increases in short-term interest rates
(including IOER rate) raise cost of funding portfolio and lower
net interest income and remittances to Treasury.
We examine how interest rate changes will affect the Fed’s
assets and income using a probability-based stress test.
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What is a Probability-Based Stress Test?
Conventional stress test: How much bank capital remains
in specific, arbitrary economic scenarios?
Carpenter et al. (2015) and Greenlaw et al. (2013, GHHM)
apply this method to the Fed’s balance sheet. They project
the Fed’s assets and income assuming ±100 basis points
parallel shifts in yield curve.
But how to weigh scenarios without probabilities?
Probability-Based Stress Test: We use a dynamic term
structure model to generate distributional interest rate
forecasts and attach probabilities to specific portfolio
outcomes.
Our Results: We find fairly low probabilities of large losses
on Fed’s portfolio and of large interest income shortfalls.
5 / 37
Outline
1
Introduction
2
The Shadow-Rate Term Structure Model
3
Probability-Based Stress Test of Fed’s Portfolio
4
Probability-Based Stress Test of Fed’s Income
5
Conclusion
6 / 37
Modeling Interest Rates Near the Zero Lower Bound
Standard Gaussian term structure models do not restrict
interest rates to be nonnegative!
To account for the ZLB, Black (1995) proposed a shadow
rate, st , that may be negative, while the observed short rate,
rt , is truncated:
rt = max{st , 0}.
This nonlinearity is computationally burdensome.
Option-based approximation of Krippner (2013):
P t (τ ) = Pt (τ ) − CtA (τ, τ ; 1).
Christensen and Rudebusch (2015a) combine this with the
arbitrage-free Nelson-Siegel (AFNS) model to deliver a
tractable shadow-rate model.
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The Arbitrage-Free Nelson-Siegel (AFNS) Model
Proposition: Given the risk-free rate
rt = Lt + St
and Q-dynamics of factors Xt = (Lt , St , Ct )


 
  Q  
θ1
0 0 0
Lt
dLt
 dSt  =  0 λ −λ   θQ  −  St  dt +ΣdWtQ ,
2
0 0 λ
Ct
dCt
θ3Q
where Σ is a constant matrix, then zero-coupon yields have
the popular Nelson-Siegel factor structure
yt (τ ) = Lt +
1 − e−λτ 1 − e−λτ
A(τ )
St +
− e−λτ Ct −
.
λτ
λτ
τ
This defines the AFNS model class derived in Christensen,
Diebold, and Rudebusch (2011).
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The Shadow-Rate AFNS Model Class (1)
Christensen and Rudebusch (2015a) derive shadow-rate
AFNS class of models with shadow rate
st = Lt + St ,
and unchanged AFNS Q-dynamics.
The instantaneous shadow forward rate is
ft (τ ) = Lt + e−λτ St + λτ e−λτ Ct + Af (τ ),
where Af (τ ) is an analytical function of model parameters.
The nonnegative instantaneous forward rate is
1 h f (τ ) i2 f (τ ) 1
t
t
f t (τ ) = ft (τ )Φ
+ ω(τ ) √ exp −
,
ω(τ )
2 ω(τ )
2π
where ω(τ ) is a deterministic function of model parameters.
9 / 37
The Shadow-Rate AFNS Model Class (2)
The yield-to-maturity is defined the usual way as
y t (τ )
=
=
1
τ
Z
t+τ
1
τ
Z
t+τ
f t (s)ds
t
t
h
f (s) 1 h f (s) i2 i
1
t
t
ft (s)Φ
+ ω(s) √ exp −
ds.
ω(s)
2 ω(s)
2π
This is the measurement equation in the Kalman filter.
Since yields are nonlinear functions of the state variables, we
use the standard extended Kalman filter.
10 / 37
The Real-World Dynamics
Christensen and Rudebusch (2015b) apply the shadow-rate
AFNS model to weekly U.S. Treasury yields (1985-2014).
They favor these model P-dynamics:

 
10−7
dLt
 dSt  =  κP21
dCt
0
0
κP22
0


 
 

dWtL,P
0
0
Lt


κP23   θ2P  −  St  dt+Σ  dWtS,P  ,
Ct
κP33
θ3P
dWtC,P
where Σ is a diagonal matrix.
This is the transition equation in the Kalman filter.
We use this “B-CR model” for our analysis.
11 / 37
The B-CR Model: Further Details
Key strengths of B-CR model:
It fits yield curve quite well.
It can replicate the compression in short- and
medium-term Treasury yield volatility since 2009.
Its short rate forecasts have been accurate and fairly
closely match federal funds futures.
We estimate B-CR model with daily zero-coupon Treasury
bond yields from the Gürkaynak, Sack, and Wright (2007)
database.
We use 11 maturities from 3 months to 30 years starting in
January 2, 1986. Three estimation sample end dates.
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100
Tests of Parameter Restrictions
60
40
20
Likelihood ratio test
80
Independent−factor AFNS model
CR model
Independent−factor B−AFNS model
B−CR model
99% quantile in chi sq. distribution, df = 6
0
99% quantile in chi sq. distribution, df = 4
1996
1998
2000
2002
2004
2006
2008
2010
2012
2014
2016
In AFNS models, we test the significance of parameter
restrictions with standard likelihood ratio tests, while we
use quasi likelihood ratio tests for the B-AFNS models.
The CR and B-CR models are again sufficiently flexible to
capture the variation in the data.
13 / 37
5
Observed and Fitted Yield Curves (1/13 & 6/14)
3
2
0
1
Rate in percent
4
Fitted yield curve, January 2, 2013
Observed yields, January 2, 2013
Fitted yield curve, June 25, 2014
Observed yields, June 25, 2014
0
5
10
15
20
25
30
Time to maturity in years
The model provides close fit to entire cross section of
yields and gives good bond pricing.
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Test of Model: Value Fed’s Treasury Portfolio
Number of Securities
237
Value in billions as of June 25, 2014
Face value Bloomberg
Model
2,284
2,470
2,473
We consider the future cash flows for all 237 nominal
Treasury securities in Fed’s portfolio on June 25, 2014.
We value each payout stream using the model-implied
yield curve to obtain the model-implied market value.
This model value of the entire Treasury portfolio differs
from the market value according to Bloomberg bond
prices by less than $3 billion.
This is a strong test of model fit as these raw bond prices
were not used in model estimation.
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0
−1
Pricing error per 100 dollar notional
1
Illustration of Treasury Pricing Performance
0
5
10
15
20
25
30
Time to maturity in years
Illustration of pricing errors in dollars per $100 notional
across bond maturities.
As of June 25, 2014, the pricing errors are limited in size.
16 / 37
P-Based Stress Test of Fed’s Treasury Portfolio
We project the value of the Fed’s Treasury securities for
three years.
We analyze two start dates: January 2, 2013, and June
25, 2014.
We fix Fed’s Treasury portfolio as of each start date.
We use the B-CR model estimated as of the start date to
generate Treasury yield curve projections.
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Model-Based Portfolio Value Projections
We use the estimated B-CR model to generate Treasury yield
curve projections and assign them probabilities.
Procedure:
Estimate parameters and state variables from the B-CR
model as of the start date.
Simulate N = 10,000 sample paths for the state variables.
Convert the simulated state variables into full yield curves
and calculate the corresponding Treasury portfolio values
along each path (i.e. value all future cash flows).
Examine the resulting portfolio value distributions.
18 / 37
Short Rate Projections as of June 25, 2014 (1)
8
6
0
2
4
Rate in percent
10
12
B−CR model, median
B−CR model, 95 percentile
B−CR model, 5 percentile
BC, consensus forecast
BC, top 10 average forecast
BC, bottom 10 average forecast
ED options, median
ED options, 95 percentile
ED options, 5 percentile
2014
2016
2018
2020
2022
2024
The figure shows short-rate distributions from the B-CR
model, 3-m LIBOR outcomes implied by options on
eurodollar futures, and federal funds rate forecasts from
the Blue Chip survey.
19 / 37
12
Short Rate Projections as of June 25, 2014 (2)
8
6
4
0
2
Rate in percent
10
30−year Treasury yield
10−year Treasury yield
5−year Treasury yield
1−year Treasury yield
Overnight federal funds rate
B−CR model, median
B−CR model, 90% confidence band
1985
1995
2005
2015
2025
The figure shows the short-rate distribution from the
B-CR model with a comparison to past Treasury yields
and overnight federal funds rates.
20 / 37
1900
1700
1800
Realizations
50 percentile
25 percentile
10 percentile
5 percentile
1 percentile
1600
Portfolio value in billions of dollars
2000
Treasury Portfolio Valuations as of Jan. 2, 2013
1500
Face value of Treasury portfolio
2013
2014
2015
2016
Shown are lower percentiles of the Fed’s Treasury
portfolio value distribution from B-CR model projections.
As of January 2, 2013, the chance of the Treasury
portfolio value going below par is less than 5 percent.
21 / 37
2500
2300
2100
1900
1700
Portfolio value in billions of dollars
2700
Treasury Portfolio Valuations as of June 25, 2014
Realizations
^
Face value of Treasury portfolio
50 percentile
25 percentile
10 percentile
5 percentile
1 percentile
2015
2016
2017
Shown are lower percentiles of the Fed’s Treasury
portfolio value distribution from B-CR model projections.
As of June 25, 2014, the chance of the Treasury portfolio
value going below par is less than 25 percent.
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4
3
2
1 percentile
5 percentile
50 percentile
90% confidence band
Fitted yields, June 25, 2014
0
1
Rate in percent
5
6
Underlying Yield Curves
0
5
10
15
20
25
30
Time to maturity in years
Projected yield curves that produce the 1st , 5th , and 50th
(i.e., median) percentiles of the Treasury portfolio values
by mid-2017.
23 / 37
P-Based Stress Test Including Fed’s MBS Holdings
MBS portfolio is complex, diverse, hard to value.
We convert Fed’s MBS portfolio into ten-year equivalent
U.S. Treasury securities.
As of June 25, 2014, Fed held 68,557 separate MBS,
which had a total face value of $1,664 billion and an
assumed duration of 5.8 years.
According to the B-CR model, the ten-year Treasury
par-coupon bond had a yield of 2.6% and a duration of
8.8 years.
Thus, we replace the MBS holdings with $1,093 billion of
ten-year Treasuries with 2.6% coupon.
As of Jan. 2, 2013, the MBS holdings represented $271.2
billion in ten-year equivalents.
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3000
Fed’s Securities Portfolio in Ten-Year Equivalents
2000
1500
1000
0
500
Billions of dollars
2500
Total SOMA portfolio
Treasury securities
Agency MBS
2009
2010
2011
2012
2013
2014
2015
Fed’s portfolio measured in ten-year equivalents—a
measure sensitive to both size and duration.
25 / 37
2200
1800
1900
2000
2100
Realizations
50 percentile
25 percentile
10 percentile
5 percentile
1 percentile
Face value of augmented portfolio
1700
Portfolio value in billions of dollars
2300
Treasury & MBS Portfolio Valuations — Jan. 2, 2013
2013
2014
2015
2016
Shown are lower percentiles of the Fed’s Treasury & MBS
portfolio value distribution from B-CR model projections.
As of January 2, 2013, the chance of the Treasury & MBS
portfolio value going below par is less than 10 percent. 26 / 37
3600
Realizations
3400
Face value of augmented portfolio
2800
3000
3200
v
2600
Portfolio value in billions of dollars
3800
Treasury & MBS Portfolio Valuations — Jun. 25, 2014
50 percentile
25 percentile
10 percentile
5 percentile
1 percentile
2015
2016
2017
Shown are lower percentiles of the Fed’s Treasury & MBS
portfolio value distribution from B-CR model projections.
As of June 25, 2014, the chance of the Treasury & MBS
portfolio value going below par has increased to 1-in-3.
27 / 37
P-Based Stress Test of Fed’s Income
To generate projections of the Fed’s income, we use B-CR
model and GHHM accounting framework.
We use GHHM baseline assumptions with three changes:
We assume no securities sales.
We use asset purchase path through 2014 from NY Fed
Primary Dealer Survey of June 2014.
We set the path for the interest paid on excess reserves
(IOER) equal to the short rate from model simulation, but
with a minimum of 25 basis points.
We simulate the B-CR model (estimated as of December 31,
2013) to generate interest rate scenarios through 2020.
28 / 37
120
Distributional Forecast of Fed’s Interest Expenses
80
60
40
0
20
Billions of dollars
100
Median
90% confidence band
2014
2015
2016
2017
2018
2019
2020
Forecast horizon
At longer forecast horizons, the estimated B-CR model
allows for significant variation in the overnight rate.
Thus, the Fed is exposed to significant uncertainty about
funding costs and projected interest expenses.
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0
120
Fed’s Payments to the U.S. Treasury
−20
−30
Billions of dollars
−50
−40
80
60
40
Median
5 percentile
1 percentile
−60
20
0
Billions of dollars
100
−10
Median
90% confidence band
2014
2015
2016
2017
2018
Forecast horizon
2019
2020
2014
2015
2016
2017
2018
2019
2020
Forecast horizon
Left figure shows Fed’s remittances to the Treasury,
which will vary due to variation in interest expenses.
Right figure shows about a 5 percent chance that
remittances will halt. Then Fed accrues a “deferred
asset” that peaks in 2018.
30 / 37
5
Probability Distribution of Peak Deferred Asset
3
2
Peak deferred asset larger than $10 billion: 4.9%
0
1
Probability in percent
4
No deferred asset: 92.7%
0
20
40
60
80
100
Peak deferred asset in billions of dollars
Distribution shows 92.7% of no deferred asset, and 4.9%
chance of a peak deferred asset in excess of $10 billion.
31 / 37
120
Projected Fed Remittances to U.S. Treasury
80
60
40
20
0
Billions of dollars
100
Realized remittances to the Treasury
1990−2007 trend of remittances
2008−2020 trend projection of remittances
Median, 2014−2020 projections
90% confidence band, 2014−2020 projections
1990
1995
2000
2005
2010
2015
2020
32 / 37
20
15
10
0
5
Probability in percent
25
30
Cumulative Remittances to Treasury Net of Trend
0
100
200
300
400
500
600
700
Cumulative remittances net of trend in billions of dollars
Probability distribution for cumulative Fed remittances
2008-2020 minus projected trend (from 1990-2007). QE
boosted remittances above trend.
33 / 37
Conclusion
We introduce probability-based stress tests and argue
that attaching likelihoods to adverse outcomes is an
important addition to the debate.
We use a shadow-rate term structure model that respects
the zero lower bound to generate Treasury yield curve
projections.
We find that the Fed’s potential losses over the next
several years are in most cases relatively small.
We generate comprehensive projections of the Fed’s
future income and find a very small chance of a
temporary halt of the remittances to the Treasury.
34 / 37
Short Rate Projections as of July 29, 2015 (1)
8
6
0
2
4
Rate in percent
10
12
B−CR model, median
B−CR model, 95 percentile
B−CR model, 5 percentile
BC, consensus forecast
BC, top 10 average forecast
BC, bottom 10 average forecast
ED options, median
ED options, 95 percentile
ED options, 5 percentile
2015
2017
2019
2021
2023
2025
The figure shows short-rate distributions from the B-CR
model and 3-m LIBOR outcomes implied by options on
eurodollar futures as of July 29, 2015.
The Blue Chip ff rate forecasts are from June 2015.
35 / 37
12
Short Rate Projections as of July 29, 2015 (2)
8
6
4
0
2
Rate in percent
10
30−year Treasury yield
10−year Treasury yield
5−year Treasury yield
1−year Treasury yield
Overnight federal funds rate
B−CR model, median
B−CR model, 90% confidence band
1985
1995
2005
2015
2025
The figure shows the short-rate distribution from the
B-CR model with a comparison to past Treasury yields
and overnight federal funds rates as of July 29, 2015.
Note the considerable model-implied variation.
36 / 37
3200
3400
2016
2017
2018
3000
Face value of augmented portfolio
2800
v
2600
2200
Portfolio value in billions of dollars
2000
1800
50 percentile
25 percentile
10 percentile
5 percentile
1 percentile
2400
2400
2200
Face value of Treasury portfolio
v
1600
Portfolio value in billions of dollars
2600
Updated Portfolio Projections as of July 29, 2015
50 percentile
25 percentile
10 percentile
5 percentile
1 percentile
2016
2017
2018
Lower percentiles of the Fed’s portfolio values from
model-based projections as of July 29, 2015.
Due to recent yield declines, the balance sheet risk has
declined slightly—despite the final portfolio expansion
and the sizeable dispersion in model-implied yield curves.
37 / 37