Term Structures of Asset Prices and Returns

David Backus (NYU), Nina Boyarchenko (NY Fed∗ ), & Mikhail Chernov (UCLA) (With thanks to Ian Martin and Stan Zin)

Stockholm School of Economics | June 4, 2015

∗ Fed

disclaimer | June 1, 2015

Look for entropy

Paul Samuelson (“Gibbs in economics,” 1989): I have limited tolerance for the perpetual attempts to fabricate for economics concepts of “entropy.”

1 / 58

Look for logarithms I

Think about log mt,t+1 rather than mt,t+1

I

Sums more user-friendly than products log(mt,t+1 rt,t+1 ) = log mt,t+1 + log rt,t+1 more user-friendly than mt,t+1 rt,t+1 log(mt,t+1 mt+1,t+2 ) = log mt,t+1 + log mt+1,t+2 more user-friendly than mt,t+1 mt+1,t+2

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Excess returns

1 Monthly excess log returns in dollars: log rt,t+1 − log rt,t+1

Asset S&P 500 Fama-French (small, low) Fama-French (small, high) Pound Sterling 5-year bond 10-year bond

Mean 0.0040 −0.0030 0.0090 0.0035 0.0015 0.0019

Standard Deviation

Skewness

Excess Kurtosis

0.0556 0.1140 0.0894 0.0316 0.0190

−0.40 0.28 1.00 −0.50 0.10

7.90 9.40 12.80 1.50 4.87

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Term structure data: US nominal

Mean Yields and Forward Rates

7.5 7.0 forwards 6.5 6.0

yields

5.5 5.0

0

20

40

60 80 Maturity in Months

100

120

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Mean Yield Difference

Mean Yields

Term structure data: US nominal and real 5 4 3 2 1 0 1 2 0 4.0 3.5 3.0 2.5 2.0 1.5 0

nominal real 20

40

60

80

100

120

60 80 Maturity in Months

100

120

nominal minus real 20

40

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Term structure data: other assets relative to US nominal −3

2

x 10

1.5 1

Risk premium

0.5 0 −0.5 −1 −1.5 SEK NZD Inflation Dividends

−2 −2.5 −3 0

20

40

60 Horizon

80

100

120

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Where we’re headed

What makes these term structures different? Plan of attack I

Entropy: dispersion in the pricing kernel

I

Risk premiums: entropy bound

I

Term structure: pricing kernel dynamics

I

Coentropy: risk premiums revisited

I

Other term structures: cash flow dynamics, more coentropy

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Entropy

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Entropy I

Entropy is a measure of dispersion: for rv x > 0 L(x) ≡ log E (x) − E (log x) ≥ 0

I

Invariant to scale: L(αx) = L(x) for α > 0

I

Lognormal example: if log x ∼ N (κ1 , κ2 ), then log E (x) = κ1 + κ2 /2 E (log x) = κ1 L(x) = (κ1 + κ2 /2) − κ1 = κ2 /2

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Cumulants I

Cumulant generating function (cgf) of rv y ∞ X
k(s; y ) = log E e sy = κj s j /j! | {z } j=1 mgf

I

Cumulants are close relatives of moments mean = κ1 variance = κ2 3/2

skewness = κ3 /κ2

excess kurtosis = κ4 /κ22 I

If y is normal: k(s; y ) = κ1 s + κ2 s 2 /2

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Entropy and cumulants I

Cumulants of y = log x k(s; log x) = log E (e

s log x

) =

∞ X

κj (log x)s j /j!

j=1 I

Entropy and cumulants (set s = 1) L(x) = k(1; log x) − E (log x) =

∞ X

κj (log x)/j!

j=2

=

κ2 (log x)/2! + κ3 (log x)/3! + κ4 (log x)/4! + · · · | {z } | {z }

(log)normal term

high-order cumulants

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Entropy of a stationary stochastic process

I

Conditional entropy defined for conditional distribution Lt (xt+1 ) = log Et (xt+1 ) − Et (log xt+1 )

I

We define entropy as the mean E [Lt (xt+1 )]

I

Connected to entropy for unconditional distribution L(xt+1 ) = E [Lt (xt+1 )] +L[Et (xt+1 )] {z } | entropy

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Risk premiums: the entropy bound

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Entropy bound (Alvarez-Jermann) I

Returns satisfy the pricing relation Et (mt,t+1 rt,t+1 ) = 1

I

1 Entropy bound: maximize Et (log rt,t+1 − log rt,t+1 )

I

Maximization leads to the bound risk premium?

z }| { 1 Et (log rt,t+1 − log rt,t+1 ) ≤ Lt (mt,t+1 ) 1 E (log rt,t+1 − log rt,t+1 ) ≤ E [Lt (mt,t+1 )] | {z } entropy

I

High return is log rt,t+1 = − log mt,t+1

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Hansen-Jagannathan bound

I

HJ bound: maximize Sharpe ratio

I

Maximization leads to the bound SRt

1 1 ≡ Et (rt,t+1 − rt,t+1 )/Vart (rt,t+1 − rt,t+1 )1/2

≤ Vart (mt,t+1 )1/2 /Et (mt,t+1 ) I

High return is rt,t+1 =

1 + Vart (mt,t+1 )1/2 mt,t+1 − Et (mt,t+1 ) − Et (mt,t+1 ) Vart (mt,t+1 )1/2

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Stan’s “never a dull moment” machine I

Entropy of pricing kernel
L(m) = log E e log m − E (log m) = k(1; log m) − E (log m) =

∞ X

κj (log m)/j!

j=2 I

Stan’s entropy machine (but ask about Lukacs) L(m) =

κ2 (log m)/2! + κ3 (log m)/3! + κ4 (log m)/4! + · · · {z } | {z } |

(log)normal term I

high-order cumulants

Kraus and Litzenberger revisited?

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Why is this entropy? I

Humpy Dumpty (in “Through the Looking Glass”) “When I use a word,” Humpty Dumpty said, “it means just what I choose it to mean — neither more nor less.”

I

Hans-Otto Georgii (quoted by Hansen and Sargent): When Shannon had invented his quantity and consulted von Neumann on what to call it, von Neumann replied: “Call it entropy. It is already in use under that name and, besides, it will give you a great edge in debates because nobody knows what entropy is anyway.”

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Why is this entropy? I

Notation: states z have (true) probabilities π(z)

I

Risk-neutral probabilities π ∗ π ∗ (z) = π(z)m(z)/p 1 m(z) = p 1 π ∗ (z)/π(z) p 1 = E (m) (1-period bond price)

I

Entropy (aka “relative entropy” or “Kullback-Leibler divergence”) L(m) = L(π ∗ /π) = E log(π/π ∗ ) (π ∗ = π ⇒ L(m) = 0, risk premiums = 0)

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Vasicek model I

Pricing kernel log mt,t+1 = log β + xt + λwt+1 with {wt } iid, mean zero, variance one, and cgf k(s)

I

Conditional entropy Lt (mt,t+1 ) = k(λ) = λ2 κ2 /2! + λ3 κ3 /3! + λ4 κ4 /4! + · · ·

I

Entropy: the same (maximum risk premium is constant)

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State-dependent price of risk I

Pricing kernel log mt,t+1 = log β + xt + (λ0 + λ1 xt )wt+1 with {wt } ∼ NID(0, 1), k(s) = s 2 /2

I

Conditional entropy Lt (mt,t+1 ) = k(λ0 + λ1 xt ) = (λ0 + λ1 xt )2 /2

I

Entropy E [Lt (mt,t+1 )] = E [(λ0 + λ1 xt )2 /2]

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Power utility I

Consumption growth gt,t+1 = ct+1 /ct iid

I

Pricing kernel log mt,t+1 = log β − α log gt,t+1 (Vasicek with xt = 0, λ = −α, and wt+1 = log gt,t+1 )

I

Yaron’s bazooka Lt (mt,t+1 ) = k(−α) = (−α)2 κ2 /2! + (−α)3 κ3 /3! + (−α)4 κ4 /4! + · · ·

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Power utility: the bazooka −3

Cumulant

2

x 10

0 −2

2

3

4

5

6

7

8

Contribution

Contribution

−3

4

x 10

risk aversion α=2

2 0

2

3

4

5

6

0.1

7

8

risk aversion α=10

0.05 0

2

3

4

5 Order j

6

7

8

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Power utility: the bazooka 0.4

Entropy of Pricing Kernel L(m)

0.35 0.3 0.25 disasters

0.2 0.15 0.1 0.05 0 0

normal

equity premium

booms 2

4

6 Risk Aversion α

8

10

12

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Term structure: pricing kernel dynamics

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The idea

I

In an iid world, entropy is proportional to time interval

I

Deviations from proportionality reflect pricing kernel dynamics

I

Detectable from mean yields and forward rates

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Bond prices, yields, and forward rates I

Bond price: ptn is price at t of a claim to one (dollar?) at t + n

I

Bond yield: ytn = −n−1 log ptn

I

Forward rate: ftn = log(ptn /ptn+1 ) ⇒ ytn =

I

One-period return:

j−1 j=1 ft

Pn

n+1 n log rt,t+1 = log(pt+1 /ptn+1 ) ⇒ E (log r n+1 ) = E (f n )

I

Cross sections reflect pricing kernel dynamics

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Bond pricing fundamentals I

Markov environment with state variable x

I

Bond pricing is recursive   p n (xt ) = Et m(xt , xt+1 )p n−1 (xt+1 ) starting with p 0 = 1

I

Equivalent to   p n (xt ) = Et m(xt , xt+1 )m(xt+1 , xt+2 ) · · · m(xt+n−1 , xt+n )

I

Definitions give us yields y n (xt ) and forward rates f n (xt )

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Entropy and the term structure I

Entropy over n periods mt,t+n = mt,t+1 mt+1,t+2 · · · mt+n−1,t+n Lt (mt,t+n ) = log Et (mt,t+n ) −Et (log mt,t+n ) {z } | log ptn =−nytn

L(n) ≡ E [Lt (mt,t+n )] = −nE (y n ) − nE (log mt,t+1 ) I

Two measures of horizon dependence H(n) ≡

n−1 L(n) | {z }

avg over n periods

− L(1) |{z}

= −E (ytn − yt1 )

one period

F (n) ≡ L(n + 1) − L(n) − L(1) = −E (ftn − ft0 )

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Entropy and the term structure

I

The iid benchmark: {mt,t+1 } iid ⇒ L(n) = n L(1) H(n) = 0 F (n) = 0

I

Also: yields and forwards constant, same at all maturities

I

Anything different from this reflects dynamics in m

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Vasicek model: dynamic structure I

Pricing kernel log mt,t+1 = log β + xt + λwt+1 xt+1 = ϕxt + σwt+1 x is (persistent or long-run) risk, λ is price of risk

I

Moving average representation log mt,t+1 = log β + xt + λwt+1 = log β + λwt+1 + σwt + σϕwt−1 + · · · | {z } xt

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Vasicek model: entropy

I

Pricing kernel dynamics inherited from x

I

Term structure of entropy L(1) = k(λ) L(2) = k(λ) + k(λ + σ) L(3) = k(λ) + k(λ + σ) + k(λ + σ + σϕ)

I

What makes this non-iid?

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Vasicek model: parameter values I

Short rate yt1 = ft0 = − log Et (mt,t+1 ) = −[log β + k(λ)] − xt

I

I

Choose I

(ϕ, σ) = (0.98, −0.006) match variance and autocorrelation

I

w normal ⇒ k(s) = s 2 /2

I

λ = 0.088 matches mean forward spread E (f n − f 0 ) (ask how this works)

Features I

σ and λ must have opposite signs for curve to slope up

I

λ much greater than σ in absolute value

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Vasicek model: mean forward spreads 2.5 Vasicek model

Mean Forward Spread

2.0 1.5 1.0 0.5 0.0

0

dots are data 20

40

60 80 Maturity in Months

100

120

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Vasicek model: moving average coefficients 0.003 Moving Average Coefficients

= 0.088 0.002 0.001 0.000 0.001

0

2

4 Lag

6

8

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Vasicek model: entropy 0.5

Entropy

0.4 0.3 iid benchmark 0.2 Vasicek model

0.1 0.0

0

20

40 60 80 Time Horizon in Months

100

120

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Coentropy: risk premiums revisited

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The idea

I

Expected excess returns differ across assets

I

Reflects dependence of pricing kernel and cash flows

I

We measure dependence with coentropy

I

Extend shortly to long time horizons

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Coentropy I

Coentropy is a measure of dependence: for x1 , x2 > 0 C (x1 , x2 ) ≡ L(x1 x2 ) − L(x1 ) − L(x2 )

I

I

Features I

Invariant to scaling

I

Equals zero if x1 and x2 are independent

Related to (joint) cgf k(s1 , s2 ) = log E (e s1 log x1 +s2 log s2 ) C (x1 , x2 ) = k(1, 1) − k(1, 0) − k(0, 1) | {z } | {z } | {z } x1 x2

x1

x2

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Coentropy (continued) I

If log x = (log x1 , log x2 ) is normal, coentropy = covariance

I

Can also be much different

I

Example: Poisson mixture (“jump process”)

I

I

Poisson jumps: probability e −ω ω j /j! of j = 0, 1, 2, . . ..

I

Conditional on j, log x ∼ N (jθ, j∆)

Properties Cov(log x1 , log x2 ) = ω(θ1 θ2 + δ12 ) C(x1 , x2 ) = E2C2E

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Coentropy and covariance 1.8 Coentropy and Covariance

1.6 1.4 1.2 1.0 0.8 covariance

0.6 0.4

coentropy

0.2 0.0

0

1

2 3 Poisson intensity ω

4

5

40 / 58

Coentropy and excess returns I

Consider a claim to the cash flow gt,t+1

I

Return is cash flow over price

I

Invariance to scaling implies Lt (rt,t+1 ) = Lt (gt,t+1 ) Ct (mt,t+1 , rt,t+1 ) = Ct (mt,t+1 , gt,t+1 )

I

Expected excess returns (“risk premiums”)

1 Et (log rt,t+1 − log rt,t+1 ) = −Lt (gt,t+1 ) − Ct (mt,t+1 , gt,t+1 ) 1 E (log rt,t+1 − log rt,t+1 ) = − E [Lt (gt,t+1 )] − E [Ct (mt,t+1 , gt,t+1 )] {z } | {z } | entropy

coentropy

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KLV model (streamlined version) I

Add another disturbance to Vasicek log mt,t+1 = log β + xt + λ1 w1t+1 + λ2 w2t+1 xt+1 = ϕxt + σw1t+1 (w1t , w2t ) ∼ NID(0, I )

I

Stir in some cash flow growth log gt,t+1 = log γ + θxt + η1 w1t+1 + η2 w2t+1

I

Entropy and coentropy E [Lt (mt,t+1 )] = (λ21 + λ22 )/2 E [Lt (gt,t+1 )] = (η12 + η22 )/2 E [Ct (mt,t+1 , gt,t+1 )] = λ1 η1 + λ2 η2 42 / 58

KLV model: numerical example I

Choose (ϕ, σ, λ1 ) = (0.98, −0.0006, 0.088) as before to fit yields/forwards

I

Choose (η1 , η2 ) = (−0.005, −0.050) to match I

Variance of excess return on equity (0.05)

I

Correlation of excess returns on equity and bonds (0.1)

I

Choose λ2 = 0.097 to match equity premium

I

Results (monthly): I

Bond premium (10 years): 0.002

I

Equity premium: 0.004 = 0.005 (coentropy) – 0.001 (entropy)

I

Entropy of m: 0.009 (upper bound)

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Other term structures: cash flow dynamics

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The idea

I

Consider claims to currencies, equity indexes, dividends, ...

I

How do their term structures compare?

I

The time horizon of coentropy

I

Explorations with the KLV model

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Prices, yields, and forward rates I

Let pbtn be price at t of claim to cash flow growth gt,t+n

I

Term structure ybtn = −n−1 log pbtn fbtn = log(b ptn /b ptn+1 ) n+1 n log rbt,t+1 = log pbt+1 − log pbtn+1 + log gt,t+1

E (log rbn+1 ) = E (f n + log g ) I

Forward price qtn = pbtn /ptn

⇒

n−1 log qtn = ytn − ybtn log q n+1 − log q n = f n − fbn t

t

t

t

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Pricing fundamentals I

Pricing is recursive   pbn (xt ) = Et m(xt , xt+1 )g (xt , xt+1 )b p n−1 (xt+1 )   b t , xt+1 )b = Et m(x p n−1 (xt+1 ) b t , xt+1 ) = m(xt , xt+1 )g (xt , xt+1 ) with pb0 = 1 and m(x

I

Definitions give us yields ybn (xt ) and forward rates fbn (xt )

I

Think of this as a change of units: dollars to yen

I

Empirical strategy changes: we observe g

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Entropy, coentropy, and term structures

I

Entropy and coentropy b t,t+n ) = Lt (mt,t+n gt,t+n ) Lt (m = Ct (mt,t+n , gt,t+n ) + Lt (mt,t+n ) + Lt (gt,t+n ) Lmb (n) = Lm (n) + E [Ct (mt,t+n , gt,t+n )] + E [Lt (gt,t+n )] = Lm (n) + Cmg (n) + Lg (n) | {z } | {z } coentropy

I

entropy

Same connection to mean yields and forward rates as before

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Term structure data: other assets relative to US nominal −3

2

x 10

1.5 1

Risk premium

0.5 0 −0.5 −1 −1.5 SEK NZD Inflation Dividends

−2 −2.5 −3 0

20

40

60 Horizon

80

100

120

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KLV model I

Model log mt,t+1 = log β + xt + λ1 w1t+1 + λ2 w2t+1 xt+1 = ϕxt + σw1t+1 log gt,t+1 = log γ + θxt + η1 w1t+1 + η2 w2t+1

I

Transformed pricing kernel long-run risk

b t,t+1 log m

z }| { = (log β + log γ) + (1 + θ) xt + (λ1 + η1 ) w1t+1 + (λ2 + η2 ) w2t+1 | {z } | {z } price of lr risk

I

price of iid risk

Roles of: η1 , λ2 , η2 , θ

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KLV model: currencies I

Model log mt,t+1 = log β + xt + λ1 w1t+1 + λ2 w2t+1 xt+1 = ϕxt + σw1t+1 log gt,t+1 = log γ + θxt + η1 w1t+1 + η2 w2t+1

I

Currencies: θ ≈ 0, η12 + η22 ≈ 0.033 , λ2 ≈ 0 (for now)

I

Then (η1 , η2 ) control coentropy I

(η1 , η2 ) = (0, 0.3): coentropy is zero

I

(η1 , η2 ) = (0.3, 0): coentropy is positive

I

(η1 , η2 ) = (−0.3, 0): coentropy is negative

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KLV model: currencies 10 9 Mean Forward Rates

8

negative coentropy

USD benchmark

7 6 5

zero coentropy

4 3 2 1 0

positive coentropy 20

40

60 80 Maturity in Months

100

120

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KLV model: currencies 3.5

Mean Forward Spreads

3.0 2.5

negative coentropy

USD benchmark

2.0 1.5 1.0

positive coentropy

0.5 0.0

0

20

40

60 80 Maturity in Months

100

120

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KLV model: equity I

Model log mt,t+1 = log β + xt + λ1 w1t+1 + λ2 w2t+1 xt+1 = ϕxt + σw1t+1 log gt,t+1 = log γ + θxt + η1 w1t+1 + η2 w2t+1

I

Reminder: transformed pricing kernel long-run risk

b t,t+1 log m

z }| { = (log β + log γ) + (1 + θ) xt + (λ1 + η1 ) w1t+1 + (λ2 + η2 ) w2t+1 | {z } | {z } price of lr risk

I

price of iid risk

Roles of: λ2 + η2 , η1 (−0.005), θ (±0.25)

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KLV model: equity 3.0

Mean Forward Spreads

2.5 theta = 0.25

2.0

theta = 0

1.5 theta = -0.25

1.0 0.5 0.0

0

20

40

60 80 Maturity in Months

100

120

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KLV model: equity

Mean Forward Differentials

4.8 theta = 0.25

5.0 5.2

theta = 0

5.4 5.6 5.8 0

theta = -0.25 20

40

60 80 Maturity in Months

100

120

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Where were we?

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Summary and open questions

Summary I

Significant variation in average term structures across assets

I

Connected to entropy and coentropy

I

Large quantitative effects in simple models (still no bazooka!)

Open questions I

What would you do with these ingredients?

I

Would predictability interest you?

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