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Optimal Outlooks
Narayana Kocherlakota
Disclaimer and Acknowledgements
Disclaimer: I am not speaking for others in the Federal Reserve System.
Acknowledgements: I thank Doug Clement, David Fettig, Terry Fitzgerald,
Ron Feldman, Thomas Tallarini, and Kei-Mu Yi for their comments.
Need for Outlooks
• A policymaker needs to make a decision today.
• The current decision results in random future net losses to society.
• Hence, the policymaker’s decision depends on his or her outlook about
those net losses.
Question
What’s the appropriate notion of an outlook for this policymaker?
Answer
• The needed outlook is not a statistically motivated predictive density ...
• But rather an asset-price-based risk-neutral probability density (RNPD).
Intuition
• From an ex ante perspective, resources may be more valuable in one state
than in another state.
• Optimal decisions should reflect these relative resource valuations.
• RNPDs are derived from financial market prices.
• Hence, an outlook based on an RNPD does reflect the relative values of
resources in different states.
• But an outlook based on a statistical forecast does not.
Outline
1. General Policy Problem
2. Risk-Neutral Probabilities
3. Example: Inflation-Targeting
4. Conclusions
GENERAL POLICY PROBLEM
Choice Problem
• Policymaker (P) chooses an action a.
• The result of the action next period depends on the realization of x.
— The random variable x has realizations {xn}N
n=1.
• The outcome (a, x) results in a welfare loss of L(a, x) dollars.
— The loss L(a, x) may be positive or negative.
Possible Losses
• When P chooses an action a, there is a vector of possible social losses:
(L(a, xn))N
n=1
• Dollars in different states are really different goods.
• Hence, each choice of a results in a distinct bundle of different goods.
• How should P compare these bundles?
Simple Fruit Analogy
• I face a choice between giving up two baskets of fruit:
— A apples and B bananas
— OR A’ apples and B’ bananas
• I need a way to combine apples and bananas together.
— Should I just add the number of apples and bananas?
— Should I estimate CES preferences over apples/bananas?
Using Prices
• Right approach: How much will it cost me to replace the lost fruit?
• Hence, I need to compare:
pAA + pB B
vs. pAA0 + pB B 0
• This comparison requires the use of appropriate market prices.
Replacement Cost Approach
• If P chooses a, then society suffers a random loss L(a, x).
• By buying a portfolio with random payoff L(a, x), P can replace the losses
incurred by the action a.
• Hence, the value of that portfolio is the current (replacement) cost of
taking action a.
• P should choose a so as to minimize this cost.
• This comparison requires the use of appropriate market prices.
RISK-NEUTRAL PROBABILITIES
State Prices
• If P chooses a, then society loses L(a, xn) if x = xn.
• How much would it cost today to reimburse society for the loss in that
state?
• To answer this question, we need to know qn - the current price of a dollar
received in the event that x = xn.
— The vector (qn)N
n=1 is the vector of state prices.
• Given q, it would cost:
N
X
qnL(a, xn)
n=1
to reimburse society for the losses incurred with action a.
PN
• P should choose a so as to minimize n=1 qnL(a, xn).
Risk-Neutral Probabilities
• We don’t affect decisions if we divide qn by a constant.
• Define:
∗ =
qn
PN
qn
m=1 qm
• q ∗ is called the risk-neutral probability density (RNPD) of x.
∗ is nonnegative for all n.
— Probability means: q ∗ sums to one and qn
Risk-Neutral and "True" Probabilities
• The RNPD q ∗ of x is not the same as the "true" probability density of x.
— And what exactly is the "true" probability density of x?
• q ∗ reflects asset traders’ aversion to risk.
• And q ∗ reflects asset traders’ assessments of the likelihood of x.
E*
• For any function φ of x, define:
E ∗(φ(x)) =
N
X
∗ φ(x )
qn
n
n=1
• P can optimally choose a by minimizing:
E ∗(L(a, x))
• If L is differentiable with respect to a:
E ∗{
∂L ∗
(a , x)} = 0
∂a
Verbal Summary
• Standard: Policymaker’s optimal choice sets the outlook for La equal to
zero.
• Novel: The appropriate notion of the outlook is given by E ∗.
• Intuitively, policymaker makes choices so as to balance losses across states
of the world.
• The relevant trade-offs are governed by state prices, not statistical forecasts.
Aside: Endogeneity of State Prices
• Above: I’ve treated q ∗ as exogenous to P.
• More realistic: Risk-neutral probability density q ∗ depends on a.
• Then, P’s problem is to choose a to minimize:
N
X
n=1
∗ (a)L(a, x )
qn
n
• Suppose P ignores endogeneity and chooses a∗ so that:
∂L ∗
∗
E [ (a , xn)] = 0
∂a
• Result: This choice is nearly optimal as long as this second moment:
∂ ln q ∗(a∗)
∗
∗
Cov (L(a , x),
)
∂a
is sufficiently small.
• Note: This second moment is calculated using the RNPD q ∗(a∗).
EXAMPLE:
INFLATION-TARGETING
Model of Inflation-Targeting
• Consider a hypothetical central bank (CB) with a single mandate: inflation
target π.
• CB chooses accommodation a that, next period, results in:
— inflation rate π = (a + x)
— where x is random
• Sticky prices imply that there is an efficiency loss if π differs from the
target π.
• The gap |π − π∗| generates an approximate dollar loss:
κ(π − π)2
• That is, the CB’s loss function is well approximated by:
L(a, x) = κ(a + x − π)2
First Order Condition
• The CB chooses a to minimize:
E ∗(a + x − π)2
• This results in the first-order condition:
E ∗(π) = π
• The inflation-targeting CB ensures that the outlook for π is kept near π.
• Standard result - except the relevant outlook is given by E ∗, not E.
Intuition
• E ∗(π) can be measured with inflation break-evens.
— on TIPS bonds or on zero coupon inflation swaps
• These break-evens imply that E ∗(π) is generally larger than (usual measures of) E(π).
• Keeping E ∗(π) equal to π will result in E(π) being less than π.
• Why is this desirable?
• E ∗(π) > E(π) ⇒ state prices tend to be high when inflation is high.
• This means that π > π is more costly to society than π < π.
• Hence, optimal monetary policy should lead to E(π) being lower than π.
CONCLUSIONS
RNPDs and Predictions
• FAQ: Do RNPDs forecast the future better than statistical models?
• Similar: Did RNPDs in 2006 reveal the coming asset price corrections?
• My point today is that these are the wrong questions for policymakers to ask.
Financial Market Data and Decisions
• Policymakers form future outlooks so as to make current decisions with
future outcomes.
• Optimal decisions trade off benefits/costs in future states of the world.
• The trade-off should not be based on ex ante (or ex post!) assessments of
the states’ probabilities.
• Instead, the trade-off should be based on the ex ante relative values of
resources in those states.
Hence, the relevant outlook for a policymaker is an RNPD.
Implementation Challenges
• Decision-making using RNPDs is not necessarily easy.
— Need to determine appropriate financial proxy.
— Even then: Available options may not cover longer horizons or extreme
tail events.
• Nothing new: Good decisions are always based on a mix of good judgment,
good data, and good modeling choices.
BUT:
The right goal is to model/estimate RNPDs, not statistical forecasts.
Ninth District Activities
• Minneapolis Fed’s Banking Group uses options data to compute RNPDs.
• They report the results on the public website for a wide range of assets.
— Gold, silver, wheat, S&P 500, exchange rates, etc.
• They report and archive the results on a biweekly basis.
• See http://www.minneapolisfed.org/banking/assetvalues/index.cfm.