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Debt as Safe Asset: Mining the Bubble 03.a. xx Markus K. Brunnermeier Sebastian Merkel Yuliy Sannikov Princeton and Stanford Virtual Finance Workshop 2020-12-07

Debt as Safe Asset: Mining the Bubble...Debt as Safe Asset: Mining the Bubble 03.a. xx Markus K. Brunnermeier Sebastian Merkel Yuliy Sannikov Princeton and Stanford Virtual Finance

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  • Debt as Safe Asset:Mining the Bubble03.a. xx

    Markus K. BrunnermeierSebastian Merkel

    Yuliy SannikovPrinceton and Stanford

    Virtual Finance Workshop2020-12-07

  • How much government debt can the market absorb? At what interest rate? Is there a limit, a β€œDebt Laffer Curve”?What is the impact on inflation?When can governments run a deficit without ever pa-

    ying back its debt, like a Ponzi scheme, and nevertheless individual citizens’ transversality conditions hold?What is a safe asset? What are its features? Retrading?Why is government debt a safe asset? When do you lose safe asset status?Why is there debt valuation puzzle for US, Japanese? How do we have to modify representative agent asset

    pricing and the FTPL?2

    Questions of our times

  • Valuating Government Debt Think of a representative agent holding all gov. debt His cash flow is primary surplus

    β„¬π‘‘π‘‘β„˜π‘‘π‘‘

    = 𝐸𝐸𝑑𝑑 βˆ«π‘‘π‘‘βˆž πœ‰πœ‰π‘ π‘ πœ‰πœ‰π‘‘π‘‘

    𝑇𝑇𝑠𝑠 βˆ’ 𝐺𝐺𝑠𝑠 𝑑𝑑𝑑𝑑 + 𝐸𝐸𝑑𝑑 (πœ‰πœ‰π‘‘π‘‘= SDF) … but Japan primary surplus was negative for 50 out of 60 years Can surpluses be negative forever? Yes, if gov. debt is safe asset

    How to rescue the FTPL? 3

  • Asset Price = E[PV(cash flows)] + E[PV(service flows)]dividends/interest convenience yield

    4

    What’s a Safe Asset?

  • Asset Price = E[PV(cash flows)] + E[PV(service flows)]dividends/interest convenience yield

    5

    What’s a Safe Asset?

    0

    CF

    0

    CF

    0 0

    CFCF

    00

    CFCF

    BA

    BA

    Portfolio of

    Safe asset

    Cash flowasset

    shocks

    shocks

    …

    …

    …

  • Asset Price = E[PV(cash flows)] + E[PV(service flows)]dividends/interest convenience yield

    Value come from re-trading

    7

    What’s a Safe Asset?

    0

    CF

    0

    CF

    0 0

    CFCF

    00

    CFCF

    BA

    BA …

    …

    …

  • Asset Price = E[PV(cash flows)] + E[PV(service flows)]dividends/interest convenience yield

    Value come from re-trading Insures by partially

    completing markets

    Can be β€œbubbly” = fragile

    8

    What’s a Safe Asset?

    0

    CF

    0

    CF

    0 0

    CFCF

    00

    CFCF

    BA

    BA …

    …

    …

  • Asset Price = E[PV(cash flows)] + E[PV(service flows)]dividends/interest convenience yield

    2 𝛽𝛽s 𝛽𝛽𝑐𝑐𝑐𝑐 > 0 𝛽𝛽𝑠𝑠𝑐𝑐 < 0

    1. Good friend analogy (Brunnermeier Haddad, 2012) When one needs funds, one can sell at stable price… since others buy Idiosyncratic shock: Partial insurance through retrading - low bid-ask spread Aggregate (volatility) shock: Appreciate in value – negative 𝛽𝛽 = πœ”πœ”π›½π›½π‘π‘π‘π‘ + (1 βˆ’ πœ”πœ”)𝛽𝛽𝑠𝑠𝑐𝑐 < 0

    2. Safe Asset Tautology Safe asset is a bubble from aggregate perspective - fragility

    Other service flows: collateral constraint, double-coincidence of wants 9

    Safe Asset Pricing Equation, 2 𝛽𝛽𝑑𝑑, Fragility

  • Model with Capital + Safe Asset Each heterogenous citizen Μƒπš€πš€ ∈ [0,1]

    𝐸𝐸 ∫0∞ π‘’π‘’βˆ’πœŒπœŒπ‘‘π‘‘ log π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€ 𝑑𝑑𝑑𝑑 s.t.

    Each citizen operates one firm Output π‘¦π‘¦π‘‘π‘‘οΏ½ΜƒοΏ½πš€ = π‘Žπ‘Žπ‘‘π‘‘π‘˜π‘˜π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    Physical capital π‘˜π‘˜π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    π‘‘π‘‘π‘˜π‘˜π‘‘π‘‘

    οΏ½ΜƒοΏ½πš€

    π‘˜π‘˜π‘‘π‘‘οΏ½ΜƒοΏ½πš€ = (Ξ¦ πœ„πœ„π‘‘π‘‘οΏ½ΜƒοΏ½πš€ βˆ’ 𝛿𝛿)𝑑𝑑𝑑𝑑 + οΏ½πœŽπœŽπ‘‘π‘‘π‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    Aggregate risk:οΏ½πœŽπœŽπ‘‘π‘‘ , π‘Žπ‘Žπ‘‘π‘‘, β„Šπ‘‘π‘‘ exogenous process with aggregate shock 𝑑𝑑𝑍𝑍𝑑𝑑

    Financial Friction: Incomplete markets: citizens cannot trade claims on 𝑑𝑑 οΏ½π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    11

    A LA L

    A LA L

    Gov. debtMoney

    Net

    wor

    th

    π‘˜π‘˜π‘‘π‘‘οΏ½ΜƒοΏ½πš€π‘›π‘›οΏ½ΜƒοΏ½πš€

    π‘‘π‘‘π‘›π‘›π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    π‘›π‘›π‘‘π‘‘οΏ½ΜƒοΏ½πš€= βˆ’

    π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    π‘›π‘›π‘‘π‘‘οΏ½ΜƒοΏ½πš€π‘‘π‘‘π‘‘π‘‘ + π‘‘π‘‘π‘Ÿπ‘Ÿπ‘‘π‘‘β„¬ + 1 βˆ’ πœƒπœƒπ‘‘π‘‘οΏ½ΜƒοΏ½πš€ π‘‘π‘‘π‘Ÿπ‘Ÿπ‘‘π‘‘

    𝐾𝐾,οΏ½ΜƒοΏ½πš€ πœ„πœ„π‘‘π‘‘οΏ½ΜƒοΏ½πš€ βˆ’ π‘‘π‘‘π‘Ÿπ‘Ÿπ‘‘π‘‘β„¬

  • Model with Capital + Safe Asset Each heterogenous citizen Μƒπš€πš€ ∈ [0,1]

    𝐸𝐸 ∫0∞ π‘’π‘’βˆ’πœŒπœŒπ‘‘π‘‘ log π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€ 𝑑𝑑𝑑𝑑 s.t.

    Each citizen operates one firm Output π‘¦π‘¦π‘‘π‘‘οΏ½ΜƒοΏ½πš€ = π‘Žπ‘Žπ‘‘π‘‘π‘˜π‘˜π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    Physical capital π‘˜π‘˜π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    π‘‘π‘‘π‘˜π‘˜π‘‘π‘‘

    οΏ½ΜƒοΏ½πš€

    π‘˜π‘˜π‘‘π‘‘οΏ½ΜƒοΏ½πš€ = (Ξ¦ πœ„πœ„π‘‘π‘‘οΏ½ΜƒοΏ½πš€ βˆ’ 𝛿𝛿)𝑑𝑑𝑑𝑑 + οΏ½πœŽπœŽπ‘‘π‘‘π‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    Aggregate risk:οΏ½πœŽπœŽπ‘‘π‘‘ , π‘Žπ‘Žπ‘‘π‘‘, β„Šπ‘‘π‘‘ exogenous process with aggregate shock 𝑑𝑑𝑍𝑍𝑑𝑑

    Financial Friction: Incomplete markets: citizens cannot trade claims on 𝑑𝑑 οΏ½π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    12

    A LA L

    A LA L

    Gov. debtMoney

    Net

    wor

    th

    π‘˜π‘˜π‘‘π‘‘οΏ½ΜƒοΏ½πš€π‘›π‘›οΏ½ΜƒοΏ½πš€

    π‘‘π‘‘π‘›π‘›π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    π‘›π‘›π‘‘π‘‘οΏ½ΜƒοΏ½πš€= βˆ’

    π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    π‘›π‘›π‘‘π‘‘οΏ½ΜƒοΏ½πš€π‘‘π‘‘π‘‘π‘‘ + π‘‘π‘‘π‘Ÿπ‘Ÿπ‘‘π‘‘β„¬ + 1 βˆ’ πœƒπœƒπ‘‘π‘‘οΏ½ΜƒοΏ½πš€ π‘‘π‘‘π‘Ÿπ‘Ÿπ‘‘π‘‘

    𝐾𝐾,οΏ½ΜƒοΏ½πš€ πœ„πœ„π‘‘π‘‘οΏ½ΜƒοΏ½πš€ βˆ’ π‘‘π‘‘π‘Ÿπ‘Ÿπ‘‘π‘‘β„¬

  • Taxes, Bond/Money Supply, Gov. Budget Government policy Instruments Government spending β„Šπ‘‘π‘‘πΎπΎπ‘‘π‘‘ Proportional tax πœπœπ‘‘π‘‘π‘˜π‘˜π‘‘π‘‘ on capital Nominal government debt supply

    𝑑𝑑ℬ𝑑𝑑ℬ𝑑𝑑

    = πœ‡πœ‡π‘‘π‘‘β„¬π‘‘π‘‘π‘‘π‘‘

    Nominal interest rate 𝑖𝑖𝑑𝑑 Government budget constraint (BC)

    πœ‡πœ‡π‘‘π‘‘β„¬ βˆ’ π‘–π‘–π‘‘π‘‘οΏ½πœ‡πœ‡π‘‘π‘‘π΅π΅:=

    ℬ𝑑𝑑 + β„˜π‘‘π‘‘πΎπΎπ‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ β„Šπ‘‘π‘‘π‘ π‘ π‘‘π‘‘β‰”

    = 0

    Assume here: Gov. chooses πœ‡πœ‡β„¬, 𝑖𝑖; while πœπœπ‘‘π‘‘ adjusts to satisfy (BC) Goods market clearing:

    𝐢𝐢𝑑𝑑 + β„Šπ‘‘π‘‘πΎπΎπ‘‘π‘‘ = π‘Žπ‘Žπ‘‘π‘‘ βˆ’ πœ„πœ„π‘‘π‘‘ 𝐾𝐾𝑑𝑑 13Let οΏ½π‘Žπ‘Žπ‘‘π‘‘: = π‘Žπ‘Žπ‘‘π‘‘ βˆ’ β„Šπ‘‘π‘‘

    Primary surplus (per 𝐾𝐾𝑑𝑑)

  • Real prices and returns π‘žπ‘žπ‘‘π‘‘πΎπΎπΎπΎπ‘‘π‘‘ value of physical capital

    Return π‘‘π‘‘π‘Ÿπ‘Ÿπ‘‘π‘‘πΎπΎ,οΏ½ΜƒοΏ½πš€ = π‘Žπ‘Ž(1βˆ’πœπœ)βˆ’πœ„πœ„π‘‘π‘‘

    οΏ½ΜƒοΏ½πš€

    π‘žπ‘žπ‘‘π‘‘πΎπΎ+ Ξ¦ πœ„πœ„π‘‘π‘‘οΏ½ΜƒοΏ½πš€ βˆ’ 𝛿𝛿 + πœ‡πœ‡π‘‘π‘‘

    π‘žπ‘žπΎπΎ 𝑑𝑑𝑑𝑑 + πœŽπœŽπ‘‘π‘‘π‘žπ‘žπΎπΎπ‘‘π‘‘π‘π‘π‘‘π‘‘ + οΏ½πœŽπœŽπ‘‘π‘‘π‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    π‘žπ‘žπ‘‘π‘‘π΅π΅πΎπΎπ‘‘π‘‘ real value of gov. debt ℬ𝑑𝑑/β„˜π‘‘π‘‘ = π‘žπ‘žπ‘‘π‘‘π΅π΅πΎπΎπ‘‘π‘‘

    Return π‘‘π‘‘π‘Ÿπ‘Ÿπ‘‘π‘‘π΅π΅ = (𝑖𝑖 βˆ’ πœ‡πœ‡π‘‘π‘‘β„¬

    βˆ’οΏ½πœ‡πœ‡π‘‘π‘‘β„¬

    + Ξ¦ πœ„πœ„π‘‘π‘‘ βˆ’ 𝛿𝛿𝑔𝑔=

    βˆ’ πœ‡πœ‡π‘‘π‘‘π‘žπ‘žπ΅π΅)𝑑𝑑𝑑𝑑 + πœŽπœŽπ‘‘π‘‘

    π‘žπ‘žπ΅π΅π‘‘π‘‘π‘π‘π‘‘π‘‘

    Μƒπš€πš€β€™s dynamic trading strategy of gov. bond Inflow (outflow) from selling (buying) bond Reduces (increases) future payoffs

    14

    Dividend Yield Capital gains

    βˆ’ inflation

  • Optimal real investment rate πœ„πœ„π‘‘π‘‘: (Tobin’s q)

    Optimal consumption: 𝑐𝑐𝑑𝑑 = πœŒπœŒπ‘›π‘›π‘‘π‘‘

    Optimal portfolio choice: 1 βˆ’ πœƒπœƒπ‘‘π‘‘ =π‘Žπ‘Žπ‘‘π‘‘βˆ’πœ„πœ„π‘‘π‘‘ /π‘žπ‘žπ‘‘π‘‘πΎπΎ+οΏ½πœ‡πœ‡π΅π΅

    π›Ύπ›ΎοΏ½πœŽπœŽπ‘‘π‘‘2 = 1 βˆ’ πœ—πœ—π‘‘π‘‘

    15

    Optimality and market clearings

  • Two Stationary Equilibria (for 𝐾𝐾0 = 1)

    16

    Gordon-Growth Formula Closed Form Solution

    π‘žπ‘žπΎπΎ = 1βˆ’πœπœ οΏ½π‘Žπ‘Žβˆ’πœ„πœ„πΈπΈ π‘‘π‘‘π‘Ÿπ‘ŸπΎπΎ /π‘‘π‘‘π‘‘π‘‘βˆ’π‘”π‘”

    π‘žπ‘žπΎπΎ =𝜌𝜌 + οΏ½πœ‡πœ‡π΅π΅ 1 + πœ™πœ™οΏ½π‘Žπ‘ŽπœŒπœŒ + οΏ½πœ‡πœ‡π΅π΅ + πœ™πœ™ �𝜎𝜎𝜌𝜌

    β„¬β„˜ =

    𝑑𝑑𝐸𝐸 π‘‘π‘‘π‘Ÿπ‘Ÿπ‘›π‘› /𝑑𝑑𝑑𝑑 βˆ’ 𝑔𝑔

    +1 βˆ’ πœ—πœ— 2 �𝜎𝜎2β„¬β„˜

    𝐸𝐸 π‘‘π‘‘π‘Ÿπ‘Ÿπ‘›π‘› /𝑑𝑑𝑑𝑑 βˆ’ π‘”π‘”π‘žπ‘žπ΅π΅πΎπΎπ‘‘π‘‘ =

    �𝜎𝜎 βˆ’ 𝜌𝜌 + οΏ½πœ‡πœ‡π΅π΅ 1 + πœ™πœ™οΏ½π‘Žπ‘Ž

    𝜌𝜌 + οΏ½πœ‡πœ‡π΅π΅ + πœ™πœ™ οΏ½πœŽπœŽπœŒπœŒπΎπΎπ‘‘π‘‘

    πœ„πœ„ = π‘Žπ‘Ž 𝜌𝜌+οΏ½πœ‡πœ‡π΅π΅βˆ’οΏ½πœŽπœŽπœŒπœŒ

    𝜌𝜌+οΏ½πœ‡πœ‡π΅π΅+πœ™πœ™οΏ½πœŽπœŽπœŒπœŒ

    π‘‘π‘‘π‘Ÿπ‘Ÿπ‘›π‘› = πœƒπœƒπ‘‘π‘‘π‘Ÿπ‘Ÿπ΅π΅ + 1 βˆ’ πœƒπœƒ π‘‘π‘‘π‘Ÿπ‘ŸπΎπΎ 𝜌𝜌 time preference rateπœ™πœ™ adjustment cost for investment rateοΏ½ΜŒοΏ½πœ‡π‘‘π‘‘π΅π΅ = πœ‡πœ‡π‘‘π‘‘π΅π΅ βˆ’ 𝑖𝑖 bond issuance rate beyond interest rateοΏ½π‘Žπ‘Ž = π‘Žπ‘Ž βˆ’ 𝔀𝔀 part of TFP not spend on gov.)

  • Individual Perspective π‘‘π‘‘πœ‰πœ‰π‘‘π‘‘οΏ½ΜƒοΏ½πš€/πœ‰πœ‰π‘‘π‘‘οΏ½ΜƒοΏ½πš€ = βˆ’π‘Ÿπ‘Ÿπ‘‘π‘‘π‘π‘π‘‘π‘‘π‘‘π‘‘ βˆ’ πœπœπ‘‘π‘‘π‘‘π‘‘π‘π‘π‘‘π‘‘ βˆ’ Μƒπœπœπ‘‘π‘‘οΏ½ΜƒοΏ½πš€π‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    Bond as part of a dynamic trading strategy Cash flow from selling (buying) after negative (positive) idiosyncratic shock Price β€œbond-part” of portfolio Integrate over citizens weighted by net worth share πœ‚πœ‚π‘‘π‘‘π‘–π‘– πœ‰πœ‰π‘–π‘– and πœ‚πœ‚π‘–π‘– are negatively correlated β‡’ depresses weighted SDF

    (higher discount rate)

    Aggregate Perspective 𝑑𝑑 Μ…πœ‰πœ‰π‘‘π‘‘/ Μ…πœ‰πœ‰π‘‘π‘‘ = βˆ’π‘Ÿπ‘Ÿπ‘‘π‘‘π‘π‘π‘‘π‘‘π‘‘π‘‘ βˆ’ πœπœπ‘‘π‘‘π‘‘π‘‘π‘π‘π‘‘π‘‘

    Without aggregate risk Μ…πœ‰πœ‰π‘‘π‘‘ = π‘’π‘’βˆ’π‘Ÿπ‘Ÿπ‘“π‘“π‘‘π‘‘

    Lower social discount rate + Bubble term 17

    Safe Asset Valuation Equation: 2 Perspectives

  • Individual Perspective π‘‘π‘‘πœ‰πœ‰π‘‘π‘‘οΏ½ΜƒοΏ½πš€/πœ‰πœ‰π‘‘π‘‘οΏ½ΜƒοΏ½πš€ = βˆ’π‘Ÿπ‘Ÿπ‘‘π‘‘π‘π‘π‘‘π‘‘π‘‘π‘‘ βˆ’ πœπœπ‘‘π‘‘π‘‘π‘‘π‘π‘π‘‘π‘‘ βˆ’ Μƒπœπœπ‘‘π‘‘οΏ½ΜƒοΏ½πš€π‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    Bond as part of a dynamic trading strategy Cash flow from selling (buying) after negative (positive) idiosyncratic shock Price β€œbond-part” of portfolio Integrate over citizens weighted by net worth share πœ‚πœ‚π‘‘π‘‘π‘–π‘– πœ‰πœ‰π‘–π‘– and πœ‚πœ‚π‘–π‘– are negatively correlated β‡’ depresses weighted SDF

    (higher discount rate)

    Aggregate Perspective 𝑑𝑑 Μ…πœ‰πœ‰π‘‘π‘‘/ Μ…πœ‰πœ‰π‘‘π‘‘ = βˆ’π‘Ÿπ‘Ÿπ‘‘π‘‘π‘π‘π‘‘π‘‘π‘‘π‘‘ βˆ’ πœπœπ‘‘π‘‘π‘‘π‘‘π‘π‘π‘‘π‘‘

    Without aggregate risk Μ…πœ‰πœ‰π‘‘π‘‘ = π‘’π‘’βˆ’π‘Ÿπ‘Ÿπ‘“π‘“π‘‘π‘‘

    Lower social discount rate + Bubble term 18

    Safe Asset Valuation Equation: 2 Perspectives

    β€œPartial insurance Service”

    β„¬β„˜ =

    𝑑𝑑𝐸𝐸 π‘‘π‘‘π‘Ÿπ‘Ÿπ‘›π‘› /𝑑𝑑𝑑𝑑 βˆ’ 𝑔𝑔

    +1 βˆ’ πœ—πœ— 2 �𝜎𝜎2β„¬β„˜

    𝐸𝐸 π‘‘π‘‘π‘Ÿπ‘Ÿπ‘›π‘› /𝑑𝑑𝑑𝑑 βˆ’ 𝑔𝑔

    𝜌𝜌 + 𝑔𝑔 = discount rate

    𝐸𝐸 π‘‘π‘‘π‘Ÿπ‘Ÿπ‘›π‘› /𝑑𝑑𝑑𝑑 = π‘Ÿπ‘Ÿπ‘π‘ + 𝜍𝜍𝜎𝜎 + Μƒπœπœ �𝜎𝜎

  • Individual Perspective π‘‘π‘‘πœ‰πœ‰π‘‘π‘‘οΏ½ΜƒοΏ½πš€/πœ‰πœ‰π‘‘π‘‘οΏ½ΜƒοΏ½πš€ = βˆ’π‘Ÿπ‘Ÿπ‘‘π‘‘π‘π‘π‘‘π‘‘π‘‘π‘‘ βˆ’ πœπœπ‘‘π‘‘π‘‘π‘‘π‘π‘π‘‘π‘‘ βˆ’ Μƒπœπœπ‘‘π‘‘οΏ½ΜƒοΏ½πš€π‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½ΜƒοΏ½πš€

    Bond as part of a dynamic trading strategy Cash flow from selling (buying) after negative (positive) idiosyncratic shock Price β€œbond-part” of portfolio Integrate over citizens weighted by net worth share πœ‚πœ‚π‘‘π‘‘π‘–π‘– πœ‰πœ‰π‘–π‘– and πœ‚πœ‚π‘–π‘– are negatively correlated β‡’ depresses weighted SDF

    (higher discount rate)

    Aggregate Perspective 𝑑𝑑 Μ…πœ‰πœ‰π‘‘π‘‘/ Μ…πœ‰πœ‰π‘‘π‘‘ = βˆ’π‘Ÿπ‘Ÿπ‘‘π‘‘π‘π‘π‘‘π‘‘π‘‘π‘‘ βˆ’ πœπœπ‘‘π‘‘π‘‘π‘‘π‘π‘π‘‘π‘‘

    Without aggregate risk Μ…πœ‰πœ‰π‘‘π‘‘ = π‘’π‘’βˆ’π‘Ÿπ‘Ÿπ‘“π‘“π‘‘π‘‘

    Lower social discount rate + Bubble term 19

    Safe Asset Valuation Equation: 2 Perspectives

    β€œPartial insurance Service”

    β„¬β„˜ =

    𝑑𝑑𝐸𝐸 π‘‘π‘‘π‘Ÿπ‘Ÿπ‘›π‘› /𝑑𝑑𝑑𝑑 βˆ’ 𝑔𝑔

    +1 βˆ’ πœ—πœ— 2 �𝜎𝜎2β„¬β„˜

    𝐸𝐸 π‘‘π‘‘π‘Ÿπ‘Ÿπ‘›π‘› /𝑑𝑑𝑑𝑑 βˆ’ 𝑔𝑔

    Only for s>0β„¬β„˜ =

    π‘‘π‘‘π‘Ÿπ‘Ÿπ‘π‘ βˆ’ 𝑔𝑔

    𝜌𝜌 + 𝑔𝑔 = discount rate

    𝑔𝑔 βˆ’ οΏ½ΜŒοΏ½πœ‡π΅π΅ = discount rate

    𝐸𝐸 π‘‘π‘‘π‘Ÿπ‘Ÿπ‘›π‘› /𝑑𝑑𝑑𝑑 = π‘Ÿπ‘Ÿπ‘π‘ + 𝜍𝜍𝜎𝜎 + Μƒπœπœ �𝜎𝜎

    π‘Ÿπ‘Ÿπ‘π‘ + 𝜍𝜍𝜎𝜎

  • Bubble/Ponzi Scheme and Transversality Gov. Debt is a Ponzi scheme/bubble (in aggregate perspective) Service flow – partial insurance to overcome market incompleteness

    Why does transversality condition not rule out the bubble? Individual Perspective

    High individual discount rate (low SDF) since net worth

    limπ‘‡π‘‡β†’βˆž

    𝐸𝐸 πœ‰πœ‰π‘‡π‘‡π‘›π‘›π‘‡π‘‡οΏ½ΜƒοΏ½πš€ = 0 Aggregate perspective

    Low β€œsocial” discount rate (high SDF)limπ‘‡π‘‡β†’βˆž

    𝐸𝐸 Μ…πœ‰πœ‰π‘‡π‘‡π‘›π‘›π‘‡π‘‡οΏ½ΜƒοΏ½πš€ > 0

    20

  • π‘Ÿπ‘Ÿπ‘π‘ versus 𝑔𝑔 for different οΏ½πœ‡πœ‡β„¬

    21

    π‘Ÿπ‘Ÿπ‘π‘ = Ξ¦ πœ„πœ„ βˆ’ 𝛿𝛿=𝑔𝑔

    βˆ’ οΏ½πœ‡πœ‡β„¬

    𝑔𝑔 = 1πœ™πœ™ log𝜌𝜌+οΏ½πœ‡πœ‡β„¬ 1+πœ™πœ™π‘Žπ‘ŽπœŒπœŒ+οΏ½πœ‡πœ‡β„¬+πœ™πœ™οΏ½πœŽπœŽπœŒπœŒ

    βˆ’ 𝛿𝛿

    bubbly

    π‘Žπ‘Ž = .27,β„Š =π‘Žπ‘Ž3

    , 𝛿𝛿 = .1,𝜌𝜌 = .02, �𝜎𝜎 = .25,πœ™πœ™ = 3 ,

    οΏ½πœ‡πœ‡β„¬

    When primary deficit forever 𝑑𝑑 < 0 βˆ€π‘‘π‘‘βŸΊ οΏ½ΜŒοΏ½πœ‡π΅π΅ > 0? Japan? Higher issuance rate β‡’ higher inflation tax β‡’ lower real return β‡’ π‘Ÿπ‘Ÿπ‘π‘ < 𝑔𝑔

  • Higher issuance rate, οΏ½ΜŒοΏ½πœ‡π΅π΅ β‡’ higher inflation tax But real value of bonds, β„¬β„˜, declines β‡’ lower β€œtax base”

    22

    Debt Laffer Curve

  • Flight to Safety: Comparative static w.r.t. �𝜎𝜎 Flight to safety into bubbly gov. debt π‘žπ‘žπ΅π΅ rises (disinflation) π‘žπ‘žπΎπΎ falls and so does πœ„πœ„ and 𝑔𝑔

    Similar withstochastic idiosyncratic volatility 23

  • Aggregate risk state variable: Stochastic idiosyncratic volatility: 𝑑𝑑 log οΏ½πœŽπœŽπ‘‘π‘‘ = βˆ’πœ“πœ“ log

    οΏ½πœŽπœŽπ‘‘π‘‘οΏ½πœŽπœŽ0𝑑𝑑𝑑𝑑 + 𝜎𝜎π‘₯π‘₯𝑑𝑑𝑍𝑍𝑑𝑑

    Stochastic TFP: π‘Žπ‘Žπ‘‘π‘‘ = π‘Žπ‘Ž( οΏ½πœŽπœŽπ‘‘π‘‘) s.t.𝐢𝐢𝐾𝐾

    οΏ½πœŽπœŽπ‘‘π‘‘ = 𝛼𝛼0 βˆ’ 𝛼𝛼1 οΏ½πœŽπœŽπ‘‘π‘‘ linear

    Policy (surpluses decrease in οΏ½πœŽπœŽπ‘‘π‘‘): οΏ½ΜŒοΏ½πœ‡π‘‘π‘‘β„¬ = βˆ’πœˆπœˆ0 + 𝜈𝜈1 οΏ½πœŽπœŽπ‘‘π‘‘ Individual perspective: 2 terms of valuation equation X - cash flow term around 0 X - safe asset service flow term

    dominates

    25

    Countercyclical Safe Asset

  • 𝛽𝛽𝐡𝐡,𝑐𝑐𝑐𝑐 > 0 for cash flow term (primary surplus term)

    𝛽𝛽𝐡𝐡,𝑠𝑠𝑐𝑐 < 0 for service flow term (due to risk sharing)

    26

    Countercyclical Safe Asset – 2 Betas

  • Bubbles can pop

    Able to prop up the bubble/safe-asset status by (off-equilibrium) hiking taxes (fiscal space)

    Market maker of last resort to secure low bid-ask spread 10 year US Treasury in March 2020

    Competing safe asset Interest rate policy of competing central banks β€œleast ugly horse”

    27

    Loss of Safe Asset Status

  • Asset Pricing Safe asset is different – provides service flow Risk sharing via precautionary saving and constant retrading 2 terms: cash flow + service flow Split depends on perspective (individual vs. aggregate) different discount rates 2 𝛽𝛽𝑑𝑑

    Flight to safety creates countercyclical Safe Asset Valuations negative 𝛽𝛽

    Bubble mining for government Negative primary surpluses for decades (like in Japan) But has its limits (unlike MMT)

    Bubbles can pop: Loss of flight to safe asset status Fiscal capacity to fend off + Market maker of last resort

    28

    Conclusion

    Debt as Safe Asset:οΏ½Mining the BubbleοΏ½03.a. xxοΏ½Questions of our timesValuating Government DebtWhat’s a Safe Asset?What’s a Safe Asset?What’s a Safe Asset?What’s a Safe Asset?Safe Asset Pricing Equation, 2 𝛽𝑠, FragilityModel with Capital + Safe AssetModel with Capital + Safe AssetTaxes, Bond/Money Supply, Gov. BudgetReal prices and returnsOptimality and market clearingsTwo Stationary Equilibria (for 𝐾 0 =1)Safe Asset Valuation Equation: 2 PerspectivesSafe Asset Valuation Equation: 2 PerspectivesSafe Asset Valuation Equation: 2 PerspectivesBubble/Ponzi Scheme and Transversality π‘Ÿ 𝑓 versus 𝑔 for different πœ‡ ℬ Debt Laffer CurveFlight to Safety: Comparative static w.r.t. 𝜎 Countercyclical Safe AssetCountercyclical Safe Asset – 2 BetasLoss of Safe Asset StatusConclusion


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