A while ago in two blog posts here and here I suggested many ways other than currency to get a zero interest rate if the government tries to lower rates below zero. Buy gift cards, subway cards, stamps; prepay bills, rent, mortgage and especially taxes -- the IRS will happily take your money now and you can credit it against future tax payments; have your bank make out a big certified check in your name, and sit on it, don't cash incoming checks. Start a company that takes money and invests in all these things (as well as currency).
Chris and Miles Kimball have an interesting essay exploring these ideas "However low interest rates might go, the IRS will never act like a bank." Their central point: sure that's how things work now. But with substantial negative interest rates, all of these contracts can change. It's technically possible in each case for people and businesses to charge pre-payment penalties amounting to a negative nominal rate.
Reply: Sure, in principle. Nominal claims can all be dated, and positive or negative interest charged between all dates.
But this did not happen in the US and does not happen in other countries for positive inflation and high nominal rates, despite symmetric incentives, and at rates much higher than the contemplated 3-5% or so negative rates. Yes, with large nominal rates there is pressure to pay faster, inventory cash-management to reduce people's holdings of depreciating nominal claims, but this pervasive indexation of nominal payments did not break out. The IRS did not offer interest for early payment.
More deeply, what they're describing is a tiny step away from perfect price indexing. If all nominal payments are perfectly indexed to the nominal interest rate, accrued daily, then it's a tiny change to index all prices themselves to the CPI, accrued daily. If "how much you owe me," say to rent a house, is legally, contractually, and mechanically determined as a value times e^rt, and changes day by day, then e^(pi t) is just as easy.
So, price stickiness itself would (should!) disappear under this scenario.
Price stickiness has always been a bit of a puzzle for economists. As the Kimballs speculate how easy it is to index payments to negative interest rates, so economists speculate how easy it is to index payments to inflation. Yet it seems not to happen.
So this point of view strikes me as a bit of a catch-22 for its advocates, who generally are of the frame of mind that prices and nominal contracts are sticky and that’s why negative nominal rates are a good idea to "stimulate demand" in the first place. If we can have negative nominal rates and change all these legal and contractual zero-rate promises to allow it, then prices won't be sticky any more! Conversely, I should be cheering, as it amounts to a broad push to unstick prices. That has long seemed to me the natural policy response to the view that sticky prices are the root of all our troubles. It would allow negative rates, but eliminate their need as well.
Alas, the world seems remarkably resistant to time-indexing all payments.
Thursday, April 16, 2015
Wednesday, April 15, 2015
Gdefault needs not Grexit
The little grumpy cartoon usually represents me pounding my coffee down in agreement as the WSJ exposes some idiocy. Last week, alas, I spilled my grumpy coffee in disagreement with a little part of its otherwise excellent "The case for letting Greece go."
Sure we can have an argument about whether it would be a good idea. The first 147 devaluations and currency confiscations didn't produce Singapore on the Mediterranean, but maybe the 148th will do the trick. The canard is the logical necessity of Grexit.
This is a particularly dangerous canard too. Greece is undergoing a slow motion bank run. Greeks are wisely taking their euros out of Greek banks and either holding cash or taking it abroad. So, how to Greek banks give them euros without selling all their assets -- loans and Greek government bonds? Answer, they get the money from the Greek central bank, which gets the euros from the ECB. The ECB is getting antsy about funding not just Greek government debt, but the whole Greek banking system.
Sooner or later Greeks will translate all this central banker speak about "capital controls" "liquidity management" and so forth to "there is a good chance that tomorrow morning your bank account will be frozen or converted to Drachmas." Then the run of all time starts and the whole thing unravels.
How do you stop that from happening? By shouting from the rooftops that the currency remains the euro, no matter if the government defaults on its loans to the IMF. At least we can shout from the rooftops that changing currencies is a separate decision, and that stiffing the IMF does not imply the logical necessity of grabbing Greek bank accounts.
To be sure the article gets much right. It's main thesis: Letting Greece default might be the right thing to do
Thursday marks another deadline in Greece’s struggle to avoid default, as a €450 million payment to the International Monetary Fund comes due. Athens says it will meet this obligation, but sooner or later Prime Minister Alexis Tsipras and his government will miss a payment to someone if it doesn’t agree with creditors on a new bailout. An exit from the euro would then be a real possibility.Please can we stop passing along this canard -- that Greece defaulting on some of its bonds means that Greece must must change currencies. Greece no more needs to leave the euro zone than it needs to leave the meter zone and recalibrate all its rulers, or than it needs to leave the UTC+2 zone and reset all its clocks to Athens time. When large companies default, they do not need to leave the dollar zone. When cities and even US states default they do not need to leave the dollar zone. A common currency means that sovereigns default just like large financial companies. (Yes, a bit of humor in the last one.)
Sure we can have an argument about whether it would be a good idea. The first 147 devaluations and currency confiscations didn't produce Singapore on the Mediterranean, but maybe the 148th will do the trick. The canard is the logical necessity of Grexit.
This is a particularly dangerous canard too. Greece is undergoing a slow motion bank run. Greeks are wisely taking their euros out of Greek banks and either holding cash or taking it abroad. So, how to Greek banks give them euros without selling all their assets -- loans and Greek government bonds? Answer, they get the money from the Greek central bank, which gets the euros from the ECB. The ECB is getting antsy about funding not just Greek government debt, but the whole Greek banking system.
Sooner or later Greeks will translate all this central banker speak about "capital controls" "liquidity management" and so forth to "there is a good chance that tomorrow morning your bank account will be frozen or converted to Drachmas." Then the run of all time starts and the whole thing unravels.
How do you stop that from happening? By shouting from the rooftops that the currency remains the euro, no matter if the government defaults on its loans to the IMF. At least we can shout from the rooftops that changing currencies is a separate decision, and that stiffing the IMF does not imply the logical necessity of grabbing Greek bank accounts.
To be sure the article gets much right. It's main thesis: Letting Greece default might be the right thing to do
But if Athens won’t implement reforms that would return Greece to growth and sustainable finances, allowing the country to leave would be the least bad outcome.And if the WSJ understood that "allowing the country to default" is not the same thing as "allowing the country to leave" the case is even stronger. (Though who does this "allowing" is a bit muddy. One more subject-less sentence infects the forlorn English language of policy-speak)
No one should cheer a Greek exit, which would be a disaster for the Greeks.Yes. Yet another reason to separate sovereign default from a change of monetary units.
Greece’s main contagion threat now would be if it is bailed out again without reform.This is the article's central point, and a good one. In financial as in foreign policy, people take important lessons from discovering that threats are empty.
The strongest argument against allowing Greece to leave the euro is that it would dent the bloc’s appearance of permanence, making the euro more like a currency peg that members could leave at will.Exactly. And if we would all go back to the original Instruction Manual For the Euro, that says sovereign default can happen, just like corporate default, and does not require a change of currency, that permanence would be all the more assured.
Tuesday, April 14, 2015
Blanchard on Countours of Policy
Olivier Blanchard, (IMF research director) has a thoughtful blog post, Contours of Macroeconomic Policy in the Future. In part it's background for the IMF's upcoming conference with the charming title Rethinking Macro Policy III: Progress or Confusion?” (You can guess my choice.)
Olivier cleanly poses some questions which in his view are likely to be the focus of policy-world debate for the next few years. Looking for policy-oriented thesis topics? It's a one-stop shop.
Whether these should be the questions is another matter. (Mostly no, in my view.)
As a blogger, I can't resist a few pithy answers. But please note, I'm mostly having fun, and the questions and essay are much more serious.
Olivier cleanly poses some questions which in his view are likely to be the focus of policy-world debate for the next few years. Looking for policy-oriented thesis topics? It's a one-stop shop.
Whether these should be the questions is another matter. (Mostly no, in my view.)
As a blogger, I can't resist a few pithy answers. But please note, I'm mostly having fun, and the questions and essay are much more serious.
Financial regulation
... Where do we stand? Are some dimensions of systemic risk easier to measure (e.g., leverage in the banking sector vs. interconnectedness of banks and non-banks or risks outside the banking sector)? How should we assess the experience with stress-tests? And have we made enough progress in reducing systemic risk since the crisis, e.g., with Dodd-Frank, the Vickers commission, the Financial Stability Board, etc?
Answer: "Systemic risk" is barely defined. The idea that regulators will, this time, really really, understand risks taken by the big banks, see trouble ahead, and stop the banks from failing, is a triumph of hope over repeated experience.
The only progress -- and it's big -- is the slow realization that banks can and should issue lots more equity.
Answer: "Systemic risk" is barely defined. The idea that regulators will, this time, really really, understand risks taken by the big banks, see trouble ahead, and stop the banks from failing, is a triumph of hope over repeated experience.
The only progress -- and it's big -- is the slow realization that banks can and should issue lots more equity.
Macro Prudential Policies
... Do we have or can we develop tools to deal with the different types of risk, from high housing prices, to insufficient capital in some financial institutions, to sudden drops in liquidity in some financial markets?
Using these tools ...raises political economy issues. In a housing boom, increasing the loan to value ratio may be politically difficult. Questions: Given these issues, when should we use macro prudential tools, or should we use tougher, non contingent financial regulation? To be concrete, should we aim for variable capital ratios and decide when to adjust them, or just give up on the variable part, and aim for high but constant capital ratios?
Answer: The hubris that the Davos set will be able to figure out just the right amount of capital, and then fine-tune that month-to-month and bank-to-bank is astounding. "Political economy concerns" is putting it mildly. The IMF's "bubble" or "imbalance" is the local Congressman's boom, and he or she will be hopping mad if the Fed restricts credit to his district or pet industry in favor of another.
The fact that our regulators are still talking about liquidity betrays a fundamental confusion of individual vs. systemic risks. Liquidity is the plan, "if we lose money we'll sell assets." To who? Regulators demanding liquidity to plan for a financial crisis is like the FAA making sure everyone on the plane has enough money to buy a parachute in case of engine failure.
Answer: The hubris that the Davos set will be able to figure out just the right amount of capital, and then fine-tune that month-to-month and bank-to-bank is astounding. "Political economy concerns" is putting it mildly. The IMF's "bubble" or "imbalance" is the local Congressman's boom, and he or she will be hopping mad if the Fed restricts credit to his district or pet industry in favor of another.
The fact that our regulators are still talking about liquidity betrays a fundamental confusion of individual vs. systemic risks. Liquidity is the plan, "if we lose money we'll sell assets." To who? Regulators demanding liquidity to plan for a financial crisis is like the FAA making sure everyone on the plane has enough money to buy a parachute in case of engine failure.
Finally, it is clear that both financial regulation and macro prudential tools are likely to lead financial actors to adjust and explore ways of getting around them. Questions: In this game of cat and mouse, can the macro prudential regulators hope to win? Or will regulation and tools become increasingly complex and possibly counterproductive?
That's easy. No and Yes. Actually I'm being too pessimistic. Regulatory capture works both ways. An easy forecast: Stress-testers at the Fed will be getting lucrative salary offers to move to the private sector and help pass stress tests. Which they will increasingly do.
That's easy. No and Yes. Actually I'm being too pessimistic. Regulatory capture works both ways. An easy forecast: Stress-testers at the Fed will be getting lucrative salary offers to move to the private sector and help pass stress tests. Which they will increasingly do.
Monetary Policy
... Questions: Under the highly realistic assumption that financial regulation and macroprudential tools do not fully take care of financial stability, [Highly realistic indeed! You just answered the first set of questions as I did!] should monetary policy take financial stability into account? And if so, how? Can the interest rate or other monetary policy tools reduce financial risk? How should macro prudential tools and monetary policy be coordinated? Should they both be under the responsibility of the central bank?
Let's remember that the crash of 1929 was, at least in the standard history, sparked by the Fed trying to restrain what they saw as the bubble in the stock market.
If this is the case, and central banks have tools which can have effects on very specific sectors of the economy, can they retain full independence?
No. In a democracy, independence comes with limited authority. The financial central planner cannot and will not long stay independent.
If this is the case, and central banks have tools which can have effects on very specific sectors of the economy, can they retain full independence?
No. In a democracy, independence comes with limited authority. The financial central planner cannot and will not long stay independent.
The zero ... lower bound on the interest rate set by central banks was thought to be a theoretical curiosum, unlikely to happen, and, in any case, easy to combat if reached. If reached, central banks could, through announcements of future monetary policy, increase expected inflation and achieve large negative interest rates. We have learned that this was simply wishful thinking. The zero lower bound could be reached, inflation expectations are not easy to manipulate, and it may take a very long time to exit.
Three cheers. Wow, Olivier, who wrote one of the most influential calls for announcements of higher inflation targets, looks at the data and calls it "wishful thinking." Bravo.
.. Quantitative Easing,... Questions: ...should central banks eventually return to the traditional mode of intervening at the short end of the market, or should they continue to buy and sell longer maturity sovereign or corporate bonds? Should the balance sheets of central banks return to their pre-crisis size, or remain permanently larger? If the central bank intervenes along the yield curve, how should monetary policy and debt management by the Treasury be combined?
Three cheers. Wow, Olivier, who wrote one of the most influential calls for announcements of higher inflation targets, looks at the data and calls it "wishful thinking." Bravo.
.. Quantitative Easing,... Questions: ...should central banks eventually return to the traditional mode of intervening at the short end of the market, or should they continue to buy and sell longer maturity sovereign or corporate bonds? Should the balance sheets of central banks return to their pre-crisis size, or remain permanently larger? If the central bank intervenes along the yield curve, how should monetary policy and debt management by the Treasury be combined?
Large balance sheet, interest-paying reserves, open to everyone. Some crisis interventions reveal very desirable permanent states of affairs. Stop fooling around with direct intervention in long-term debt, mortgage-backed security markets, and don't follow other central banks to buying and selling stocks, foreign exchange, etc.
Fiscal Policy
Fiscal Policy
... Questions: What is a dangerous level of debt? That which markets doubt you can repay. Seriously, if you're growing fast with a good long run plan for containing expenditures and raising revenue without ruinous taxation, a lot. If not, a lot less. ... What do we know about confidence effects? You mean statements by officials that "engender confidence?" Go back to the Romans, burn incense at the Temple of Jupiter. More seriously, we've learned that speaking loudly with no stick doesn't work. ...Should the old idea of the fiscal golden rule, the separation of a current and of a capital account, be resurrected? Separating two sides of an accounting identity sounds like an interesting golden rule. I think it would be golden to separate the current account and capital account I run down at the apple store -- they give me stuff, I don't have to give them money. Olivier surely has something more sophisticated in mind, and I'm revealing I'm a rube at this policy-speak coded language.
Most observers agree that the fiscal stimulus early in the crisis was instrumental in limiting the decrease in output. I'm glad he said "most" not "all"....
Capital inflows, exchange rate management and capital controls
The crisis has reinforced the notion that international capital flows can be very volatile, with emerging markets being particularly vulnerable. Back to previous comment. Capital can try to flow, but unless goods flow in the other direction, all it does is to lower prices. Unless you can pass a rule to get rid of accounting identities. See above. Policy makers have responded with a panoply of tools, from capital controls A polite word for expropriation to macro prudential measures aimed at shaping flows, What a lovely little policy-ese phrase and FX intervention. .... And what does the experience since the crisis say about the optimal opening of the capital account, even in the long run? Translated to English, back to the de-globalized protectionist world. If capital can't flow, neither can goods.
The International Monetary and Financial System
.... Questions: ... Should we reexamine the rules of the game for exchange rates? How can we improve on the process of sovereign debt restructuring?
As Olivier's essay moves on, and gradually reverts to the obfuscatory Orwellian prose of the international policy world, I get more and more animated. I mean just who is this "we?" Who is going to tell you you're not allowed to buy euros for your vacation this summer ("capital controls"), tell your bank not to give you a loan ("macro-produential policy"), decide how many billions to siphon from your pocket to the owners of large banks ("recapitalization" "process of sovereign debt restructuring"), not allowed to expand your business in a new country ("macro prudential measures aimed at shaping flows") and so forth? When there even is a "we," unlike most sentences with no subjects, like "the optimal opening of the capital account."
What should be the role of international forums such as the G20?
Aha, now I get it.
As Olivier's essay moves on, and gradually reverts to the obfuscatory Orwellian prose of the international policy world, I get more and more animated. I mean just who is this "we?" Who is going to tell you you're not allowed to buy euros for your vacation this summer ("capital controls"), tell your bank not to give you a loan ("macro-produential policy"), decide how many billions to siphon from your pocket to the owners of large banks ("recapitalization" "process of sovereign debt restructuring"), not allowed to expand your business in a new country ("macro prudential measures aimed at shaping flows") and so forth? When there even is a "we," unlike most sentences with no subjects, like "the optimal opening of the capital account."
What should be the role of international forums such as the G20?
Aha, now I get it.
Thursday, April 2, 2015
The sources of stock market fluctuations
How much do dividend-growth vs. discount-rate shocks account for stock price variations?
An under-appreciated point occurred to me while preparing for my Coursera class and to comment on Daniel Greewald, Martin Lettau and Sydney Ludvigsson's nice paper "Origin of Stock Market Fluctuations" at the last NBER EFG meeting
The answer is, it depends the horizon and the measure. 100% of the variance of price dividend ratios corresponds to expected return (discount rate) shocks, and none to dividend growth (cash flow) shocks. 50% of the variance of one-year returns corresponds to cashflow shocks. And 100% of long-run price variation corresponds to from cashflow shocks, not expected return shocks. These facts all coexist
I think there is some confusion on the point. If nothing else, this makes for a good problem set question.
The last point is easiest to see just with a plot. Prices and dividends are cointegrated. Prices correspond to dividends and expected returns. Dividends have a unit root, but expected returns are stationary. Over the long run prices will not deviate far from dividends. So 100% of long-enough run price variation must come from dividend variation, not expected returns.
Ok, a little more carefully, with equations.
A quick review:
The most basic VAR for asset returns is \[ \Delta d_{t+1} = b_d \times dp_{t}+\varepsilon_{t+1}^{d} \] \[ dp_{t+1} = \phi \times dp_{t} +\varepsilon_{t+1}^{dp} \] Using only dividend yields dp, dividend growth is basically unforecastable \( b_d \approx 0\) and \( \phi\approx0.94 \) and the shocks are conveniently uncorrelated. The behavior of returns follows from the identity, that you need more dividends or a higher price to get a return, \[ r_{t+1}\approx-\rho dp_{t+1}+dp_{t}+\Delta d_{t+1}% \] (This is the Campbell-Shiller return approximation, with \(\rho \approx 0.96\).) Thus, the implied regression of returns on dividend yields, \[ r_{t+1} = b_r \times dp_{t}+\varepsilon_{t+1}^{r} \] has \(b_r = (1-\rho\phi)+0 = 1-0.96\times0.94 = 0.1\) and a shock negatively correlated with dividend yield shocks and positively correlated with dividend growth shocks.
The impulse response function for this VAR naturally suggests "cashflow" (dividend) and "expected return" shocks, (d/p). (Sorry for recycling old points, but not everyone may know this.)
Three propositions:
But
Why are returns and p/d so different? Current cash flow shocks affect returns. But a shock to dividends, when prices rise at the same time, does not affect the dividend price ratio. (This is the essence of the Campbell-Ammer return decomposition.)
The third proposition is less familiar:
This is related to a point made by Fama and French in their Equity Premium paper. Long run average returns are driven by long run dividend growth plus the average value of the dividend yield. The difference in valuation -- higher prices for given set of dividends -- can affect returns in a sample, as higher prices for a given set of dividends boost returns. But that mechanism can't last. (Avdis and Wachter have a nice recent paper formalizing this point.) It's related to a similar point made often by Bob Shiller: Long run investors should buy stocks for the dividends.
A little more generality as this is the new bit.
\[ p_{t+k}-p_t = dp_{t+k}-dp_t + \sum_{j=1}^{k}\Delta d _{t+j} \] \[ p_{t+k}-p_t = (\phi^{k}-1)dp_t + \sum_{j=1}^{k}\phi^{k-j} \varepsilon^{dp}_{t+j} + \sum_{j=1}^{k} \varepsilon^d _{t+j} \] \[ var(p_{t+k}-p_t) = \frac{(1-\phi^{k})^2}{1-\phi^2} \sigma^2(\varepsilon^{dp}) + \frac{(1-\phi^{2k})}{1-\phi^2} \sigma^2(\varepsilon^{dp}) + k\sigma^2(\varepsilon^d) \] \[var(p_{t+k}-p_t) = 2\frac{(1-\phi^{k})}{1-\phi^2} var(\varepsilon^{dp}_{t+1}) + k var(\varepsilon^d_{t+j})\] So you can see the last bit takes over. It doesn't take over as fast as you might think. Here's a graph using sample values,
At a one year horizon, it's just about 50/50. The dividend shocks eventually take over, at rate 1/k. But at 50 years, it's still about 80/20.
Exercise for the interested reader/finance professor looking for problem set questions: Do the same thing for long horizon returns, \( r_{t+1}+r_{t+2}+...+r_{t+k} \) using \(r_{t+1} = -\rho dp_{t+1} + dp_t + \Delta d_ {t+1} \) It's not so pretty, but you can get a closed form expression here too, and again dividend shocks take over in the long run.
Be forewarned, the long run return has all sorts of pathological properties. But nobody holds assets forever, without eating some of the dividends.
Disclaimer: Notice I have tried to say "associated with" or "correspond to" and not "caused by" here! This is just about facts. The facts have just as easy a "behavioral" interpretation about fads and bubbles in prices as they do a "rationalist" interpretation. Exercise 2: Write the "behavioralist" and then "rationalist" introduction / interpretation of these facts. Hint: they reverse cause and effect about prices and expected returns, and whether people in the market have rational expectations about expected returns.
An under-appreciated point occurred to me while preparing for my Coursera class and to comment on Daniel Greewald, Martin Lettau and Sydney Ludvigsson's nice paper "Origin of Stock Market Fluctuations" at the last NBER EFG meeting
The answer is, it depends the horizon and the measure. 100% of the variance of price dividend ratios corresponds to expected return (discount rate) shocks, and none to dividend growth (cash flow) shocks. 50% of the variance of one-year returns corresponds to cashflow shocks. And 100% of long-run price variation corresponds to from cashflow shocks, not expected return shocks. These facts all coexist
I think there is some confusion on the point. If nothing else, this makes for a good problem set question.
Ok, a little more carefully, with equations.
A quick review:
The most basic VAR for asset returns is \[ \Delta d_{t+1} = b_d \times dp_{t}+\varepsilon_{t+1}^{d} \] \[ dp_{t+1} = \phi \times dp_{t} +\varepsilon_{t+1}^{dp} \] Using only dividend yields dp, dividend growth is basically unforecastable \( b_d \approx 0\) and \( \phi\approx0.94 \) and the shocks are conveniently uncorrelated. The behavior of returns follows from the identity, that you need more dividends or a higher price to get a return, \[ r_{t+1}\approx-\rho dp_{t+1}+dp_{t}+\Delta d_{t+1}% \] (This is the Campbell-Shiller return approximation, with \(\rho \approx 0.96\).) Thus, the implied regression of returns on dividend yields, \[ r_{t+1} = b_r \times dp_{t}+\varepsilon_{t+1}^{r} \] has \(b_r = (1-\rho\phi)+0 = 1-0.96\times0.94 = 0.1\) and a shock negatively correlated with dividend yield shocks and positively correlated with dividend growth shocks.
Three propositions:
- The variance of p/d is 100% risk premiums, 0% cashflow shocks
But
- The variance of returns is 50% due to risk premiums, 50% due to cashflows.
Why are returns and p/d so different? Current cash flow shocks affect returns. But a shock to dividends, when prices rise at the same time, does not affect the dividend price ratio. (This is the essence of the Campbell-Ammer return decomposition.)
The third proposition is less familiar:
- The long-run variance of stock market values (and returns) is 100% due to cash flow shocks and none to expected return or discount rate shocks.
This is related to a point made by Fama and French in their Equity Premium paper. Long run average returns are driven by long run dividend growth plus the average value of the dividend yield. The difference in valuation -- higher prices for given set of dividends -- can affect returns in a sample, as higher prices for a given set of dividends boost returns. But that mechanism can't last. (Avdis and Wachter have a nice recent paper formalizing this point.) It's related to a similar point made often by Bob Shiller: Long run investors should buy stocks for the dividends.
A little more generality as this is the new bit.
\[ p_{t+k}-p_t = dp_{t+k}-dp_t + \sum_{j=1}^{k}\Delta d _{t+j} \] \[ p_{t+k}-p_t = (\phi^{k}-1)dp_t + \sum_{j=1}^{k}\phi^{k-j} \varepsilon^{dp}_{t+j} + \sum_{j=1}^{k} \varepsilon^d _{t+j} \] \[ var(p_{t+k}-p_t) = \frac{(1-\phi^{k})^2}{1-\phi^2} \sigma^2(\varepsilon^{dp}) + \frac{(1-\phi^{2k})}{1-\phi^2} \sigma^2(\varepsilon^{dp}) + k\sigma^2(\varepsilon^d) \] \[var(p_{t+k}-p_t) = 2\frac{(1-\phi^{k})}{1-\phi^2} var(\varepsilon^{dp}_{t+1}) + k var(\varepsilon^d_{t+j})\] So you can see the last bit takes over. It doesn't take over as fast as you might think. Here's a graph using sample values,
At a one year horizon, it's just about 50/50. The dividend shocks eventually take over, at rate 1/k. But at 50 years, it's still about 80/20.
Exercise for the interested reader/finance professor looking for problem set questions: Do the same thing for long horizon returns, \( r_{t+1}+r_{t+2}+...+r_{t+k} \) using \(r_{t+1} = -\rho dp_{t+1} + dp_t + \Delta d_ {t+1} \) It's not so pretty, but you can get a closed form expression here too, and again dividend shocks take over in the long run.
Be forewarned, the long run return has all sorts of pathological properties. But nobody holds assets forever, without eating some of the dividends.
Disclaimer: Notice I have tried to say "associated with" or "correspond to" and not "caused by" here! This is just about facts. The facts have just as easy a "behavioral" interpretation about fads and bubbles in prices as they do a "rationalist" interpretation. Exercise 2: Write the "behavioralist" and then "rationalist" introduction / interpretation of these facts. Hint: they reverse cause and effect about prices and expected returns, and whether people in the market have rational expectations about expected returns.
Monday, March 30, 2015
Adam Davidson on Immigration
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| Illustration by Andrew Rae, source New York Times |
Adam Davidson has a very nice New York Times Magazine article, "Debunking the Myth of the Job-Stealing Immigrant", in favor of "radically open borders."
Here's how a top professional journalist and writer puts together the central argument, so much more cleanly than I can do it:
So why don’t we open up?
The chief logical mistake we make is something called the Lump of Labor Fallacy: the erroneous notion that there is only so much work to be done and that no one can get a job without taking one from someone else. It’s an understandable assumption. After all, with other types of market transactions, when the supply goes up, the price falls. If there were suddenly a whole lot more oranges, we’d expect the price of oranges to fall or the number of oranges that went uneaten to surge.Needless to say the "lump of labor" fallacy pervades politics, policy, and popular discussion on more than immigration. But Adam doesn't bother with the 100 other fallacies.
But immigrants aren’t oranges. It might seem intuitive that when there is an increase in the supply of workers, the ones who were here already will make less money or lose their jobs. Immigrants don’t just increase the supply of labor, though; they simultaneously increase demand for it, using the wages they earn to rent apartments, eat food, get haircuts, buy cellphones. That means there are more jobs building apartments, selling food, giving haircuts and dispatching the trucks that move those phones. Immigrants increase the size of the overall population, which means they increase the size of the economy. Logically, if immigrants were “stealing” jobs, so would every young person leaving school and entering the job market; countries should become poorer as they get larger. In reality, of course, the opposite happens.
Most anti-immigration arguments I hear are variations on the Lump of Labor Fallacy. That immigrant has a job. If he didn’t have that job, somebody else, somebody born here, would have it. This argument is wrong, or at least wildly oversimplified. But it feels so correct, so logical. And it’s not just people like my grandfather making that argument. Our government policy is rooted in it.
The single greatest bit of evidence disproving the Lump of Labor idea comes from research about the Mariel boatlift, a mass migration in 1980 that brought more than 125,000 Cubans to the United States. According to David Card, an economist at the University of California, Berkeley, roughly 45,000 of them were of working age and moved to Miami; in four months, the city’s labor supply increased by 7 percent. Card found that for people already working in Miami, this sudden influx had no measurable impact on wages or employment. His paper was the most important of a series of revolutionary studies that transformed how economists think about immigration. Before, standard economic models held that immigrants cause long-term benefits, but at the cost of short-term pain in the form of lower wages and greater unemployment for natives. But most economists now believe that Card’s findings were correct: Immigrants bring long-term benefits at no measurable short-term cost.
A beautiful stylistic choice: Adam's antagonist is ... his grandfather. That lets Adam have an anonymous, sympathetic antagonist, who is slowly changing his mind in Adam's favor. Adam doesn't have to pick on a particular individual or set of individuals with complex opinions; he doesn't resort to the horrible vague antagonist, "some think;" and he avoids the usual partisan politics and vilification of so much political blogging and editorial writing.
Students: notice concrete not abstract words. "Using the wages they earn to rent apartments, eat food, get haircuts, buy cellphones." Not "Using earned income to demand goods and services."
Thursday, March 26, 2015
A New Structure for U. S. Federal Debt
A new paper by that title, here.
I propose a new structure for U. S. Federal debt. All debt should be perpetual, paying coupons forever with no principal payment. The debt should be composed of the following:
Nominal perpetuities are a nice way to condense the hundreds of outstanding issues into one, which should increase their liquidity a good deal.
Indexed perpetuities are a cleaner way to implement today's tips.
The tax free analysis is maybe the most interesting. I put together a little tax clientele model with some interesting results. No, issuing tax free debt is not a present to rich people. By attracting the high tax clientele back to Treasury debt, we should see lower net (after tax) interest costs to the Treasury.
I have a nice implementation of Treasury swaps too, that might open them up a lot.
Comments welcome. It's a bit long because it responds to a previous round of comments, so if you're bubbling over with what's wrong with the proposals, do check that I haven't already answered your comment.
I propose a new structure for U. S. Federal debt. All debt should be perpetual, paying coupons forever with no principal payment. The debt should be composed of the following:
- Fixed-value, floating-rate debt: Short-term debt has a fixed value of $1.00, and pays a floating rate. It is electronically transferable, and sold in arbitrary denominations. Such debt looks to an investor like a money-market fund, or reserves at the Fed.
- Nominal perpetuities: This debt pays a coupon of $1 per bond, forever.
- Indexed perpetuities: This debt pays a coupon of $1 times the current consumer price index (CPI).
- Tax free: Debt should be sold in a version that is free of all income, estate, capital gains, and other taxes. Ideally, all debt should be tax free.
- Variable coupon: Some if not all long-term debt should allow the government to vary the coupon rate without triggering legal default.
- Swaps: The Treasury should manage the maturity structure of the debt, and the interest rate and inflation exposure of the Federal budget, by transacting in simple swaps among these securities.
Economists have long dreamed of interest-paying money. It fulfills Milton Friedman’s (1969) optimal quantity of money without deflation. Paper money is free to produce, so the economy should be satiated in liquidity...If the Treasury offers what are essentially interest-paying reserves, then we don't have to argue about the size of the Fed's balance sheet, ON RRP, etc.
Our economy invented inside interest-paying electronic money in the form of money market funds, overnight repurchase agreements, and short-term commercial paper, and found it useful. But that money failed, suffering a run in the 2008 financial crisis. Treasury-provided interest-paying electronic money is immune from conventional runs. Money market funds 100% backed by fixed-value Treasury debt cannot suffer a run...
By analogy, in the 19th century, the Treasury provided coins. Banks issued notes. Notes were convenient, being a lot lighter than coins. But there were repeated runs and crises involving bank notes. The U.S. government issued paper money, which might inflate, but cannot suffer conventional default or a run. That money eventually drove out private banknotes, and that source of financial crises was permanently ended. (Crises involving demand deposits did not end, but here the U.S. tried a different policy response, deposit insurance and risk regulation. It has not worked as well.)
In the 21st century, the Treasury has exactly the same natural monopoly in providing default-free and run-free electronically-transferable interest-paying money to private parties. It should do so.
Nominal perpetuities are a nice way to condense the hundreds of outstanding issues into one, which should increase their liquidity a good deal.
Indexed perpetuities are a cleaner way to implement today's tips.
The tax free analysis is maybe the most interesting. I put together a little tax clientele model with some interesting results. No, issuing tax free debt is not a present to rich people. By attracting the high tax clientele back to Treasury debt, we should see lower net (after tax) interest costs to the Treasury.
I have a nice implementation of Treasury swaps too, that might open them up a lot.
Comments welcome. It's a bit long because it responds to a previous round of comments, so if you're bubbling over with what's wrong with the proposals, do check that I haven't already answered your comment.
Tuesday, March 24, 2015
Jumps and diffusions
I learned an interesting continuous time trick recently. The context is a note, "The fragile benefits of endowment destruction" that I wrote with John Campbell, about how to extend our habit model to jumps in consumption. The point here is more interesting than that particular context.
Suppose one time series \(x\), which follows a diffusion, drives another \(y\). In the simplest example, \[dx_t = \sigma dz_t \] \[ dy_t = y_t dx_t. \] In our example, the second equation describes how habits \(y\) respond to consumption \(x\). The same kind of structure might describe how invested wealth \(y\) responds to asset prices \(x\), or how option prices \(y\) respond to stock prices \(x\).
Now, suppose we want to extend the model to handle jumps in \(x\), \[dx_t = \sigma dz_t + dJ_t.\] What do we do about the second equation? \(y_t\) now can jump too. On the right hand side of the second equation, should we use the left limit, the right limit, or something in between?
The usual answer is to use the left limit. We generalize the model to jumps this way: \[dx_t = \sigma dz_t+ dJ_t \] \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] where \(y_{t_{-}}\) denotes the left limit.
That approach has some weird properties however. Suppose \(y_{t_-}=1\), and \(dJ_t=1\). Then \(y_t\) jumps to \(y_t=2\). But suppose there are two jumps of size 1/2, one at time \(t\) and one at time \(t+\varepsilon\). Now \(y\) jumps up to 1.5 after the first jump, and then jumps another \(1.5 \times 0.5 = 0.75\), ending up at \(y_{t+\varepsilon} =2.25\). Two half jumps have a different response than one full jump.
Suppose instead we extend the original model to jumps by taking the jump limit of a continuous process. Imagine that we observe realizations of \(\{dz_t\}\) that get closer and closer to a jump in \(dx_t\), and let's find what happens to \(y_t\). The general solution to the first set of equations is \[ y_{t+\Delta} = y_t e^{(x_{t+\Delta}-x_t - \frac{1}{2}\sigma^2\Delta)}\] so, in the limit \(\Delta \rightarrow 0\) that \(x_t\) takes a jump of size \(dJ_t\), the jump-limit of a continuous movement is \[ dy_{t} \equiv y_t -y_{t_-} = y_{t_-}(e^{dx_{t}}-1) = y_{t_-}\sigma dz_t + y_{t_-}e^{dJ_t}\] rather than \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] So, the left-limit method produced a response to a jump that was different from the response to a continuous process arbitrarily close to a jump. For example, the left-limit approach can produce a negative \(y_t\), but this method, like the diffusion process, cannot fall below zero. This method also produces a response to two half jumps that is the same as the response to a full jump.
As you can see, the difference is whether the state variable \(y_t\) gets to change during the jump. In the left-limit approach, the same \(y_{t_-}\) gets applied to the whole jump. In the continuous-limit version, \(y_t\) implicitly gets to move while the jump in \(x_t\) is moving.
A nonlinear function of a jump is a little novel, but there's nothing wrong with it, and it exists in the continuous time literature. We don't see it that often, because when you're only studying one series it's easier to just change the distribution of the jump process instead. This question occurs when you can see both series x and y and you want to model the relationship between them.
Which is right?
Which extension to jumps is correct? Both are mathematically correct. There is nothing wrong with writing down a model in which the response to a jump is different from the response to continuous movements arbitrarily close to jumps. The answer depends on the economic situation.
For example, consider models with bankruptcy constraints. Agents who can continuously adjust their investments may always avoid bankruptcy in a diffusion setting. If we extend such a model to jumps with the continuous limit approach, implicitly preserving the investor's ability to trade as fast as asset prices change even in the jump limit, we will preserve bankruptcy avoidance in face of a jump in prices. However, if we model portfolio adjustment to jumps with the left-limit generalization, agents may be forced in to bankruptcy for price jumps.
Sometimes, one introduces jumps precisely to model a situation in which prices can move faster than agents can adjust their portfolios, so agents may be forced to bankruptcy. Then the left-limit generalization is correct. But if one wants to extend a model to jumps for other reasons, while avoiding bankruptcy, negative consumption, negative marginal utility (consumption below zero or below habits), violations of budget constraints, feasibility conditions, borrowing constraints, and so forth, then one should choose a generalization in which the jump gives the same result as the continuous limit.
Similarly, when extending option pricing models to jumps, one may want to model the jump in such a way that investors cannot adjust portfolios fast enough. Then the left-limit extension is appropriate, and investors must hold the jump risk. But one may wish to accommodate jumps in asset prices to better fit asset price dynamics while maintaining investor's ability to dynamically hedge. Then the nonlinear extension is appropriate, maintaining the equivalence between jumps and the limiting diffusion.
A little more general treatment
A little more generally, suppose \[ dx_t = g dt + \sigma dz_t \] \[dy_t = \mu(y_t) dt + \lambda(y_t)dx_t.\] We want to add \(dJ_t\) to the first equation. The left-limit approach is \[dy_t = \mu(y_{t_-}) dt + \lambda(y_{t_-})dx_t \] If there is a jump \(dJ_t\), \(y\) moves by an amount \[\frac{1}{\lambda(y_{t_-})}dy_t \equiv \frac{1}{\lambda(y_{t_-})}(y_t - y_{t_-}) = dx_t .\] The limit of a continuous movement solves the differential equation \[\int_{y_{t_-}}^{y_t} \frac{1}{\lambda(\xi)}d\xi = dx_t\] Again, you see the crucial difference, whether the state variable gets to move "during" the jump. We can write this as a differential, by writing the solution to this last differential equation as \[y_t-y_{t_-}=f(x_t-x_{t_-};y_{t_-})\] and then \[dy_t = \mu(y_{t_-}) dt + f(dx_t;y_{t_-})=\mu(y_{t_-}) dt + \lambda(y_{t_-})\sigma dz_t+f(dJ_t;y_{t_-})\]
So, you don't have to extend the model to jumps with the left-limit approach, and you don't have to swallow the idea that a jump has a different response than an arbitrarily close continuous-sample-path movement. The last equation shows you how to modify the model to include jumps in a way that preserves the property that the jump has the same effect as its continuous limit.
The point
Why a blog post on this? I asked a few continuous-time gurus, and none of them had seen this issue before. If someone knows where this has all been worked out with proper is dotted and ts crossed, I would like to know and cite it properly. (I would think the literature on option pricing with jumps had done it, but I couldn't find a reference.) Or perhaps it hasn't been done and someone wants to do it. I'm not good enough at the technical aspects of continuous time to write this with the right precision and generality.
And it's a cool trick that may be useful to someone outside of the narrow context that we had for it.
Update:
Perhaps the right application is stock prices and option prices. When stock prices jump, someone must have studied the case that option prices move by the same amount the Black-Scholes formula gives for the same size stock price movement. Does anyone have a citation to that case?
Suppose one time series \(x\), which follows a diffusion, drives another \(y\). In the simplest example, \[dx_t = \sigma dz_t \] \[ dy_t = y_t dx_t. \] In our example, the second equation describes how habits \(y\) respond to consumption \(x\). The same kind of structure might describe how invested wealth \(y\) responds to asset prices \(x\), or how option prices \(y\) respond to stock prices \(x\).
Now, suppose we want to extend the model to handle jumps in \(x\), \[dx_t = \sigma dz_t + dJ_t.\] What do we do about the second equation? \(y_t\) now can jump too. On the right hand side of the second equation, should we use the left limit, the right limit, or something in between?
The usual answer is to use the left limit. We generalize the model to jumps this way: \[dx_t = \sigma dz_t+ dJ_t \] \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] where \(y_{t_{-}}\) denotes the left limit.
That approach has some weird properties however. Suppose \(y_{t_-}=1\), and \(dJ_t=1\). Then \(y_t\) jumps to \(y_t=2\). But suppose there are two jumps of size 1/2, one at time \(t\) and one at time \(t+\varepsilon\). Now \(y\) jumps up to 1.5 after the first jump, and then jumps another \(1.5 \times 0.5 = 0.75\), ending up at \(y_{t+\varepsilon} =2.25\). Two half jumps have a different response than one full jump.
Suppose instead we extend the original model to jumps by taking the jump limit of a continuous process. Imagine that we observe realizations of \(\{dz_t\}\) that get closer and closer to a jump in \(dx_t\), and let's find what happens to \(y_t\). The general solution to the first set of equations is \[ y_{t+\Delta} = y_t e^{(x_{t+\Delta}-x_t - \frac{1}{2}\sigma^2\Delta)}\] so, in the limit \(\Delta \rightarrow 0\) that \(x_t\) takes a jump of size \(dJ_t\), the jump-limit of a continuous movement is \[ dy_{t} \equiv y_t -y_{t_-} = y_{t_-}(e^{dx_{t}}-1) = y_{t_-}\sigma dz_t + y_{t_-}e^{dJ_t}\] rather than \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] So, the left-limit method produced a response to a jump that was different from the response to a continuous process arbitrarily close to a jump. For example, the left-limit approach can produce a negative \(y_t\), but this method, like the diffusion process, cannot fall below zero. This method also produces a response to two half jumps that is the same as the response to a full jump.
As you can see, the difference is whether the state variable \(y_t\) gets to change during the jump. In the left-limit approach, the same \(y_{t_-}\) gets applied to the whole jump. In the continuous-limit version, \(y_t\) implicitly gets to move while the jump in \(x_t\) is moving.
A nonlinear function of a jump is a little novel, but there's nothing wrong with it, and it exists in the continuous time literature. We don't see it that often, because when you're only studying one series it's easier to just change the distribution of the jump process instead. This question occurs when you can see both series x and y and you want to model the relationship between them.
Which is right?
Which extension to jumps is correct? Both are mathematically correct. There is nothing wrong with writing down a model in which the response to a jump is different from the response to continuous movements arbitrarily close to jumps. The answer depends on the economic situation.
For example, consider models with bankruptcy constraints. Agents who can continuously adjust their investments may always avoid bankruptcy in a diffusion setting. If we extend such a model to jumps with the continuous limit approach, implicitly preserving the investor's ability to trade as fast as asset prices change even in the jump limit, we will preserve bankruptcy avoidance in face of a jump in prices. However, if we model portfolio adjustment to jumps with the left-limit generalization, agents may be forced in to bankruptcy for price jumps.
Sometimes, one introduces jumps precisely to model a situation in which prices can move faster than agents can adjust their portfolios, so agents may be forced to bankruptcy. Then the left-limit generalization is correct. But if one wants to extend a model to jumps for other reasons, while avoiding bankruptcy, negative consumption, negative marginal utility (consumption below zero or below habits), violations of budget constraints, feasibility conditions, borrowing constraints, and so forth, then one should choose a generalization in which the jump gives the same result as the continuous limit.
Similarly, when extending option pricing models to jumps, one may want to model the jump in such a way that investors cannot adjust portfolios fast enough. Then the left-limit extension is appropriate, and investors must hold the jump risk. But one may wish to accommodate jumps in asset prices to better fit asset price dynamics while maintaining investor's ability to dynamically hedge. Then the nonlinear extension is appropriate, maintaining the equivalence between jumps and the limiting diffusion.
A little more general treatment
A little more generally, suppose \[ dx_t = g dt + \sigma dz_t \] \[dy_t = \mu(y_t) dt + \lambda(y_t)dx_t.\] We want to add \(dJ_t\) to the first equation. The left-limit approach is \[dy_t = \mu(y_{t_-}) dt + \lambda(y_{t_-})dx_t \] If there is a jump \(dJ_t\), \(y\) moves by an amount \[\frac{1}{\lambda(y_{t_-})}dy_t \equiv \frac{1}{\lambda(y_{t_-})}(y_t - y_{t_-}) = dx_t .\] The limit of a continuous movement solves the differential equation \[\int_{y_{t_-}}^{y_t} \frac{1}{\lambda(\xi)}d\xi = dx_t\] Again, you see the crucial difference, whether the state variable gets to move "during" the jump. We can write this as a differential, by writing the solution to this last differential equation as \[y_t-y_{t_-}=f(x_t-x_{t_-};y_{t_-})\] and then \[dy_t = \mu(y_{t_-}) dt + f(dx_t;y_{t_-})=\mu(y_{t_-}) dt + \lambda(y_{t_-})\sigma dz_t+f(dJ_t;y_{t_-})\]
So, you don't have to extend the model to jumps with the left-limit approach, and you don't have to swallow the idea that a jump has a different response than an arbitrarily close continuous-sample-path movement. The last equation shows you how to modify the model to include jumps in a way that preserves the property that the jump has the same effect as its continuous limit.
The point
Why a blog post on this? I asked a few continuous-time gurus, and none of them had seen this issue before. If someone knows where this has all been worked out with proper is dotted and ts crossed, I would like to know and cite it properly. (I would think the literature on option pricing with jumps had done it, but I couldn't find a reference.) Or perhaps it hasn't been done and someone wants to do it. I'm not good enough at the technical aspects of continuous time to write this with the right precision and generality.
And it's a cool trick that may be useful to someone outside of the narrow context that we had for it.
Update:
Perhaps the right application is stock prices and option prices. When stock prices jump, someone must have studied the case that option prices move by the same amount the Black-Scholes formula gives for the same size stock price movement. Does anyone have a citation to that case?
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