I have a puzzle for you.

Last week, Prof. Krugman linked to a post of mine from last summer on inflation swaps. This reminded me of a mistake I made that I have been meaning to correct for some time.

…by examining the market rate for inflation swaps (â€œyâ€, in the notation above), we obtain the pure, unadulterated market expectation for the rate of inflation for the next N years.

That is not quite true, and explaining how and why is the point of this post.

Meet Joseph Haubrich:

(Click on the picture for a larger photo. Would you believe I own the same tie?)

Dr. Haubrich is a senior researcher at the Cleveland Fed. You may remember him from such classics as “Swaps and the Swaps Yield Curve”, which I cited some time ago. More recently, he co-authored “A New Approach to Gauging Inflation Expectations” and “Inflation: Noise, Risk, and Expectations”, where he talks about using market data to model inflation expectations.

Why should that be difficult? If inflation swaps are just a straight bet on the CPI, why can we not gauge the market’s inflation expectations directly from the market prices of such swaps?

Suppose I give you a simple lottery ticket that works like this: Tomorrow morning, I will flip a fair coin. If the coin comes up heads, I will give $1 to the holder of the ticket. If the coin comes up tails, I will give nothing to the holder of the ticket. Suppose you decide to sell that ticket now. How much do you think you could get for it on the open market?

The answer is “somewhat less than 50 cents”. Rational people are *risk averse*, which simply means they prefer a certain 50 cents to a random payment that is only worth 50 cents on average. (True, irrational people overpay for lottery tickets all the time. But that is a topic for another post.)

The difference between 50 cents and what the market will pay for my lottery ticket is called the *risk premium* associated with the ticket. If I were inventing the term, I would probably have called it a “risk discount”, because I am always thinking about the downside of things… But “risk premium” it is.

In an inflation swap, one party is promising to make a known, fixed payment in the future, while the other party is promising to make a payment based on unknown future events (specifically, the future CPI). Therefore, because of risk aversion, the going market price for such fixed payments — called the “break-even rate” — should understate the market’s expectation for the future rate of inflation.

How did I do, Dr. Haubrich?

The problem is that the break-even rate includes a risk premium. This risk premium means that

the break-even rate overstates expected inflationand that changes in the break-even rate might arise from changes in the risk premium, not changes in expected inflation.

Hm. Dr. Haubrich says I have it backwards. And he does not look like the kind of person with whom I want to argue.

To re-cap the argument:

- In an inflation swap, one party offers a fixed future payment, which is known today. The other party offers a future payment based on the future CPI, which is unknown today.
- When two rational investors agree to trade a certain payment for an uncertain payment, the certain payment will always be less than the expected value of the uncertain payment.
- Therefore, the party paying the fixed side of the inflation swap will offer less than his/her expected rate of inflation.

Is Dr. Haubrich wrong? Or is one of the above statements wrong? Or is some unstated assumption wrong?

The answer… Next time.

(Hint: When is a fixed payment not a fixed payment?)

I have the impression that your argument does not limit itself to the sole case of the inflations swaps. Let us consider the case of a Forward Rate Agreement (FRA).

To re-cap the argument:

1. In a LIBOR FRA, one party offers a fixed future payment, which is known today. The other party offers a future payment based on the future LIBOR, which is unknown today. (True, the two payments will be actually made at the beginning of the LIBOR time span instead of at its end, and this will be accounted for by a discount based on that LIBOR which is not yet known, but, from an actuarial calculus point of view, it is as if the two payments were made at the maturity of the LIBOR and without discount : so one fixed versus one variable)

2. When two rational investors agree to trade a certain payment for an uncertain payment, the certain payment will always be less than the expected value of the uncertain payment.

3. Therefore, the party paying the fixed side of the LIBOR FRA will offer less than his/her expected LIBOR rate.

But then, it applies also to LIBOR swaps, as those can be reconstructed by the means of a sequence of LIBOR FRAs. So the argument leads us to claim that LIBOR swaps rates underestimate the expectations of future LIBOR.

But then, it also applies to OIS, so the argument leads us to claim that the OIS rates actually underestimate the expectations of future effective fed funds rates.

Is it plausible ? And what could be the order of magnitude the underestimation ? My answerâ€¦ next time, too.

I guess it depends if “fixed” is in real or nominal terms. “Fixed” has so many connotations…

I’ll take a stab.

You’re example about the coin refers to them *winning* $1. That is not always the case, what about a (forced game) of lose $1 or nothing. What is the fair value of that bet for a slightly risk-averse person? My guess, *pay* the host about 51 cents. Assumption #2: “the certain payment will always be less than the expected value of the uncertain payment” It can be a certain payment or receivable. I like to think of swaps as paying/receiving the difference between the fixed and floating so higher inflation that agreed fixed rate means *paying* on the floating cpi side, while lower inflation means *receiving* cash.

The “risk premium” can be thought of as a volatility adjustment. ie, there has been great swing in inflation expectations recently meaning this risk premium has increased. You will notice in figure #2 of the 2nd link (10-year inflation risk premium chart) that this risk premium has increased over during the financial crisis and there was a bit of a general decrease 2001 to 2008. Know what else behaved like that … the VIX.

To use the same example, what if, the payment was unknown (not a known $1 received or paid) but was now expected to be more like $20 received or paid. The difference between expected actual expectation and risk-averse figure will increase from a couple cents to a couple bucks -> *increase in the risk premium*.

I also think that risk premium in regards to fed actions is generally for higher inflation. ie, most foreign investors believe the fed will err on the side of caution when combating deflation resulting in higher inflation rates (remember those Chinese university students laughing at Geitner when he said US-$ assets are safe – http://www.reuters.com/article/idUSPEK14475620090601). So while, I think inflation will average 0.5%, I think there is some *upside* risk (due to fed’s general policy keeping rates too low – maybe because of its dual mandate of full employment?????) to my estimate b/c of about 15 bps so, as a risk-averse investor I think its fair to receive fixed 66bp. The Cleveland Fed has attempted to back out this 15 bps to get the true inflation expectation of 50 bps. I think we’ve found the cost of “Greenspan’s put.” I think this is why Haubrich generalizes to the case of *positive* inflation risk premium.

Think of how often inflation has been above the fed’s target (cumulatively) and how many times its been below.

Now if Volcker were back in charge … maybe the sign on that risk premium shifts to a negative

ps, in the3rd last paragraph, my example with numbers: 50bp + 15bp is 65 bps, not 66, sorry.

Risk premia could go either way in this context, although the observed nominal term yield (e.g., the 5 year Treasury zero coupon yield) already embeds the same risk premium.

But you also have to worry about the correlation between the real interest rate and the inflation rate. And I think there’s also a convexity effect to worry about as well, namely that the expectation of the compounded inflation rate (which appears in the calculation of the inflation leg) is not the same as the compounded expected inflation rate (which appears in the fixed leg).

Several moving parts, and it’s not obvious – to me anyway – that they should net out to 0.