Inflation Derivatives: A Bird's Eye View
This article provides a brief overview of inflation, discusses inflation derivatives, explores an analytical exercise involving the calculation of yield on UK bonds and shows a pricing comparison in output between FINCAD® XL and Bloomberg Finance LP*. FINCAD XL v9.1 also offers analytics coverage for inflation curve building and inflation swaps.
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Brief Overview of Inflation
Inflation is defined as the rate of change of prices over a unit of time. More succinctly,
where, P and π are price levels and resultant inflation rates at the indexed time points.
The primary reason inflation continues to be an attractive tool for policymakers is the potential tradeoff between inflation and unemployment - as evidenced in the statistical relationship called the Phillips curve. In this simple world, as inflation increases unemployment decreases and vice-versa. A standard reasoning is that the amount of credit in the economy increases (whether due to money supply levels being relaxed or looser rate-targeting regimes), there is a lagged increase in the price levels due to increased aggregate demand. It would then seem that central banking systems which increase inflationary levels will have lower levels of unemployment and presumably high output. But, casual empiricism reveals otherwise. A more extensive scrutiny of the purported tradeoff reveals that several leaps of faith need to be made in order to sustain belief in such a simplistic model of inflation and real output (via employment).
In the post-Lucas (1972) world, it is widely expected that inflation itself can be broken down into two components:
πt = πtexpected + πtunexpected
In such a view of inflation, πtexpected refers to the component of inflation that is expected by the participants in the marketplace. As the old adage has it about 'fooling all the people all the time', πtexpected contributes to no changes in the aggregate output, because agents in an economy optimally readjust their outputs. (A more technical term is that the production function has a zero-homogeneity condition.) It is the πtunexpected that produces a transient Phillips curve like tradeoff.
The ability to use inflation as an output increasing mechanism is intimately tied to the central bank's ability to ''surprise'' the market. Connected to this ability to surprise is the credibility factor attributed to the central bank's independence from the fiscal authority. Standard literature on game theory and behavioral economics abound with examples of subtle forms of dynamic and time inconsistency (i.e., the central banker's objective criterion is likely to change from period t - k to present period t -- thus the inconsistency).
In essence, it is the unexpected part of inflation that one seeks to hedge with derivatives inflation. In reality, of course, such a neat division is hardly possible - and most instruments are indexed to a general inflation indicator itself.
Inflation Derivatives
Indexed debt issuance has been known at least since 1742, when Massachusetts issued debt on silver on the London Exchange. In 1780, during the American War of Independence, wages of soldiers were indexed on a basket of goods (five bushels of corn, sixty-eight and four seventh pounds of beef, ten pounds of sheep wool and sixteen pounds of sole leather) (Deacon and Derry 1998). Analogously, a debt issued on the average levels of inflation observed in the economy is a relatively straightforward issue. The earliest issuer of inflation linked gilts was in the UK (1981), and in more recent times the US Treasury's TIPS program (1998). The spiraling market share of inflation-linked debt in the post 2003 era can be traced in parts to fears regarding loose monetary policy pursued by the central banks, crude and energy price levels and the reliance of pension fund holders on real-returns as cohorts of baby-boomer generation begin their retirement.
With the rigorous modeling of inflation-linked derivatives, at least, the following theoretical and market-specific issues which need to be addressed (Hughston 1998, 2004):
Theoretical:
- Non-existence of index-linked bonds in certain states of the world - thus the idea of sequential trades need to be addressed
- Illiquidity in the market of bonds of certain specifications
Mechanistic:
- The existence of coupon bonds indexed to inflation, as opposed to zero coupon bonds solely.
- Most payoffs on the termination date are pre-fixed months prior to the maturity date itself. The exact number of months changes between jurisdictions.
Standard ways to trade inflation includes portfolios with overweighted amounts of nominal bonds and underweighted amounts of inflation bonds - thus implicitly taking position on the inflation rates (and thus underlying rates as issued by the banks). Inflation swaps, unlike inflation indexed bonds, allows the user to create customized trades that enable them to take positions on the inflation itself. A clear bottleneck in trading inflation swaps in the past has been the liquidity constraints when initiating or unwinding positions in customized deals.
A FINCAD Calculation of Yield on UK Gilts
There are two types of yield for UK gilts.
1) DMO Yield (debt management office): This is calculated without regard to inflation indices. This takes the market price and calculates the yield assuming no inflation (just like a normal bond). In this sense, one could call it the “real yield".
The above equation is solved for using a standard root finding algorithm - after taking into account the conventions etc.
2) The Nominal Yield: The calculation adjusts the cash flows in the future according to your inflation assumptions, as usual, the algorithm finds out what yield is needed to get the current market price. In this case, the futures values are adjusted for inflation.
Here, IN is the inflation index defined on the DMO website.
3) The reason why changing the inflation results in changing nominal yield is the adjustment mechanism mentioned above. The inflation index is the ratio of the projected Retail Price Index (RPI) to the current RPI. The projections are calculated by using the assumed annual inflation via:
Using FINCAD to Benchmark
One can use FINCAD's inflation-linked, fixed income analytics to calculate price, yield and risk statistics.
FINCAD risk analytics allows the user to use pre-specified templates to price inflation linked bonds, calculate yields and calculate explicit cash flows for Australia, Canada, France, Italy, South Africa, Sweden, United Kingdom and United States. A more general template allows users to value for other countries as well.
For e.g., for UK inflation-linked bonds with a 3-month indexation lag, one can price aaBond_UK_Index_3MLag_p( ) and aaBond_UK_Index_3MLag_y( ).
References:
Deacon, A and Derry, A. 1998. Inflation-Indexed Securities (Prentice Hall Europe)
Hughston, L.P. 1998. Inflation Derivatives, Unpublished Manuscript.
Lucas, Robert E., Jr. 1972. Expectations and the neutrality of money. Journal of Economic Theory, 4 (April): 103-24.
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