# short-term break-even inflation rate .short-term break-even inflation rate ... final remarks •we

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• Short-Term Break-Even Inflation RateGustavo Araujo and Jos Vicente

Banco Central do Brasil Depep

So Paulo, August 11, 2017

• The views expressed in this work are those of the authors and do not necessarily reflect those of the Banco Central do Brasil or its members.

• Motivation

Market prices reflect market expectations

Some advantages over surveys: real time updates, no need for incentives, ...

How to extract inflation expectations from market prices?

Look at the differential between nominal and real interest rate (BEIR: break-even inflation rate)

• Motivation

In Brazil, financial institutions have been trading inflation in the short-term through the NTN-Bs (inflation-indexed bonds) and the DAP (the IPCA Coupon futures contract)

For example, with a long position in DI1 futures contracts and a short position in DAP contracts, an investor assume a long position in inflation.

Therefore, the Brazilian financial market has information about inflation.

• Problems to Extract the BEIR

There are two problems in the construction of the BEIR curve (for example, in Anbimas calculation):

1) Imperfect Indexing: The face value of the NTN-Bs is adjusted on the fifteenth day of each month using the inflation rate of the previous month.

2) ANBIMA's interpolation methodology (based on Svensson, 1995) does not consider the inflation seasonality. Svensson only considers 2 curvatures, but there must be

several curvatures due to the seasonality.

• Problem 1: Imperfect Indexing

The price of any zero-coupon bond (P) at t with maturity in T is equal to its face value (FV) in Tdiscounted by the nominal interest rate (R):

The same applies to an inflation-linked bond:

)1( ,Tt

Tt

R

FVP

) (1)1(

) (1

,,

15,15

TtTt

Ttt

tr

FVP

• Problem 1: Imperfect Indexing

Therefore, the IPCA coupon and the real interest rate are not equal.

) (1)1(

) (1

) )(1 (1)1(

) )(1 (1

,15,

,15

,1515,,

15,,15

TTTt

ttt

TTTtTt

Ttttt

tr

FV

r

FVP

The IPCA coupon (C_IPCA ) is the solution of the following equation:

Thus,)_1(

,Tt

tt

IPCAC

FVP

) (1

) (1)1(_1

,15

,15,

,

tt

TTTt

Tt

rIPCAC

• Example Imperfect Indexing

Month M

Month M, 15th

Month M+1

Month M+1, 15th

Today Maturity

R=14.25% per year = 1.1163% for the 30-day periodE() = 0.40% for the 30-day period

Therefore:

r = 0.7135% for the 30-day period = 8.91% per year

• Example Imperfect Indexing

Month M

15/M

Month M+1

15/M+1

Today Maturity

E() = 0.25% E() = 0.16%

year)per 7.74%( periodday -30afor 0.6231%1.25%)0 (1

.16%)0 (1)1(_

rIPCAC

%49.01.6231%)0 (1

.1163%)1 (1

BEIR The Expectation of Inflation is 0.40% !

The real interest rate is 8.91% per year

• Methodology

15,15** 1 TttT BEIRFVFV

t* is the date of the last known FV FVt* is the last known FV and considers inflation up to

15 days earlier

For a Zero-Coupon Bond:

)1(

1

,

15,15**

Tt

Ttt

tR

BEIRFVP

1)1(

*

,

15,15*

t

Ttt

TtFV

RPBEIR

• Example (December 7, 2016)

1)PR (1

16/16/11

12/7/1617/15/516/7/12

2017/30/42016/1/11

FV

BEIR

Assume there is a zero-coupon NTN-B.

The shortest NTN-B expires on 5/15/2017

The last released IPCA refers to October (the IPCA from November was not released)

The last known FV is from 11/16/2016

R can be extracted from DI1 futures contracts or fixed-rate bonds.

Although the trading date of the bond is 12/07/2016, we extract the BEIR from November 2016 to April 2017

• Now assume that the NTN-B is not a zero-coupon bond. The NTN-B does not have intermediary coupons anymore.

1%)956301,21(

)PR (1

16/16/11

12/7/1617/15/516/7/12

2017/30/42016/1/11

FV

BEIR

Example (December 7, 2016)

• Example (November 7, 2016)

The shortest NTN-B expires on 5/15/2017 and pays coupon on 11/16/2016

The FV10/15/2016 is known (The October IPCA was not released yet).

P11/7/2016 of a zero-coupon bond can be calculated by bootstrapping through the DAP contract and the NTN-B with coupons (the DAP contract is essential for the methodology)

1)PR (1

16/15/10

11/7/1617/15/516/7/11

2017/30/42016/1/10

FV

BEIR

• Problem 2 Seasonality

To compute monthly inflation, we want to consider not only the seasonality, but also the current information that may influence short-term inflation.

We use the distribution implicit in the forecasts of the Focus Survey (TOP 5 Short-Term).

*Focus is a survey conducted by the Central Bank of Brazil among professional forecasters

• Empirical Exercises

Sample: from November 2014 to March 2017 We compared the errors (in relation to the observed IPCA) of the BEIR with

the errors of the Focus forecasts (Aggregate Median and of the Short-term TOP 5).

We did not calculate the BEIR from 5/2015 to 1/2016, because in this period there was no NTN-B without intermediary payments and the DAP contract still had no liquidity.

We did six exercises:

(i) the first Friday closest to the IPCA release;

(ii) the second Friday closest to the IPCA release;

(iii) the fourth Friday closest to the IPCA release;

(iv) the eighth Friday closest to the IPCA release;

(v) the first critical date closest to the IPCA release; and

(vi) the second critical date closest to the IPCA release.

• Empirical Exercise

Matched-pair t-test for the difference between the means of absolute errors in relation to the observed IPCA.

BEIR minus Top 5 ST BEIR minus Median

Forecast Date Observations Mean Difference t-statistic Mean Difference t-statistic

1st Friday closest to the IPCA release 20 0.003% 0.33 -0.003% -0.30

2nd Friday closest to the IPCA release 20 -0.002% -0.15 -0.007% -0.66

4th Friday closest to the IPCA release 17 -0.005% -0.38 -0.028% -1.99

8th Friday closest to the IPCA release 14 -0.009% -0.76 -0.019% -1.45

1st critical date closest to the IPCA release 20 0.010% 1.11 0.005% 0.52

2nd critical date closest to the IPCA release 16 0.007% 0.72 -0.025% -1.93

• 0.00%

0.05%

0.10%

0.15%

0.20%

0.25%

0.30%

0.35%

0.40%

0.45%

0.50%

11/16 12/1 12/16 12/31 1/15 1/30 2/14 3/1 3/16 3/31

Inflation Forecasts (Focus) and Break-even Inflation Rate - March 2017

Focus Top 5 Short Term Focus Median BEIR

On March 28, 2017, Aneel approved a reduction of 10% (in average) in electric energy tariff, valid only for April (reimbursement for an extra charge that occurred in the past), negatively impacting the expectation of the IPCA for that month.

The BEIR of April responded immediately, dropping by 21 basis points from March 27 to March 28.

The median FOCUS forecast fell by only 1 basis point in the same period.

• Final Remarks

We develop a parsimonious methodology for the extraction of BEIR that addresses two problems (which are more relevant in the short term):

Imperfect Indexing; and Seasonality of inflation.

The short-term BEIR is competitive in relation to Focus

The BEIR is updated continuously

• For more details, see theWP 460

in the BC website

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