khrishnamoorthysooben fixed income...

Fitting linkers into a portfolio

Khrishnamoorthy SOOBEN

Fixed Income Strategist

+44 (0)20 7676 7713

1

Contents

� Efficient frontier analysis

� Using historical data

� Forward looking approach: bet on expected return, volatility and

correlation

� Finding value in linkers – common strategies

� Linkers vs nominal bonds

2

Historical return and risk

Source: Calculations from SG Fixed Income & Forex Research

� Definitions:

� Nominal bonds and linkers data are computed from total return Barclays Capital Euro Indices (France). Money market returns are based on 1 month Euribor rates. Finally, equity returns are derived from a total return MSCI Equity index for France. Data are from 1999.

� Return: annualised average monthly total return

� Risk: annualised standard deviation of monthly total returns

Money Market Nominal bonds Linkers Equities

Historical Return 3.1% 4.3% 5.6% 9.3%

Historical Risk 0.3% 3.3% 4.2% 17.6%

Correlations Money Market Nominal bonds Linkers Equities

Money Market 1 0.12 0.03 -0.24

Nominal bonds 0.12 1 0.77 -0.35

Linkers 0.03 0.77 1 -0.28

Equities -0.24 -0.35 -0.28 1

3

Ex-post efficient frontier

� We minimise the portfolio risk for a

given return, using historical

returns, risks and correlation.

Efficient frontier derived from

historical performance

Source: SG Fixed Income & Forex Research

Efficient Frontier

3%

4%

5%

6%

7%

8%

9%

10%

0% 3% 6% 9% 12% 15% 18%Risk

Portfolio Return

4

Ex-post portfolio components

� What does 1999 - 2007 history tell us, based on our chosen assets and indices?

� Nominal bonds may not offer the worst return

AND risk, but positive correlation with other

asset classes implies a 0% weight throughout

� Optimal weight of equity increases steadily for

higher risk and return portfolios, thanks to high

historical return and sufficiently positive

correlation vs other assets

� Except for very low return levels, ex-post

portfolios are essentially made up of linkers

and equity

� Conclusion: Historical analysis is, by definition, backward looking => useful but potentially irrelevant. A forward looking approach is more appropriate.

Portfolio components derived from

historical performance


Portfolio Components

0%

20%

40%

60%

80%

100%

0% 3% 6% 9% 12% 15% 18%

Risk

Weights

Money Market Nominal BondsLinkers Equities

5

Anticipating returns, risks and correlations


� This set of return, risk and correlation is based on a combined historical and theoretical analysis.

� Assumption 1: Equities show negative correlation vs. nominal bonds (-0.1%) and linkers (-0.3%)

� Assumption 2: Equities show positive correlation vs. nominal bonds (+0.1%) and linkers (+0.3%)

Money Market Nominal bonds Linkers Equities

Expected Return 3.5% 4.8% 4.5% 7.5%

Expected Risk 0.5% 7.0% 5.0% 18.0%

Expected Correlations Money Market Nominal bonds Linkers Equities

Money Market 1 0.15 0.05 -0.15

Nominal bonds 0.15 1 0.55 -0.1/+0.1 Correlation

Linkers 0.05 0.55 1 -0.3/+0.3 assumptions

Equities -0.15 -0.1/+0.1 -0.3/+0.3 1

6

Correlation assumptions are crucial

� We now minimise risk for a given

level of return, using expected

return, risk and correlation under

two different assumptions.

� The grey frontier is built assuming negative

correlation between stocks on the one hand,

nominal bonds and inflation-linked bonds

(ILBs) on the other.

� The red frontier is built assuming positive

correlation.

� As expected the grey frontier is the highest,

since the negative correlation increases the

diversification potential.

Efficient frontiers under different

correlation assumptions


3%

4%

5%

6%

7%

8%

0% 5% 10% 15% 20%

-ve Corr Equities vs Linkers & Nominal Bonds

+ve Corr Equities vs Linkers & Nominal Bonds

Expected Return

Risk

7

Moving the efficient frontier - Assumption 1

� We work under assumption 1:

� negative correlation between stocks on the

one hand, nominal bonds and inflation-linked

bonds (ILBs) on the other.

� ILBs exhibiting higher de-correlation to stocks

than nominal bonds to stocks.

� Adding ILBs to the portfolio moves the

efficient frontier up, except for extreme risk

levels.

� Under that set of risk/return assumptions, ILBs

win a very large weighting in the portfolio.

Linkers provide diversification benefits

under assumption 1


3%

4%

5%

6%

7%

8%

0% 5% 10% 15% 20%

Linkers included Linkers excluded

Expected Return

Risk

8

Portfolio weights – Assumption 1

� The first set of assumption delivers a very high

weighting to ILBs for portfolio return volatility

around 5%.

� Such weighting may be difficult to achieve on big

portfolios because of the relatively small size of

the ILB market.

� One would need to run optimization under an

additional constraint in that case (capping the

weighting of ILBs).

Linkers win a significant share of the

portfolio under assumption 1



0%

20%

40%

60%

80%

100%

0% 5% 10% 15% 20%

Risk

Weights


9

Moving the efficient frontier - Assumption 2

� We work under assumption 2:

� positive correlation between stocks on the one

hand, nominal bonds and inflation-linked

bonds (ILBs) on the other.

� ILBs exhibiting higher correlation to stocks

than nominal bonds to stocks.

� Adding ILBs to the portfolio hardly moves the

efficient frontier. Yet ILBs win a significant

weighting in portfolios showing a limited risk.

That weight peaks at 31%.

Negligible diversification benefit from

linkers under assumption 2


3%

4%

5%

6%

7%

8%

0% 5% 10% 15% 20%

Linkers included Linkers excluded

Expected Return

Risk

10

Portfolio weights - Assumption 2

� Although ILBs do not greatly

improve the risk/return outlook

under that set of assumptions,

they still win a significant

weighting.

Linkers still win a significant share of the

portfolio under assumption 2



0%

20%

40%

60%

80%

100%

0% 5% 10% 15% 20%

Risk

Weights


11

Stocks vs. bonds and ILBs (1)

� We call Rs, Ri and Rr the nominal total return of stocks, bonds and ILBs (real bonds). The above results partially depend on where you fix ϕϕϕϕ(Rs, Ri) relatively to ϕϕϕϕ(Rs, Rr).

� ϕ(Rs, Ri) = Cov (Rs, Ri) / (σRs * σRi)

� ϕ(Rs, Rr) = Cov (Rs, Rr) / (σRs * σRr)

� Cov (Rs,i) = Cov (Rs, r+BEIR) = Cov (Rs, r) + Cov (Rs, BEIR)

� Assuming Cov (Rs, BEIR) < 0, then Cov (Rs,i) < Cov (Rs, r).

� As in most cases σi > σr then ϕ(Rs, i) < ϕ(Rs, r)

� This also means ϕϕϕϕ(Rs, Ri) > ϕϕϕϕ(Rs, Rr)

� In other words, if stock returns are negatively correlated to inflation expectations, then ILBs show a smaller positive correlation or larger negative correlation to stocks than nominal bonds do. And that is good news (ILBs add to diversification).

• Eg: ϕ(Rs, Ri) = 0.25, ϕ(Rs, Rr) = 0.15 or ϕ(Rs, Ri) = -0.15, ϕ(Rs, Rr) = -0.25

12


� The result above is intuitive. If both stocks and nominal bonds react badly to increasing inflation expectations (or to a risinginflation risk premium), then that will tend to increase their correlation - ILBs will increase diversification.

� However, it is not clear whether stocks will be negatively correlated to inflation.

� Correlation has been negative in the last twenty years, but not in the

last seven years.

� There is no definite theoretical answer either. The falling inflation

trends have generally been good news to the economy and equities

in the last 20 years. Yet an excessive fall in inflation, potentially

leading to deflation, would clearly be bad news for stocks -

correlation between inflation and stock market returns would

become positive in that case.

13


� Deciding ex-ante for the nominal and relative levels of ϕϕϕϕ(Rs, Ri) and

ϕϕϕϕ(Rs, Rr) is one difficult task. In the above calculations, we used

� one friendly example ( ϕ(Rs, Ri) > ϕ(Rs, Rr) )� ILBs clearly add

diversification to the portfolio

� and one unfriendly example � ILBs still gain a significant weighting for

limited amount of risks.

� NB1: Deciding for ex-ante correlation is an important task, but no more than

deciding for ex-ante returns and volatility. Assessing relative valuation of

stocks and bonds is a key step.

� NB2: We run calculations in the nominal world. Yet it makes sense,

especially for pension funds, to work in the real world. Doing so would

generally increase the volatility of money market instruments relatively to

volatility of ILBs, and increase the weighting of ILBs for low levels of

portfolio risk.

Finding value in linkers

Common strategies and technical

considerations

15

Common linker strategies

� Outright positions in linkers: going long or short an inflation

linked bond => position in real yield space

� Breakeven positions: going long or short an inflation linked

bond against a nominal bond => position in inflation space

� Strategies in real yield and breakeven space can be put on as

curve trades (e.g steepening or flattening, butterfly etc…),

cross market trades (e.g French vs European inflation).

16

Strategies in real yield space

� Outright positions in linkers, i.e., in real yield space usually express a view on both nominal yields AND inflation breakevens

� Consider an investor who has a bullish view on 5-year yields (i.e. 5-year yields expected to fall). The classic position to be built from this view would be to go long a 5-year nominal bond. Now consider three possibilities:

� The investor does not have a view on 5-year inflation breakevens. Then going long the 5-year nominal

bond would be the best strategy.

� The investor believes that 5-year inflation breakevens are too low and will increase. In this case, going

long a 5-year linker is a more attractive alternative to the classic position. A long position in the linker

actually allows the investor to position for BOTH his view on nominal yields and inflation

breakevens. NB: Carry/forward levels should be precisely taken into consideration in real yield

strategies (refer to slides on carry)

� The investor believes that 5-year inflation breakevens are too high and will fall. The investor would then

go long the nominal bond or consider a position in breakeven space (next section).

� Strategies in real yields may suit specific needs too. For example, an outright long position on a linker in real yield space may suit the hedging needs of a pension fund, irrespective of its view on yields or inflation breakevens.

17

Strategies in breakeven space

� An inflation breakeven strategy involves trading linkers against nominal bonds. The breakeven inflation rate (BEIR) is defined as the difference between the nominal yield and the real yield.

� Bets in breakeven space can be of two types:

� For “buy and hold” investors, the relative performance between nominal bonds and ILBs

will depend on actual average inflation throughout the remaining life of the bond relatively

to BEIR at time of purchase.

� Trading breakevens over a shorter period implies a view on the dynamics of the BEIR

(forwards should be taken into account). Future actual inflation will drive the relative

performances of the nominal bond and linker.

� Views on the dynamics of the BEIR are usually related to views on future inflation and inflation expectations. Other factors may intervene: impact from other markets (e.g. commodities markets), changes in regulations (e.g. taxes), bond supply etc…

18

Carry on linkers


� Carry is a key technical aspect in linker strategies

Real yield BEIR1Mth Carry

ILB3Mth Carry ILB

6Mth Carry

ILB1Mth Fwd BEIR 3Mth Fwd BEIR 6Mth Fwd BEIR

OATei 3% Jul 2012 2.088 221.0 11.73 13.48 4.90 209.85 208.43 216.79

OATei 1.6% Jul 2015 2.132 219.1 7.35 8.41 3.19 212.13 211.32 216.48

OATei 2.25% Jul 2020 2.200 221.1 4.97 5.74 2.43 216.54 216.05 219.49

OATei 3.15% jul 2032 2.191 230.7 3.11 3.57 1.48 227.86 227.74 230.12

OATei 1.8% jul 2040 2.155 233.6 2.28 2.58 1.00 231.58 231.54 233.36

BTANei 1.25% Jul 2010 2.037 224.4 18.09 20.88 6.79 207.28 204.94 218.54

OATi 3% Jul 2009 2.196 208.1 18.47 23.01 11.26 190.99 187.13 198.16

OATi 1.6% Jul 2011 2.260 202.9 9.57 11.70 6.04 194.11 192.36 197.70

OATi 2.5% Jul 2013 2.252 205.3 6.68 8.02 3.96 199.15 198.10 202.02

OATi 1% Jul 2017 2.289 206.3 3.97 4.76 2.43 202.70 202.18 204.56

OATi 3.4% Jul 2029 2.258 223.6 2.34 2.76 1.35 221.64 221.54 223.24

BUNDei 1.5% Apri l 2016 2.198 210.1 6.75 7.84 3.24 203.71 202.83 207.33

BTPei 1.65% Sep 2008 2.006 229.3 44.60 55.48 19.34 186.79 177.10 213.12

BTPei 0.95% Sep 2010 2.139 221.4 17.38 20.44 8.01 205.02 202.62 215.15

BTPei 1.85% Sep 2012 2.205 216.6 11.22 13.15 5.60 206.29 205.10 212.76

BTPei 2.15% Sep 2014 2.231 217.6 8.38 9.79 4.27 209.89 209.06 214.79

BTPei 2.1% Sep 2017 2.325 219.2 6.20 7.36 3.60 213.65 213.17 217.44

BTPei 2.35% Sep 2035 2.460 232.8 2.83 3.42 1.91 230.48 230.53 232.76

GGBei 2.9% Jul 2025 2.414 224.3 4.11 4.96 2.70 220.75 220.62 223.69

GGBei 2.3% Jul 2030 2.466 232.3 3.30 4.02 2.29 229.46 229.33 231.77

19

Inflation seasonality

Source: SG Quantitative Research and Fixed Income & Forex Research

� An accurate estimation of the seasonality of a linker’s underlying inflation index is key for inflation carry estimations

0.994

0.9965

0.999

1.0015

1.004

D J F M A M J J A S O N D- 0 .4%

- 0 .2%

0 .0%

0 .2%

0 .4%

J F M A M J J A S O N D

Deme tr a Dumm ie s

MoM a d ju s tme n ts

Average seasonality adjustments for

HICP for the 96-06 periodEMU inflation seasonality coefficient

multiplier

20

Carry and P&L


1M P&L of long OATi 2009 BEIR trade

when 1M carry is positive

1M P&L of short OATi 2009 BEIR trade

when 1M carry is negative

-30

-20

-10

0

10

20

30

Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Mar-07

Gain

Loss

bp

-30

-20

-10

0

10

20

30

Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Mar-07

Gain

Loss

bp

21

ILB market increasingly efficient

� It is remarkable that the P&L of long

BEIR positions (trade locked for 1

month in our case) has declined

steadily over the past few years.

� This suggests that the market is now

efficient in pre-adjusting to carry

conditions. Making systematic profits

from carry is no longer possible –

unless you get a superior CPI

forecasting model.

3-month standard deviation of P&L on

long BEIR OATi 2009 trade


0

4

8

12

16

May-02 May-03 May-04 May-05 May-06

Linkers vs Nominal bonds

Relative volatility and correlation

The inflation risk premium

23

Fisher – linking nominal and real rates

� The fundamental equation assessing the link between nominal

rates and real rates has been postulated be the American

economist Irving Fisher (1867-1947):

(1+Nominal rate)=(1+ Break-even inflation rate)*(1+real rate)

Nominal rate = real rate + break-even inflation rate + (break-even inflation rate * real rate)

Neglecting 2nd order terms, we get: Nominal rate = real rate + break-even inflation rate

Nominal rate = real rate + expected inflation + risk premium

� The break-even inflation rate (BEIR) is made of expected

inflation and a risk premium covering inflation forecast

uncertainties and the difference in the liquidity of the nominal

and the inflation bond

24

Fisher – what it means for volatility

� i = r + e + p

� with i nominal bond yield

� r real bond yield

� e inflation expectation

� p inflation risk premium

� i = r + BEIR

� with Break-Even Inflation Rate BEIR = e + p

� Var (i) = Var (r) + Var (BEIR) + 2 * Cov (r,BEIR)

� Var(i) > Var(r) unless Cov (r, BEIR) < - 0.5 * Var (BEIR)

i.e. unless Correl (r, BEIR) < -0.5 * STDV (BEIR) / STDV (r)

� Unless real yields and inflation expectation + premium exhibit a large negative correlation, nominal yields will be more volatile than real bond yields.

25

Real and nominal yield volatility


30

50

70

90

110

Sep-02 Mar-04 Sep-05 Mar-07

26-wk STDEV real yield OATei 201226-wk STDEV nominal yield OAT 2012

bp

30

45

60

75

90


26-wk STDEV real yield OATei 203226-wk STDEV nominal yield OAT 2032

bp

Real and nominal yield volatilities are currently very low and close

� We look at annualised volatility of nominal and real bond yields

26

Yield volatility vs Price volatility


� When deciding for volatility through the asset allocation process, we need to look at (annualised) volatility of total returns instead of yield volatility.

� Importantly, the clean (full) invoice price of an ILB is obtained by multiplying the clean (full) price by an index ratio. One could be concerned that this index adds to the volatility of an ILB. Actually it does not -as the left-hand chart shows. That is because the clean price and the index ratio are negatively correlated (due to immediate price reaction to CPI release and lag in price indexation).

0

20

40

60

80

100

120

Sep-00 Nov-02 Jan-05 Mar-07

26-wk STDEV real yield OATi 200926-wk STDEV nominal yield OAT 2009

bp

0

1

2

3

4

5

6

00 01 02 03 04 05 06 07

Vol. of Clean Price OATi 09

Vol. of (Clean P * Index Ratio) OATi 09

Vol. of Clean Price OAT Apr 09

Rolling 12Mth volatility, %

27

Yield beta (sensitivity)

� Above we discussed correlation

between real yields and BEIR - key to

relative volatility between nominal and

real bond yields.

� ϕ(r, BEIR) = Cov (r,BEIR) / (σr * σBEIR)

� Now we look at the yield beta, which

tells us about the sensitivity of real bond

yields to nominal bond yields.

� β = Cov (i,r) / Var (i)

� We calculate yield beta by regressing

changes in real yields vs. changes in nominal

yields.

� NB: When deciding for correlation through

the asset allocation process, we use

correlation between total returns, not

yields.

1-month beta (daily data) between real

and nominal yields


0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

2005 2006 2007

BTPei 2010 OATei 2015 OATei 2032

28

The role of beta

� Guessing the beta is key because:

� Correlation between nominal bonds and ILBs is instrumental in the asset

allocation process; beta and correlation are closely related: ϕ(i, r) = β (i,

r) * σi / σr

� The beta will decide of the duration of the ILB sub-portfolio

� Effective duration of ILB = Modified duration of ILB * yield beta

� Which duration should you attach to ILBs, i.e. what beta to choose?

Looking at historicals can be a useful starting point. However, betas

may be unstable. Ideally, the choice of beta should reflect the

investors’ expectations about the relative moves in nominal and

real yields within the expected market context.

29

Relative performance

� Under- or out-performance of ILBs relatively to nominal bonds is

often discussed in terms of Break-even Inflation rates (BEIR). This

is misleading. If one attaches a 0.5 beta to an inflation-linked bond

in one’s portfolio, one should expect that BEIR widen in a bear

market, and narrow in a bull market. If one assumes that long-term

yield beta is 0.5, then:

� ILB out-performance:

� beta > 0.5 in a bull bond market

� beta < 0.5 in a bear bond market

� ILB under-performance

� beta < 0.5 in a bull bond market

� beta > 0.5 in a bear bond market

30

Directionality of betas (1)


0

1

2

3

4

Sep-00 Apr-02 Nov-03 Jun-05 Jan-07

0.2

0.4

0.6

0.8

1

Yield of OATi 2009

26-week yield beta

%

1.3

1.6

1.9

2.2

2.5

2.8


0.5

0.7

0.9

1.1

1.3

1.5

Yield of OATei 2032

26-week yield beta

%

� Are betas directional? The answer is not simple!

31

Directionality of betas (2)

� Short history suggests that betas tend to be higher when bond yields fall. Our view is actually that betas will tend to be high when nominal bonds yields move towards extreme levels (low or high).

� Low yields: Following Fisher, a decline in nominal bond yields comes from either falling real bond yields or falling BEIR. If one assumes yield beta constant at 0.5, BEIR fall as much as real bond yields through that process. Yet BEIR tend to quickly meet support, which pushes betas up:

• “Sticky” inflation in Eurozone - ECB has generally struggled to keep inflation below 2% since 1999.

• In the US, when the Fed cut rates to record lows to erase the deflation risk, this has prevented a dramatic fall in BEIR.

� High yields: Inversely, if nominal bond yields rise due to a positive shock on final demand, the central bank makes it clear it will fight inflation - that tends to cap BEIR, which pushes yield betas up.

32

The inflation risk premium

� In the long run, assuming inflation expectations are correctly priced

in the market, ILBs are expected to under-perform nominal bonds.

That is because ILBs protect the investor against the inflation risk.

Nominal bonds pay an inflation risk premium.

� Inflation risk premium: excess expected real yield of a nominal

bond over an equivalent inflation-linked bond.

� Inflation risk premium: expected saving on interest payments

realized by issuing an inflation-indexed bond rather than a nominal

bond.

� This relation is unbiased if the inflation expectations priced in

nominal bonds are rational.

33

Portfolio investment and the inflation risk

� Consider the position of an investor in 10-year Treasuries carrying

a nominal yield of 5%. Assume that the investor’s inflation

expectation for the coming decade is 2% per year, and therefore

that the expected annual real yield is 3%. Consider two situations

(we assume that the inflation rate has a normal distribution)

� E.g 1 - standard deviation of inflation is 3% .There is a 16%

probability that the inflation rate is above 5% (2% + 1 standard

deviation), and therefore that the real return actually obtained

from the bond is negative.

� E.g 2 - standard deviation of inflation is 1% . The probability

falls to just 0.1%. In other words, the division of the standard

deviation by three divides the probability of a negative real return

by 117.5.

� This difference in risk has to be incorporated in the bond’s price.

34

What is driving the inflation premium?

� Since we cannot precisely extract inflation expectations from

market data, it is difficult to assess the value of the inflation risk

premium.

� Intuitive thoughts about the inflation risk premium:

� The premium is growing with inflation volatility. Gain in central

bank credibility over the last twenty years has cut the premium.

� The premium is growing with the needs for inflation protection.

As the baby-boomers get ready for retirement, the increased

demand for inflation protection may cause an increase of the

premium.

�Of course, shocks on relative supply/demand on ILBs

relatively to nominal bonds will affect the premium, too.

35

Measuring the premium (1)

� A recent ECB article, “Inflation risk premia in the term structure of

interest rates” by Peter Hördahl and Oreste Tristani, offers a review of

recent research on the inflation premium, and makes its own estimate.

� The State of the Art shows dramatically different results, depending on

the model specifications.

� Their own study, focusing on the 10-year maturity, concludes that “on

average, the inflation risk premium on euro area nominal yields was

insignificantly different from zero over the EMU sample. Nevertheless,

fluctuations around the average have been relatively small, but

statistically significant, in the 2004-2006 period.” However, “fluctuations

in the raw break-even rate have mostly reflected variations in the

inflation risk premium: adjusting for such premium, long-term inflation

expectations appear to have remained well-anchored in the euro area

from 1999 to date.” They show, in particular, that the premium tends to

grow when short term rates are low and/or the output gap rises.

36

Measuring the premium (2)

� The models unfortunately have their shortcomings.

� Those based on historical nominal and real yield series suffer from the

relatively short history of the inflation-linked market. UK series are

longer, but this market has been heavily distorted by supply and demand

considerations.

� Models using inflation-linked bonds’ price or yields fail to properly

measure the liquidity premium, especially for older data.

� Results are often dependant on the assumptions and specifications of

the models. Hördahl and Tristani themselves use output gap data in the

macro specification of the model, that are not broadly used or even

known by market participants.

� History will tell (maybe). In the long run, comparing the return (or cost) of investing (or issuing) in inflation-linked bonds to that of nominal bonds will help to measure the value of the inflation premium, but even there the results will be dependant on the assumption that inflation expectations priced in nominal bonds are rational.

37

Conclusion

� Linkers add to global portfolio diversification. The diversification

potential increases when stock returns are negatively correlated to

inflation expectations.

� The key decision in the asset allocation process is to decide about

� expected return EXPECTED PERFORMANCE

� volatility (relative valuation)

� CORRELATION

� Carry is key in linker strategies. Always look at linkers on a forward

basis.

� The relation between nominal linkers and linkers is not stable. When

assessing beta and correlations, historicals should only be the

starting point.

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