Fitting linkers into a portfolio

Fitting linkers into a portfolio Khrishnamoorthy SOOBEN Fixed Income Strategist +44 (0)20 7676 7713

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 1

Historical return and risk 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 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 2

Ex-post efficient frontier Portfolio Return 10% 9% 8% 7% 6% 5% 4% 3% Efficient frontier derived from historical performance Efficient Frontier 0% 3% 6% 9% 12% 15% 18% Risk We minimise the portfolio risk for a given return, using historical returns, risks and correlation. Source: SG Fixed Income & Forex Research 3

Ex-post portfolio components Weights 100% 80% 60% 40% 20% Portfolio components derived from historical performance 0% Source: SG Fixed Income & Forex Research Portfolio Components Risk 0% 3% 6% 9% 12% 15% 18% Money Market Nominal Bonds Linkers Equities 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. 4

Anticipating returns, risks and correlations 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 Source: SG Fixed Income & Forex Research 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%) 5

Correlation assumptions are crucial Expected Return 8% 7% 6% 5% 4% 3% Efficient frontiers under different correlation assumptions Risk 0% 5% 10% 15% 20% -ve Corr Equities vs Linkers & Nominal Bonds +ve Corr Equities vs Linkers & Nominal Bonds 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. Source: SG Fixed Income & Forex Research 6

Moving the efficient frontier - Assumption 1 Expected Return Linkers provide diversification benefits under assumption 1 8% 7% 6% 5% 4% 3% 0% 5% 10% 15% 20% Linkers included Linkers excluded Risk 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. Source: SG Fixed Income & Forex Research 7

Portfolio weights Assumption 1 Linkers win a significant share of the portfolio under assumption 1 The first set of assumption delivers a very high weighting to ILBs for portfolio return volatility around 5%. 100% 80% Portfolio Components Such weighting may be difficult to achieve on big portfolios because of the relatively small size of the ILB market. Weights 60% 40% 20% Risk 0% 0% 5% 10% 15% 20% Money Market Nominal Bonds Linkers Equities One would need to run optimization under an additional constraint in that case (capping the weighting of ILBs). Source: SG Fixed Income & Forex Research 8

Moving the efficient frontier - Assumption 2 Expected Return Negligible diversification benefit from linkers under assumption 2 8% 7% 6% 5% 4% 3% Risk 0% 5% 10% 15% 20% Linkers included Linkers excluded 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%. Source: SG Fixed Income & Forex Research 9

Portfolio weights - Assumption 2 Linkers still win a significant share of the portfolio under assumption 2 100% Portfolio Components Although ILBs do not greatly improve the risk/return outlook under that set of assumptions, they still win a significant weighting. Weights 80% 60% 40% 20% Risk 0% 0% 5% 10% 15% 20% Money Market Nominal Bonds Linkers Equities Source: SG Fixed Income & Forex Research 10

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 ϕ(rs, Ri) relatively to ϕ(rs, Rr). ϕ(rs, Ri) = Cov (Rs, Ri) / (σ Rs * σ Ri ) ϕ(rs, Rr) = Cov (Rs, Rr) / (σ Rs * σ Rr ) 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 σ i > σ r then ϕ(rs, i) < ϕ(rs, r) This also means ϕ(rs, Ri) > ϕ(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 11

Stocks vs. bonds and ILBs (2) The result above is intuitive. If both stocks and nominal bonds react badly to increasing inflation expectations (or to a rising inflation 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. 12

Stocks vs. bonds and ILBs (3) Deciding ex-ante for the nominal and relative levels of ϕ(rs, Ri) and ϕ(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. 13

Finding value in linkers Common strategies and technical considerations

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). 15

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. 16

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 17

Carry on linkers Real yield BEIR 1Mth Carry ILB 3Mth Carry ILB 6Mth Carry ILB 1Mth 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% April 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 Source: SG Fixed Income & Forex Research Carry is a key technical aspect in linker strategies 18

Inflation seasonality Average seasonality adjustments for HICP for the 96-06 period EMU inflation seasonality coefficient multiplier 0.4 % M o M a d ju s tm e n ts 1.004 0.2 % 1.0015 0.0 % 0.999-0.2 % 0.9965-0.4 % D e m e tr a Du mm ie s J F M A M J J A S O N D 0.994 D J F M A M J J A S O N D 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 19

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 bp 30 Gain 20 10 0-10 -20 Loss -30 Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Mar-07 bp 30 Gain 20 10 0-10 -20 Loss -30 Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Mar-07 Source: SG Fixed Income & Forex Research 20

ILB market increasingly efficient 16 12 8 4 3-month standard deviation of P&L on long BEIR OATi 2009 trade 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. 0 May-02 May-03 May-04 May-05 May-06 Source: SG Fixed Income & Forex Research 21

Linkers vs Nominal bonds Relative volatility and correlation The inflation risk premium

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 2 nd 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 23

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. 24

Real and nominal yield volatility Real and nominal yield volatilities are currently very low and close bp 110 90 70 50 30 Sep-02 Mar-04 Sep-05 Mar-07 26-wk STDEV real yield OATei 2012 26-wk STDEV nominal yield OAT 2012 bp 90 75 60 45 30 May-03 May-04 May-05 May-06 May-07 26-wk STDEV real yield OATei 2032 26-wk STDEV nominal yield OAT 2032 Source: SG Fixed Income & Forex Research We look at annualised volatility of nominal and real bond yields 25

Yield volatility vs Price volatility bp 120 100 80 60 40 20 0 Sep-00 Nov-02 Jan-05 Mar-07 26-wk STDEV real yield OATi 2009 26-wk STDEV nominal yield OAT 2009 6 5 4 3 2 1 0 Rolling 12Mth volatility, % 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 Source: SG Fixed Income & Forex Research 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). 26

Yield beta (sensitivity) 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 1-month beta (daily data) between real and nominal yields 2005 2006 2007 BTPei 2010 OATei 2015 OATei 2032 Source: SG Fixed Income & Forex Research 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) / (σ r * σ BEIR ) 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. 27

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) * σ i / σ r 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. 28

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 29

Directionality of betas (1) % 4 3 Yield of OATi 2009 26-week yield beta 0.2 0.4 2.8 2.5 2.2 % Yield of OATei 2032 26-week yield beta 0.5 0.7 0.9 2 0.6 1.9 1.1 1 0.8 1.6 1.3 0 Sep-00 Apr-02 Nov-03 Jun-05 Jan-07 1 1.3 May-03 May-04 May-05 May-06 May-07 1.5 Source: SG Fixed Income & Forex Research Are betas directional? The answer is not simple! 30

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. 31

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. 32

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. 33

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. 34

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. 35

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. 36

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 volatility CORRELATION EXPECTED PERFORMANCE (relative valuation) 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. 37