# Inflation Protected Investment Strategies

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Inflation-Hedging Properties of Different Asset Classes

#### 2.1. Stocks

#### 2.2. Bonds

#### 2.3. Commodities

#### 2.4. Gold

#### 2.5. Real Estate

#### 2.6. Data

## 3. Markov-Switching Model

**θ**, we use the findings of [36].2 After the parameters have been estimated, the filtered probabilities:

#### 3.1. Markov-Switching Model for the S&P 500 Index

#### 3.2. Markov-Switching Model for the Inflation Rate

#### 3.3. Markov-Switching Model for Inflation Rate and S&P 500

- Market calm
- –
- Regime 1: Inflation low, market calm
- –
- Regime 2: Inflation in transition, market calm
- –
- Regime 3: Inflation high, market calm

- Market turbulent-bearish
- –
- Regime 4: Inflation low, market turbulent-bearish
- –
- Regime 5: Inflation in transition, market turbulent-bearish
- –
- Regime 6: Inflation high, market turbulent-bearish

- Market turbulent-bullish
- –
- Regime 7: Inflation low, market turbulent-bullish
- –
- Regime 8: Inflation in transition, market turbulent-bullish
- –
- Regime 9: Inflation high, market turbulent-bullish

## 4. Optimal Portfolios

#### 4.1. Regime-Dependent Optimal Portfolios

**Optimization problem “Mean Variance (MV)”:**

**Optimization problem “Inflation Protection (IP)”:**

**x**represents the portfolio weights, $r(\mathbf{x})$ the portfolio return, I the inflation rate, $\overline{r}(\mathbf{x})$ the expected portfolio return and $\overline{I}$ the expected inflation rate. Obviously, (MV)is the classical mean-variance approach. The Optimization problem (IP)is the modified optimization problem and mainly driven by the aspect that an investor might be interested in a portfolio that behaves similar to the inflation rate whenever inflation is high, but will be rather unhappy with a match with inflation when inflation is low. Therefore, the return component of the optimization problem considers the out-performance of the inflation rate, whereas the risk component is driven by events where inflation has been higher than the portfolio return.

- Draw randomly in which regime ${s}_{0}$ we are at the beginning of the simulation. The probabilities are taken from the column sums of the matrix described in Table 3 divided by the sum of all entries.
- Draw randomly, in which regime ${s}_{1}$ we are after the next time step, given that we have been in ${s}_{0}$ before. Here, the conditional probabilities are calculated from the according row of the matrix described in Table 3.
- Denote by ${\widehat{r}}_{{s}_{1}}$ and ${\widehat{\Sigma}}_{{s}_{1}}$ the average return and covariance matrix of all observations that ended up in ${s}_{1}$. Furthermore, take the average return ${\widehat{r}}_{{s}_{0},{s}_{1}}$ of all observations that changed from ${s}_{0}$ to ${s}_{1}$ as an expert view. Combine this expert view with the average return ${\widehat{r}}_{{s}_{1}}$ by setting:$${\overline{r}}_{{s}_{0},{s}_{1}}=w\xb7{\widehat{r}}_{{s}_{0},{s}_{1}}+(1-w)\xb7{\widehat{r}}_{{s}_{1}}$$
- Goto Step 2 as long as the desired number of simulations is not reached.

#### 4.2. Monthly Time Horizon

#### 4.3. Longer Time Horizons

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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^{1.}For our empirical analysis, it is necessary to let all indices start at August 1974. Data for the bond index are available since December 1989, for the real estate index since January 1987 and the S&P 500 total return index since January 1988. Therefore, we performed a back calculation of these three indices via a linear regression, e.g., the bond index is regressed on its available historical data using factors with a longer empirical availability, such as U.S. Treasury Constant Maturities middle rates for 1, 3, 5 and 10 years and the BofA Merrill Lynch U.S. Corporate & Government Master Index. The resulting linear function and the older data of the factors are used to calculate back the bond index for the times when it is not available. Accordingly the real estate index is calculated back based on a regression with the U.S. Price Index of new One-Family Houses Under Construction (not seasonally adjusted - nadj), the US-Datastream Real Estate Investment Services (U.S.-DS Real Est Inv,Svs) Price Index and the National Association of Real Estate Investment Trusts (NAREIT) Total Return Index. The total return stock index was calculated back based on a regression with the corresponding S&P 500 price index.^{2.}The implementation of the presented Markov-switching model has been performed in R. The package HiddenMarkov (see http://cran.fhcrc.org/web/packages/HiddenMarkov/index.html) applies the Baum–Welch algorithm described in [36].^{3.}The algorithm is applied to avoid inconsistencies in the case of a low number of observed changes from one state to another. Step 3 is usually done before the simulation for all states ${s}_{0},{s}_{1}$ to fill a table from which the average returns are then taken within the simulation process.

**Figure 1.**Classification of the S&P 500 regimes based on two Markov-switching models with two states each. The approach of [12] detects a calm state (green), a turbulent state with mainly negative returns (bearish, red) and a turbulent state with mainly positive returns (bullish, yellow).

**Figure 2.**Classification of the inflation regimes based on a Markov-switching model with three states. We detect a calm inflation state (green), a high inflation state (red) and a transition inflation state (yellow).

**Figure 4.**Overview of selected (IP) portfolios over time assuming that the current regime will also be the regime of the next month.

**Figure 5.**Overview of selected (MV) portfolios over time assuming that the current regime will also be the regime of the next month.

**Table 1.**Sample characteristics of the monthly returns of the S&P 500 price index in the three states derived from the two two-state Markov-switching models [12].

S&P 500 | Calm | Turbulent | Turbulent Positive | Turbulent Negative |
---|---|---|---|---|

Mean (in %, ann.) | 10.50 | −2.92 | 12.78 | −19.89 |

Standard deviation (in %, ann.) | 13.39 | 20.44 | 15.36 | 22.18 |

Skewness | −0.08 | −0.50 | 0.20 | −0.49 |

Kurtosis | 0.15 | 0.87 | 0.52 | 0.48 |

**Table 2.**Sample characteristics of the monthly inflation rate in the three states derived from the one three-state Markov-switching model.

Inflation | State 1 | State 2 | State 3 |
---|---|---|---|

Mean (in %, ann.) | 2.68 | 3.58 | 8.14 |

Standard deviation (in %, ann.) | 0.95 | 0.92 | 1.32 |

Skewness | −1.77 | 0.60 | −0.77 |

Kurtosis | 15.32 | 5.87 | 1.67 |

From/to | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

1 | 112 | 4 | 1 | 6 | 0 | 0 | 2 | 0 | 0 |

2 | 7 | 138 | 5 | 0 | 1 | 0 | 1 | 1 | 0 |

3 | 1 | 5 | 41 | 0 | 0 | 4 | 0 | 0 | 1 |

4 | 4 | 1 | 0 | 18 | 2 | 1 | 7 | 0 | 1 |

5 | 0 | 3 | 0 | 0 | 11 | 0 | 0 | 4 | 0 |

6 | 0 | 1 | 4 | 1 | 1 | 11 | 1 | 1 | 3 |

7 | 2 | 0 | 0 | 8 | 0 | 0 | 34 | 1 | 0 |

8 | 0 | 1 | 0 | 0 | 3 | 1 | 0 | 6 | 1 |

9 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 4 |

Regime | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Inflation | 0.21 | 0.31 | 0.74 | 0.27 | 0.14 | 0.52 | 0.26 | 0.12 | 0.53 |

Stocks | 0.93 | 1.06 | 0.54 | −1.08 | −0.39 | −0.39 | 1.75 | 2.18 | 4.25 |

Real Estate | 0.27 | 0.30 | 0.35 | 0.32 | 0.32 | 0.33 | 0.19 | 0.63 | 0.24 |

Gold | −0.32 | 0.49 | 2.16 | 1.31 | 0.94 | 1.16 | 0.52 | 0.74 | 2.22 |

Bonds | 0.49 | 1.03 | 0.66 | 0.86 | 0.70 | 1.05 | 0.66 | −0.91 | −3.51 |

Com. | 0.34 | 0.66 | 1.89 | 0.58 | −1.92 | 0.59 | 1.29 | 1.47 | −2.54 |

Regime | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Inflation | 0.28 | 0.24 | 0.36 | 0.16 | 0.26 | 0.39 | 0.28 | 0.36 | 0.58 |

Stocks | 2.79 | 3.04 | 3.28 | 3.31 | 3.89 | 3.68 | 3.47 | 3.81 | 3.88 |

Real Estate | 1.25 | 0.07 | 0.11 | 0.15 | 0.12 | 0.13 | 0.54 | 1.29 | 0.17 |

Gold | 4.57 | 5.01 | 9.45 | 3.94 | 4.27 | 7.40 | 5.16 | 5.99 | 6.86 |

Bonds | 1.62 | 5.84 | 5.62 | 1.85 | 4.12 | 5.55 | 2.67 | 4.20 | 7.49 |

Com. | 6.29 | 4.58 | 4.87 | 4.84 | 5.96 | 5.57 | 5.62 | 9.20 | 10.72 |

Regime | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Stocks | −2.02 | 9.50 | 2.09 | 2.59 | 17.95 | 5.25 | 1.49 | −35.96 | 41.55 |

Real Estate | −0.21 | −6.09 | −20.67 | −21.12 | 17.90 | −1.48 | −7.78 | −26.10 | −14.22 |

Gold | 24.94 | −0.72 | 20.56 | 47.58 | 30.37 | 14.99 | 12.03 | −26.39 | 2.13 |

Bonds | −5.08 | 0.99 | 1.57 | −6.90 | 28.80 | 1.56 | −7.08 | 2.12 | −39.91 |

Com. | 66.35 | 33.25 | 5.80 | 54.97 | 39.84 | 21.61 | 63.05 | 75.89 | 62.07 |

Regime | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Portfolio Weights | |||||||||

Stocks | 66.22 | 62.94 | 2.16 | 0.00 | 17.84 | 0.00 | 72.18 | 69.88 | 92.68 |

Real Estate | 0.00 | 4.01 | 0.20 | 0.00 | 29.55 | 77.99 | 0.00 | 18.79 | 0.00 |

Gold | 0.00 | 7.51 | 18.88 | 47.02 | 21.23 | 0.00 | 0.00 | 1.53 | 7.32 |

Bonds | 33.78 | 17.84 | 7.17 | 52.98 | 31.39 | 22.01 | 9.90 | 0.00 | 0.00 |

Com. | 0.00 | 7.70 | 71.60 | 0.00 | 0.00 | 0.00 | 17.92 | 9.80 | 0.00 |

Portfolio characteristics | |||||||||

Mean | 0.78 | 0.94 | 1.82 | 1.07 | 0.47 | 0.70 | 1.56 | 1.80 | 4.10 |

SD | 1.87 | 2.32 | 4.18 | 2.09 | 1.45 | 1.90 | 2.79 | 2.80 | 3.54 |

Inflation | 0.21 | 0.31 | 0.74 | 0.27 | 0.14 | 0.52 | 0.26 | 0.12 | 0.53 |

**Table 8.**Optimal regime-dependent portfolios (in %) for problem (Inflation Protection (IP) (monthly data).

Regime | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Portfolio Weights | |||||||||

Stocks | 95.51 | 70.22 | 0.00 | 0.00 | 6.87 | 0.00 | 86.84 | 42.56 | 87.84 |

Real Estate | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 12.92 | 0.00 | 4.88 | 0.00 |

Gold | 0.00 | 0.00 | 22.61 | 100.00 | 50.08 | 0.00 | 0.00 | 7.86 | 12.16 |

Bonds | 4.49 | 19.82 | 29.85 | 0.00 | 43.05 | 63.12 | 0.00 | 0.00 | 0.00 |

Com. | 0.00 | 9.96 | 47.55 | 0.00 | 0.00 | 23.96 | 13.16 | 44.71 | 0.00 |

Portfolio characteristics | |||||||||

Mean | 0.91 | 1.01 | 1.59 | 1.31 | 0.47 | 1.36 | 1.69 | 1.68 | 4.01 |

SD | 2.65 | 2.61 | 3.98 | 3.94 | 2.46 | 4.13 | 3.15 | 4.31 | 3.35 |

Inflation | 0.21 | 0.31 | 0.74 | 0.25 | 0.14 | 0.52 | 0.26 | 0.12 | 0.53 |

**Table 9.**Overview of mean, standard deviation and the risk measures Sharpe ratio, Omega and maximum drawdown (in %; the benchmark is the inflation rate) over the whole time horizon for the strategies, as in Figure 6.

(IP) Portfolio | (MV) Portfolio | Stocks/Bonds | $\frac{\mathbf{1}}{\mathit{N}}$-Portfolio | |
---|---|---|---|---|

Mean | 0.85 | 0.71 | 0.79 | 0.61 |

Standard deviation | 3.05 | 2.60 | 2.60 | 2.06 |

Sharpe ratio | 16.03 | 16.82 | 14.08 | 11.39 |

Omega | 155.13 | 160.32 | 150.74 | 136.43 |

Maximum Drawdown | −31.09 | -25.14 | −24.86 | −17.80 |

**Table 10.**Overview of the performance (in %) of the (IP) and (MV) portfolios during several periods and crises.

Period | IP-Port. | MV-Port. | Stocks/Bonds | $\frac{\mathbf{1}}{\mathit{N}}$-Port. | Infl. | ||||
---|---|---|---|---|---|---|---|---|---|

Mean | SD | Mean | SD | Mean | SD | Mean | SD | ||

2nd oil crisis | 2.59 | 5.78 | 1.99 | 5.96 | 1.62 | 4.11 | 2.25 | 4.36 | 1.04 |

Early 1980s recession | 0.82 | 2.60 | 0.46 | 2.58 | 1.21 | 3.15 | 1.18 | 1.79 | 0.46 |

First Gulf War | 1.44 | 3.20 | 1.77 | 4.42 | 0.88 | 1.77 | 0.43 | 1.27 | 0.38 |

S&P 500 rally 87-00 | 1.01 | 2.50 | 0.98 | 2.21 | 0.91 | 1.87 | 0.62 | 1.41 | 0.21 |

Katrina | 0.56 | 1.28 | 1.03 | 1.40 | 0.40 | 0.57 | 0.50 | 1.45 | 0.25 |

Financial Crisis | 0.39 | 2.38 | 0.42 | 1.66 | 0.19 | 1.88 | 0.29 | 1.49 | 0.13 |

**Table 11.**Overview of Sharpe ratio, Omega and maximum drawdown (in %; the benchmark is the inflation rate) during several periods and crises for the (IP) and (MV) portfolios.

Period | IP-Portfolio | MV-Portfolio | ||||
---|---|---|---|---|---|---|

Sharpe | Omega | M.D. | Sharpe | Omega | M.D. | |

2nd oil crisis | 26.91 | 204.47 | −14.52 | 16.11 | 151.54 | −15.52 |

Early 1980s recession | 13.20 | 147.61 | −5.72 | 0.00 | 100.00 | −6.49 |

First Gulf War | 34.40 | 254.38 | −3.00 | 32.50 | 295.51 | −4.03 |

S&P 500 rally 87−00 | 27.57 | 211.59 | −10.77 | 29.92 | 231.94 | −8.40 |

Katrina | 24.12 | 192.21 | −0.87 | 54.56 | 585.72 | −0.38 |

Financial Crisis | 10.84 | 129.92 | −7.65 | 16.69 | 157.44 | −3.40 |

**Table 12.**Overview of Sharpe ratio, Omega and maximum drawdown (in %; the benchmark is the inflation rate) during several periods and crises for the stock/bond and the $\frac{1}{N}$-portfolio.

Period | Stocks/Bonds | $\frac{\mathbf{1}}{\mathit{N}}$-Portfolio | ||||
---|---|---|---|---|---|---|

Sharpe | Omega | M.D. | Sharpe | Omega | M.D. | |

2nd oil crisis | 13.77 | 138.67 | −5.25 | 27.36 | 231.75 | −9.63 |

Early 1980s recession | 23.00 | 182.64 | −3.64 | 37.59 | 266.50 | −3.14 |

First Gulf War | 28.61 | 198.55 | −3.23 | 4.53 | 111.01 | −2.23 |

S&P 500 rally 87−00 | 31.45 | 226.27 | −7.50 | 21.91 | 173.87 | −5.46 |

Katrina | 22.82 | 169.37 | −0.40 | 18.22 | 158.55 | −1.71 |

Financial Crisis | 3.15 | 108.54 | −9.17 | 11.74 | 135.26 | −3.87 |

Asset/Months | 12 | 60 | 120 | 180 | 240 |
---|---|---|---|---|---|

Portfolio Weights | |||||

Stocks | 43.83 | 41.47 | 43.13 | 43.86 | 42.48 |

Real Estate | 18.02 | 21.01 | 19.58 | 20.38 | 20.13 |

Gold | 7.95 | 6.81 | 7.76 | 6.62 | 7.53 |

Bonds | 22.63 | 23.02 | 22.32 | 22.28 | 22.98 |

Commodities | 7.57 | 7.69 | 7.21 | 6.86 | 6.87 |

Portfolio characteristics | |||||

Mean | 0.63 | 0.61 | 0.63 | 0.63 | 0.62 |

SD | 2.01 | 2.00 | 2.03 | 2.03 | 2.02 |

Asset/Months | 12 | 60 | 120 | 180 | 240 |
---|---|---|---|---|---|

Portfolio Weights | |||||

Stocks | 57.49 | 58.69 | 63.72 | 57.03 | 56.72 |

Real Estate | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Gold | 2.59 | 0.00 | 0.00 | 0.00 | 0.00 |

Bonds | 27.54 | 37.09 | 36.28 | 42.97 | 43.28 |

Commodities | 12.38 | 4.21 | 0.00 | 0.00 | 0.00 |

Portfolio characteristics | |||||

Mean | 0.71 | 0.73 | 0.75 | 0.75 | 0.74 |

SD | 2.54 | 2.77 | 2.88 | 2.89 | 2.90 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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Mahlstedt, M.; Zagst, R. Inflation Protected Investment Strategies. *Risks* **2016**, *4*, 9.
https://doi.org/10.3390/risks4020009

**AMA Style**

Mahlstedt M, Zagst R. Inflation Protected Investment Strategies. *Risks*. 2016; 4(2):9.
https://doi.org/10.3390/risks4020009

**Chicago/Turabian Style**

Mahlstedt, Mirco, and Rudi Zagst. 2016. "Inflation Protected Investment Strategies" *Risks* 4, no. 2: 9.
https://doi.org/10.3390/risks4020009