# Valuation of Index-Linked Cash Flows in a Heath–Jarrow–Morton Framework

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**X**is denoted by ${Q}_{t}\left[\mathit{X}\right]$.

**ϕ**is given by:

**ϕ**if the deflated price process ${\left({\text{\varphi}}_{t}{Q}_{t}\left[\mathit{X}\right]\right)}_{t=0,\dots ,{t}_{\text{max}}}$ is a $(\mathbb{P},\mathbb{F})$-martingale. Using the tower property of conditional expectation, we see that the definition in Equation (1) implies that deflated price processes are $(\mathbb{P},\mathbb{F})$-martingales,

#### The Heath–Jarrow–Morton Framework

## 3. Valuation of an Index-Linked Cash Flow

**I**, which is the cash flow of a real zero-coupon bond maturing at time $kT$.

**Proposition 1.**Suppose that:

## 4. Model Selection and Validation

#### 4.1. The Volatility Structure of Changes in the Forward Rates

**Table 1.**Eigenvalues (${w}_{i}$) of the sample covariance matrix of forward rate changes (${\text{\Delta}}_{t}$).

${w}_{1}$ | $13.6\xb7{10}^{-6}$ |

${w}_{2}$ | $3.9\xb7{10}^{-6}$ |

${w}_{3}$ | $1.1\xb7{10}^{-6}$ |

${w}_{4}$ | $0.8\xb7{10}^{-6}$ |

${w}_{5}$ | $0.4\xb7{10}^{-6}$ |

**Figure 1.**Eigenvectors ${e}_{1}$ (blue crosses) and ${e}_{2}$ (red circles) together with the functions ${\text{\xi}}_{1}$ (blue solid line) and ${\text{\xi}}_{2}$ (red dashed line).

#### 4.2. The Market Price of Risk and Model Validation

Number of Non-Zero ${\tilde{b}}_{i}$ | Non-Zero ${\tilde{b}}_{i}$ | Maximum Log-Likelihood |
---|---|---|

6 | ${\tilde{b}}_{0}$, ${\tilde{b}}_{1}$, ${\tilde{b}}_{2}$, ${\tilde{b}}_{3}$, ${\tilde{b}}_{4}$, ${\tilde{b}}_{5}$ | $-226.21$ |

5 | ${\tilde{b}}_{0}$, ${\tilde{b}}_{1}$, ${\tilde{b}}_{2}$, ${\tilde{b}}_{3}$, ${\tilde{b}}_{5}$ | $-226.29$ |

4 | ${\tilde{b}}_{0}$, ${\tilde{b}}_{1}$, ${\tilde{b}}_{2}$, ${\tilde{b}}_{3}$ | $-228.28$ |

3 | ${\tilde{b}}_{0}$, ${\tilde{b}}_{1}$, ${\tilde{b}}_{3}$ | $-230.92$ |

2 | ${\tilde{b}}_{0}$, ${\tilde{b}}_{1}$ | $-234.69$ |

1 | ${\tilde{b}}_{0}$ | $-243.09$ |

0 | - | $-246.55$ |

**Figure 2.**Scatter and QQ plots of the innovations for the cases with 0, 1, 2 and 3 non-zero ${\tilde{b}}_{i}$, respectively.

**Figure 3.**Auto- and cross-correlations of the innovations for the cases with 0, 1, 2 and 3 non-zero ${\tilde{b}}_{i}$, respectively.

#### 4.3. The Relation between the Consumer Price Index and the Short Rate

**Figure 4.**Cross-correlations between $\tilde{r}$ and $\tilde{q}$ and time series plots of ${\tilde{r}}_{t}^{4}$ (blue crosses) and ${\tilde{q}}_{t-1}^{4}$ (red circles).

## 5. Model-Based Valuation

**X**today is $\mathbb{E}\left[{\text{\varphi}}_{kT}{X}_{kT}\right]$. However, in practice, one often assumes independence between the interest rates used for discounting and the claims payments, and under this assumption, today’s value is $\mathbb{E}\left[{\text{\varphi}}_{kT}\right]\mathbb{E}\left[{X}_{kT}\right]$.

**Table 3.**Values of $\mathbb{E}\left[exp\left\{{D}_{T}\right\}\right]$ with ${\text{\mu}}_{{D}_{T}}=cT$.

T | $c=-0.04$ | $c=-0.02$ | $c=0.0$ | $c=0.02$ | $c=0.04$ |
---|---|---|---|---|---|

1 | 0.961 | 0.980 | 1.000 | 1.020 | 1.041 |

2 | 0.923 | 0.961 | 1.000 | 1.041 | 1.083 |

5 | 0.819 | 0.905 | 1.000 | 1.105 | 1.221 |

10 | 0.670 | 0.819 | 1.000 | 1.221 | 1.492 |

**Table 4.**Market rates (in percent) of Swedish Consumer Price Index (CPI)-linked government bonds on 28 November 2014.

Name | Years to Maturity | Average Rate |
---|---|---|

RGKB 3105 | $1.0$ | $-0.168$ |

RGKB 3107 | $2.5$ | $-0.569$ |

RGKB 3102 | $6.0$ | $-0.505$ |

RGKB 3108 | $7.5$ | $-0.329$ |

RGKB 3109 | $10.5$ | $-0.048$ |

RGKB 3104 | $14.0$ | $-0.220$ |

**Table 5.**Mean and confidence interval of ${\text{\eta}}_{4T}\left(\text{\lambda}\right)$ and valuation ratios for the market prices of risks 0 and ${\text{\lambda}}_{\text{belief}}$.

T | ${\text{\eta}}_{4T}\left(0\right)$ | $\widehat{{\text{\eta}}_{4T}\left(\text{\lambda}\right)}$ | 95% CI of ${\text{\eta}}_{4T}\left(\text{\lambda}\right)$ | ${\text{\eta}}_{4T}\left({\text{\lambda}}_{\text{belief}}\right)$ |
---|---|---|---|---|

1 | 1.000 | 1.011 | (1.004, 1.019) | 0.994 |

2 | 0.999 | 1.028 | (1.007, 1.049) | 0.983 |

5 | 0.991 | 1.059 | (0.965, 1.160) | 0.925 |

10 | 0.935 | 0.884 | (0.638, 1.194) | 0.780 |

**Figure 8.**Valuation ratios ${\text{\eta}}_{4T}\left(\text{\lambda}\left(\text{\varphi}\right)\right)$ (left) and forward rate curves in five years for the minimum and maximum valuation ratios (right). The blue solid line, the red dashed line, the green dot-dashed line and the magenta dotted line correspond to $T=1,2,5,10$, respectively. The black thick solid and dashed lines represent the market prices of risk $\widehat{\text{\lambda}}$ and ${\text{\lambda}}_{\text{belief}}$, respectively.

## 6. Conclusions and Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

**Proof.**Proof of Proposition 1

## References

- T. Møller, and M. Steffensen. Market-Valuation Methods in Life and Pension Insurance. Cambridge, UK: Cambridge University Press, 2007. [Google Scholar]
- M.V. Wüthrich, H. Bühlmann, and H. Furrer. Market-Consistent Actuarial Valuation, 2nd ed. Heidelberg, Germany: Springer-Verlag Berlin Heidelberg, 2010. [Google Scholar]
- M.V. Wüthrich, and M. Merz. Financial Modeling, Actuarial Valuation and Solvency in Insurance. Heidelberg, Germany: Springer-Verlag Berlin Heidelberg, 2013. [Google Scholar]
- M.J. Brennan, and E.S. Schwartz. “The pricing of equity-linked life insurance policies with an asset value guarantee.” J. Financ. Econ. 3 (1976): 195–213. [Google Scholar] [CrossRef]
- H. Bühlmann. “New Math for Life Actuaries.” ASTIN Bull. 32 (2002): 209–211. [Google Scholar] [CrossRef]
- M. De Felice, and F. Moriconi. “Market based tools for managing the life insurance company.” ASTIN Bull. 35 (2005): 79–111. [Google Scholar] [CrossRef]
- P. Løchte Jørgensen. “On accounting standards and fair valuation of life insurance and pension liabilities.” Scand. Actuar. J. 5 (2004): 372–394. [Google Scholar] [CrossRef]
- T. Møller. “Hedging equity-linked life insurance contracts.” N. Am. Actuar. J. 5 (2001): 79–95. [Google Scholar] [CrossRef]
- S.-A. Persson, and K.K. Aase. “Valuation of the minimum guaranteed return embedded in life insurance products.” J. Risk Insur. 64 (1997): 599–617. [Google Scholar] [CrossRef]
- A. Grosen, and P. Løchte Jørgensen. “Fair valuation of life insurance liabilities: The impact of interest rate guarantees, surrender options, and bonus policies.” Insur. Mathe. Econ. 26 (2000): 37–57. [Google Scholar] [CrossRef]
- M. Steffensen. “Surplus-linked life insurance.” Scand. Actuar. J. 1 (2006): 1–22. [Google Scholar] [CrossRef]
- A.J. Tanskanen, and J. Lukkarinen. “Fair valuation of path-dependent participating life insurance contracts.” Insur. Mathe. Econ. 33 (2003): 595–609. [Google Scholar] [CrossRef]
- P. Hilli, M. Koivu, and T. Pennanen. “Cash-flow based valuation of pension liabilities.” Eur. Actuar. J. 1 (2011): 329–343. [Google Scholar] [CrossRef]
- J.-P. Schmidt. “Market-consistent valuation of long-term insurance contracts: Valuation framework and application to German private health insurance.” Eur. Actuar. J. 4 (2014): 125–153. [Google Scholar] [CrossRef]
- M. Buchwalder, H. Bühlmann, M. Merz, and M.V. Wüthrich. “Valuation portfolio in non-life insurance.” Scand. Actuar. J. 2 (2007): 108–125. [Google Scholar] [CrossRef]
- D. Diers, M. Eling, C. Kraus, and A. Reuß. “Market-consistent embedded value in non-life insurance: How to measure it and why.” J. Risk Financ. 13 (2012): 320–346. [Google Scholar] [CrossRef]
- D. Heath, R. Jarrow, and A. Morton. “Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation.” Econometrica 60 (1992): 77–105. [Google Scholar] [CrossRef]
- A.J.G. Cairns. Interest Rate Models: An Introduction. Princeton, NJ, USA: Princeton University Press, 2004. [Google Scholar]
- D. Filipović, and S. Tappe. “Existence of Lévy term structure models.” Financ. Stoch. 12 (2008): 83–115. [Google Scholar] [CrossRef]
- D. Filipović, S. Tappe, and J. Teichmann. “Term structure models driven by Wiener processes and Poisson measures: Existence and positivity.” SIAM J. Financ. Math. 1 (2010): 523–554. [Google Scholar] [CrossRef]
- R. Jarrow, and Y. Yildirim. “Pricing Treasury Inflation Protected Securities and Related Derivatives using an HJM Model.” J. Financ. Quant. Anal. 38 (2003): 337–358. [Google Scholar] [CrossRef]
- L. Hughston. Inflation Derivatives. Working Paper; London, UK: Merrill Lynch and King’s College, 2007. [Google Scholar]
- Z. Eksi-Altay, and D. Filipović. “Pricing and hedging of inflation-indexed bonds in an affine framework.” J. Comput. Appl. Math. 259 (2014): 452–463. [Google Scholar] [CrossRef]
- R. Aïd, O. Féron, N. Touzi, and C. Vialas. “An arbitrage-free interest rate model consistent with economic constraints for long-term asset liability management.” Bank. Mark. Invest. 116 (2012): 4–19. [Google Scholar]
- D. Brigo, and F. Mercurio. Interest Rate Models—Theory and Practice, 2nd ed. Heidelberg, Germany: Springer-Verlag Berlin Heidelberg, 2007. [Google Scholar]
- R. Rebonato. Modern Pricing of Interest-Rate Derivatives: The LIBOR Market Model and Beyond. Princeton, NJ, USA: Princeton University Press, 2002. [Google Scholar]
- R. Rebonato. Term-Structure Models: A Review. Technical Report; Edinburgh, UK: QUARC—Royal Bank of Scotland, 2003. [Google Scholar]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alm, J.; Lindskog, F.
Valuation of Index-Linked Cash Flows in a Heath–Jarrow–Morton Framework. *Risks* **2015**, *3*, 338-364.
https://doi.org/10.3390/risks3030338

**AMA Style**

Alm J, Lindskog F.
Valuation of Index-Linked Cash Flows in a Heath–Jarrow–Morton Framework. *Risks*. 2015; 3(3):338-364.
https://doi.org/10.3390/risks3030338

**Chicago/Turabian Style**

Alm, Jonas, and Filip Lindskog.
2015. "Valuation of Index-Linked Cash Flows in a Heath–Jarrow–Morton Framework" *Risks* 3, no. 3: 338-364.
https://doi.org/10.3390/risks3030338