# Best-Estimates in Bond Markets with Reinvestment Risk

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bond Market with Reinvestment Risk

**Assumption**

**1.**

**x**denote an L-dimensional and $\mathbb{F}$-adapted process ${x}={\left({{x}}_{t}\right)}_{t=0,...,n}$ with

**x**defines a portfolio strategy. We assume that

**x**is ${\mathcal{L}}_{2}$-admissible, meaning that it satisfies ${\sum}_{\ell =1}^{L}{x}_{\ell ,t-1}\phantom{\rule{3.33333pt}{0ex}}P(t,t-1+\ell )\in {\mathcal{L}}_{2}\left({\mathcal{F}}_{t}\right)$ for all t.

**x**at time t. ${\mathscr{H}}_{t}$ contains the one-period hedgeable claims, that is the payoffs at time t that can be attained by an ${\mathcal{F}}_{t-1}$-measurable portfolio strategy ${{x}}_{t-1}$. Note that claim $P(t,t+L)$ cannot be perfectly replicated from $t-1$ to t by an ${\mathcal{F}}_{t-1}$-measurable portfolio strategy ${{x}}_{t-1}$ and bonds with payoffs $P(t,t),...,P(t,t-1+L)$ because time to maturity $L+1$ cannot be purchased at time $t-1$.

**Definition**

**1.**

**Lemma**

**1**

**Definition**

**2**

## 3. Best-Estimate Prices in Multifactor Vasiček Models

#### 3.1. The Multifactor Vasiček Bond Market Model

#### 3.2. The Aggregate Market Span-Deflator

**Lemma**

**2**

**Lemma**

**3.**

#### 3.3. Best-Estimate Bond Prices

**Theorem 1**Under the discrete time multifactor Vasiček model, the best-estimate price at time t of a bond that matures at time $t+\ell $, for $\ell \ge L+1$, is given by

**Example**

**1.**

#### 3.4. Hedging the Long-Term Bond

**Proposition**

**1.**

**Example**

**2**

## 4. Numerical Illustrations

#### 4.1. Numerical Illustration 1

Parameters | Parameter set 1 | Parameter set 2 |
---|---|---|

k | $[0.1360,0.2000]$ | $[0.1360,0.5500]$ |

b | $[0.0045,0.0005]$ | $[0.0045,0.0005]$ |

g | $[0.0080,0.0052]$ | $[0.0080,0.0123]$ |

λ | $[8,15]$ | $[8,15]$ |

${y}_{0}$ | $[0.0050,-0.0025]$ | $[0.0050,-0.0025]$ |

**Figure 1.**Correlations and yield curves for Parameter sets 1 and 2. (

**a**) Correlation between bond prices for Parameter set 1; (

**b**) Correlation between bond prices Parameter set 2; (

**c**) Yield curves obtained using no-arbitrage bond prices.

#### 4.1.1. Numerical Illustration 1: Parameter Sets

#### 4.1.2. Numerical Illustration 1: Projected Yield Curve

**Figure 2.**No-arbitrage and best-estimate projected yield curves. (

**a**) Parameter set 1; (

**b**) Parameter set 2.

**Table 2.**Differences between the best-estimate and the no-arbitrage yields for different numbers L of traded bonds, for Parameter sets 1 and 2. Entries 0.0000 and -0.0000 indicate positive and negative numbers whose absolute value is less than ${10}^{-8}$.

Difference between Best-Estimate and No-Arbitrage Yields $(\times {\mathbf{10}}^{-\mathbf{4}})$ | ||||||||
---|---|---|---|---|---|---|---|---|

Time to | ||||||||

Maturity | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Parameter set 1 | ||||||||

$L=2$ | $-0.0497$ | $-0.1355$ | $-0.2475$ | $-0.3779$ | $-0.5211$ | $-0.6727$ | $-0.8294$ | $-0.9887$ |

$L=3$ | 0 | $-0.0004$ | $-0.0016$ | $-0.0037$ | $-0.0069$ | $-0.0112$ | $-0.0167$ | $-0.0234$ |

$L=4$ | 0 | 0 | $-0.0000$ | $-0.0000$ | $-0.0000$ | $-0.0001$ | $-0.0003$ | $-0.0005$ |

Parameter set 2 | ||||||||

$L=2$ | $-0.4996$ | $-1.2757$ | $-2.2378$ | $-3.3359$ | $-4.5347$ | $-5.8052$ | $-7.1227$ | $-8.4663$ |

$L=3$ | 0 | $-0.0001$ | $-0.0023$ | $-0.0064$ | $-0.0115$ | $-0.0170$ | $-0.0220$ | $-0.0263$ |

$L=4$ | 0 | 0 | $0.0000$ | $0.0005$ | $0.0017$ | $0.0037$ | $0.0066$ | $0.0105$ |

#### 4.1.3. Numerical Illustration 1: Sensitivity of the Best-Estimate Yield Curve to **λ**

**λ**is proportional to the market risk premium for the associated factor. In this section, we study the impact of increasing the risk premium associated with the first factor. We consider that there are $L=3$ traded bonds in a market governed by Parameter set 1 with different values of

**λ**to assess its influence on the spread between the best-estimate and the no-arbitrage yield curve. The results are presented in Figure 3.

**Figure 3.**Difference between the best-estimate and no-arbitrage yield for times to maturity ℓ from 4 to 10, $L=3$, for different market risk premium parameter λ. (

**a**) ${\lambda}_{2}=0$; (

**b**) ${\lambda}_{2}=15$.

#### 4.1.4. Numerical Illustration 1: Sensitivity to the Number of Factors

Parameters | Parameter set 3 | Parameter set 4 |
---|---|---|

k | $[0.1360,0.1750,0.0500,0.4000]$ | $[0.1360,0.5500,0.2500,0.4500]$ |

b | $[0.0055,0.0005,0.0005,0.0005]$ | $[0.00375,0.0005,0.0005,0.0010]$ |

g | $[0.0070,0.0042,0.0050,0.0015]$ | $[0.0070,0.0075,0.0050,0.0045]$ |

λ | $[8,15,5,5]$ | $[8,15,5,5]$ |

${y}_{0}$ | $[0.003,-0.00025,0.00025,0.00025]$ | $[0.003,-0.00025,0.00025,0.00025]$ |

**Figure 4.**Correlations and yield curves for Parameter sets 3 and 4. (

**a**) Correlation between bond prices for Parameter set 3; (

**b**) Correlation between bond prices for Parameter set 4; (

**c**) Yield curves obtained using no-arbitrage bond prices.

**Figure 5.**No-arbitrage and best-estimate projected yield curves. (

**a**) Parameter set 3; (

**b**) Parameter set 4.

**Table 4.**Differences between the best-estimate and the no-arbitrage yields for different numbers L of traded bonds, for Parameter sets 3 and 4.

Difference between Best-Estimate and No-Arbitrage Yields $(\times {\mathbf{10}}^{-\mathbf{4}})$ | ||||||||
---|---|---|---|---|---|---|---|---|

time to | ||||||||

maturity | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Parameter set 3 | ||||||||

$L=2$ | $0.0028$ | $\phantom{-}0.0174$ | $\phantom{-}0.0499$ | $\phantom{-}0.1040$ | $\phantom{-}0.1822$ | $\phantom{-}0.2855$ | $\phantom{-}0.4141$ | $\phantom{-}0.5679$ |

$L=3$ | 0 | $\phantom{-}0.0014$ | $\phantom{-}0.0063$ | $\phantom{-}0.0174$ | $\phantom{-}0.0367$ | $\phantom{-}0.0664$ | $\phantom{-}0.1078$ | $\phantom{-}0.1615$ |

$L=4$ | 0 | $\phantom{-}0$ | $\phantom{-}0.0001$ | $\phantom{-}0.0006$ | $\phantom{-}0.0016$ | $\phantom{-}0.0034$ | $\phantom{-}0.0063$ | $\phantom{-}0.0100$ |

Parameter set 4 | ||||||||

$L=2$ | $-0.1397$ | $-0.4049$ | $-0.7877$ | $-1.2766$ | $-1.8562$ | $-2.5098$ | $-3.2208$ | $-3.9738$ |

$L=3$ | 0 | $-0.0033$ | $-0.0146$ | $-0.0372$ | $-0.0729$ | $-0.1222$ | $-0.1845$ | $-0.2589$ |

$L=4$ | 0 | 0 | $-0.0003$ | $-0.0010$ | $-0.0026$ | $-0.0053$ | $-0.0094$ | $-0.0149$ |

#### 4.2. Numerical Illustration 2: Life Insurance Inspired

#### 4.2.1. Numerical Illustration 2: Parameter Set

Parameters | Parameter set 5 |
---|---|

k | $[0.1600,0.5214,0.2728]$ |

b | $[0.0060,0.0005,0.0005]$ |

g | $[0.0060,0.0064,0.0042]$ |

λ | $[7.8704,13.8290,4.6956]$ |

${y}_{0}$ | $[0.0079,0.0005,0.0005]$ |

#### 4.2.2. Numerical Illustration 2: Projected Yield Curve

**Table 6.**Differences between the best-estimate and the no-arbitrage yields for different sets of traded bonds, for Parameter set 5.

Difference between Best-Estimate and No-Arbitrage Yields $(\times {\mathbf{10}}^{-\mathbf{6}})$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Time to | ||||||||||

Maturity | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

$\mathcal{M}=\{1,10\}$ | $-1.2626$ | $-3.9880$ | $-8.1578$ | $-13.6365$ | $-20.2351$ | $-27.7488$ | $-35.9780$ | $-44.7389$ | $-53.8692$ | $-63.2290$ |

$\mathcal{M}=\{1,2,10\}$ | $-0.3343$ | $-1.0648$ | $-2.1920$ | $-3.6848$ | $-5.4973$ | $-7.5781$ | $-9.8757$ | $-12.341$ | $-14.9308$ | $-17.6051$ |

$\mathcal{M}=\{1,5,10\}$ | $-0.1594$ | $-0.5152$ | $-1.07332$ | $-1.8229$ | $-2.7437$ | $-3.8115$ | $-5.0009$ | $-6.2870$ | $-7.6467$ | $-9.0592$ |

$\mathcal{M}=\{1,2,5,10\}$ | $-0.0007$ | $-0.0009$ | $-0.0009$ | $-0.0012$ | $-0.0023$ | $-0.0046$ | $-0.0081$ | $-0.0130$ | $-0.0194$ | $-0.0273$ |

## 5. Conclusions

## 6. Proofs

**Proof of Lemma**

**3.**

**Proof of Theorem**

**1.**

**Proof of Proposition**

**1.**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- M.V. Wüthrich. “An Academic View on the illiquidity premium and market-consistent valuation in insurance.” Eur. Actuar. J. 1/1 (2011): 93–105. [Google Scholar] [CrossRef]
- J. Bierbaum, H. Bartel, N. Dennstedt, T. Dillmann, W. Engel, M. Keller, K. Musialik, T. Pauls, N. Quapp, and J. Winter. “Practical valuation of long-term guarantees in inactive financial markets.” Eur. Actuar. J. 4 (2014): 101–124. [Google Scholar] [CrossRef]
- M. Martin. “Assessing the model risk with respect to the interest rate term structure under Solvency II.” J. Risk Financ. 14/3 (2013): 200–233. [Google Scholar] [CrossRef]
- M. Dahl. “A discrete-time model for reinvestment risk in bond markets.” ASTIN Bull. 37/2 (2007): 235–264. [Google Scholar] [CrossRef]
- D. Stefanovits, and M.V. Wüthrich. “Hedging of long term zero coupon bonds in a market model with reinvestment risk.” Eur. Actuar. J. 4/1 (2014): 49–75. [Google Scholar] [CrossRef]
- S. Happ, M. Merz, and M.V. Wüthrich. “Best-estimate claims reserves in incomplete markets.” Eur. Actuar. J. 5 (2014): 55–77. [Google Scholar] [CrossRef]
- European Commission. “Framework Solvency II Directive (Directive 2009/138/EC).” 2009. Avaliable online: http://eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:L:2009:335:0001:0155:en:PDF (accessed on 15 July 2015).
- H. Föllmer, and M. Schweizer. “Hedging by sequential regression: an introduction to the mathematics of option trading.” ASTIN Bull. 18/2 (1988): 147–160. [Google Scholar] [CrossRef]
- S. Malamud, and E. Trubowitz. “The structure of optimal consumption streams in general incomplete markets.” Math. Financ. Econ. 1/2 (2007): 129–161. [Google Scholar] [CrossRef]
- S. Malamud, E. Trubowitz, and M.V. Wüthrich. “Market consistent pricing of insurance products.” ASTIN Bull. 38/2 (2008): 483–526. [Google Scholar] [CrossRef]
- S. Malamud, E. Trubowitz, and M.V. Wüthrich. “Indifference pricing for CRRA utilities.” Math. Financ. Econ. 7/3 (2013): 247–280. [Google Scholar] [CrossRef]
- A. C̆erný, and J. Kallsen. “Hedging by sequential regressions revisited.” Math. Financ. 19/4 (2009): 591–617. [Google Scholar] [CrossRef]
- H. Föllmer, and A. Schied. Stochastic Finance: An Introduction in Discrete Time, 3rd ed. Berlin, Germany: de Gruyter, 2011. [Google Scholar]
- D. Brigo, and F. Mercurio. Interest Rate Models-Theory and Practice: With Smile, Inflation and Credit. Berlin, Germany: Springer, 2007. [Google Scholar]
- M.V. Wüthrich, and M. Merz. Financial Modeling, Actuarial Valuation and Solvency in Insurance. New York, NY, USA: Springer, 2013. [Google Scholar]
- K.B. Nowman. “Estimation of one-, two-and three-factor generalized Vasicek term structure models for Japanese interest rates using monthly panel data.” Appl. Financ. Econ. 21/14 (2011): 1069–1078. [Google Scholar] [CrossRef]

© 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**

MacKay, A.; Wüthrich, M.V.
Best-Estimates in Bond Markets with Reinvestment Risk. *Risks* **2015**, *3*, 250-276.
https://doi.org/10.3390/risks3030250

**AMA Style**

MacKay A, Wüthrich MV.
Best-Estimates in Bond Markets with Reinvestment Risk. *Risks*. 2015; 3(3):250-276.
https://doi.org/10.3390/risks3030250

**Chicago/Turabian Style**

MacKay, Anne, and Mario V. Wüthrich.
2015. "Best-Estimates in Bond Markets with Reinvestment Risk" *Risks* 3, no. 3: 250-276.
https://doi.org/10.3390/risks3030250