Pensions with Stochastic Inflation and
Interest Rate
Xiaojing Shen
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics Author: Xiaojing Shen Student nr: 11595094
Email: yeq.sxj@gmail.com Date: July 9, 2018
Supervisor: Prof. Dr. M.H. (Michel) Vellekoop Second reader: Prof. Dr. R.J.A. (Roger) Laeven Supervisor: MSc. Joeri Potters
Abstract
This paper studies assets portfolio management for inflation-indexed pension obligations under interest rate risk and inflation risk. We ob-tained valuation models for inflation-linked obligations, nominal zero coupon bonds and inflation-linked bonds by assuming stochastic pro-cesses of inflation index, expected inflation rate and nominal interest rate. Assets portfolios for hedging are established by immunizing sensi-tivities and/or convexities on assets side and liabilities side. A numeric analysis on setting up assets portfolios and comparing hedging results are then performed and displayed.
Keywords
Stochastic nominal interest rate; Stochastic inflation; Inflation-indexed pension; Inflation-linked bonds; Asset-liability management
Preface vi
1 Introduction 1
1.1 Case Background . . . 1
1.2 Research Question . . . 1
2 Literature Review 3 2.1 Financial Market with Stochastic Interest Rate and Inflation . . . 3
2.2 Asset-Liability Management with Stochastic Interest Rate and Inflation 4 3 Current Assets and Liabilities 7 3.1 The Insurer’s Current Assets Portfolio and Liabilities. . . 7
3.2 The Insurer’s Current Hedging and Evaluation Method . . . 8
3.2.1 Input Data . . . 8
3.2.2 Formulas . . . 9
3.3 Further Elaboration of the Inflation-linked Pensions and Assets Instruments 10 3.3.1 Cost of Assets Portfolio . . . 10
3.3.2 Exposed Inflation Index . . . 10
3.3.3 Certain Percentage of Inflation Compensation . . . 11
3.3.4 Cap and Floor Structure. . . 11
3.3.5 Liquidity Issue . . . 11
3.3.6 Credit spread . . . 11
3.3.7 Inflation-linked Instruments . . . 12
4 An Extension of BX’s Model - Stochastic Inflation and Nominal In-terest Rate 14 4.1 The Financial Market: Index and Ratios . . . 15
4.2 Inflation-linked Pension Obligations . . . 16
4.3 Assets Portfolio . . . 18
4.3.1 Nominal zero-coupon Bond . . . 19
4.3.2 Inflation-linked Bond. . . 19
5 Hedging Strategy 21 5.1 The Insurer’s Current Practice . . . 21
5.2 Proposed Alternative Approaches . . . 22
5.2.1 Approach 1 . . . 22 5.2.2 Approach 2 . . . 23 6 Numerical Results 29 6.1 Model Fit . . . 29 6.1.1 Parameters . . . 29 6.1.2 Model Validation . . . 31 6.2 Portfolio Setup . . . 31 iv
6.2.1 Approach 1: Matching with Present Value, Expected Inflation Rate Sensitivity and Nominal Interest Rate Sensitivity . . . 32
6.2.2 Approach 2: Matching with Expected Inflation Rate Sensitivity, Nominal Interest Rate Sensitivity and Monetary Convexity . . . 33
6.3 Hedging Quality Analysis . . . 34
7 Conclusion 36
Appendix 37
I would like to express my sincere gratitude to all kinds of helps from the people who supported me through my studies, and in particular: to my university supervisor Prof. Michel Vellekoop for his insightful su-pervision and advising; to my supervisor Joeri Potters from the insur-ance company for his plenty advices and comments; to the colleagues for sharing useful feedbacks and information; to my classmates for spending one of the most important time periods in our life together; to my husband and parents, for their love and constant support.
Introduction
1.1
Case Background
This paper conducted a study on the inflation risk hedging for inflation-linked pensions at the request of one of the largest insurance companies in the Netherlands. This in-surance company (we call “the insurer”) owns several inflation-linked pension products whose obligations grow with certain percentages of the Consumer Price Index (CPI). Compared with other available pension products which the pension participants bear inflation risk, the promised indexation on inflation switches inflation risk to the insurer. When compared with the guaranteed indexation on pension benefits, the inflation-linked obligations bear the risk of stochastic inflation. Hence, the insurer seeks proper asset allocation strategies to hedge the interest rate risk as well as inflation risk.
To manage those inflation-linked obligations, the insurer initially allocated its assets on several inflation-linked notes, which had customized structures (such as specific Con-sumer Price Index, certain percentages of inflation compensation, specific cap and floor structure on inflation rate) to fit with pension contracts, offered by some banks. Later on, with the change of financial markets, the assets portfolio did not follow the liabilities perfectly. Thus, assets portfolio adjustment was required. When doing the adjustment, the insurer added inflation-linked bonds into the assets portfolio. Recently, some of the inflation-linked notes went mature. The insurer would like to check whether they need to do reinvestment after the maturity. At the meantime, it was noticed that the inflation-linked pensions were facing “over-hedged”1 problem. In this case, the insurer launched a research on hedging strategy for inflation-linked pensions with stochastic inflation and interest rate.
1.2
Research Question
For inflation-linked pensions, the insurer aims to hedge against interest rate risk and inflation risk simultaneously with available financial instruments in the market. In other words, the insurer’s hedging target is to immunize the risk of stochastic interest rate and inflation on an inflation-indexed asset-liability portfolio. According to the hedging target, the research objectives in this article are listed as below.
Firstly, one of the most important objectives is to set up a valuation model under stochastic interest rate and inflation for the inflation-linked obligations and correspon-dent asset instruments. The content regarding model construction is mainly in Chapter
4. 1
Over-hedged means the dollar duration value of assets portfolio over that of liabilities exceeds a certain threshold.
Secondly, another objective is to establish a dynamic hedging portfolio. As the inflation-linked notes, which account most assets portfolio, are illiquid, the insurer’s current hedging strategy is rather static and restricted. Hence, the insurer wants to improve the current hedging strategy by using a more active and dynamic method. In this article, stochastic interest rate and inflation will be applied to asset-liability management anal-ysis so that a dynamic hedging portfolio can be established to immunize both inflation and interest rate sensitivity. Chapter5 discusses this part.
Thirdly, a comparison of the hedging performances between the proposed dynamic hedging portfolio and the insurer’s current hedging portfolio is needed. The insurer would like to see the evaluation on both dynamic hedging portfolio and current hedging portfolio, so that they will be able to tell, under stochastic interest rates and inflations, which method delivers better hedging results. Therefore, an analysis of hedging results will be performed under different time-step movements for both methods. This analysis is illustrated in the section 6.3.
Literature Review
2.1
Financial Market with Stochastic Interest Rate and
Inflation
The researches regarding stochastic interest rate risk have been deepened and applied in practice continuously since Merton (1975) (11). The studies combining inflation risk, however, are still under development. The practices and strategies of inflation hedging, especially for insurance companies, have not been well developed yet. However, as a long run investor, the insurance companies actually have huge demands on mature studies of inflation risk and effective approaches to inflation risk controlling. In the late 1990s, Santomero et al. (1999) (16) studied several annuities obligations with inflation of UK and US on individual perspective. They evaluated potential asset instruments includ-ing inflation-linked bonds to protect the risk from inflation. Campbell and M. Viceira (2001) (4) gave the concept to invest in long-term inflation-linked bond to mimic similar long-term consumption. This concept is close to the practice of the research question in this article. Campbell and M. Viceira (2001) (4) and the extension Campbell et al. (2003) (3) applied AR(1) model to monitor log real interest rate and log expected rate of inflation when they analyzed assets allocation under inflation.
Brennan and Xia (2002) (2) analyzed the assets allocation problem of a long run but finite-lived investor who can only invest in nominal assets (i.e. stock, cash and nomi-nal bonds). It applied diffusion process on the stock price level and Ornstein-Uhlenbeck process on the expected rate of inflation and real risk-less interest rate with constant the risk premia. The research attempted to get the optimal assets allocation by maximizing two different utility of total wealth.
The conditions our insurer is facing is similar to Brennan and Xia (2002) (2) consid-ered. First, the insurer is a typical long run but finite-lived investor. Second, the pension obligations are exposed to the risk of stochastic interest rate and inflation. Therefore, Brennan and Xia’s (2002) (2) gives a nice reference when we generate the pricing model for the insurer’s case.
However, there are also differences between Brennan and Xia (2002) (2) and the in-surer’s case. First of all, Brennan and Xia (2002) (2) applied utility function to generate optimal consumption, and thus optimal assets allocation. In the insurer’s case, the obli-gation cash flows without inflation (CFtN L) are determined by the insurer’s internal models. It is not necessary to generate an optimal consumption of cash flows and to determine a risk aversion level. Hence there is no need to introduce utility functions in this article.
Secondly, Brennan and Xia (2002) (2) assumed that only nominal assets were
able to be invested, while in the insurer’s case, real assets (e.g. inflation-linked bonds) are used to construct hedging portfolio. This change leads to a different starting point of analyses. In Brennan and Xia (2002) (2), the nominal value of assets and liabilities are determined (the nominal asset cash flow is 1 of zero-coupon bond; the nominal li-ability cash flow is the optimal consumption) in the model and a real discount factor is used so that the “real return” of the portfolio is analyzed. On the insurer’s side, the originally determined cash flows of obligations are the “real cash flows” (we will use CFN L
t to represent it in the model later on). The actual payoff of the pension should
be the obligation cash flows without inflation plus the compensation of inflation (i.e. obligation cash flows with inflation CFtL). Therefore, the nominal value of cash flows is stochastic which are generated by adding inflation indexation on the “real cash flows” (CFtN L). On the assets portfolio side, to get the better match off of inflation hedging, inflation-linked instruments (in this article, inflation-linked bonds “ILB”) are introduced to replace stock (i.e. real assets also available). When being considered in utility func-tion and risk aversions, the hedging objective of the insurer leads to a risk aversion approaching positive infinity. As a result, there will be no allocation on equities. The determined cash flows of inflation-linked bonds are also the “real cash flows” with fixed coupon payment each year and 1 at maturity. Nominal cash flows will be generated by adding inflation compensation on them. Besides, nominal bonds (we use zero-coupon bonds “ZCB”) are kept to fulfill interest rate hedge. The analysis of the zero-coupon bond shall keep the same as Brennan and Xia (2002) (2) did as only nominal cash flows are determined.
Thirdly, Brennan and Xia (2002) (2) introduced real pricing kernel (Mtreal). By ap-plying Fisher’s Equation, it is not hard to switch between nominal interest rate and real interest rate with the expected inflation rate. However, in practice, the actual rela-tionships among those three rates does not perfectly fit Fisher’s Equation. Munk et al. (2004) (13) set nominal interest rate as the factor and it turns out that inflation does not affect the nominal term structure. Chiarella et al. (2007) (5) considered valuation in both real and nominal term, Chiarella et al. (2007) believed that it was not wise to use real pricing kernel in a world without real rate. This assumption requires a “frictionless and efficient transactions”. In addition, Chou et al. (2011) (6) and Slipsager (2018) (17) also gave researches with nominal interest rate process. They provided a nice reference for us as the insurer is doing the valuation in nominal terms although both pension obligations and assets portfolio deliver a real payoff. We will introduce nominal pricing kernel (Mnom
t ) rather than the real one as Brennan and Xia (2002) (2) did to analysis
the price of assets and liabilities. The reason will be elaborated in Chapter4.
2.2
Asset-Liability Management with Stochastic Interest
Rate and Inflation
Regarding to assets allocation, Santomeroet al. (1999) (16), Campbell and M. Viceira (2001) (4), Munk et al. (2004) (13), Chiarella et al. (2007) (5), Jong (2008) (7), Chou et al. (2011) (6), Kmak and Lim (2014) (10) and many other researches later on intro-duced utility functions as Brennan and Xia (2002) (2) did. Some of them constructed consumption streams while others generated optimal portfolios with different risk aver-sions. Recently, Mkaouar et al. (2017) (12) adopt the model introduced by Chiarella et al. (2007) (5) and determined the optimal asset allocation by maximizing the utility function. Different from Brennan and Xia (2002) (2) which generated an optimal con-sumption stream, Mkaouar et al. (2017) (12) only maximize the utility on the real value of asset portfolio to determine the weight on each asset component. In the insurer’s case, the pension obligation is treated as determined beforehand. And the target is to hedge interest and inflation risk of indexed obligation. Therefore, it is not necessary to
introduce utility function and generate optimal consumptions. The determined pension obligation can be treated as given consumptions and risk reversion (γ) of the hedging strategy, as no equity will be included, can be taken as γ → ∞.
J Pan and Q Xiao (2017a; 2017b) (14) (15) did optimal asset-liability management analysis under stochastic interest rate and inflation risks. They did it in two methods. One is applying the expected utility maximization framework and the other is using mean-variance analysis. J Pan and Q Xiao (2017a) (14) generated a surplus function using assets portfolio minus liabilities. It considered real surplus by dividing inflation pricing level from nominal asset portfolio and liability. They maximized the expected value of real surplus at time T by constraining of initial surplus larger than 0. J Pan and Q Xiao (2017b) (15) apply the mean-variance framework to manage asset and liability. In the mean-variance framework, they minimized the variance of real wealth (assets minus liabilities). However, in our model, we consider only the nominal surplus. Siegel and Waring (2004)(1) implemented inflation duration and real interest rate dura-tion matching to hedge pension fund obligadura-tions. As the duradura-tion of an asset portfolio is a linear combination of the duration of all asset components with their weights respec-tively, it is easy to implement duration matching and thus form the hedging portfolio. The concept of expected inflation rate duration and nominal interest rate duration will also be introduced in this article. However, Siegel and Waring (2004)(1) applied Fisher’s Equation and argued the inflation sensitivity (i.e. inflation duration) of inflation-linked bonds should be zero. It concluded that the hedging portfolio for a pension fund with full inflation indexation is to invest entirely of TIPS1. Many other kinds of research reach a similar conclusion. Campbell and Viceira (2001) (4) proved also that for the rather risk-averse investors, long-term inflation-indexed bonds in the portfolio increases significantly. In Brennan and Xia (2002), although they assumed only nominal assets were available in the market, they also suggested that a conservative investor should use a mix of nominal bonds to replicate the return of inflation-linked bonds. Chou et al. (2011) (6) and Chiarella et al. (2007) (5) both stated that the infinitely risk-averse investor would prefer to invest all her wealth in inflation-indexed bonds, maturing at the investment horizon. However, compared with Siegel and Waring (2004)(1), they didn’t give any clue about the tenor to choose for the inflation-linked bonds. In Siegel and Waring (2004)(1), on the other hand, by setting inflation duration of inflation-linked bonds to be zero, real interest rate duration of inflation-linked bonds to be equal to its nominal interest rate duration, solution was suggested to the tenor, which is able to satisfy the condition that the nominal interest rate duration of assets equaled to that of liabilities. In the real world, if the Fisher’s Equation does not hold perfectly, this result cannot be true. In this article, Fisher’s Equation is not applied, thus the assets portfolios are set up under a more realistic condition.
Han and Hung (2012) (18) focused on the optimal investment problem for the de-fined contribution (DC) pension plan with the risks of interest rate and inflation. They reached a conclusion that the inflation-indexed bond is indispensable for the pension plan to hedge against the inflation risk. And it is also crucial to provide a downside protection with the annuitants.
Gajek et al. (2005) (8) discussed classical immunization method. They stated the method to immunize the first and second order sensitivity on risk factors, particularly for in-terest rate risk. In this article, their concept of immunization will be borrowed when establishing a hedging portfolio. And the method they used will be extended to the inflation rate risk.
Further studies such as Hosseinzadeh (2017) (9) considered asset-liability management with correlation among economic cycle, inflation rate, and interest rate. It assumed stochastic correlation among each investing instrument (fixed income, commodity, and equity). We are not going to go as deep as Hosseinzadeh (2017) (9) did, but it is nice to refer to if one would like to extend our model and consider inconstant correlation among the innovations of all the processes.
Current Assets and Liabilities
3.1
The Insurer’s Current Assets Portfolio and Liabilities
The pension obligation cash flows without the impact of inflation (CFtN L), which shall be similar to the cash flows of normal pension products, could be generated via certain models with risk factors such as mortality rate, longevity rate, expected wages increas-ing rate, inflation etc. The cash flows are available by runnincreas-ing the insurer’s separate model systems. The actual payoff of the inflation-linked pensions should be the summa-tion of those obligasumma-tions (CFtN L) and the uncertain inflation compensations on those obligations. Therefore, having been given those pension obligations, the insurer is facing long-term uncertainty on Consumer Price Index change and interest rate change. The stochastic changes on CPI level and interest rate will give variation to the estimated value on pension liabilities.
The different Consumer Price Index data that the pensions linked to is rather regional re-lated. In current pension obligations, most pension accounts are indexed with HICPxT1,
which is one of the consumer price indexes of the European region. While the rest part of the pension accounts is compensated by the increase of Dutch inflation (CPI issued by CBS2).
Besides, the scale of inflation compensation for each inflation-linked pension account is not identical as well. Most of the pensions provide 100% indexation, while the others give a compensation of a certain percentage less than 1. Regarding the range of inflation compensation, each pension contract states respective maximum inflation rate it will cover.
On the assets portfolio side, the assets allocation on inflation-linked instruments en-abled the insurer a certain extend of management on assets and liabilities. The insurer is currently investing on linked notes and linked bonds. The inflation-linked notes, which are customized, have the corresponding characteristics imitating the features of pension contracts. The features include the following items: which Consumer Price Index the pension follows, what the percentage of indexation is, and what the maximum inflation rate the pension would compensate. The inflation-linked bonds, on the other hand, are all linked with HICPxT, have 100% indexation and have no cap of the maximum inflation rate. The inflation-linked notes only provide periodic inflation compensation while the inflation-linked bonds give cumulated inflation compensation. This can be observed in the formulas of assets payoffs in 3.2.2.
1
The Harmonised Index of Consumer Prices Exclude Tobacco 2The Central Bureau for Statistics of the Netherlands
3.2
The Insurer’s Current Hedging and Evaluation Method
The insurer tried to match the total dollar duration of assets and liabilities. The total present value and dollar duration on both the investments side and the obligations side are calculated and compared quarterly with market estimated inflation and interest rate term structures. A certain level of threshold was set up to control the nominal interest rate risk. When the difference of dollar duration between assets and liabilities breaches that certain threshold, adjustment on assets portfolio should be executed to eliminate the breach.
3.2.1 Input Data
a. Pension Obligation Cash Flows without inflation (CFt+iN L, i= 1, ..., N) The obligation cash flows without inflation are estimated and generated by the insurer’s separate model. In the insurer’s current hedging analysis, the cash flows generated on each valuation day (”t”) are taken to be deterministic.
The insurer assumes the payment days of pension obligation cash flows are on the last day of each year, ignoring the actual date of cash flows occur.
b. Inflation Term Structure
The insurer is using deterministic expected inflation term structure estimated by the market data (inflation-linked bonds and inflation-linked swaps). The expected inflation term structure comes from the market value of inflation zero rates (πt+i) with an
extrap-olation method where the inflation rates will be extrapolated to the long-term target inflation 2% (i.e. the ECB3 long-run target inflation ratio).
c. Nominal Interest Rate Term Structure
The insurer is using latest nominal interest rate term structure estimated by market value. The term structure is constructed with zero interest rate (Rt+i) of each year with
i from 1 to 20 and long-run target interest rate under economic framework assumption.
d. Credit Spread of Bonds and Notes (st)
The most updated credit spread of each bond and notes are obtained from the market.
e. Economic Terms of Bonds and Notes According to the Term Sheet Principal (P ), coupon rate (c%), cap and floor (πup, πdown), percentage of indexation
(ω%), issue date (I), coupon payment date (t), maturity date (M ), consumption price level (base index) on start day (ΠI) are all defined in the term sheets of the
inflation-linked bonds and inflation-inflation-linked notes.
f. CPI level on evaluation day (Πt)
HICPxT or CBS Dutch inflation index as of valuation day (“t”) should also be obtained from the market.
3.2.2 Formulas
The insurer is using a static and discrete method to calculate the present value and dollar duration of assets portfolio and pension obligations.
a. Present Value of Pension Obligations
As mentioned above, the obligation cash flows are valued on a yearly basis. The formula being used to calculate the present value at each valuation day (“t”) of all the pension obligations is:
P VtL=
N
X
i=1
CFt+iN L∗ [1 + ω% ∗ max(min(πt+i, πup), πdown)]i
(1 + Rt+i)i
, (3.1)
where N is the largest tenor of estimated pension obligation cash flows. The N insurer applies is 80.
b. Present Value of Inflation-linked Bonds (ILB)
The insurer values assets on monthly basis. That is, it generates monthly zero inflation rate as well as monthly zero interest rate and values the payoffs on different months. Here, to simplify the problem, we will consider yearly valuation as obligations do. There-fore, the formula the insurer is using to calculate the present value of inflation-linked bonds valuing at each valuation day is:
P VtILB= M X i=1 P ∗ c% ∗Πt ΠI ∗ (1 + πt+i) i (1 + Rt+i+ s)i +P ∗ Πt ΠI ∗ [1 + max(πM, 0)] M (1 + RM + s)M , (3.2) where M is the maturity day of the inflation-linked bond.
c. Present Value of Inflation-Linked Notes (ILN)
The formula the insurer is using to calculate the present value of inflation-linked notes valuing at each valuation day is:
P VtILN =
M
X
i=1
P ∗ [1 + c% + ω% ∗ max(min(πt+i, πup), πdown)]i
(1 + Rt+i+ s)i
+ P
(1 + RM + s)M
(3.3) where M is the maturity day of the inflation-linked note.
According to the contract, the final returned principal amount of the inflation-linked note is not indexed by inflation index and the coupons every year are compensated by the inflation increase on that year (Πt+i+1
Πt+i ). However, the insurer assumes the coupons are re-invested in the same asset. Therefore, the equation 3.3is applied.
d. Dollar Duration (DD) DDtL= N X i=1 P VtL∗ (t + i) (1 + Rt+i)/(1 + πt+i) DDILBt = M X i=1 P VtILB∗ (t + i) (1 + Rt+i)/(1 + πt+i) + s DDILNt = M X i=1 P VtILN ∗ (t + i) (1 + Rt+i)/(1 + πt+i) + s (3.4)
where (1 + Rt+i)/(1 + πt+i) is the estimated real zero rate.
3.3
Further Elaboration of the Inflation-linked Pensions
and Assets Instruments
To narrow the research question for this article, some features of inflation-linked pen-sions and assets instruments should be elaborated and explored more deeply.
3.3.1 Cost of Assets Portfolio
The funds available to hedge the pension obligations are not the main concern in this article. Currently, the present value of the insurer’s assets portfolio is much larger than that of the liabilities. As long as the assets portfolio proposed has less or similar assets cost, it will not be a problem. Therefore, this article will mainly focus on the valuation model and assets portfolio setting up to replicate pension obligations rather than the optimal asset allocation under the budget constraint.
3.3.2 Exposed Inflation Index
As mentioned in 3.1, some inflation-linked pensions are exposed to EU inflation (i.e. HICPxT index level) while the others are indexed with Dutch inflation (i.e. CPI issued by CBS). An ideal hedging usually means to construct a portfolio with the instru-ments of the identical underlying inflation index. However, there are no specific Dutch inflation-linked instruments available in the liquid market. Doing OTC transactions is possible. The insurer invested in some customized inflation-linked notes which offer the same indexation percentage and matched underlying inflation index.
When doing an evaluation, the insurer applies an expected inflation term structure on both European inflation-linked and Dutch inflation-linked pensions cash flows. This expected inflation term structure comes from the market data of inflation-linked instru-ments. Therefore, it only indicates the European inflation. The expected inflation term structure for Dutch inflation is not available.
Considering the high correlation (0.9946 from 1996 to 2017) between HICPxT price level and CBS Dutch CPI, we will introduce one stochastic inflation model only and we will allocate assets on HICPxT linked instruments to hedge all the pensions. That is, we are under the assumption that HICPxT price level and CBS Dutch CPI are fully correlated.
3.3.3 Certain Percentage of Inflation Compensation
Besides, not all the pensions are 100% indexed on inflation. A small part of the inflation-linked pensions has certain percentages (e.g. 85.2%) of compensation on inflation. ω% is used to represent this percentage. To match with the certain percentages, the insurer previously chose customized inflation-linked notes as the hedging instrument. Those notes have the same percentages of compensation on the same underlying CPIs. By applying the delta-gamma hedging strategy, choosing the assets with exact same com-pensation percentage is not necessary. As long as the inflation sensitivity and convexity on both investments side and obligations side are the same, the inflation risk is immu-nized no matter what the compensation percentage is. Therefore, it is feasible to get rid of customized inflation-linked notes considering the different percentage of inflation compensation.
3.3.4 Cap and Floor Structure
There are cap and floor structures on inflation-linked pensions. On one hand, all the pensions are protected by deflation (i.e. a floor of inflation rate 0%), which means the pension participants will not get a negative impact on their pensions even though defla-tion happens. On the other hand, each pension has a specific rate of maximum infladefla-tion compensation (i.e. a cap inflation rate: cap%). If the inflation rate goes above the cap, the pension participants will not get exceeding compensation.
On the assets side, all inflation-linked bonds and notes have deflation protections. The linked bonds don’t have a cap on inflation compensation while the inflation-linked notes do. This implies that the inflation-inflation-linked bonds give more prudent manage-ment on inflation risk hedging. Therefore, getting rid of inflation-linked notes will not lead to more risk if the insurer switch the assets portfolio to inflation-linked bonds. However, the inflation-linked swaps do not have deflation protection. As a result, if the insurer considers hedging with linked swaps, it is necessary to include inflation-linked options (e.g. inflation-inflation-linked zero floors) at the meantime to hedge deflation risk. This will also be discussed in section3.3.7.
In our model set up in 4, the cap, and floor structure are ignored. The model can be extended by adding the embedded option.
3.3.5 Liquidity Issue
The financial market for inflation-linked instruments is not as advanced and liquid as it is for nominal interest rate instruments. The development of the inflation-linked bonds market in the EU is even slower compare to the markets in the US and UK. But since the insurer’s inflation-linked pension liability portfolio is rather small (comparing with normal pension products), we take the market of inflation-linked instruments has suf-ficient liquidity. The research could be further extended by considering the condition of a less liquid market, so that it can still be applied even when the inflation hedging demand is huge.
3.3.6 Credit spread
According to the insurer’s current valuation formula in 3.2.2, the present value of inflation-linked instruments is not only depended on the expected inflation rate and
nominal interest rate but also depended on the credit spread of the bond issuer. In this article, however, it is assumed that the inflation-linked bonds are default free. As a result, the credit spread is not included in the valuation model we obtain. The model can be promoted by including stochastic credit spread simultaneously with stochastic interest rate and inflation.
3.3.7 Inflation-linked Instruments
Having considered replicating inflation compensation, the insurer may choose to invest in inflation-linked instruments including inflation-linked bonds (ILB), inflation-linked swaps (ILS), inflation-linked options (ILO) etc. The European market for those instru-ments are not as liquid as nominal assets, but it is relatively liquid. Although in this article, we only take inflation-linked bonds as the real assets to hedge the real liabilities, it is nice to illustrate the alternative instruments the insurer can use. One can extend the research by modeling those alternative instruments and including them into the assets portfolio. Below we state the cash flows structure of each inflation-linked instrument (we assume all principals equal to 1).
a. Inflation-linked Notes (ILN)
As mentioned above, to hedge with specific underlying CPI or certain inflation index-ation ratio, the insurer invested on several inflindex-ation-linked notes. As notes were cus-tomized and offered by banks, in principle they can have any specific cash flow struc-tures negotiated by two counter-parties. All the notes the insurer invested has similar pay-offs however. The pay-offs of the inflation-linked notes the insurer holds look like below.
The coupon cash flows of inflation-linked notes are:
Coupon P aymentILNn = c% + ω% ∗ M in[M ax[0%, Πn Πn−1
− 1], cap%],
where c% is the fixed coupon rate determined when the note was issued; ω% is the inflation compensation percentage mentioned in subsection3.3.3; cap% is the maximum compensation on inflation stated in 3.3.4 ; n = 1, 2, ..., N, which represent the time period; N is the tenor of the notes determined on the contract; Πn is the consumer
pricing level at time period n.
The maturity redemption happens on the maturity day T for inflation-linked notes is:
M aturity redemptionILN = 1.
The inflation-linked notes have certain disadvantages. First, it is not liquid in the market. The insurer therefore cannot adjust its hedging portfolio actively. Second, the transac-tion cost of such customized instruments is higher than the standard instruments. Fur-thermore, to match up with the cumulating inflation indexed obligations, the coupon payments on each period should be reinvested immediately into the same inflation-linked instrument. But obviously it is not feasible.
b. Inflation-linked Bonds (ILB)
Inflation-linked bonds offer cumulated inflation compensation on their principals. The link with the real yields allows the inflation-linked bonds to effectively incorporate ex-plicit real returns into a portfolio. The bonds also offer protection on deflation which
provide good fit to the pension structures just as the inflation-linked notes. As they are standard and tradable in the market, they are more liquid compared with inflation-linked notes.
The coupon cash flows of inflation-linked bonds are: Coupon paymentILBn = c% ∗Πn
ΠI
,
where c% is the fixed coupon rate; n = 1, ..., N , represents the coupon payment period; Πnis the inflation index level at time n; ΠIis the “base index”, which indicates inflation
index level when the bond was issued, hence ΠI+1= Π1.
The maturity redemption for inflation-linked bond is: M aturity redemptionILB = max[ΠT
ΠI
, 100%],
c. zero-coupon Inflation-linked Swap (ILS)
As of inflation-linked zero-coupon swap, it has no deflation protection. The market of inflation-linked zero-coupon swap is liquid. It has a simple cash flow on maturity day:
At maturity : ΠN ΠI
− (1 + s%)N
in which s% is the Swap rate determined when the Swap was issued and it can also be a floating ratio such as (LIBOR + Spread).
By investing on inflation-linked Swaps, one switches the floating indexation (indexed with inflation) to a fixed indexation (i.e. fixed swap rate s%) or another floating index-ation (e.g. LIBOR + Spread).
There are Year-on-Year swaps which have similar pay-offs every year.
d. Zero Option (example: Inflation-linked Zero Cap) (ILO)
As mentioned above, since inflation-linked swap does not protect against deflation, one should construct the asset portfolio with inflation-linked option to hedge deflation risk. We take inflation-linked zero cap as an example. The cash flow of inflation-linked zero cap looks like:
At maturity : M ax[0%,ΠN ΠI
− (1 + c%)N], in which, c% is the cap ratio of the inflation-linked zero cap.
The inflation buyer normally need to pay a upfront or forward premium for the op-tion. There are also Year-on-Year Swaps and options which give periodical settlements base on the pricing level (Πt) on that period.
An Extension of BX’s Model
-Stochastic Inflation and Nominal
Interest Rate
As mentioned in section2.1, Brennan and Xia (2002) (2) did asset allocation for a long run but finite-lived investor under stochastic inflation and interest rate condition. The insurer shares the same condition with that in BX’s 1 world. The insurer is a typical long run but finite investor and the inflation-linked pensions held by the insurer have the same risk drivers of stochastic interest rate and inflation. Therefore, in this section, we will apply the valuation methodology introduced by BX to generate our financial markets and relevant prices of the assets and liabilities.
However, some changes should be made to the BX’s model. Instead of modeling real in-terest rate as BX did, we would model with nominal inin-terest rate and generate nominal pricing kernel. There are several reasons for supporting us to do so.
Firstly, the purpose of the research is different. BX assumed there were only nomi-nal assets available in the market, and the real pricing kernel was used to calculate the real price of assets. When they do the asset allocation, real interest rate risk and infla-tion risk were treated as two risk drivers. However, in the insurer’s case, real assets were introduced as the main instruments to replicate inflation-indexed obligations (CFtL), which aims to establish hedging portfolio for the nominal value of real assets and lia-bilities. Intuitively it is preferred to use a nominal pricing kernel to evaluate nominal prices of those assets and pension obligations.
Secondly, by applying the nominal pricing kernel, it is not necessary to apply Fisher Equation. Therefore, assumptions for the Fisher Equation can be released. As discussed in section2.1, it is no longer required to set the restrict of the relationship among the nominal interest rate, the real interest rate and the expected inflation rate.
Thirdly, some of the pensions are only compensated with a certain percentage of infla-tion (please refer to equainfla-tion (4.8)). Hence, the present value of the obligations cannot be computed easily as the pension obligation cash flows without inflation indexation (CFtN L) discounted by real pricing kernel. Transforming the real pricing kernel for partially compensated obligations does not make much sense. While applying inflation pricing kernel and nominal interest rate pricing kernel instead is easier to interpret. Moreover, the insurer is now using a static method to do a quarterly valuation in which
1
BX indicates Brennan and Xia (2002) (2). This abbreviation will be used in this chapter for sim-plicity
a deterministic expected yield curve is applied. The market yield curve of real interest rate is not available, while the yield curve of nominal interest rate is.
Therefore, it is more reliable for the insurer to use nominal pricing kernel rather than constructing a real rate.
Apart from setting different interest rate process, some other changes have been made compared with BX’s model. As mentioned above, we assume the real assets are available and there is ample liquidity in the market. In addition, stocks are not suitable hedging instruments according to the insurer’s risk appetite. Therefore, they are not included in the financial market analysis as well as the assets portfolio.
4.1
The Financial Market: Index and Ratios
We assume Consumer Price Index level Πt follows a geometric Brownian motion, in
which the volatility of price level σΠis constant and the expected inflation rate πtfollows
an Ornstein-Uhlenbeck process. σΠ can be taken as the magnitude of the unexpected
inflation shock. The volatility of the expected inflation rate σπ is assumed to be constant
as well. The process can be illustrated as below: dΠt
Πt
= πtdt + σΠdZtΠ (4.1)
dπt= α(π − πt)dt + σπdZtπ (4.2)
where π describes the long-run mean of the rate of expected inflation, and α is the degree of mean-reversion.
As for the interest rate process, as Munk et al. (2004) (13) and Slipsager (2018) (17) did, a nominal interest rate process is introduced to our model. We assume the nominal risk-free interest rate (Rt) also follows an Ornstein-Uhlenbeck process. The innovation
term of the nominal interest rate process is correlated with that of expected inflation rate process, (i.e. dZtπ and dZtR are correlated that dZtRdZtπ = ρRπdt). The correlation
ρRπ is assumed to be constant. Therefore, we have:
dRt= κ(R − Rt)dt + σRdZtR (4.3)
where R is the long-run mean of nominal interest rate. κ describes the degree of mean-reversion. σR represents the nominal interest rate volatility and is assumed to be
con-stant.
The nominal pricing kernel (Mtnom) which represents time value on nominal interest rate is stated as below. Although the insurer applies a different framework of discount-ing rates dodiscount-ing the valuation, we assume an identical pricdiscount-ing kernel in our model.
dMtnom Mtnom = −Rtdt + λRdZ R t + λπdZtπ+ λvdZtv = −Rtdt + ΛR 0 dZtnom+ λvdZtv (4.4)
The drift of the pricing kernel is the negative value of the instantaneous nominal risk-free rate Rt. And we apply constant risk factors λR, λπ and λv associated with innovations
dZtR, dZtπ and dZtv. dZtv is orthogonal to dZtnom= (dZtR, dZtπ)0. ΛR= (λR, λπ)
0
. The risk parameter λRand λπ shall be estimated by various nominal bonds and inflation-linked
(17) estimations. dZtv is not hedgable in the market and therefore λv cannot be
esti-mated by market data.
Similarly to the nominal pricing kernel, we can write consumption price index Πtwith:
dΠt Πt = πtdt + σΠdZtΠ = πtdt + ξRdZtR+ ξπdZtπ+ ξvdZtv = πtdt + ξR 0 dZtnom+ ξvdZtv (4.5) in which ξR= (ξR, ξπ) 0 .
We borrow the statements regarding “complete market” and “incomplete market” from BX. When the expected inflation rate is not observable in the market and can only be generated from realized Consumer Price Index, then the change of expected inflation rate (dπt) is perfectly depended on the change of Consumer Price Index (dΠΠtt). In that
case, the innovation of unexpected risk inflation given in equation 4.1will be fully cor-related with the innovation in the expected rate of inflation given by equation4.2. That is dZtπdZtΠ = dt, in which ρπΠ = 1. It is considered as “complete market” condition.
Under “complete market” condition, ξR = ξv = 0, so that the change on Consumer
Price Index can be indicated fully on the expected inflation rate πt, and the risk dZtΠ
can be explained by dZtπ.
However, generally speaking, the expected inflation rate is not fully correlated with stochastic Consumer Price Index. This can be observed from the parameter estimations in previous literature, where ρπΠ were estimated to be close to 0 instead of 1. Therefore,
the “incomplete market” condition is more suitable to the real financial market. Under “incomplete market” condition, the innovation of stochastic Consumer Price Index pro-cess dZtΠis written as the linear combination of innovation vector dZtnomand orthogonal innovation dZtv as we do in equation 4.5.
Apply Itˆo’s lemma to dΠt and dMt, we get:
Msnom Mnom t = exp[ Z s t −Rτ−1 2(Λ R0ρRΛR+ λ v2)dτ + Z s t ΛR0dZτnom+ Z s t λvdZτv] (4.6) Πs Πt = exp[ Z s t πτ − 1 2(ξ R0 ρRξR+ ξv2)dτ + Z s t ξR0dZτnom+ Z s t ξvdZτv], (4.7) where ρR= 1 ρRπ ρRπ 1 .
4.2
Inflation-linked Pension Obligations
As mentioned in Chapter3, the pension obligation cash flows without inflation (CFtN L) are estimated and generated by the insurer’s separate models. As our research mainly focuses on inflation and interest rate hedge, in this section, we will assume the obliga-tion cash flows without inflaobliga-tion are deterministic. But in Chapter5 when we propose hedging strategy, this assumption will be extended and the impact of realized Consumer Price Index level on pension obligation cash flows will be considered further.
The actual obligation cash flows (CFL
t ) are determined by the sum of obligation cash
flows without inflation and the inflation compensation in that period. The inflation compensation in one period can be calculated as a constant compensation percentage (ω%, which is stated in each pension contract) multiplies by the inflation rate within
a certain range (i.e. a cap and floor of the inflation ratio. The cap and floor structure are predetermined in pension contracts). To simplify our research, we will ignore the structure of the cap and floor on obligation cash flows when generating the valuation model. The result of data analysis will not be changed by such elimination since the inflation ratio has never grown outside the cap. For further study, one may take the cap and floor the structure into consideration to complete the analysis.
As a result, the actual pension obligation cash flows (CFL
t+i) at time t + i (i ∈ [0, T − t])
without a cap and floor structure will be:
CFt+iL = CFt+iN L∗ (1 + ω% ∗ πi∗)i, or CFt+iL = CFt+iN L∗ (1 + ω% ∗ ((Πt+i
Πt
)1i − 1)))i,
(4.8)
in which ω% is a promised constant compensation percentage of inflation, πi∗ is the i time period zero inflation rate from time t satisfying (1+π∗i)i= (1+πt)(1+πt+1) · · · (1+
πt+i−1), and Πtis the price level at time t.
If the promised compensation percentage ω = 1, then the actual obligation cash flows would be simplified as:
CFt+iL = CFt+iN L∗ (1 + πi∗)i, or CFt+iL = CFt+iN L∗ Πt+i
Πt
(4.9)
The total present value at time t of all obligation cash flows CFτN L (τ ∈ [t, T ])2 will be: P VtL= T X τ =t Et[CFτL∗ Mτnom Mtnom] = T X τ =t Et[CFτN L∗ Πτ Πt ∗ M nom τ Mtnom], (4.10) in which Et[·]3 is the expectation in real-world standing at the time “t”.
After derivation (refer to Appendix 7), we have
P VtL=
T
X
τ =t
CFτN L∗ exp{Anom(t, τ ) − Bnom(t, τ )Rt+ C(t, τ )πt}, (4.11)
where, Bnom(t, τ ) = 1 κ(1 − e −(τ −t)κ ) C(t, τ ) = 1 α(1 − e −(τ −t)α ) Anom(t, τ ) = [Bnom(t, τ ) − (τ − t)] ∗ [R +σR κ ζR] − [C(t, τ ) − (τ − t)] ∗ [π + σπ α ζπ] − σ 2 R 4κ3[2κ(B nom(t, τ ) − (τ − t)) + κ2Bnom2(t, τ )] − σ 2 π 4α3[2α(C(t, τ ) − (τ − t)) + α 2C2(t, τ )] −σRσπρRπ ακ [(τ − t) − B nom(t, τ ) − C(t, τ ) +1 − e−(α+κ)(τ −t) α + κ ] + (ξRλR+ ξπλπ+ ξuλu) ∗ (τ − t) (4.12) 2
t represents current time; T represents time of cash flows in most remote future 3
The expectations in this article are for the real world, which is corresponding to the one for risk neutral world.
In the expression of Anom(t, τ ) (4.12), ζ
R and ζπ represent market risk premium of the
nominal interest rate and the expected inflation rate, which satisfy, ζR= λR+ λπρRπ+ ξR+ ξπρRπ
ζπ = ξRρRπ+ ξπ+ λRρRπ+ λπ
Chiarella et al. (2007) (5) assumed the coefficient of last term in Anom(t, τ ) (4.12) (here is the term “ξRλR+ ξπλπ+ ξuλu”) to be ξ0, which satisfied Rt= ξ0+ rt+ πt4 . When
inflation-linked bonds are included in investment set, Chiarella et al. (2007) (5) stated ξ0 = −λΠσΠ, where λΠ is the risk premium of Consumer Price Index level. That is:
ξ0 = ξRλR+ ξπλπ + ξuλu
= −λΠσΠ
(4.13)
For the present value of each period cash flow, dτP VtL
τP VL t
= µ(t, τ )dt − Bnom(t, τ )σRdZtR+ C(t, τ )σπdZtπ (4.14)
As for µ(t, τ ), please refer to equation 7.4in Appendix7.
The pension obligations pricing model (equation 4.11) shows that the present value of inflation-linked pension cash flows depends on two factors, the determined pension obligation cash flows without inflation (CFN L
t ), and a combination function consists of
the nominal risk-free interest rate (Rt) and the expected inflation rate (πt). The
stochas-tic differential equation 4.14illustrates that the present value of pension obligations on each period are associated with both innovations of ZR
t and Ztπ. If we apply Fisher’s
equation5 to transform the combination of the nominal interest rate and the expected inflation rate to the real interest rate, we will reach the conclusion that full inflation compensations on pension obligation cash flows are able to eliminate the expected in-flation part in the nominal interest rate. In other words, the inin-flation-linked cash flows only bear real interest rate risk. This result is consistent with the model obtained for the price of a real bond in BX and Mkaouar et al. (2017) (12) etc. By releasing the restriction of Fisher’s equation, we keep both nominal interest rate shock and inflation shock as the influence on the shocks of pension obligations.
4.3
Assets Portfolio
When it comes to choosing the financial instruments, the insurer prefers liquid asset such as inflation-linked bonds and inflation-linked swaps instead of customized notes. In the scope of this article, the complicated asset portfolio that insurer currently holds will not be considered. Instead, portfolios by using nominal zero-coupon bonds (ZCB) with different maturity dates and inflation-linked bonds (ILB) will be established to hedge both inflation and interest exposure. The inflation-linked bonds with proper tenor are introduced to hedge inflation risk of the pension obligations. And the zero-coupon bonds with different maturities will also be applied to hedge remained interest rate risk. Inflation-linked swaps may also be included to the model to deliver multiple choices to the insurer’s hedging portfolio. It should be kept in mind that inflation-linked swaps do not have deflation protection as inflation-linked bonds do. Therefore, the insurer should decide whether to bear deflation risk or to apply inflation-linked option (i.e.
4
In this equation and in the following article, rt represents instantaneous real interest rate. 5R
inflation-linked floor) to hedge deflation risk when using inflation-linked swaps as hedg-ing instruments.
4.3.1 Nominal zero-coupon Bond
The nominal price of nominal zero-coupon bonds at time t, paying out face value 1 at maturity T, would be:
PZCB(t, T ) = Et[1 ∗
MTnom Mnom
t
] (4.15)
After calculation (see Appendix7), we get,
PZCB(t, T ) = exp[Anom∗(t, T ) − Bnom(t, T )Rt], (4.16)
where, Anom∗(t, T ) = − R(T − t) + RBnom(t, T ) − σ 2 R 4κ3[2κ(B nom(t, T ) − (T − t)) + κ2Bnom2(t, T )] −σR κ (λR+ λπρRπ)[(T − t) − B nom(t, T )]
Equation 4.16indicates that the price of nominal zero-coupon bond is only affected by the nominal interest rate.
Applying Itˆo’s lemma, we have, dPZCB(t, T1) PZCB(t, T 1) =[∂A nom∗(t, T 1) ∂t − ∂Bnom(t, T1) ∂t Rt − Bnom(t, T1)κ(R − Rt) + 1 2B nom2(t, T 1)σ2R]dt − Bnom(t, T 1)σRdZtR (4.17) 4.3.2 Inflation-linked Bond
The present value of an inflation-linked coupon bond (ILB) at time t with coupon rate c, nominal principal 1 and maturity T6 is:
PILB(t, T ) = Et[ T X τ =t c ∗Πτ ΠI ∗M nom τ Mnom t + 1 ∗ ΠT ΠI ∗M nom T Mnom t ] = Et[ T X τ =t c ∗ Πt ΠI ∗ Πτ Πt ∗ M nom τ Mtnom + Πt ΠI ∗ΠT Πt ∗ M nom T Mtnom] (4.18)
in which, I is the issue day of the bond and ΠI is the base index. We take time “t”
as the valuation day (i.e. current time), and on that day, all market data is available. Therefore, Πt
ΠI is known.
According to the derivation (refer to Appendix7), we get:
PILB(t, T ) =c ∗ Πt ΠI ∗ T X τ =t
exp{Anom(t, τ ) − Bnom(t, τ )Rt+ C(t, τ )πt}
+ Πt ΠI
∗ exp{Anom(t, T ) − Bnom(t, T )R
t+ C(t, T )πt}
(4.19)
6
Although we are using the same notation of maturity (T) for both zero-coupon bond and inflation-linked bond, it should be noted that the maturity dates of inflation-inflation-linked bond and zero-coupon bond are different
Equation 4.19 illustrates the price of inflation-linked bond is subjected to the nominal interest rate (Rt), the expected inflation rate (πt) and the realized Consumer Price
In-dex (Πt).
In section 4.2 pension obligations analysis, the pension cash flows without inflation (CFtN L) are taken as deterministic. On each valuation day, CFtN L can be generated by the insurer’s separate model. Actually, these cash flows have already included realized CPI impact, which can be represented as Πt
ΠI here. By assuming CF
N L
t to be
determinis-tic, we are blurring the dynamic impact of realized CPI shock and only analysis changes of expected inflation rate on the cash flows. Correspondingly, the realized CPI shock impact on inflation-linked bonds should also be treated as determined. And therefore, the price of inflation-linked bonds will only fluctuate with nominal interest rate (Rt) and
expected inflation rate (πt). In hedging strategy section (Chapter 5), we will consider
both scenarios (scenarios with and without unexpected inflation risk) when establishing assets portfolio.
And, we also obtain SDE function of each cash flow for inflation-linked bonds, dτP VtILB
τP VILB t
= µΠ(t, τ )dt − Bnom(t, τ )σRdZtR+ C(t, τ )σπdZtπ+ σΠdZtΠ, (4.20)
in which µΠ(t, τ ) can be find from equation 7.5in Appendix7.
Similar to the pension obligation cash flows, the inflation-linked bond is impacted both by the innovation of inflation Ztπ and interest rate ZtR. Apart from that, a realized Con-sumer Price Index innovation also affects the price of inflation-linked bonds. This result makes sense as inflation-linked bonds receive full inflation compensation. Meanwhile, it is subject to nominal interest rate risk. Comparing to the result of inflation-linked bonds, the zero-coupon bonds are only related with Rt. The SDE function of zero-coupon bond
Hedging Strategy
5.1
The Insurer’s Current Practice
Currently, the assets portfolio for hedging used by the insurer consists of several inflation-linked notes and inflation-inflation-linked bonds. The notes and bonds have different coupon rates, maturity dates, inflation cap structures respectively, so as to match cash flows of inflation-linked pension obligation with different characteristics. With all assets fully allocated in inflation-linked instruments, the insurer does a sensitivity analysis on nomi-nal interest rate only, to let the dollar duration on assets side and pension liabilities side be roughly matched. A certain threshold was set and the difference of dollar duration between assets and liabilities are monitoring within that threshold.
However, three main issues remain to be solved. First, the sensitivity of inflation is ignored. Whether the assets portfolio performs a good hedging on inflation risk is not monitored. By fully investing in inflation-linked instruments, the insurer thinks its infla-tion risk is eliminated. However, this practice is proved to be improper in5.2.2. Second, the second order derivative of the asset-liability portfolio over nominal interest rate is ignored. In the other words, the convexities of the assets side and liabilities side are not matched accordingly. Third, investing in inflation-linked notes brings about the re-investment issue on the coupon receiving. The inflation-linked notes only give inflation compensation on yearly basis instead of cumulated basis (refer to section 3.3.7). Only by re-investing all coupon received with the same instruments during the notes holding period can the insurer get a similar inflation-linked payoff as pension obligations do. Furthermore, there are some other limitations of the current hedging strategy. First of all, the dollar duration matching currently is only applied to a parallel shift of the whole term structure, no time buckets division or key rate duration matching is applied. If the evaluation is performed by time buckets, a huge mismatch will happen because the maturities of assets, which the insurer holds currently, is highly concentrated. In the scope of our study, time buckets division is not considered as well. One can take such scope as the practice that sets the whole time period as one bucket. And the results we obtain for one bucket can be easily applied in the condition of more buckets. Secondly, the present value of assets is much larger than the present value of pension obligations, which means the current hedging portfolio is quite costly. If an alternative assets port-folio with less present value is available, the insurer can benefit from a reduced assets occupation. Moreover, the inflation-linked notes are customized, namely, illiquid. When new hedging plan is implemented, the inflation-linked notes have to be taken as a legacy and be kept till maturity.
5.2
Proposed Alternative Approaches
In this section, we will propose two approaches alternatively to manage the asset-liability portfolio. One is to match the present value, inflation sensitivity and nominal interest rate sensitivity on assets and liabilities sides using one inflation-linked bond and two zero-coupon bonds. In this approach, after the assets portfolio is obtained, the convex-ity pattern will be checked by comparing the convexconvex-ity of assets to that of liabilities. The other approach is to match the inflation sensitivity, nominal interest rate sensitivity and nominal interest rate monetary convexity accordingly but regardless of present value matching. This approach is closer to the practice of the insurer on hedging inflation-linked pensions. As mentioned in section 5.1, the insurer currently only control the nominal interest rate sensitivity regardless of the present value of its assets portfolio. Besides, in the second approach, we will further discuss the strategy of matching with unexpected inflation risk.
5.2.1 Approach 1
To match with present value and the sensitivity on expected inflation rate and nom-inal interest rate, we will construct our assets portfolio with two zero-coupon bonds PZCB(t, T1, Rt), PZCB(t, T2, Rt) and one inflation-linked bond PILB(t, T3, Rt, πt). We
assume the weight of asset allocation on those two nominal bonds and one inflation-linked bond at time t are wt,1, wt,2 and wt,3. Hence, to match with the preset value on
assets side and liabilities side, we have:
wt,1+ wt,2+ wt,3 = 1 (5.1)
Meanwhile, we will let the sensitivity of the expected inflation rate and the nominal interest rate on the assets side and liabilities to be equal. That is:
wt,1∗ DR1 + wt,2∗ D2R+ wt,3∗ DILBR = DRL
wt,1∗ Dπ1 + wt,2∗ D2π+ wt,3∗ DπILB= DπL
(5.2) Here, DR is the notation for Duration, while DπILB and DπL satisfy:
DILBπ = 1 PILB(t, T 3, Rt, πt) ∗∂P ILB(t, T 3, Rt, πt) ∂π DπL= 1 P VL(t, R t, πt) ∗ ∂P V L(t, R t, πt) ∂π
Record the equations 4.11 and 4.19 we obtain in chapter 4, where P VZCB(t, T, Rt),
P VL(t, Rt, πt) and PILB(t, T3, Rt, πt) have the expressions:
P VL(t, Rt, πt) = T
X
τ =t
CFτN L∗ exp{Anom(t, τ ) − Bnom(t, τ )R
t+ C(t, τ )πt} PILB(t, T, Rt, πt) =c ∗ Πt ΠI ∗ T X τ =t
exp{Anom(t, τ ) − Bnom(t, τ )Rt+ C(t, τ )πt}
+ Πt ΠI
∗ exp{Anom(t, T ) − Bnom(t, T )R
t+ C(t, T )πt}
PZCB(t, T, Rt) = exp[Anom∗(t, T ) − Bnom(t, T )Rt],
The present value of inflation-indexed pension cash flows P VL(t, Rt, πt) and the price
of inflation-linked bond PILB(t, T
well as instantaneous expected inflation rate. While PZCB(t, T1, Rt), PZCB(t, T2, Rt)
changes only with nominal interest rate. Therefore, ∂PZCB(t, T1, Rt) ∂πt = ∂P ZCB(t, T 2, Rt) ∂πt = 0
Hence D1π and Dπ2 equal to zero.
After solving functions5.1 and 5.2, we have:
wt,3= DLπ Dπ ILB wt,1= DRL − wt,3D3R− DR2 + wt,3D2R D1R− DR 2 wt,2= DRL − wt,3D3R− DR1 + wt,3D1R D2R− DR 1 (5.3) 5.2.2 Approach 2
Considering the insurer’s current practice and the main issues mentioned in5.1, we pro-pose to get rid of inflation-linked notes by investing in inflation-linked bonds and nominal zero-coupon bonds. Meanwhile, we propose to do inflation sensitivity and nominal inter-est rate sensitivity matching simultaneously when setting up assets portfolio. In other words, the assets portfolio, consisting inflation-linked bonds and nominal zero-coupon bonds, will be established to immunize the sensitivity of interest rate risk, expected inflation rate risk and realized CPI shock on pension obligation cash flows. Moreover, we will consider monetary convexity matching on both the assets side and liabilities side to immunize second order nominal interest rate sensitivity.
In subsection A below, only nominal interest rate sensitivity matching and expected inflation rate sensitivity matching will be considered, while the unexpected inflation shock will be ignored. As discussed in section 4.2, the pension obligation cash flows without inflation (CFtN L) which including realized CPI shocks at time t is taken as de-terministic. Correspondingly, we assume the shocks of realized CPI on inflation-linked bond to be deterministic. As a result, only nominal interest rate risk and expected in-flation rate risk will be covered. This scope will be adopted in numeric analysis section
6.2when establishing assets portfolio with actual data of pension cash flows.
Most researchers would prefer optimizing certain utility functions for asset-liability management under stochastic interest rate and inflation. Some other researchers chose mean-variance method so that the efficient frontier is introduced and analyzed. Here, we will construct a surplus function (S(t)) which equals the difference between the present values of assets portfolio and pension obligation cash flows. In the classical immuniza-tion method discussed in Gajek et al. (2005) (8), one should make the sensitivity of interest rate of the surplus function to be zero and, at the same time make sure the monetary convexity on the surplus function equal to or larger than zero. This concept will be implemented in our approach as well. We will make the sensitivity of nominal interest rate, expected inflation rate and unexpected CPI shock on Stto be immunized
while also monitor the monetary convexity on St at the same time.
In subsection B below, we will not only consider the sensitivity matching of nomi-nal interest rate and expected inflation rate as we do in subsection A, but also consider sensitivity hedging on unexpected inflation shock (dZtΠ), i.e. change on Consumer Price Index level (Πt). However, this scope will not be covered in the numerical analysis when
establishing assets portfolio in part6.This is because as for calculation of CPI level sen-sitivity on pension obligations, we need more details regarding the models generating CFtN L, which now is assumed to be deterministic.
A. Hedging Strategy without considering Unexpected Inflation Risk Considering only nominal interest rate risk and expected inflation risk, we will construct our asset portfolio with two zero-coupon bonds PZCB(t, T1, Rt), PZCB(t, T2, Rt) and one
inflation-linked bond PILB(t, T3, Rt, πt). We assume the asset volume allocate on two
nominal bonds and one inflation-linked bond at time t are ηt,1, ηt,2, ηt,3. Hence, we write
the surplus function as St= At− Lt
= ηt,1PZCB(t, T1, Rt) + ηt,2PZCB(t, T2, Rt) + ηt,3PILB(t, T3, Rt, πt)
− P VL(t, Rt, πt)
(5.4)
The present value of inflation-indexed cash flows P VL(t, Rt, πt) and the bond price
PILB(t, T3, Rt, πt) vary with nominal interest rate as well as instantaneous expected
inflation rate. While PZCB(t, T1, Rt), PZCB(t, T2, Rt) vary only with nominal interest
rate. Therefore, ∂PZCB(t, T 1, Rt) ∂πt = ∂P ZCB(t, T 2, Rt) ∂πt = 0
We set first derivative of surplus function (S(t, πt, Rt)) over πt to be zero. That is:
0 = ∂S(t, πt, Rt) ∂πt = ηt,1 ∂PZCB(t, T1, Rt) ∂πt + ηt,2 ∂PZCB(t, T2, Rt) ∂πt + ηt,3 ∂PILB(t, T3, Rt, πt) ∂πt −∂P V L(t, R t, πt) ∂πt = ηt,3 ∂PILB(t, T3, Rt, πt) ∂πt −∂P V L(t, R t, πt) ∂πt (5.5)
Function5.5 gives the result that:
ηt,3 = ∂P VL t (t, Rt, πt) ∂πt /∂P ILB(t, T 3, Rt, πt) ∂πt (5.6)
The nominal interest rate sensitivity of surplus function (S(t, πt, Rt)) must satisfy:
0 = ∂S(t, πt, Rt) ∂Rt = ηt,1 ∂PZCB(t, T1, Rt) ∂Rt + ηt,2 ∂PZCB(t, T2, Rt) ∂Rt + ηt,3 ∂PILB(t, T3, Rt, πt) ∂Rt −∂P V L(t, R t, πt) ∂Rt (5.7)
And we want ηt,1 and ηt,2 satisfying, ∂2S(t, πt, Rt) ∂R2 t = ηt,1 ∂2PZCB(t, T1) ∂R2 t + ηt,2 ∂2PZCB(t, T2) ∂R2 t + ηt,3 ∂2PILB(t, T3, Rt, πt) ∂R2 t −∂ 2P VL(t, R t, πt) ∂R2 t ≥ 0 (5.8)
If equation 5.8 does not hold, some other instruments such as interest rate swaptions should be introduced to hedge nominal interest rate convexity. In section 6.2, in order to obtain exact hedging portfolios, we take ∂2S(t,πt,Rt)
∂R2
t = 0 and get the result: η2 = ∂2P VL(t,R t,πt) ∂R2 t − ηt,3∂2PILB(t,T3,Rt,πt) ∂R2 t ∂2PZCB(t,T 2) ∂R2 t −∂PZCB(t,T1,Rt) ∂Rt ∂PZCB(t,T 2,Rt) ∂Rt − ( ∂P VL(t,R t,πt) ∂Rt − ηt,3 ∂PILB(t,T 3,Rt,πt) ∂Rt ) ∗ ∂PZCB(t,T 1,Rt) ∂Rt ∂2PZCB(t,T 2) ∂R2 t − ∂PZCB(t,T1,Rt) ∂Rt ∂PZCB(t,T 2,Rt) ∂Rt η1 = ∂P VL(t,Rt,πt) ∂Rt − ηt,3 ∂PILB(t,T3,Rt,πt) ∂Rt − ηt,2 ∂PZCB(t,T2,Rt) ∂Rt ∂PZCB(t,T 1,Rt) ∂Rt (5.9)
To elaborate deeper of equation5.7, we insert ηt,3 into5.7and get:
ηt,3 ∂PILB(t, T3, Rt, πt) ∂Rt = ∂P V L t (t, Rt, πt) ∂Rt (5.10) which states that theoretically the nominal interest rate sensitivity of inflation-linked bonds with volume ηt,3 can cover all nominal interest rate sensitivity of pension
obli-gations P VtL. This result indicates that we may simply get rid of nominal zero-coupon bonds (i.e. ηt,1 = ηt,2 = 0) as the inflation-linked bond can fulfill nominal interest rate
sensitivity matching already.
This result seems to be reasonable when reviewing the equation. According to the equa-tions5.5 and 5.10, the expected inflation rate risk is hedged by inflation-linked bonds, while the nominal interest rate risk is eliminated simultaneously by that inflation-linked bonds. On the other hand, the interpretation is also reasonable. As the pension obliga-tions and inflation-linked bonds are both fully indexed with inflation, there is no inflation risk on their cash flows. The pension participants receive inflation-indexed pensions so that they pass the inflation risk to the insurer. In the meantime, the insurer invests its assets on inflation-linked bonds so as to get rid of the inflation risk accordingly. The only instrument bearing inflation risk in our portfolio is the nominal zero-coupon bonds. Since the liabilities bear no inflation risk, it is sensible to set the proportion of zero-coupon bonds to be zero. This inference is consistent because an inflation-linked bond with a certain maturity matched with the duration of pension obligations is the ideal instrument providing a perfect hedge. Many previous pieces of research reached such conclusion as well including Brennan and Xia (2002) (2) and Jong (2008) (7). However, it is true only when the Fisher Equation holds. Having been holding full-inflation-indexed assets (real assets) and liabilities (real liabilities), the investors still face the real interest rate risk. If the Fishers Equation holds, keeping the expected infla-tion rate remains unchanged, the shock on real interest rate will bring the same shock on the nominal interest rate. Therefore, the inflation-linked bonds can perfectly hedge
the sensitivity of expected inflation risk and nominal interest risk simultaneously. But, if the Fishers Equation does not hold, this is not the case. In the numeric analysis in section6.2, we observed the fact that after obtaining asset allocation of inflation-linked bonds (η3) by matching the duration of the expected inflation rate, the nominal interest
rate sensitivity of liabilities does not equal to that of the inflation-linked bond portfo-lio. That is, equation 5.10 does not hold. This fact implies that the hedging practice the insurer currently taking, which is fully investing in inflation-linked instruments and matching nominal interest sensitivity rather than real interest rate sensitivity, is a bit improper. In our proposal, we will consider the remaining nominal interest rate sensi-tivity after hedging with inflation-linked bond and we will immunize it by investing in zero-coupon bonds.
B. Hedging Strategy considering Unexpected Inflation Risk
In equation 4.17 and 4.20 of section 4.3, we obtain the result that zero-coupon bonds are only subjected to nominal interest rate innovation, while inflation-linked bonds are subjected to three innovations (nominal interest rate, expected inflation rate and CPI level) simultaneously. Therefore, the asset portfolios with those two kinds of instruments have the function to hedge nominal interest rate risk, expected inflation rate risk and unexpected CPI shock. On the liabilities side, the result in section 4.3 (equation4.14) shows that pension obligations CFtL are subjected to the change on nominal interest rate and expected inflation rate. However, in actual data analysis, it is better to keep in mind that the pension obligation cash flows without inflation (CFtN L) generated by insurer’s separate model may also have sensitivity on the expected inflation rate and CPI level. Therefore, analyzing sensitivity on change of the Consumer Price Index level is also necessary.
By including sensitivity analysis on unexpected inflation risk (dΠt), the assets
port-folio will be composed by two zero-coupon bonds PZCB(t, T1, Rt), PZCB(t, T2, Rt) and
two inflation-linked bonds PILB(t, T3, Rt, πt, Πt), PILB(t, T4, Rt, πt, Πt). We assume the
asset volume allocated on two nominal bonds and two inflation-linked bonds at time t are ηt,1, ηt,2, ηt,3 and ηt,4. Hence, the surplus function (S(t)) which equals the difference
between the present values of assets portfolio (At) and pension obligations (Lt) will be:
St= At− Lt = ηt,1PZCB(t, T1, Rt) + ηt,2PZCB(t, T2, Rt) + ηt,3PILB(t, T3, Rt, πt, Πt) + ηt,4PILB(t, T4, Rt, πt, Πt) − P VL(t, R t, πt, Πt) (5.11)
The present value of pension obligation cash flows, CFL(t, τ, Rt, πt, Πt) (τ ∈ [t, T ]),
and the price of inflation-linked bond, PILB(t, T3, Rt, πt, Πt), vary as nominal interest
rate, expected inflation rate and realized Consumer Price Index level. While the price of zero-coupon bond, PZCB(t, T
1, Rt) and PZCB(t, T2, Rt), varies as nominal interest rate
only. Therefore, ∂PZCB(t, T1, Rt) ∂Πt = ∂P ZCB(t, T 2, Rt) ∂Πt = 0, ∂PZCB(t, T1, Rt) ∂πt = ∂P ZCB(t, T 2, Rt) ∂πt = 0
