#### Abstract

We study the problem of pricing an inflation adjusted annuity in a forward rates market with jumps. Since the market will be incomplete, we use the minimal -martingale measure which we use for computing discounted expectations. We give explicit results for together with explicit results for the price of the annuity.

#### 1. Introduction

Inflation derivatives are a new feature in the derivatives markets with active trading starting in the 1980’s. Their origin has been necessitated by the risk of inflation eroding medium to long-term maturity securities. In economies with anticipation of high inflation, it makes sense to consider the real value of future assets as opposed to the nominal value. According to the simplified Fisher’s model, the real interest rate equals the nominal interest rate minus the inflation rate. It is only in cases where inflation rate is negligible that these two rates almost give the same rate of return. Pricing annuities differ from pricing options in that an annuity makes periodic or continuous payments usually up to a certain redemption date which could be predetermined or which is contingent on the life of the policy holder. An option, when it matures and if its maturity is beneficial to its holder, will make a single payment, the redemption value of the option. An option can also expire worthless. Annuities do not have this option of expiring worthless. In elementary Actuarial analysis, the premium to be paid for an annuity is the expected present value (EPV) of cash flows. As a result, we then find striking similarities between computing option prices and insurance premiums.

Motivated by the work [1] and references therein, in this paper, we find the premium of a nonlife-based annuity in a market where bond prices are driven by jump processes. We emphasize here that the inclusion of life contingency will not bring much deviation of this result because market movements are independent of the future lifetime of a policyholder. Due to incompleteness caused by the introduction of jumps, we consider and give explicitly the minimal -martingale measure studied in [2].

We characterize our problem as one similar to interest rate and foreign exchange derivatives (see [3] for more). Naturally the premium of an inflation protected annuity is equal to the premium of an ordinary annuity plus some “penalty” which results from the extra benefit of the policy holder being cushioned for inflation. This paper concentrates on finding explicit results of this extra “penalty”.

The paper is organized as follows: the following section looks at the mathematical preliminaries. For this part, we use the notation in [4]. We also refer the reader to the same text for more detailed analysis of Lévy processes. Section 3 looks at the assets price dynamics. We give the Consumer Price Index (CPI) as the ratio of the nominal bond to the real bond. This approach is similar to foreign exchange derivatives where we consider the nominal bond as representing the *local* currency, the real bond as representing the *foreign* currency, and then the CPI being the exchange rate of foreign currency to local currency. A high value of exchange rate indicates a weak local currency with respect to the foreign currency. In the same vein, high inflation indicates high-interest rates which may not stimulate production due to high cost of production. Of interest in Chapter 3 is our version of Fisher’s equation (see [5]) which is an extended version that incorporates volatility coefficients and jump terms.

In Section 4, we give the main result which ends with the premium of a non-life annuity adjusted for inflation being given explicitly.

#### 2. Mathematical Preliminaries

Consider a filtered probability space where is the filtration such that the Lévy-Itô processes discussed in this paper are adapted.

Given a Lévy process , the jump of is . The number of jumps, which occur before or at time and of a size contained in some Borel set , can be counted by the measure called the Poisson random measure. It turns out that For times , we denote the differential form of by . The set function is called the Lévy measure of and we require that .

The measure is called the compensated Poisson random measure. In this paper, we will be considering jumps of sizes greater than −1. Therefore, we have the relationship (see [4]), The Lévy-Itô process that we will consider in this paper is of the form where is a standard Brownian motion and the coefficients and satisfy the necessary growth conditions (see [4]) for which (2.3) has a unique strong solution .

The unique characteristic function is given by the Lévy-Khintchine formula (see [4] or [6] for more).

Frequently we will be referring to the Lévy-Itô formula for this Lévy process. We refer the reader to ([4]) for a more general definition. We will, for the purpose of this paper, restrict ourselves to a two-dimensional version of the definition.

*Definition 2.1. *Let be an Itô-Lévy process of the form
where *α*:, , and are adapted processes such that the integrals exist. Let and put . Then
where is column vector . Here the represents transposition of a matrix.

The Itô-Lévy formula above shall be useful in the next chapters.

#### 3. Asset Dynamic Models

##### 3.1. The Bond Price Process

Suppose that the real() and nominal() forward rates are given jointly by

We know that the price of a zero coupon nominal (real) bond is given by

Proposition 3.1. *Let and . Define also . Assume that , and saisfy the necessary regularity conditions for applying Fubini’s theorem. Then is given by
*

*Proof. *Simplifying and using the Fubini's theorem, we have
Therefore, we have
Let and . Define also
and assume that .

Then from the previous expression we have,

##### 3.2. The CPI Process

As we said before, the Consumer Price Index (CPI), , is taken as the ratio of the nominal bond price to the real bond price. By Lévy-Itô formula on , we have where Note that the solution of (3.8) is or alternatively where .

*Definition 3.2 (Fisher’s inflation model). *The rate of inflation at time is the drift process of the CPI, so that
which is essentially our version of the Fisher equation.

In particular, if the volatility coefficients are zero and there are no jumps in the bond price dynamics, then the Fisher’s equation says that the real interest rate is approximately equal to the nominal rate less inflation rate. High inflation coupled with low nominal rates may result in negative returns and no incentives for depositors to keep their money in the bank. This will also affect pensioners whose future payments in the form of annuities will be eroded by inflation. Keeping inflation low is one of the main tasks of the central bank of any country since hyperinflation brings with itself also economic and social chaos.

In our case, we see that volatility and jumps can further reduce the real rate to far lower than the original rate given by Fisher. It is thus crucial to offer annuities that are inflation adjusted in that case.

Theorem 3.3 (Girsanov Theorem; see [4]). *Suppose that there exists a process and which are -adapted and such that
**
and such that the process
**
is well defined and satisfies where expectation is taken with respect to the original probability measure . Define the probability measure on by . Then *(1)* is a local martingale with respect to , *(2)*the process given by is a -Brownian motion, *(3)*the random measure measure given by is a compensated Poisson random measure in the sense that is a local -martingale provided the condition .**For a proof, we just consider the Itô-Lévy formula on and proceed as in ([4]).*

We have to observe here that for , (3.14) will result in infinitely many solutions and as such the market of nominal (real) bonds is incomplete. We do have an infinite number of equivalent martingale measures . Let denote the set of all equivalent martingale measure for .

Observe that, with respect to , we have

Then with respect to , we have given by where and as before and Consequently, the solution of (3.17) is or alternatively where .

#### 4. Pricing an Inflation Adjusted Annuity

Pricing in incomplete markets requires one to choose one of the equivalent martingale measures in . The most popular measures are minimal martingale measure which reduces the squared variance between the payoff of a contingent claim and the terminal value of a portfolio of stocks and bond chosen to hedge the claim. Alternatively the minimal relative entropy measure has been widely suggested as one of the best measures “closest” in some sense to the original probability . In this paper, we will consider the minimal -martingale measure.

*Definition 4.1 ((Minimal -martingale measure) (see [2])). *Let and let be arbitrary but fixed. For , the minimal -martingale measure (qMMM) is the equivalent martingale measure which minimizes the -divergence

Corollary 4.2 (see [2]). *Let and let , then
*

The above corollary tells us that consists only of deterministic and time independent pair which minimizes the -divergence.

Theorem 4.3. *Let and . Assume that the Girsanov parameters of Theorem 3.3 are time-independent and constant, that is, assume that and . Then the minimal -martingale measure is given by the optimal Girsanov pair where is a solution of the equation
**
and and are related by
*

*Proof. **First observe that (4.4) is actually (3.14) with ** replaced with ** and ** replaced with **. Next, we need also the following proposition which can be found as an exercise in [4]:*

Proposition 4.4. *Suppose that is deterministic and satisfies some regularity conditions. Then
**
The proof is trivial. We now go back to the proof of Theorem 4.3. **Assume that and , then due to independence of the Brownian motion and the poisson random measure, one has
**
Therefore is found by solving the following deterministic optimization problem, which is a particular case of Problem in [2]:
**
It was proved in [2] that a general solution exists under certain conditions. We conjecture here that the necessary conditions for a solution exist. We the strive to find the optimal solution explicitly.**Let and . Thus is a solution of
**
Define the Lagrangian
**
then one gets the following:
*

The result comes from solving these equations.

As a special case, we now let given by denote the -Brownian motion while given by denote the compensated Poisson random measure.

In that case, with respect to , we have
An inflation protected annuity pays if inflation (as measured by the CPI) exceeds a certain threshold . The payout at any time will be given by a payout function given by
where is the *nominal* annuity and is the *notional*. Assume that this payment is for a fixed period where is a predetermined limiting age for the policy holder. In most cases is a random variable with a given distribution. For nonlife policies, this may not be necessary. Assume further that the nominal interest rate is constant over time. Then, the premium to be paid for such an annuity at any time is given by
In order to simplify this value, we may want to consider the fact that and are constant (or at best time dependent) and as such we are interested in the value given by
where . The following result follows

Theorem 4.5. *Let the CPI be given by (3.8) and given by (4.12).**Let for some constant .*(a)*Suppose that**
Then the price of an inflation protected annuity is
**
where
*(b)*If on the other hand**
then
**
where, is a constant,
**
and is a solution of the equation
*

Note that the second expression of (4.16) and (4.19) represents the “penalty” charged to the investor for protection against inflation. We also assume that payments are not linked to an inflation index with a time lag, which is a common scenario in real life. We also should point out here that the assumption on part (a) of the theorem that will not result in a unique equivalent martingale measure because (3.14) depends on and . However, given by (4.11) will be a -geometric Brownian motion.

*Proof. * (a) By Fubini’s Theorem on expectations, it is enough to consider
Note that for all and
If , then
Now with respect to , we have
Therefore, by direct integration or otherwise, (4.22) easily simplifies to
with
(b)If , consider the discounted value function , then by the one-dimensional Lévy-Itô formula, we have
The discounted value function is a -martingale if the term equals zero, or alternatively, if
Then is a solution of the following boundary value problem:
where .

We assume a solution of the form then by substitution into
we get where
Therefore and if we let , then the terminal conditions imply
Suppose that , then there exists a unique solution to this equation.

Then .

*Remark 4.6. *Note that if and are constants, then one can explicitly simplify the integral of Theorem 4.5 since for any , then and which increases with term to maturity. Therefore, inflation adjusted annuities of this nature have capital losses in that the price charged at issue or purchase is more than redemption value. This applies to the case of course where we consider the price as a function of maturity time .

#### 5. Conclusions

Indexation helps to adjust future earnings in line with the cost of living at that time. Usually the income is indexed by reference to the values of inflation with some time lag. This is done to allow for publishing the RPI values. In that case proper adjustment must be made to the premium. In this paper we have managed to get explicit results to the price of inflation indexed annuities without time lag. We looked at non-life-based annuities and conjecture that there is not much value added by considering life based annuities.

#### Acknowledgments

The author would like to thank staff in the School of Management Studies at UCT for useful comments during a presentation of this paper at its draft stages. Last but not least, the author is grateful to the reviewers for their useful comments to the first draft of the paper. This paper was funded by the University of Cape Town Research Grant 461091.