CONTINGENT LIABILITY SWAP Alvarado Alfaro Ricardo / Iglesias Nava David • Organization(s) ADMINISTRACIÓN DE RIESGOS / PRICEWATERHOUSECOOPERS • Address Ricardo Alvarado Office. Bosques de Ciruelos 278, Bosques de las Lomas, Delegación Miguel

Hidalgo, México D.F. C.P. 11700 David Iglesias Office. Mariano Escobedo 573, Col. Rincón del Bosque, Delegación Miguel

Hidalgo, México, D.F. C.P. 11580 • Telephone Ricardo Alvarado Office. (0052) 55-5246-4216 David Iglesias Office. (0052) 55-5263-6000 • Fax : including country and area code Ricardo Alvarado (0052) 55-5246-4237 David Iglesias (0052) 55-5263-6010 • E-mail

ricardo.alvarado@adrisa.com.mx / david.iglesias@mx.pwc.com • Abstract In exchange for a financial risk an Insurance Company would take the survivorship risk of the retirees group of another firm. Under this instrument both counter parties would swap the risk that does not form part of their core business and would accept a contingency which can be controlled better. • Keywords Pensions Contingent Liabilities Swaps Financial Instruments Financial Risks Derivatives

I N D E X Executive Summary .......................................................................................................... 1 Section I. Introduction I.1 SWAP’s ................................................................................................................... 2 I.2 Contingent Liabilities .............................................................................................. 5 Section II. Contingent Liability SWAP II.1 Concept.................................................................................................................... 8 II.2 Methodology............................................................................................................ 9

Counter parties involved in the SWAP Contingent Liability SWAP structure process Transference Risk determination Possible Cash Flow for the Contingent Liability SWAP (Results) II.3 Conclusions and Recommendations........................................................................ 26 Bibliography....................................................................................................................... 27 Endnotes ............................................................................................................................. 29

1

EXECUTIVE SUMMARY

With the development of the derivative products, new tools have arisen to reduce the financial risks of the companies. One of these tools is the SWAP, where the participants of the market use the comparative advantages that possess in their respective business agreeing to carry out exchanges of money. For many companies that offer a pension plan benefit, consisting in monthly pensions to their pensioners and that enjoy a good financial situation, it is easy to create a pension fund to cover these obligations. They have the comparative advantage, respect to other institutions, that the interest rate achieved in pension funds is better than those achieved in other kind of funds. However, even though the level of interest rate could be good and the financial obligations could be financed, there is still a problem, it is possible that the retirees group survive longer than the estimated period with the mortality charts, causing an under finance of their funds as regard to its recognized liabilities. Due to this fact, there is a common interest for having an actuarial financial instrument that helps to face the future behavior of the survivorship of the group of retirees, keeping in good level the financial balance of the pension plan. On the other hand, we have insurance companies which are expert entities in managing the survival of a group of policyholders. The insurance company has the comparative advantage over other kind of companies of knowing how to manage the risk of survival. For these institutions there is a common interest to count with a financial instrument that could increase the insured groups giving a higher level of premiums. By using a SWAP of contingent liabilities, there is a link between the firm and the insurance company where each one can transfer to its counterpart the administration of the risk less controllable inside the main focus of their businesses, i.e., a company transfers to the insurance company the risk of the survivorship of the group of retirees and in exchange the firm shares with the insurance company the interest rate risk of the pension fund. This work contains the bases and the necessary methodology to be able to develop a Contingent Liability Swap, which is designed to manage risks of both entities and obtain a financial benefit.

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INTRODUCTION

I.1 SWAP’s The SWAPs have arisen like a powerful tool to reduce financial risks (volatilities of interest rates, exchange rates), to limit risk of prices, to manage flows of money, to take advantage of all the varieties of the imperfections of the market, etc. A SWAP is a contractual agreement contained in a document between two counter parties to exchange periodic payments. In the agreement it is established the assets to be exchanged (that can or cannot be equal), the currency agreed to each one of the parts (that can or cannot be equal), the interest rate to each one of the parts (that can be fixed or floating), the time limit of the coupons (or the dates in which the payments are going to be carried out) and another kind of conditions arose in the negotiation among the parts. How does it work? All the types of swaps are built based on the same basic structure. The two counter parties agree to exchange something on the base of certain assets or goods called notional. The notional or assets can or can not be exchanged. A swap can consider an exchange of notional, two exchanges of notional, a series of exchanges of notional or no exchange of them. In a general form, the swaps provide an exchange of notional at the beginning of the contract and another exchange of notional at the end, in such a way that the swap starts in an effective date that is known like the beginning date of the contract and finishes in a date known as date of maturity or closing date, establishing a period between these two dates that is known as the period of the swap. On their period, the payments by services are going to be made in intervals as it is specified in the contract of the swap that governs the relation between the two counter parties and generally these payments are carried out in monthly, annually, semiannually or quarterly, depending on the nature of good or assets to be exchanged, calling dates of payment to the current dates in which the exchanges of payments occur. The payments of services begin to be carried out from the effective date of the contract and finish in the date of its maturity establishing that the notional exchanged can be equal or different. Among the exchanges of notional, the counter parties carry out the payments of periodic services to a fixed price (or rate) by the use of the notional of the second counterparty. This fixed price is called coupon of the swap. The payments for services of the second counterparty are carried out in a specific market, price (or rate) by the use of the assets or notional of the first counterpart. These two payments are named legs or sides of the swap. Modifying the terms properly or adding special negotiations, this simple structure of a swap can have hundreds of variants to be adjusted to the specific needs of those concerned.

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Usually it is very difficult to agree a swap contract between two users. A more efficient structure involves a financial intermediary that serves as counterpart among those interested. This counterpart is known as dealer. The income for the dealer comes from the spread that exists in the coupons of the swap. A simple diagram of the flows that are presented among the counter parties can be seen in the following plan: The swaps have its economical foundation in two basic principles: comparative advantage and risk management. The swaps arise because there is a comparative advantage due to market imperfections and there are different financial prices assigned according to the credit quality of the participants in the markets assigned. Example. Many types of Swap contracts exist. The most common one is that of interest rate where the counter parties exchange a fixed rate by a variable rate in relation to an amount called notional. Example. Consider an interest rate swap with the following characteristics: Notional $100,000,000 Fixed Interest Rate 4.5% Variable Interest Rate According to LIBOR 30 days LIBOR 30 days (May 1, 2006) 3% Agreement in the number of days 360 Date of contract May 1, 2006 Beginning of contract May 1, 2006 End of contract May 1, 2010 Frequency of payments Annually

Counterpart 1

Counterpart 2

Notional agreed by the counterpart 1 to give to the counterpart 2

Figure 1. Flow of Resources

Notional agreed by the counterpart 2 to give to the counterpart 1

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In this kind of contract, one of the counter parties agrees to pay a flow of annual cash during a period of five years calculated as the multiplication of an interest rate of 4.5% by the notional of 100,000,000 and receives in exchange a flow of annual cash calculated as the multiplication of the LIBOR rate to 30 days (variable rate) by the same notional. The first payment occurs at the end of the initial period and the counter parties only exchange the net amount of the position (difference among both flows of cash). This cycle is repeated up to the final payment, this means that it is carried out up to the expiration of the contract. The payments of fixed interest are determined according to the following formula.

××=

Basist

rNP xx

Where: xP = Payment of fixed interest

N = Amount or Notional xr = Fixed Interest Rate

t = Number of days to achieve the period evaluated. Basis = Agreement in the number of days 360 or 365. The payments of variable interest are determined according to the following formula:

××=

Basist

rNP vv

vP = Payment of variable interest

N = Amount or Notional vr = Forward Interest Rate for the period t

t = Number of days to achieve the period evaluated. Basis = Agreement in the number of days 360 or 365. The counterpart that pays fixed rate will be benefited if the rate LIBOR of 30 days increase for above the 4.5% annual, but will lose if the LIBOR is smaller than 4.5%. If we suppose that the rate LIBOR to 30 days will be of 3.5%, 4%, 4.75%, 5% and 5.5%, during the next 5 years, the following chart illustrates the flows of cash that are presented in this example, since the point of view of the one that pays the fixed rate:

Date Days Variable Variable Fixed Fixed NetRate* Leg Rate Leg Flow

01-May-06 366 3.00% 4.50%01-May-07 365 3.50% 3,050,000.00 4.50% 4,575,000.00 1,525,000.00- 01-May-08 365 4.00% 3,548,611.11 4.50% 4,562,500.00 1,013,888.89- 01-May-09 365 4.75% 4,055,555.56 4.50% 4,562,500.00 506,944.44- 01-May-10 366 5.00% 4,815,972.22 4.50% 4,562,500.00 253,472.22 01-May-11 365 5.50% 5,083,333.33 4.50% 4,575,000.00 508,333.33

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In this example we see that the counterpart that pays the fixed rate is the debtor the first three years and the creditor the last two years, as a result of the increment in the rate LIBOR to 30 days. * It is important to mention that for purposes of valuation, the rates forward are calculated. I.2 Contingent Liabilities. Contingent liabilities are those benefits offered to the workers which you do not know if they are going to be paid or not; if they will be paid, there is the question of when are they going to be paid or which is going to be the total amount or how many payments are going to be made. For example, in the private pension schemes, how many payments are going to be paid to each retired person is an unknown fact because it is an uncertain event that depends on the number of years that each person lives. Likewise in the trading of a medical expenses insurance it is not known if the insurance is going to be used or how much is going to be the total amount of claims; this happens because we can not predict when anyone is going to be sick or the kind of sickness. How the contingent liability is originated and who owns the liability? In this study we will only be focused in the labor liabilities originated from private pension schemes, to the date of retirement, established by companies in favor of their employees as a recognition to the years of service worked. At the date of retirement, the benefit offered by a pension scheme can be a lump sum payment, or a series of installments (generally monthly payments) determined with the formula established in the plan or as a result of the savings accumulated by the periodic contributions carried out in favor of an employee during its active life. When this benefit is offered in a series of installments, there are generally two options for the sponsor of the plan: ü In the first one, the company can transfer the total amount of the liability for their

pensioners to another company (generally an insurance company), for which the company will have to pay a premium equivalent to the liabilities of its pensioners. The insurance company agrees to pay the pensions needed and the company transfers any type of liability resulting from the payment of pensions.

ü In the second one, the company decides to pay the value of the pensions to its retired

personnel being willing to assume the risk of survivorship from this group, i.e. the risk that the pensioner lives more years of what suggests the demographic experience of the expectation of life. In this case, the company would enter to the business of the survival of people with certain disadvantages because most of the companies neither have experience in it nor have a large group of employees to believe that the mortality experience will behave as expected.

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Keeping in mind that the survival is not the natural business of almost any kind of company (excepting Life Insurance Companies) the recommendation is to transfer this kind of liabilities. Nevertheless for those businesses that decide to run the risk of the survival, the objective of this paper is to propose an actuarial financial instrument to benefit the counter parties involved. Calculus of the Contingent Liability. For the calculation of the contingent liability, first of all, we determine if the pension plan is of defined contribution or definite benefit. In case of a defined contribution plan, the contingent liability is equal to the total amount of contributions performed during the active life of an employee. In the case of the defined benefits plan, for determining the annual cost of a pension scheme, there are many actuarial methods of which the most commonly used is the method of credit projected unit that considers those benefits that the employee has accumulated in a direct relation with the years of service rendered to the company. This means that the normal cost of the plan is equal to the actuarial value of the benefit accumulated each year by the employees. The calculation of the contingent liability is equal to the present value of all the benefits acquired, for the active employees as well as for the retired. If it is assumed that the benefit to the retirement age will be paid yearly during the time that the retiree lives, the following formula defines this obligation for a participant with x years old:

=xVPOP( )( ) ( )k

rrkk

smkk

ddkk

vtk

r

xk

xkTxxkk FqFqFqFqvp

ykyx

B )()()()()(´´

+++−−∑

=

−− for rx ≤

Where:

=xVPOP Present Value of the contingent liability for a x years old person.

=kB Function of benefits based on the seniority, salary, etc. in the pension plan to age x.

=−)(T

xxk p Probability of a person to survive in the company, of age x to age k, where the multiple decrements could be by death, dismissal, voluntary resignation, retire, incapacity.

=−xkv Discount factor of age x to age k =)(t

kq Probability of dismissal in age k.

=)(dkq Probability of disability in age k.

=)(mkq Probability of death in age k.

=)(rkq Probability of voluntary resignation in age k.

=kv F Value of the benefit for termination in age k.

=kd F Value of the benefit for disability in age k.

=kr F Value of the benefit for retire in age k.

=ks F Value of the benefit for death in age k.

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In the case of the retired personnel, the formula is the following: =rVPOJ )( m

rarB && for rx ≥

Where: =rVPOJ Present Value of the contingent liabilities for an only person at the age of retire r.

=rB Amount of pension to the retirement date r. =)(m

ra&& Present Value to age r, of an annuity paid m times a year, which the only decrement is death. In this paper we will only deal with the present value of the contingent liabilities for retirees.

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CONTINGENT LIABILITY SWAP II.1 Concept There are a variety of swap transactions that can cover the needs of the participants in diverse ways. Our work is focused to the creation of a contingent liability swap which is explained over. On one side, we can find a company that has established a pension scheme of defined benefit in favor of its employees and that pays monthly or periodic pensions. This company may be interested in transfer the survival risk to which it is exposed by the survivorship of the retiree personnel. On the other side, we can find a counterpart that will be an insurance company interested in extending the group it covers over its life insurance brand, accomplishing in this way a risk reduction by the increase of the covered group (dispersing the risk over a greater universe of insured persons). The company assumes the survivorship risk of the retirees paying pensions over any deviation of their life expectative, therefore having to disburse the money that lacked in the estimations for the payment of the pensions and causing an effect that can affect the company balance sheet in a significant way. We know that the insurance companies have experience in the management of survivorship of different groups of policyholders for which a comparative advantage exists that the company can obtain from the insurance company. In a similar way, the insurance company needs to reach a certain level of insured persons in order to achieve a considerable certainty over the expectative of the mortality charts, even though the insurance company needs to cover in some way a premium due to the risk that will be covered. In accordance with the previous explanation, we can observe an opportunity where the company could be interested in giving some type of contingent payment to the Insurance Company in exchange for transferring the survivorship contingent risk of the retiree group. Additionally, we have though in the way of paying this survivorship premium and this would be a financial risk of the company over the pension fund that it could be transferred to the insurance company in exchange for the transference of the survivorship risk. So, we can see that the insurance company would be able to obtain a comparative advantage of the business that would be financial performance of the pension fund. Given the previous circumstances we have developed a financial-actuarial instrument called Contingent Liability SWAP where an insurance company agrees to assume a survivorship risk in exchange for some interest rate risk of the Company who has the group of retirees.

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II.2 Methodology Once it has been explained the general concept of the Contingent Liability SWAP, along this section we will develop the methodology of its operation, including the resultant flows of this actuarial financial instrument. In this manner we divide this section in the following:

1. Counterparts involved in the process of the SWAP 2. Contingent Liability SWAP structure 3. Transference Risk determination 4. Possible Cash Flow for the Contingent Liability SWAP (Results)

1. Counterparts involved in the process of the SWAP It exists a general interest among the companies that decide to pay by themselves the value of the pensions to their retiree personnel to include an actuarial financial instrument that could help them to confront the future behavior of these liabilities that varies mainly in function of the survivorship of the group of retirees and in function of the interest rate performance of the resources funded to pay these liabilities. It is important to stand out that yearly, these companies should determine the total amount of liabilities and carry out the necessary contribution to reach an adequate and sufficient funding level. Even though the company has certain discretionally on the necessary investments in order to obtain an adequate interest rate to cover the liabilities, the risk of the survivorship is found completely out of any available alternative under its control. On the other hand, there are the life insurance companies where the mission of their business is focused to the survival risks dispersion on specific groups. The life insurance companies just as any another company also have different type of risks but the elemental difference situates in the fact that the liabilities of these businesses are mostly contingent. Considering these two types of institutions we have on one side, a company that has the need to transfer the survivorship risk of its group of retiree employees and on the other side, an insurance company that seeks for bigger groups at their policies and higher certainty over the future mortality expectative. It is worth to mention that the insurance company, due to its line of business, counts on a level of policyholders with similar characteristics to those of the group of retirees of the company with the distinction that the number of participants is significantly greater which permits the dispersion of the risk and a better adjustment to the mortality curve. Through a Contingent Liability SWAP, a bond between the company and the life insurance company is established where each one manages to transfer to its counterpart the administration of the risk less controllable inside the main focus of their business. For the company this means the transference of the management for the survivorship of the group of retirees -that forms part of the core business of an insurance company- and, on the other hand, the insurance company can be benefited by the transference of an interest rate that could provide the business extra premiums in exchange for assuming the risk of survival from this group.

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For the insurance company the enlargement of its group submitted to the experience of mortality offers a greater certainty on the decision of the mortality chart what drifts in a better decision of the premiums of its products in which the survival is involved. In conclusion, in the Contingent Liability SWAP two entities will participate with a bilateral cash flow that will be explained in detail in the section of Generalities of the SWAP. 2. Generalities of the SWAP Contingent Liability SWAP structure The Contingent Liability SWAP is structured through a contract between the two entities which have already been mentioned:

1. A company with a group of retirees. 2. An insurance company with policyholders group with similar characteristics to those of

the group of retirees of the counterpart. As in any contract of this type some specifications should exist where the characteristics of the agreement can be established, (liabilities and rights of each part that participates in the SWAP) and these are the following:

1. Participants (Company and Life Insurance Company). 2. Start date of the contract. 3. Closing date of the contract. 4. Composition of the Group of Retirees of the company. 5. Discount Rate to which the contingent liability is calculated. 6. Amount of the total contingent liability of the group of retirees as of the start date of the

contract. 7. Amount of the total estimated contingent liability of the group of retirees as of the

closing date of the contract.

Company with a group

or retirees

Life Insurance Company with a group of policy

holders under a life coverage

Survivorship risk transference

Interest rate risk transference

Cash

Flow

Figure 2. Risk Transference

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8. Amount of the total estimated contingent liability of the group of retirees as of the closing date of the contract considering the mortality of the group during the period of the contract.

9. Survivorship Risk of the group of retirees that it is desired to transfer. 10. Interest rate over the discount rate that offers an equivalent performance to the total

survivorship risk that represents the group of retirees. Each one of these concepts are self-explanatory over the following section where we should determine the risks to transfer, nevertheless, with more understandable purposes through the Contingent Liability SWAP, the following diagram show the flows of risk and of resources implied in this type of contract. Date of Start of the Contract: June 30, 2005

For this diagram we assume the following: ü The contingent liability of $100 of the 10 retirees was calculated using a long term rate

of discount of 3.5%. ü The pension fund, according to the macroeconomic panorama will generate in the

following year a rate between the 3.5% and the 5.5%. As it can be observed, the company X has a risk that no matter the core of its business, it is not related to its activity and it is called “survivorship of the group of retirees”. An insurance company Y that has a large group of policyholders with the same characteristics can disperse the survivorship risk of the group of retirees of the company X, nevertheless, it is necessary to establish a compensatory account – premium- in favor of the insurance company.

Figure 3. Practical Example

Company X10 retirees

Contingent Liability $ 100

Life Insurance Company YGroup of 1000 Insured persons

with similar characteristics to the retirees group

Company X has the survivorship risk of 10 retirees

The existing pension of $100 provides a variable interes rate.

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According to the previous, it is possible to establish a relation between the insurance company Y and the company X so that the insurance company would absorb the risk of survivorship of the group of retirees in exchange for the fact that the company X will share a part of the interests that the pension fund grants over the period. The previous problem can be expressed mathematically according to the following thing:

Sea ( ) =dxVPOTJ ∑ =

nrVPOJ ∑

n

mrarB )( && Sum of the present value of the contingent liabilities

in the year x for the n retirees using a discount rate d .

Then, the example data would be: ( ) =%5.32005VPOTJ ∑ =

10rVPOJ 100

Additionally, if we suppose that we have a fund to cover the liabilities of the retirees and that the contributions, withdrawals and interests are carried out at the end of the period, the value of the fund can be written like:

xIxWxCxFPFOJxFPFOJ ±−+−= 1

Where:

=xFPFOJ Pension Fund to cover the retires liabilities over the period x

=xC Contributions over the period x

=xW Withdrawals over the period x

=xI Interests over the period x

The interest rate r accomplishes in the period can be obtained from the previous formula as

1

1

−

+−−−=

xFPFOJxWxCxFPFOJxFPFOJ

r

The interest rate achieved is expected to be at least equal to the rate of discount. Nevertheless, in practice it can happen that the interest rate is smaller or it can be that the interest rate is way above the expected percentage. If the interest rate is smaller, then the company will not pay anything to the insurance company. In the contrary case, if the interest rate earned by the fund is higher that the discount rate established in the contract, the company is going to pay the interest rate agreed (the amount earned from the interest rank above the discount rate). It is important to clarify that if the interest rate exceeds the maximum interest rate fixed in the rank at the contract there will not be any further payments for the insurance company, this exceeding interest shall remain in the fund.

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The variables involved in the decision of the total amount to pay by the company to the insurance company are the following:

r , Interest rate achieved in the period:. r ′ , Interest rate to pay in the period. d , Rate of discount to bring to present value the obligations.

maxr , Maximum rate of interest agreed to pay in the period according to the macroeconomic panorama.

Then according to the data of the example, it is expected that the rate to be paid will be found between the following:

maxrrd ≤′≤

%5.5%5.3 ≤′≤ r Estimated scenario at the closing date of the contract: June 30th, 2006

Actuarially it is predicted that by the end of the year the company will have a liability of $94.5 due to a group of 9.5 retirees, we know that this setting is impossible to reach since the company will finalize the year with 9 (maybe less) or with 10 retirees. In this manner, the maximum risk of survivorship is that the group in its totality reach one more year of life and this risk is quantified in $1.5 ($96 of 10 retirees less $94.5 of 9.5 retirees). In the case that by the end of the year the company ends up with 9 retirees, the insurance company will not pay anything to the company. On the other side there is the compensation for the insurance company, like this flow was commented previously, it will be calculated with relation to the performance of the pension fund that exists.

Figure 4. Possible Liabilities at the closing date

Company XAlternative I. 9.5 retirees

Liability $ 94.5Alternative II. 10 retirees

Liability $ 96

Insurance Company YGroup of 1000 Insured personas with similar

characteristics to the retiree group.

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We know that the balance of the fund starts the year with $100, in simple terms, we expect to receive among $3.5 and $5.5 of interest in the year. This estimation gives us a rank of $2 that can be shared with the insurance company. Thus, the insurance company will receive in exchange for the risk of survivorship quantified in $1.5 the margin of interest rate that generates this amount of interest above the initial rate of 3.5% (the discount rate used for the calculus of the contingent liability). In such a way that if the fund generates a smaller rate than 3.5% the insurance company will not receive any premium in exchange for the risk of survival but if the fund generates more than the 5% in a year the insurance company will receive up to 1.5 by the risk of the period established in the contract and any surplus will remain in the fund. In conclusion, assuming that the fund had a performance in the rank established in the contract, the cash flow would be able to be represented in the following way:

Figure 5

Figure 6

Company X10 retirees

Contingent Liability $ 96

Insurance Company Y

Alternative I. The entire group of retirees reach one more year of age.

The fund produced interests of $4.7 over the year, i.e., $1.2 above

the discount rate of 3.5%.

Pays $1.2

Pays $1.5

Company X9 retirees

Contingent Liability$ 92

Insurance Company Y

Alternativa II. Only 9 retirees reach the next year.

The fund produced interests of $4.7 over the year, i.e., $1.2 above

the discount rate of 3.5%.

Pays $1.2

Pays $0.0

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Mathematically the previous can be established as: a) To determine the total to payment from the insurance company Y to the company X

Let ( )%5.32006VPOTJ be the Contingent Liability of the retiree group as of June 30th

2006 using a discount rate of 3.5%. ( ) =%5.3

2006VPOTJ 94.5 for 5.9=m retirees.

Where m , is the number of retires that it is expected to reach the following year or period by the closing date. Subsequently the upper and lower limits for the contingent liability should be established. The upper limit is established in function to the following natural number to the calculated value of m and it is named as supn . The lower limit shall be given as a

function of the expected number of retires by the closing date [ ]n and it is named as infn . The contingent liabilities should be recalculated for the upper limit as for the lower one.

The present value for the upper limit is called as: ( )%5.3sup

2006VPOTJ and for this

example it is equivalent to 96 for a total of 10sup =n retirees.

The contingent liability for the lower limit is called as: ( )

=%5.3inf

2006VPOTJ 92 for a

total of 9inf =n retirees. Therefore the quantity that would have to disburse the insurance company Y (Maximum Risk by Survival,) to cover the risk of survivorship of the group of retirees of the company X, would be according to the following thing:

( ) ( )%5.32006

%5.3sup2006 VPOTJVPOTJMRS −= if supnn =

0=MRS if infnn ≤

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b) To determine the total payment from the company X to the insurance Company Y

Let r be the interest rate obtained in the period. Let r ′ be the interest rate to be paid by the end of the period in accordance with the macroeconomic perspectives Let d be the discount rate for the period. Let =maxr be the maximum interest rate fixed in the SWAP contract for the period. It is expected that the fund grants interest between d y maxr , this is, it is expected that

maxrrd ≤′≤ Then:

%5.5%5.3 ≤′≤ r The amount to pay from the company X (Maximum Risk of Rate,) to the insurance company Y is determined as the difference among the interest rate agreed and the rate of discount multiplied by the value of the fund at start of the period, according to the following thing:

0=MRT if dr ≤ ( )drFPFOJMRT −= *2005 if maxrrd ≤≤ , where drr −=′ ( )drFPFOJMRT −= max2005 * if maxrr ≥ , where maxrrr −=′

Over the following sections the selection of the rates to use in the contract is explained in detail as well as possible scenarios and practical examples of flows among the counterparts by the end of the contract. 3. Transference Risk determination In any type of SWAP the parts involved should determine and quantify the risks to transfer with the objective that a comparative advantage exists and that the clauses comply with the expectations and needs of each part. In the Contingent Liability SWAP can be identified two different risks to transfer that are:

a) The survivorship of the group of retirees. b) The interest rate reached by the fund that covers the contingent liabilities of the group

of retirees. Thus, subsequently the methodology for the quantification of these risks is presented so that they can be comparable to establish the relation between the two parts through the Contingent Liability SWAP.

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a) Quantification of the survival risk for the group of retirees In order to be able to determine the impact of the survivorship of the group of retirees it is necessary to set the mortality chart as well as the interest rate that will be used to calculate the total contingent liability as of the start date of the SWAP. The procedure to calculate this liability is:

1. We suppose a group of retirees that it is composed from 5 persons of different ages who receive different amounts of pension.

Chart 1. Group of Retirees

Retiree Age at 30/06/ 2006 Sex* Monthly

Pension 1 75 M $10,000 2 70 F $15,000 3 65 M $20,000 4 60 F $15,000 5 55 M $10,000

*M. Male, F. Female

2. It is necessary to determine the annuity for each retiree according to his age, in order to

simplify the calculation we will assume a life monthly pension increased annually by inflation and we will use the Mexican Experience Mortality Chart of the Social Security published by the National Commission of Insurances of Mexico in the year 2000. Using the annuity with a real rate of discount of the 3.5% (above inflation) we can determine the Total Contingent Liability of the company simply by multiplying the total of the monthly pension of the retirees by the corresponding annuity as it is shown in the following chart. It is important to mention that the date that will be used for the decision of the liability will be that of termination of the contract SWAP. For these effects a period of 1 year will be used in such a way that the total liability will be quantified as of June 30th, 2006

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Chart 2. Contingent Liability of the Retirees Group at the start date of the contract

Age at 30/06/ 2006 Sex* Monthly

Pension (1) Annuity (2) Total Liability (3) = (1) * (2)

1 75 M $10,000 95.64 $956,385 2 70 F $15,000 144.55 $2,168,197 3 65 M $20,000 147.15 $2,943,034 4 60 F $15,000 186.04 $2,790,553 5 55 M $10,000 187.70 $1,877,026

Total Liability $10,735,194

3. Finally the risk that intends to be transferred will be quantified. The maximum risk in terms of survivorship for the company with the group of retirees would be the case that the total population reach one more year of age, this would mean that the group did not behave in accordance with the survival expectations of the mortality chart.

In the following chart are the total liabilities estimated as of the future date that would constitute the end of term of the SWAP contract. Subsequently this liability will be diminished according to the mortality for the period of the contract for each participant and in this manner the obligation is determined as of the end of the contract under a specific mortality chart. Chart 3. Contingent Liability of the Retirees Group at the closing date of the contract

Age at 30/06/ 2006 Sex*

Monthly Pension

(2)

Annuity (2)

Total Liability (3) = (1) *

(2)

Survival Probability

(4)

Total Liability considering mortality

(5)=(3)*(4) 1 75 M $10,000 95.64 $956,385 0.95491 $913,262 2 70 F $15,000 144.55 $2,168,197 0.98381 $2,133,093 3 65 M $20,000 147.15 $2,943,034 0.98440 $2,897,122 4 60 F $15,000 186.04 $2,790,553 0.99328 $2,771,801 5 55 M $10,000 187.70 $1,877,026 0.99288 $1,863,661 $10,735,194 0.985443 $10,578,940

*There can be contracts where the company only wants to transfer a portion of the risk and not necessarily the case when the whole group reach the next period.

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The result of this exercise shows us that if the group of retirees survive in its totality the contingent liability would reach 10.73 million pesos, nevertheless, if the group of retirees behaves according to the experience of the chosen mortality then the contingent liability of the company would reach only 10.57 million pesos. As conclusion, the survival of the group of retirees represents for the company a possible cost for the following year of $156,255 that would be possible to disperse or to eliminate if a group large enough could absorb this cost or in other words compensate this type of deviations. Summary:

Starting date for the SWAP contract June 30th, 2005 Closing date for the SWAP contract June 30th, 2006 Retirees at June 30th, 2005 5 Liability at the closing date $10,735,194 Estimated Retirees at June 30th, 2006 4.90 Liability at the closing date considering mortality of the period $10,578,940

Maximum Cost for Survivorship $156,255 b) Quantification of the risk of the interest rate

Once it has been determined the maximum possible risk to transfer from the company to the insurance company, it is necessary to find the risk on the interest rate that will compensate to the insurance company by assuming the risk of survival of the group of retirees. On this point it is necessary to determine two interest rates:

1. The rate to which the contingent liabilities of the company are discounted. It is very important to stand out that this rate should be agreed between the two parts since it is related to the amount that will compensate the acceptance of the survivorship risk from the insurance company and that it is explained in the following paragraph. In the example a real rate of the 3.5% annual was used.

2. The second interest rate to set is the one that the insurance company will receive in

exchange for the risk of survival that is assuming. This interest rate will be determined like the rate above the discount rate mentioned in the previous point that grants an amount identical to the survivorship risk quantified that it is assuming the insurance company.

For the decision of this rate the first step consists of quantifying the liability at the start date of the SWAP contract, that for this example would be June 30, 2005.

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Chart 4. Retiree Group Example 1

Age at 30/06/ 2005 Sex* Monthly

Pension (1) Annuity (2) Total Liability (3) = (1) * (2)

1 74 M $10,000 100.15 $1,001,503 2 69 F $15,000 149.30 $2,239,465 3 64 M $20,000 147.15 $2,943,034 4 59 F $15,000 190.39 $2,855,868 5 54 M $10,000 191.90 $1,919,007

Total Liability $10,958,877 In accordance with the previous, we know that the fund should generate a minimum interest rate of 3.5% above inflation for the 10.95 million mentioned before. Commonly it would be expected that this pension fund would generate a greater interest rate than 3.5%, in the case of Mexico this rate represents a very conservative assumption. In this manner, it is necessary to determine the interest rate above the 3.5% that would generate an identical interest to $156,255 that represents the maximum cost for the survivorship of the group of retirees from the company. For this example the interest rate that would generate this performance would be 1.48% and according to this the contract of SWAP it would be given to the insurance company any performance that the pension fund obtains between the 3.5% to the 4.98% in such a way that if the fund generates an interest rate over 4.98% this interests would remain in the fund and if a smaller rate is obtained the insurance company will receive the resources corresponding to the interests obtained over the basis of the 3.5% set initially for the quantification of the contingent liability.

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4. Possible cash flows for the Contingent Liability SWAP (Results) Using a real group of retirees of a Mexican company, we present a practical example that will be carried out showing the possible flows resulting from the SWAP contract.

SWAP Contract Starting date for the SWAP contract June 30th, 2005 Closing date for the SWAP contract June 30th, 2006 Retirees at June 30th, 20054 10 Discount Rate 3.5% Liability at the closing date $15,868,768 Estimated Retirees at June 30th, 2006 9.65 Liability at the closing date considering mortality of the period $15,591,814

Maximum Cost for Survivorship $276,954 Interest rate above the discount rate used to compensate the Insurance Company

1.76%

To calculate the possible resulting cash flows we will set some possible result combinations:

Result of Survivorship for the Retiree Group

by the closing date

Performance of the fund

Cash Outflow from the company

Cash Outflow from the

Insurance Company

Scenario 1 Retirees at closing: 10 Interest Rate: 5.5% YES YES Scenario 2 Retirees at closing: 10 Interest Rate: 4.5% YES YES Scenario 3 Retirees at closing: 10 Interest Rate: 3.0% NO YES Scenario 4 Retirees at closing: 9 Interest Rate: 5.5% YES NO Scenario 5 Retirees at closing: 9 Interest Rate: 4.5% YES NO Scenario 6 Retirees at closing: 9 Interest Rate: 3.0% NO NO

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Company X10 retirees

Fund interest rate exceeded the contractual rate of 5.26%

Insurance Company YGroup of retirees of the company

X exceeded the expected survivorship

Scenario I. Retirees at closing: 10Interest rate of the Fund 5.5%

The fund obtained interests over the periodo of $865,119, this means it exceeded the contractual

compensation of $276,954 equivalent to an interest rate of 1.76%

Pays $276,954

Pays $276,954

The whole group of retirees survived to the end of the period with a cost of

$276,954

Company X10 retirees

Fund interest rate did not reached the contractual rate of 5.26%

Insurance Company YGroup of retirees of the company

X exceeded the expected survivorship

Scenario II. Retirees at closing : 10Interest rate of the Fund 4.5%

The fund obtained interests over the periodo of $707,753, this means it exceeded the contractual

compensation by $157,221 equivalent to an interes rate of 1.00%

Pays $157,221

Pays $276,954

The whole group of retirees survived to the end of the period with a cost of

$276,954

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Company X10 retirees

Fund interest rate did not reached the minimum rate

Insurance Company YGroup of retirees of the company

X exceeded the expected survivorship

Scenario III. Retirees at closing : 10Interest rate of the Fund 3.0%

The fund obtained interests over the periodo of $471,763, this means it did not reached the

discount rate of 3.5%

Pays $0

Pays $276,954

The whole group of retirees survived to the end of the period with a cost of

$276,954

Company X9 retirees

Fund interest rate exceeded the contractual rate of 5.26%

Insurance Company YGroup of retirees of the company X did not exceeded the expected

survivorship

Scenario IV. Retirees at closing : 9Interest rate of the Fund 5.5%

The fund obtained interests over the periodo of $865,119, this means it exceeded the contractual

compensation by $276,954 equivalent to an interes rate of 1.76%

Pays $276,954

Pays $0

The retirees group showed a higher mortality than expected

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Company X9 retirees

Fund interest rate did not reached the contractual rate of 5.26%

Insurance Company YGroup of retirees of the company X did not exceeded the expected

survivorship

Scenario V. Retirees at closing : 9Interest rate of the Fund 4.5%

The fund obtained interests over the periodo of $707,753, this means it exceeded the contractual

compensation by $157,221 equivalent to an interes rate of 1.00%

Pays $157,221

Pays $0

The retirees group showed a higher mortality than expected

Company X9 retirees

Fund interest rate did not reached the minimum rate

Insurance Company YGroup of retirees of the company X did not exceeded the expected

survivorship

Scenario VI. Retirees at closing : 9Interest rate of the Fund 3.5%

The fund obtained interests over the periodo of $471,763, this means it did not reached the discount

rate of 3.5%

Pays $0

Pays $0

The retirees group showed a higher mortality than expected

25

The cash flows resulting from the different scenarios are:

Result of Survivorship for the Retiree Group

by the closing date

Performance of the fund

Cash Inflow for

the company

Cash Inflow for the

Insurance Company

Scenario 1 Retirees at closing: 10 Interest Rate: 5.5% $0 $0 Scenario 2 Retirees at closing: 10 Interest Rate: 4.5% $119,7321 $0 Scenario 3 Retirees at closing: 10 Interest Rate: 3.0% $276,954 $0 Scenario 4 Retirees at closing: 9 Interest Rate: 5.5% $0 $276,954 Scenario 5 Retirees at closing: 9 Interest Rate: 4.5% $0 $157,221 Scenario 6 Retirees at closing: 9 Interest Rate: 3.0% $0 $0

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II.3 Conclusions and Recommendations

Through the SWAP of Contingent Liabilities it is possible to establish a contract between a company and an insurance company with the objective to exchange during a specific period two different risks to the ones that are hold by both parts. As it could be observed in the practical example, they exist an infinity of possible results of flows among the contractors of the SWAP presenting a fair contract without any significant advantage for none of the participants of the contract. The present work had as purpose to propose an alternative solution to a problematic presented under a changing economic environment with diverse fields of improvement.

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BIBLIOGRAPHY

Author(s): Cabriada Martínez, Erika María. Title of article (in a journal) or chapter (in a book): --- Title of book or journal: Thesis: El uso de swaps en el nuevo régimen de inversión de las SIEFOREs aplicación de técnicas de recomposición de portafolios y su contabilidad Date of publication: 2002. Volume or edition number: --- Author(s): Hernández García, Adela Verónica. Title of article (in a journal) or chapter (in a book): --- Title of book or journal: Thesis: Utilización de los swaps como mecanismos de cobertura de la renegociación de la deuda del gobierno federal con la banca comercial Date of publication: 1994. Volume or edition number: --- Author(s): Castro Fritz, Yerom Title of article (in a journal) or chapter (in a book): --- Title of book or journal: Thesis: El mercado de los SWAPS cálculo y manejo de tasas equivalentes Date of publication: 1993. Volume or edition number: --- Author(s): Marshall, John F.John Francis / Kenneth R. Kapner Title of article (in a journal) or chapter (in a book): Introduction, Chapter 1, Chapter 2, Chapter 4. Title of book or journal: Understanding swaps Date of publication: 1993. Volume or edition number: XVI, ISBN 0471308277 Author(s): Davis, E. Philip Title of article (in a journal) or chapter (in a book): --- Title of book or journal: Pension funds : retirement-income security, and capital markets Date of publication: 1995. Volume or edition number: ISBN 0198288808 Author(s): Society of Actuaries Title of article (in a journal) or chapter (in a book): Course P-360U complete Title of book or journal: principles of pension valuation I-U.S. Date of publication: 1999. Volume or edition number: USA, SOA, SERIES

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Author(s): Society of Actuaries Title of article (in a journal) or chapter (in a book): Course P- P-363 complete Title of book or journal: pension funding vehicles Date of publication: May 1999. Volume or edition number: USA, SOA, SERIES Author(s): Hall, Gordon M. Title of article (in a journal) or chapter (in a book): ---- Title of book or journal: Mercer handbook of Canadian pension and benefit plans Date of publication: North York, Ontario : CCH Canadian Limited, 1996. Volume or edition number: 11th. edition, ISBN 1551417340 Author(s): William H. Aitken Title of article (in a journal) or chapter (in a book): ---- Title of book or journal: A problem-solving approach to pension funding and valuation Date of publication: Winsted, Connecticut : ACTEX Publications, 1996 Volume or edition number: 2nd ed, ISBN 1566982006 Author(s): Anderson, Arthur W. Title of article (in a journal) or chapter (in a book): ---- Title of book or journal: Pension mathematics of actuaries Date of publication: Winsted : ACTEX Publications, 1992 Volume or edition number: 2nd ed, ISBN 0936031107 Author(s): Society of Actuaries. Title of article (in a journal) or chapter (in a book): Course P-462C complete Title of book or journal: principles of pension valuation II, and accounting standards for pension plans-Canada Date of publication: 1999 Volume or edition number: USA, SOA, SERIES Author(s): Winklevoss, Howard E. Title of article (in a journal) or chapter (in a book): Course P-462C complete Title of book or journal: Pension mathematics : with numerical illustrations Date of publication: Philadelphia, Pa. : Pension Research Council, Wharton School of the University of Pennsylvania : University of Pennsylvania Press, c1993 Volume or edition number: xxxii, 0812231961 Author(s): Stephen A. Ross / Randolph W. Westerfield/ Jeffrey F. Jaffe Title of article (in a journal) or chapter (in a book): Chapter 25 Title of book or journal: Corporate Finance Date of publication: 1999 Volume or edition number: Fifth Edition

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Author(s): Richard Brealey / Stewart Myers Title of article (in a journal) or chapter (in a book): Part 8 Chapter Twenty Six Title of book or journal: Principles of Corporate Finance Date of publication: 2000 Volume or edition number: Sixth Edition Author(s): John C. Hull Title of article (in a journal) or chapter (in a book): Swaps Title of book or journal: Options, futures & other derivatives Date of publication: 2003 Volume or edition number: Fifth Edition Author(s): Rendleman, Richard J. Title of article (in a journal) or chapter (in a book): Swaps Title of book or journal: options, futures, and swaps Date of publication: 2002 Volume or edition number: ISBN 0631215905 Author(s): Boyle, Phelim P. Title of article (in a journal) or chapter (in a book): Swaps Title of book or journal: Derivatives : the tools that changed finance Date of publication: 2001 Volume or edition number: ISBN 189933288

ENDNOTES 1. When the contributions, withdrawals and interests are made or earned over the period it should be considered the date when they have been made to obtain the effective interest rate. 2. In order to understand the competitive advantage please refer to the introduction. 3. Survival probability weighted. 4. The retiree group used for the example is:

Birth Monthly Retiree Sex Date Pension

1 M 14-May-41 $11,006 2 M 24-Nov-42 $53,825 3 M 26-Feb-37 $5,611 4 M 25-Ago-28 $1,355 5 F 1-Jun-36 $14,890 6 M 7-May-41 $7,601 7 M 7-Ago-24 $1,355 8 M 3-Mar-36 $8,523 9 M 13-Ago-25 $1,355 10 M 2-Nov-44 $5,870