FIXED-COUPON-RATE PRINCIPAL-AT-RISK LONGEVITY BOND WITH-INTER-PATH PRINCIPAL DEDUCTIBILITY VALUATION PROCEDURE

VALUATION METHODOLOGY

1. The aim is to structure and value an in-force custom principal-at-risk longevity bond to hedge against longevity risk.
2. Suppose we have a portfolio of N pensioners aged x, on which we pay V dollars at payment times before adjusting for cost inflation and we want to structure a longevity bond to hedge against longevity risk over the next T years.
3. We calculate the average payment per pensioner by dividing V by N which we denote as AVP, this becomes our payment per surviving pensioner. If the cost inflation for escalating payments is set at an annual rate of h, the value of the avarage payment per surviving pensioner after i time-steps from now is calculated as AVP*(1+h)^(i/f), where f is payment frequency per year, * is the multiplication operator and ^ is the power operator.
4. Suppose we decide on structuring a principal-at-risk longevity bond that pays non-mortality related coupons at a fixed rate of r convertible f times a year. To set the notional principal P(i) at time step i from now that ensures payments cover our corhort payments under the assumption that they all survive(non-mortality related) we simply solve the following equation for P(i): P(i)*r/f = N*AVP*(1+h)^(i/f). The result is P(i) = (f/r)*N*AVP*(1+h)^(i/f).
5. The P(i)'s set at the out-set are notional except for the terminal value of P(i) which is P(n). P(n) is the terminal principal on which all principal deductions are applied, the notional principals at other time-steps are simply used to calculate the value of deduction options if survival index threshold is exceeded and the payment time matches principal deduction time. The deducted value at principal deduction time i is P(i)*MAX[S(i)-S(ti),0] where S(ti) is the threshold survival index agreed at the outset, and S(i) is the realised survival index at time step i for a life aged x at creation of the principal-at-risk longevity bond. It is important to note that the valuation model used by this system does not assume uniform distribution of principal deduction times over the life of the bond. Instead, the assumption is that principal deductions occur at maturity, and then at points of equal successive time periods starting from maturity. The principal deduction times are also assumed to occur at coupon payment times, that is, the time period between successive principal deduction times must be an integer multiple of the time between coupon payments. As an example, if the number of outstanding principal deduction times on a 5 year principal at risk longevity bond is 1, coupon payment frequency per year is 2, principal deduction will occur at 5 years maturity time only. If on the afore-mentioned bond, the number of outstanding principal deduction times is 2 and constant period between succesive principal deduction times is 1.5 years, principal deduction will occur at 5 years point, and 3.5 years point. If, on the other hand, you enter the constant period between principal deductions as 1.2, the system will reject the value since it not an integer multiple of 0.5, the time between successive coupon payments.
6. Suppose we want to value a newly issued principal-at-risk longevity bond with q principal deduction times. To value the deduction options, we simply assume that expected survival probabilities are realised based on the reference population mortality tables, then value deduction options using a methodology similar to that applied to interest rate cap valuation. If the present values of the deduction options are OP(1), OP(2), ...., OP(q), and P(n)*D(n) is the present value of terminal principal, where D(n) is the applied discount factor, the present value of deducted terminal principal DTP is calculated as: DTP = P(n)*D(n)-OP(1)-OP(2)-....-OP(q). The other portion of the bond is non-mortality related which is valued using the usual zero volatility valuation approaches, and added to DTP to get the value of the principal-at-risk longevity bond.
7. To value an in-force principal at risk longevity bond that was created sometime ago, the system first tests if the number of outstanding principal deduction times is less than the original number of principal deduction times at time of creation of the bond, or current time matches principal deduction time. If the number of outstanding principal deduction times is less than the original number of principal deduction times for an in-force bond or current time matches principal deduction time, it means we are past at least one principal deduction time and you will be prompted to enter the accumulated value of prior and current time principal deductions. To avoid double counting, the system automatically sets current time option value to zero if current time matches principal deduction time since its value is already included in the accumulation account. When principal is deducted, its value is simply accumulated according to the terms agreed at the outset and is not applied to notional principals. This ensures that the coupon payments from the bond are non-mortality related. Instead, the accumulated value of principal deductions is added to the total present value of outstanding deduction options and deducted from the present value of terminal principal to get DTP. The value of the in-force principal-at-risk longevity bond is calculated by adding the value of non-mortality related payments to DTP.
8. Although the valuation described in this application relates to pensioners, it can be applied to other life annuitants.
9. Click the view-page-by-view-page instructions below to value the principal-at-risk longevity bond.

Financial solutions home page

1. Log into your account or if you are a new user, register your account and enable two factor authentication.
2. Click on the [Longevity bond-FXPAR] button, this will take you to the Dynamic-text-blocks parameters page.

Dynamic text-blocks parameters page

1. Select to enter flat or time-varying yield curve at indicated times.
2. Select to use system-default survival indices or your own-custom reference population survival indices.
3. Select to enter time-varying or flat payment(un-adjusted for cost-inflation) per surviving pensioner at coupon payment times.
4. Select to enter time-varying or flat survival index volatility at principal deduction times.
5. Select to enter threshold survival indices or value at-the-money-options bond. If you choose the at-the-money options bond, the threshold survival indices used in option valuation are set by the system to equal expected survival indices at principal deduction times.
6. Enter the original number of outstanding coupons at time of initial creation/structuring of the longevity bond.
7. Enter the current number of outstanding coupons.
8. Enter original number of outstanding principal deduction times at initial creation/structuring of the longevity bond.
9. The current number of outstanding principal deduction times will be calculated internally by the system.
10. Enter the time till next coupon payments in years.
11. Enter coupon payment frequency per year.
12. Enter constant period in years between successive principal deduction times.
13. Enter pensioner age at time of initial creation/structuring of the longevity bond in years.
14. Enter the number of same-age pensioners at time zero.
15. Click the submit button, and this will take you to the array populating page.

Array populating page

1. Enter zero rates convertible coupon-payment-frequency times per year at indicated times.
2. If prompted, enter survival indices at indicated times.
3. Enter payment per surviving pensioner(un-adjusted for cost inflation) at indicated payment times.
4. Click the submit button, and this will take you to the Principal-at-risk longevity bond valuation parameters page.

Principal-at-risk longevity bond valuation parameters page

1. If prompted, select to enter choice of sex for default mortality tables.
2. Enter the name of reference entity.
3. Enter the name of currency of valuation.
4. Enter the nominal coupon rate rate convertible swap-payment-frequency times per year. The coupon rate is used by the system to calculate the value of notional principal used in principal deductions options calculations. As an example, if the coupon rate convertible 2 times per year is entered as 0.05, the cost-inflation factor at a particular future principal deduction time is 1.0404, the number of pensioners at time zero(current-time) is 200, and the payment per surviving pensioner is 100, the notional principal used in calculation of principal deduction option value at this particular future time is calculated as 200*100*1.0404*2/0.05 = 832,320, that is, the coupon rate is used to scale-up the notional principal to a value such that the coupon payments will cover the inflation-adjusted payments assuming all pensioners at time zero(current-time) survive.
5. Enter future annual cost-inflation rate assumption applied to payments.The cost inflation rate assumption can be used in two equivalent ways. As a simple example, suppose you structure a three year longevity bond that is supposed to match yearly payments per survivor of 1000 escalated by cost inflation rate of 2%. You can choose to enter flat payment per survivor of 1000, and inflation rate assumption of 0.02. The system will calculate payment per survivor for year 1, year 2, and year 3 payments as 1000*1.02, 1000*(1.02)^2, and 1000*(1.02)^3, which results in three consecutive inflation-adjusted yearly payments of 1020, 1040.4, and 1061.208. Alternatively you can choose to enter time-varying payments per survivor of 1020, 1040.4, and 1061.208 and set the inflation rate assumption to 0.0. The result will be the same. If after choosing three time varying payments per survivor of 1020, 1040.4, and 1061.208, you set the cost inflation rate assumption as 0.02, the system will calculate inflation adjusted payments per survivor as 1020*1.02, 1040.4*(1.02)^2, and 1061.208(1.02)^3. This will result in three consecutive payments per survivor of 1040.4, 1082.432, and 1126.162.
6. If prompted, enter accumulated value of prior and current principal deductions.
7. The current age of same-age pensioners will be calculated internally by the system.
8. If prompted, enter threshold survival indices at indicated principal deduction times.
9. Enter volatilities of survival indices at indicated principal deduction times.
10. Click the submit button and this will take you to the output display page.

Output display page

1. You can view pricing model output together with input parameters you entered.
2. At the bottom of the page you can click on the button to create database record for the current model output. This will take you to the create database record page where you should click on the create button to create the database record.
3. After clicking on the create button to create the database record, you will be taken to the database records view page, where you can scroll vertically or horizontally to view database records including the one you just created.
4. You can filter database records according to the currency of valuation using the filter box. You can click on the details link on the extreme right of a particular database record, this will take you to the particular database record details page.