Valuation procedure - MvcFXPARMortalityBond

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

VALUATION METHODOLOGY

The aim is to structure and value an in-force custom principal-at-risk mortality bond to hedge against mortality risk.
Suppose we have a portfolio of N mortality units aged x, on which we agree to pay V dollars at payment times in the event of death before adjusting for cost inflation and we want to structure a mortality bond to hedge against mortality risk over the next T years.
We calculate the average payment per mortality unit by dividing V by N which we denote as AVP, this becomes our payment per mortality unit in the event of death. If the cost inflation for escalating payments is set at an annual rate of h, the value of the avarage payment per mortality unit 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.
Suppose we decide on structuring a principal-at-risk mortality 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 are all dead(non-mortality related in the sense that the number of dead units is fixed) 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).
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 mortality index threshold is exceeded and the payment time matches principal deduction time. The deducted value at principal deduction time i is P(i)*MAX[(1.0 - S(i))-(1.0 - S(ti)),0]=P(i)*MAX[(S(ti)- S(i)),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. The use of the words "survival index" and "mortality index" is interchangeable here because mortality index M(i) = 1.0 - S(i). 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 mortality 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.
Suppose we want to value a newly issued principal-at-risk mortality 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 floor 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 mortality bond.
To value an in-force principal at risk mortality 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 principal deductions. 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 mortality bond is calculated by adding the value of non-mortality related payments to DTP.
Click the view-page-by-view-page instructions below to value the principal-at-risk mortality bond.

Financial solutions home page

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

Dynamic text-blocks parameters page

Select to enter flat or time-varying yield curve at indicated times.
Select to use system-default survival indices or your own-custom reference population survival indices.
Select to enter time-varying or flat payment(un-adjusted for cost-inflation) per mortality portfolio unit at coupon payment times.
Select to enter time-varying or flat survival index volatility at principal deduction times.
Select to enter threshold mortality indices or value at-the-money-options bond. If you choose the at-the-money options bond, the threshold mortality indices used in option valuation are set by the system to equal expected mortality indices at principal deduction times. Mortality index as defined in this system is related to the survival index by the following formula: MortalityIndex = 1.0 - SurvivalIndex.
Enter the original number of outstanding coupons at the time of initial creation/structuring of the principal-at-risk mortality bond.
Enter the current number of outstanding coupons.
Enter the original number of outstanding principal deduction times at the time of initial creation/structuring of the principal-at-risk mortality bond.
The current number of outstanding principal deduction times is calculated internally by the system.
Enter the time till next coupon payments in years.
Enter coupon payment frequency per year.
Enter constant period in years between successive principal deduction times.
Enter age of same-age mortality units at time of initial creation/structuring of the mortality bond.
Enter the number of same-age mortality units at time of initial creation/structuring of the mortality bond.
Click the submit button, and this will take you to the array populating page.

Array populating page

Enter zero rates convertible coupon-payment-frequency times per year at indicated times.
If prompted, enter survival indices at indicated times.
Enter payment per unit of mortality portfolio(un-adjusted for cost inflation) at indicated payment times.
Click the submit button, and this will take you to the Principal-at-risk mortality bond valuation parameters page.

Principal-at-risk mortality bond valuation parameters page

If prompted, select to enter choice of sex for default mortality tables.
Enter the name of reference entity.
Enter the name of currency of valuation.
Enter the nominal coupon rate rate convertible coupon-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 mortality portfolio units at time zero(current-time) is 200, and the payment per unit of mortality portfolio 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 the same number of all same-age mortality portfolio units at time zero(current-time).
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 mortality bond that is supposed to match yearly payments per mortality unit of 1000 escalated by cost inflation rate of 2%. You can choose to enter flat payment per mortality unit of 1000, and inflation rate assumption of 0.02. The system will calculate payment per mortality unit 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 mortality unit 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 mortality unit 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 mortality unit as 1020*1.02, 1040.4*(1.02)^2, and 1061.208(1.02)^3. This will result in three consecutive payments per mortality unit of 1040.4, 1082.432, and 1126.162.
The current age of same age mortality units is calculated internally by the system.
If prompted, enter the accumulated value of prior and current principal deductions.
If prompted, enter threshold mortality indices at indicated principal deduction times.
Enter volatilities of survival indices at indicated principal deduction times. Note that, since the mortality index is related to the survival index by the formula: MortalityIndex = 1.0 - SurvivalIndex, the volatility of the mortality index is equal to the volatility of the survival index.
Click the submit button and this will take you to the output display page.

Output display page

You can view pricing model output together with input parameters you entered.
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.
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.
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.