Valuation procedure - MvcFLPARMortalityBond

FLOATING-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 portfolio units aged x, on which we 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. 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 floating 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 N 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). In this case, r is floating which also means P(i) is floating. For future valuations, we assume that forward rates are realised and set r to equal the applicable forward rate at that future time.
The P(i)'s are floating and therefore not set at the out-set, what is set at the outset is the method of calculation of P(i)'s. The P(i)'s 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 mortality 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 and apply the usual option valuation formulae similar to valuation of interest rate floor. 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-FLPAR] 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 unit of mortality portfolio 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.
Enter the original number of outstanding coupons at time of initial creation/structuring of the bond.
Enter the current number of outstanding coupons.
Enter the original number of outstanding principal deduction times at time of initial creation/structuring of the bond.
The current number of outstanding principal deduction times is internally calculated 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 mortality unit age at time of initial creation/structuring of the mortality bond in years.
Enter the number of same-age mortality portfolio 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 mortality portfolio unit(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.
If prompted, enter recent rate observation resulting in next coupon payment convertible coupon-payment-frequency times per year.
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 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 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.
If prompted, enter the accumulated value of prior and current principal deductions.
The current age of same-age mortality portfolio units is internally calculated by the system.
If prompted, enter threshold mortality indices at indicated principal deduction times.
Enter volatilities of survival indices at indicated principal deduction times. Note that the volatility of the mortality index is the same as that for the survival index since on this system mortality index is defined as: MortalityIndex = 1.0 - SurvivalIndex.
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.