1. The aim is to structure and value an in-force custom survivor-cap to hedge against longevity risk by reducing downside risk that cohort cash-outflows become higher than expected due to higher than expected survival probabilities.
  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 survivor cap to hedge against downside element of 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. Our total cohort cashflow at time step i is -N*AVP*(1+h)^(i/f)*S(i) where S(i) is the realised survival index at time step i. We structure our survivor cap to give the following cash flow at time step i : N*AVP*(1+h)^(i/f)*MAX[S(ri)-S(ti),0], where S(ti) is the survival index value agreed at the outset, and S(ri) is the realised survival index at time step i for a life aged x at creation of the survivor cap for the chosen reference population. Our aggregate cash-flow at time step i is now: -N*AVP*(1+h)^(i/f)*S(i) + N*AVP*(1+h)^(i/f)*MAX[S(ri)-S(ti),0]. If S(ri) is greater than S(ti) our aggregate cash-flow will be N*AVP*(1+h)^(i/f)*(S(ri)-S(i)-S(ti)), otherwise it will be -N*AVP*(1+h)^(i/f)*S(i). If the reference population experience matches our cohort experience, that is, S(ri)=S(i), our aggregate cashflow becomes -N*AVP*(1+h)^(i/f)*S(ti), which is known in advance if S(ri) is greater than S(i) or -N*AVP*(1+h)^(i/f)*S(i) if it is less. In most cases S(ri) and S(i) do not match but are highly positively correlated such that the variance of N*AVP*(1+h)^(i/f)*(S(ri)-S(i)-S(ti)) is very low when compared with the variance of -N*AVP*(1+h)^(i/f)*S(i), thus significantly reducing the downside element of longevity risk while retaining the ability to benefit from the upside element of longevity risk.
  5. To value the survivor cap, we simply assume that expected survival probabilities S(ri) are realised, then value the survivor caplet options in a similar way to the method used to value interest rate cap. The system gives options to use either flat survival index volatility or spot survival index volatilities. We then sum the present values of survivor caplets to get the value of the survivor cap.
  6. Click the view-page-by-view-page instructions below to value the survivor cap.

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 [Survivor cap] 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 indicated floating survival-index observation times.
  5. Select to enter threshold survival indices or value at-the-money-caplets cap. If you choose the at-the-money-caplets cap, the threshold survival indices used in option valuation are set by the system to equal expected survival indices at floating survival-index observation times.
  6. Enter the original number of outstanding payment times at time of initial creation/structuring of the survivor cap.
  7. Enter the current number of outstanding payment times.
  8. Enter the time till next caplet payments in years.
  9. Enter cap payment frequency per year.
  10. Enter age of same-age annuitants at time of initial creation/structuring of the survivor cap.
  11. Enter the number of same-age annuitants at time of initial creation/structuring of the survivor cap.
  12. Click the submit button, and this will take you to the array populating page.

Array populating page

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

Survivor cap 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 future annual cost-inflation rate assumption applied to annuitant payments. The cost inflation rate assumption can be used in two equivalent ways. As a simple example, suppose you structure a three year survivor-cap 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.
  5. Enter dealing expenses and tax.
  6. Enter profit loading. If the price of the cap including dealing expenses and tax is P, the profit loading x is such that the final price of the cap is calculated as (1.0 + x)*P.
  7. If prompted, enter threshold survival indices at indicated payment times.
  8. Enter volatilities of survival indices at indicated observation times.
  9. 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.