# Question 118 explain why the discount function for

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Question 11.8Explain why the discount function for age 65 is10vrather than10½v.Using commutation functionsThe calculation of pension EPVs can be aided by the use of commutation functions.These are not included in the standard international actuarial notation, so any symbolswe use must be defined as we go along.The idea will become clearer from someexamples.So, we will take the EPV of the lump sum we have just described, and define somesuitable commutation functions that we can use.The main step in this process is tomultiply the numerator and denominator of the EPV formula byxv, wherexis thecurrent age of the member(the “valuation age”).So, in our case, we multiply anddivide through by55vto obtain:55½56½64½6555566465555510,000EPVvrvrvrvrvlWe then define symbols to represent all the elements of the expression.
Page 14CT5-11: Pension fundsIFE: 2017 ExaminationsThe Actuarial Education CompanyWhile we could use any symbols for these (provided we define them!), we will adoptthe same notation as used in the Pension Scheme Tables.So for the denominator wedefine:xxxDv land for the numerator define:½65656565xxrxvrxCvrxNote that we have to define the function differently for the two age ranges toaccommodate the retirements occurring on the exact 65th birthday.So the EPV now looks like this:105555010,000rttEPVCDWe now tidy this up further by defining:650xrrxx ttMCand so:555510,000rMEPVDQuestion 11.9Calculate the expected present value of a lump sum benefit of £50,000 paid on normalage retirement, for a scheme member aged exactly 52, assuming that this benefit is paid:(a)regardless of the actual age at retirement(b)only if the member retires after the 64th birthday.Use the assumptions underlying the Pension Scheme Tables, and 4%painterest.
CT5-11: Pension fundsPage 15The Actuarial Education CompanyIFE: 2017 ExaminationsFrom this question note that, in general:anjxMfunction will include all decrements by causejoccurring at exact agexand in all future years after agexajxCfunction will include decrements by causejoccurring at agexand in thesingle year of age beginning at agexonly.Dividing byyD(whereyis the valuation age) will ensure that the correct discountingand survival probabilities are included.3.2Lump sums paid on other types of decrementThe other possible decrements are withdrawal, death and ill-health retirement.Assuming, as before, that all remaining active members retire “normally” on the 65thbirthday, then we know that:6565650wdiOther than this, the formulae used are of identical construction to before.So, forexample, the EPV of a lump sum of £100,000 payable immediately on death in serviceof a member currently aged 44 would be:4444100,000dMEPVDwhere:640½xxxxddxx ttdxxxDv lMCCvdnoting that thedxMfunction sums to64tx(not65x), because there is no65dCfunction to include.

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Probability theory, probability density function, Cumulative distribution function, Actuarial Education Company