## Actuarial Mathematics – Introduction to Commutation Functions

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While our generation of actuaries grew up with continuous time mathematics, my first exposure to actuarial functions and actuarial mathematics came through a chapter on commutation functions. A computational tool for easing the mathematics associated with repetitive calculations I grew so hooked to them that even today I use them as my first pass on a pricing function.

Commutation functions are a tool used by pricing and valuation actuaries. It is a means by which net single premiums and actuarial present values for various plans are determined. As compared to deriving these values from first principles, commutation functions once derived can be used to simplify and minimize the working that goes into computing these premiums, actuarial present values, etc. Through commutation functions intermediate values are tabulated and the premiums/values are expressed as functions of these intermediate values.

The commutation functions make use of the following as inputs:

- A life table which gives the values of l
_{x}at integer ages. l_{x}represents the expected number of survivors at age x out of an original cohort of lives at some starting age. - An assumption regarding a deterministic and constant interest rate per annum effective, i, that is used for discounting.

The commutation functions are defined below. Note that all summations (∑) go from y=x to ∞ (infinity):

D_{x}=ν^{x}l_{x ; }where ν=1/(1+i) è discounted lives

N_{x}=∑ D_{y} = N_{x+1}+ D_{x }è sum of discounted lives

S_{x}=∑ N_{y} = S_{x+1}+ N_{x }

C_{x}=ν^{x+1}(l_{x}– l_{x+1}) è discounted deaths

M_{x}=∑ C_{y} = M_{x+1}+ C_{x }è sum of discounted deaths

R_{x}=∑ M_{y} = R_{x+1}+ M_{x }

The summations are given to infinity above. However life tables have values up to a finite maximum age. The actual calculations for the commutation functions therefore would terminate at the highest age that is present in the life table chosen.

Examples of how commutation functions are used to determined net single premiums, actuarial present values, etc. are given below:

This is a screen shot of the hypothetical life table that we have used in building the commutation function table and the commutation functions derived in excel. Note that we have assumed that the interest rate, i, is 6% p.a.

The entries for x = 1, for example, are as follows:

ν = 1/(1+6%) = 0.94340

D_{1}=ν^{1}l_{1 }= 0.94340*97957.83 = 92,413

N_{1}= N_{2}+ D_{1} = 1487613.81 + 92413.05 = 1,580,026.86_{ }

S_{1}= S_{2}+ N_{2 }= 24266208.69 + 1,580,026.86 = 25,846,235.55

C_{1}=ν^{2}(l_{1}– l_{2}) = (0.94340^2)*(97957.83-97826.26) = 117.10

M_{1}= M_{2}+ C_{1} = 2860.47 + 117.10 = 2,977.56

R_{1}= R_{2}+ M_{1}= 114054.82+ 2977.56 = 117,032.38

**Life Insurance: Net Single Premiums **

A_{x}= Net single premium for a whole life insurance paying a unit benefit at the end of the year of death to someone who is now aged x= M_{x}/D_{x}.

For a person aged 33 now, the net single premium for a whole life insurance paying 1000 at the end of the year of death is:

1000 * A_{33}=1000*(M_{33}/D_{33}) = 1000*1624.44/13822.67 = 1000*0.11752 = 117.52

A_{x:n}= Net single premium for a n-year endowment insurance paying a unit benefit at the end of the year of death or at the end of the n-years whichever comes first, to someone who is now aged x = (M_{x}-M_{x+n}+D_{x+n})/D_{x }

For a person aged 33 now, the net single premium for a 20 year endowment insurance paying 2500 at the end of the year of death or at the end of 20-years whichever comes first is:

2500* A_{33:20} = 2500 * (M_{33}-M_{53}+D_{53})/D_{33} = 2500 * (1624.44-1127.43+4001.66)/13822.67 = 2500*0.3255 = 813.642

IA_{x}^{1}_{:n}= Net single premium for a n-year increasing term insurance payable at the end of the year of death to someone who is now aged x, where 1 is paid if the death occurs in the first year, 2 is paid is the death occurs in the second year, 3 is paid if the death occurs in the third year and so on = (R_{x}-R_{x+n}-nM_{x+n})/D_{x}

For a person aged 33 now, the net single premium for an 20-year term insurance payable at the end of the year of death, where 1 is paid if the death occurs in the first year, 2 is paid is the death occurs in the second year, 3 is paid if the death occurs in the third year and so on is:

IA_{33}^{1}_{:20} = (R_{33}-R_{53}-20M_{53})/D_{33} = (48671.94-20734.44-20*1127.43)/13822.67 = 0.3899

**Annuity: Actuarial Present Values **

a_{x}= The actuarial present value of a whole life annuity paying 1 per annum in arrears (i.e. at the end of the year), for life, to someone who is now aged x = N_{x+1}/D_{x }

For a person aged 60 the actuarial present value of a whole life annuity paying 100 per annum in arrears, for life is:

100* a_{60 }= 100*(N_{61}/D_{60}) =100 *(25181.06/2482.16) = 100 * 10.144817 = 1,014.48

ä_{x:n}= The actuarial present value of an annuity paying 1 per annum for n years where payments are made at the beginning of the year to someone who is now aged x = (N_{x}-N_{x+n})/D_{x}

For a person aged 60 the actuarial present value of a 20 year annuity paying 200 per annum at the beginning of the year is is:

200* ä_{60:20 }= 200*(N_{60}– N_{80})/D_{60} =200 *(27663.22-2183.49)/2482.16 = 200 * 10.2651 = 2,053.03

Ia_{x}= The actuarial present value of an annuity payable annually in arrears, for life, to someone now aged x of an amount of 1 in the first year, 2 in the second year and so on = S_{x+1}/D_{x }

For a person aged 60 the actuarial present value of a life annuity, payable in arrears, that pays an amount of 1 in the first year, 2 in the second year and so on is:

Ia_{60}= S_{61}/D_{60 }= 223239.29/2482.16 = 89.9375

I’m currently in those tween years btweeen 25 and 30 years of service. I think a lot about whether I should leap now, or wait for my next hiring anniversary for that extra benefit jolt. That anniversary, for me, will almost surely be after the reform (not the way *I* define reform!) legislation takes effect.I decided to try a little math. I ran a spreadsheet to see how the COLA change might affect my own pension. Warning: there’s a lot of hand-waving in this. I made two possibly unsupportable assumptions: first, the 3% flat COLA will never be revoked for retirees, even though I’ve read that this has been done by other retirement systems; and second, the increase under the new formula will average around 1.5% per year instead of 3%. This is based on the ideas that some years the increase will be zero or close to it.I also neglected to allow for the 5-year phase-in; soon I’m going to recompute the model using that.To cut to the chase, I computed that if I took the lower pension now, my monthly benefit would catch up in the 7th year after retirement. Beyond that, it would be higher under the current model.As I say, this doesn’t include the 5-year phase-in, and my assumptions are questionable. However, the fact remains that if you live long enough, and if the 3% COLA survives, it’s moderately likely that you’ll eventually catch up with and surpass the initially higher benefit you get from waiting to retire.Of course much depends on how long you wait to retire, and how much service credit you now have. Waiting 2 years rather than 1 will raise your initial monthly benefit and thus extend the catch-up date, of course. But you also won’t be getting your benefits for 2 years, which affects your lifetime total. That’s another factor I need to add to my spreadsheet.If you now have 30 years or more of SC, your benefit will increase more slowly with additional years of service. In that case, the decision is more clear cut. If I had my 30 in, I’d almost certainly jump now.I don’t know whether this actually helps, but it’s something more to consider. This is a complex and confusing situation, and at the moment nobody seems to be willing to commit to a solid answer.

The cost of living adtesjmunt is deeply flawed and does not provide adequate inflation protection for the following reasons:1.The current and proposed future COLA is a simple rather than a compound adtesjmunt. That means that the relative value of the adtesjmunt decreases (not the actual amount but the purchasing power) each year. The COLA is a fixed 3% of the initial benefit which results in an annual increase by the same amount each and every year for life. For example, if you receive a COLA of $1,000 after year one then you will receive $1,000 after year twenty of retirement as well. However, because of inflation, the $1,000 increase after year twenty will be worth far less than it did after year one. In fact, it will be worth only about one half of its initial value based on purchasing power. The bottom line is that the 3% COLA does not truly provide 3% inflation protection. Making matters worse, the proposed CPI based COLA will result in even less of an adtesjmunt. So, it is quite likely that the value of the pension received in the future will be significantly less than the initial pension benefit based on inflation.2.The proposed CPI COLA adtesjmunt has a major problem. While inflation may have averaged at or near 3% over the past twenty, it does not do so uniformly. It often rises and falls sharply. Say inflation is 5% in one year and 1% the following year. The two year CPI average would be 3%. However, the proposed COLA adtesjmunt would provide for only a maximum 3% adtesjmunt the first year and no increase for the second year. This results in a net loss of 2% and guarantees that pension purchasing power will decrease and never recover each time the annual CPI exceeds 3%.