INTRODUCTION TO ACTUARIAL STUDIES

ACST152 INTRODUCTION TO ACTUARIAL STUDIES Week 1 Lecture 1CCalculating Superannuation BalancesThe aim is to answer the question;If I put $C into an account at the end of each year, and the account earns interest at i p.a, how much will I have at the end of n years ?We can do these calculations1. Using the formula for the sum of a geometric progression2. Using a spreadsheetLets start with method 1

In this lecture we will(a) Find a shortcut formula for finding the accumulated value of regular payments of C per annum at the end of each year(b) Adjust the contributions for tax and administration fees(c) Adjust the interest rate for tax and administration feesNext lecture: Allowing for the impact of salary increases, inflation, and variable interest rates

Before we go any further we need to have a brief revision of some high school work on Geometric Progressions

Sum of a sequence of numbers which are in geometric progressionIf numbers are in a geometric progression then each number in the sequence is equal to the previous number, multiplied by a constant factorr.If the first term is denoted a, and the constant ratio is r, then the sequence isa, ar, ar2, ar3, ar4, ar5and so on ....Note that the nth term in the sequence is arn-1Examples: 2, 4, 8, 16, 32.. (a = 2 and r = 2)-3, 9, -27, 81, -243(a = -3 and r = -3)8, 4, 2, 1, , 1/4, 1/8, ..(a = 8 and r = )

Lets suppose that we want to find the sum of the first n terms in the sequence, i.e. we want to find Sn where Sn = a + ar+ ar2 +ar3+ ....arn-1(eqn 1)The easiest way to find this sum is to make a second equation by multiplying throughout by r and then subtracting the two equations.r Sn = ar+ ar2 +ar3+ .....arn(eqn2)Subtract eqn 2 from eqn 1. Most of the terms cancel out.Sn= a + ar+ ar2 +ar3+ .....arn-1r Sn= ar+ ar2 +ar3+ ..... arn-1 +arn________________________________(1-r) Sn = a - arn

and rearranging gives

or alternatively you can write

Either formula will give the same answer.

Example: (a) Find the sum of the first 8 terms of the sequence 2, 4, 8, 16, 32...

(b) Find the sum of the first 40 terms of the sequence 100, 100*1.05, 100*1.052...

Answer (a)a = 2, r = 2, n = 8

Answer (b)a = 100, r = 1.05, n = 40

Question: What happens if the value of r is 1?

Answer: Sn = n * a

Tutorial Exercise:

Use the method of Proof by Mathematical Induction to prove that

Regular SavingsLets suppose that you are going to save $C every year. (C = Contribution to Superannuation Fund)

At the end of each year you will put $C into a bank account, and you will do this for n years. [Note that when compulsory superannuation was first introduced in 1992, employers were required to make contributions annually within a few weeks of the end of the financial year. In practice most employers make contributions monthly. In ACST101 you will learn how to adjust for monthly contributions]The account earns compound interest at the rate i per annum.

Q. How much money will you have at the end of n years ?

We could do this by simply calculating the balance each year.

Iterative Formula:If the balance at time t is BtThen the balance a year later, at time t+1, will beBt+1= Bt * (1+i) + C

Balance of account at time 0 = B0 = 0(i.e the account has no money at the start of the first year)

Balance of account at time 1 = B1B1= B0 * (1+i) + C= 0*(1+i) + C = Ci.e. at the end of the year the account holds the first contribution

Balance of account at time 2 = B2B2 = B1 * (1+i) + C= C * (1+i) + C

Balance of account at time 3 = B3

B3 = B2* (1+i) + C= [C* (1+i) + C] * (1+i) + C= C *(1+i)2 + C * (1+i) + C

Balance of account at time 4 = B4B4= B3 * (1+i) + C= [C *(1+i)2 + C*(1+i) + C] *(1+i) + C= C*(1+i)3+ C* (1+i)2 + C* (1+i) + CAnd so on.You can see the pattern emerging.

The Balance of account at time n = Bn whereBn = C(1+i)n-1+ C(1+i)n-2+ . C(1+i)2 + C (1+i) + C

And writing the terms in reverse orderBn = C + C(1+i)+ . C(1+i)n-2 + C (1+i)n-1

This may be written using summation notation

The terms in this equation form a geometric progression with the initial term C, constant ratio (1+i), and n terms.So using the formula previously derived for the sum of a geometric progression,

Account at end of year n

Example: Find the accumulated value of $100 paid at the end of each year for 40 years, if the account earns interest at 5% per annum.

=

= 12,079.98

An Alternative ApproachIt is often helpful to think of each separate payment accumulating in a separate account, for n years. Find the accumulated value of each payment at time n years, and then add them together.

The first payment of $C is made at time t = 1 and earns interest for n-1 yearsAccumulated value of first payment = C (1+i) n-1The second payment of $C is made at time t = 2 and earns interest for n-2 yearsAccumulated value of second payment = C (1+i) n-2The third payment of $C is made at time t = 3 and earns interest for n-3 yearsAccumulated value of third payment = C (1+i) n-3And so on.The last-but-one payment is made at time t = n-1 and earns interest for one yearAccumulated value of last-but-one payment = C (1+i) 1The last payment is made at time t = n and earns no interestAccumulated value of last-but-one payment = C (1+i) 1If all these separate accumulated payments are added together, we get

Bn = C(1+i)n-1+ C(1+i)n-2 + . C(1+i)2 + C (1+i) + C

And writing the terms in reverse order

Bn = C + C(1+i)+ . C(1+i)n-2 + C (1+i)n-1

So this is just another method of getting to the same resultACTUARIAL NOTATION FOR ACCUMULATED VALUES

Since this type of calculation is very common in actuarial work, the actuaries got together about 100 years ago and decided to all use the same notation.

is the symbol for the accumulated value of payments of $1 per year, made at the end of each year for n years, at rate of interest i per annum compound.in arrears means at the end of the yearin advance means at the start of the year

=

Examples:Find the values of the following(a) $50 per annum payable in arrears for 20 years at the rate of interest 10% per annum(b) $200 per annum payable in arrears for 60 years at 1% per annum

Answer (a)

Answer (b)

REASONABLENESS CHECKS

Everyone makes mistakes from time to time. Therefore, whenever you do a calculation, it is important to look at the answer and say: Does this answer look reasonable ? Or is it obviously wrong?If you were earning NO interest, then the accumulated value would be just the sum of n payments of $R, i.e. n * R. If you are earning a positive rate of interest, then the accumulation should be HIGHER than n*R. So whenever you do an accumulation calculation, accumulating n payments of R each, check to make sure that your answer is higher than n * R.

Example: $200 for 60 years at 0% would be 200*60 = 12,000Our answer of 16,333.93 was larger - OKSUPERANNUATION ACCUMULATIONS

Theoretically, we could use the above formulae to estimate the accumulated value of our superannuation savings at retirement age (say age 65).

Lets suppose that your employer is complying with Australian law and paying 9% of your salary into a superannuation fund earning 8% per annum interest. Your salary is $50,000 per annum, so this means that the contribution is $4500. Assume payments are made at the end of each year. Using the formula for the accumulated value of regular payments, after 40 years you will have

4,500 at 8% = 1,165,754.33 You will be a millionaire ! Sadly, this is much too optimistic...COMPLICATIONS

Unfortunately real life is more complicated !!! We must allow for :

1. Taxes on contributions 2. Administration Fees & Insurance costs3. Taxes on investment income4. Investment management fees (asset based fees)5. Salary increases6. Inflation of the cost of living

1. Tax on Contributions

The first problem is tax on contributions. The government charges a tax of 15% on each contribution made by the employer into a super fund.

(NB In reality there are different tax rules for different types of contribution, but we will ignore this at present this is a simplified model)

Tax = 15% * 4500 = 675

Net Contribution after deducting tax = 4,500 - 675 = 3,825Accumulated value of net contributions

= 3825 at 8% = $ 990,891.18 So you wont be a millionaire, but it still looks okay.2. ADMINISTRATION FEES & INSURANCEHowever, the superannuation fund will also charge you an annual administration fee.

Different funds charge different fees, but it is likely that the fee will be at least $50 per annum. (This is probably on the low side).

Note that some funds charge a flat fee say $X each year but others charge a percentage of the amount you contribute say 1% of 4500 = $45, and some charge both a flat fee AND a percentage fee. [NB The total amount of fees paid by Australian superannuation fund members in one year is about $13 BILLION. Some people think that the superannuation funds are over-charging; we will look at this issue later in the term.]

Also, most superannuation funds buy life insurance policies for all their members.so if you die, your orphaned children will have enough money to survive and prosper.

Some also buy disability insurance for their members so that you receive a payout if you are injured or become ill and unable to work.

The cost of the insurance is deducted from your account each year. The cost will vary from fund to fund, since some funds provide a high amount of insurance and others provide a much lower amount. Lets say that you pay $150 per annum for insurance. (This is probably on the low side)

So the net contribution, after taxes, insurance, and admin fees, isGross contribution 4500 Less tax 675 Less admin fee 50 Less insurance 150 Net Contribution 3,625So your accumulation is down to

3625 at 8% = 939,079.88Note that allowing for contribution taxes, administration fees and insurance has reduced your savings by about 20%. [Down from $1,165,754.33 to 939,079.88]

3. TAX ON INVESTMENT INCOME

Sadly there is more bad news ! Investment income is also affected by taxes and fees.Investment income is taxed at 15%.

[In practice, due to various special deductions and concessions and special treatment of capital gains, the actual tax paid is usually less than 15%. The details are too complicated for first year students.]

If the gross interest rate is i And the government takes tax of t*i Then the net interest rate is i *(1-t)

Example: Suppose that a person has $100 in their account at the start of the yearInterest = 8.00Tax on interest = 15% * 8 = 1.20Net interest after tax = 8.00 1.20 = 6.80

So the net interest rate after tax is = 8% * (1-0.15) = 6.8%

Our accumulated value is now down to

3625 at 6.8% = $ 687,402.88

4. INVESTMENT MANAGEMENT FEES

Of course superannuation funds also charge you an investment management fee. This money is used to pay the investment managers who provide their expert skills in managing your investments, and also covers investment-related transaction costs such as brokerage.

This fee is an asset-based fee, i.e. it is usually a percentage of the balance of your account.

The percentage usually varies according to the type of the investment if you invest in something simple like government bonds then the rate might be just 1%, but if you invest in something which requires a lot of research and expertise, like private equity or hedge funds, it might be 2% or more.

Lets be optimistic and assume that you will only pay 1% per annum, and the fee is applied by reducing the net interest rate credited to your account. (In practice there are different methods of calculation but this is a simple approximation).

To work out the net-of-tax-and-fees interest rate, lets assume thatGross interest rate = 8%Net of tax interest rate = 6.8%And if investment management fees are 1%Net of tax and fees interest rate = 5.8%

Our accumulated value is now down to

3625 at 5.8% = $ 533, 582

OVERALL IMPACT OF FEES AND TAXES

Note that fees and taxes make an enormous difference to the outcome. In this example, the final balance after allowing for fees and taxes and insurance costs is only about 46% of initial amount. A small difference in the fees makes a big difference to the outcome.Q. How much would the accumulation be, if the fund charged an investment management fee of 1.5% , instead of 1% per annum?Answer = $471,321

Increasing the investment fee by half a percent reduced the payout by more than $60,000ADJUSTMENTS TO FINAL PAYMENT

Some funds also charge an exit fee, which is theoretically designed to cover the cost of processing the paperwork for payments. It is usually a relatively small amount.

In the past, the government would also charge a benefits tax at the end, when you took your benefit out of the fund. Fortunately, a few years ago the government decided to change the rules so that you dont have to pay tax if you are over age 60 when you take the benefit out of the fund. If you take out money before age 60, i.e. retire early, then you WILL have to pay tax.

Q. Why is there higher tax if you take money out before age 60?

Next week: Allowing for salary increases Allowing for inflation Allowing for variations in the investment income earned by the fund each year

Superannuation CalculatorOver the next two weeks you will be building your own superannuation calculator.Tute exercise: Look at ASICs superannuation calculatorHomework: Start learning EXCEL skills.Watch the EXCEL demo on iLearn in the week 1 folder and build your own version of the spreadsheet.

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