ACST152 Introduction to Actuarial Studies2015 Tutorial Questions Set 3 - SolutionsA. Calculating the impact of different risksQ1. Create a spreadsheet which will allow you to test the effect of the following events on retirement savings. In each case, find the reduction or increase in retirement savings compared to the base case. In some cases you can check your answer by doing a manual calculation, in other cases you will need to use the spreadsheet. (Later in ACST101 you will learn how to derive formula for calculating accumulations with changes occurring part-way through the period, but you dont need to know how to do this for ACST152).(a) Base Case for MaryAnn : Salary $40,000 in first year increasing at 1% p.a. on 1 January each year. Contributions of 9.5% of salary per annum at the end of each year. Tax on contributions of 15% and annual administration fees equal to $200 p.a. increasing at 1% p.a. Investment returns of 7% p.a., with investment tax at 10% of investment income and an investment management fee of 1% p.a. (charged as a deduction from the net investment returns). The worker starts work at age 20 and works until age 65 (45 years)

(b) Starting from the Base Case: Assume that MaryAnns pay is 15% lower throughout her life, i.e. her starting salary is only $34,000. What is the impact on her retirement savings?

(c) Starting from the Base Case: Assume that MaryAnn has three children and she stays at home to look after them until the youngest is at school. She makes no contributions for years 10, 11, 12, 13, 14, 15, 16, and 17. When she returns to work she has the same pay rate as if she had been working all the time. What is the impact on her retirement savings?

(d) Starting from the Base Case: Assume that during the 20th year, as a result of regulatory changes, the tax on investment income increases to 15%. What is the impact on retirement savings for MaryAnn?

(e) Starting from the Base Case: Maryann has some health problems as a result of her unhealthy lifestyle. She has to retire early, due to ill health. Her disability income insurance policy pays her living expenses for the next five years but she does not make any superannuation contributions during the period from age 60 to 65 (i.e. missing the last 5 years of contributions). What is the impact on retirement savings for MaryAnn?

(f) Starting from the Base Case: Due to changes in government regulation, the investment management fees are reduced from 1% p.a. to 0.5% p.a. (without having any effect on the investment returns). The fees drop immediately (starting in the first year). What is the impact on retirement savings for MaryAnn?(a) Solution Base CaseGross contribution in first year is 9.5% * 40000 = 3800 Net Contribution in first year after contributions tax and admin fee = (1-0.15)*3800-200 = 3030 The net contribution is assumed to increase at 1% p.a. each year (note that both salary and admin fees are increasing at 1% p.a. so the net contribution will also increase at 1% p.a.)Gross Investment return = 7% Net investment return after taxes and investment management fee = 7% *(1-0.10)-1% = 5.30% The accumulated value of the first net contribution is 3030 * 1.053^44 The accumulated value of the second contribution is 3030 * 1.01 * 1.053^43 The accumulated value of the third contribution is 3030*1.01^2 * 1.053^42 And so on..... The accumulated value of the final payment is 3030 * 1.01^44 Summing the accumulated value of all payments (which form a geometric progression with constant ratio 1.01/1.053 = 0.95916429) gives... Total accumulation at the end of 45 years = 3030 * 1.053^44 * (0.95916429^45-1)/(0.95916429-1) = 609,600 Excel Solution: See Spreadsheet (tab for Base Case)(b) Solution: for 15% salary reductionUse the same method as before, summing a GP The net contribution is the first year is (0.095*34000) * (1-0.15) - 200 = 2545.50 Total accumulation at the end of 45 years = 2545.50 * 1.053^44 * (0.95916429^45-1)/(0.95916429-1) = $512,124 Note that the 15% reduction in salary led to a 16% reduction in benefits. Excel solution: Starting from the base case spreadsheet, you should be able to simply change the initial salary to $34,000

(c) Solution for missing contributions for 8 years It is difficult to do this by manual calculation; it is easier to do this in Excel. You don't need to know how to do such a difficult question for your test - I might ask you to deduct just one or two missing contributions in a class test.If you have to do the calculation without a computer, the easiest way is to deduct the accumulated value of the missing contributions from the answer in (a).In this case we assume that the administration fees will still be deducted every year (even though she is not making any contributions) Based on the calculations in part (a), the net contribution is the first year as 3230.This increases at 1% p.a., so the next contribution in the 10th year is 3230*1.01^ 9 = 3230*1.01^9This is paid at the end of year 10 (at t = 10) and accumulates with interest to t = 45 at a net rate of 5.3% p.a.Accumulated value of net contribution in year 10 at the end of the 45th year= 3313.87 (1.053^35)So Accum10 = 3230^1.01^9 * 1.053^35 = 21,532.17Similarly Accum11 =3230 * 1.01^10 * 1.053^34Accum12 = 3230 * 1.01^11 * 1.053^33and so oneAccum17 = 3230 * 1.01^16 * 1.053^28This forms a GP with n = 8 terms, first term a = 21,532.17, constant ratio r = 1.01/1.053 So the sum of the accumulated value of the contributions made between year 10 and 17 inclusive is 149,549So the accumulated contributions will be the answer from part (a), which was 609,600, less 149,549, giving 460,051Excel solution: Starting from the base case spreadsheet, put 0 in the working column for the relevant years. This should change the contributions to 0. (a) Solution: 460,051, a reduction of 24.5% compared to the base case Starting from the Base case: Assume that Mark stays at Uni to do a PhD and does not start work until age 25. His salary when he starts is $45,000. What is the impact on his superannuation savings?

(d) Solution for Increase in Investment tax

Starting from the Base Case: Assume that during the 20th year, as a result of regulatory changes, the tax on investment income increases to 15%. What is the impact on retirement savings for MaryAnn?

Note: the question was not very clear on the timing of the change.

If the change occurs at the start of year 20, the answer isAnswer : 566,627 (from spreadsheet) a reduction of 7.0%If the change occurs at the end of year 20 (so that the new rate applies in year 21 and after), the answer is 567,832Answer : 567,832 (from spreadsheet) a reduction of 6.9%

(e) Health problems leading to early retirement

Starting from the Base Case: Maryann has some health problems as a result of his unhealthy lifestyle. She has to retire early, due to ill health. Her disability income policy pays her living expenses for the next five years but she does not make any superannuation contributions for the five year period from age 60 to 65 (i.e. missing the last 5 years of contributions). What is the impact on retirement savings for MaryAnn?Answer: $582,355 (from spreadsheet) a reduction of 4.5%

(f) Solution for reduction in investment management fees

Starting from the Base Case: Due to changes in government regulation, the investment management fees are reduced from 1% p.a. to 0.5% p.a. (without having any effect on the investment returns). The fees drop immediately (starting in the first year). Answer: $699,320 (from spreadsheet) an increase of 14.7%

B. Calculating present values for single payments, multiple payments, and fixed term annuitiesQ1. Present ValuesThe present value P is the amount which must be invested now, in order to provide a specified sum R at a specified future time t, assuming that the amount is invested to earn compound interest at i per annum.P * (1+i)t = RTherefore P = R *(1+i) - t Find the present values of the following payments:(a) Payments of $100 at time t = 1, $200 at time t = 2, $300 at time t = 3, at 5% p.a(b) Payments of $1800 at time t = 6, $1200 at time t = 8, $500 at time t = 12, at 2% p.aSolution:(a) PV = 100 * 1.05^-1 + 200 * 1.05^-2 + 300 * 1.05^-3 = 535.80Check: if you had $535.80 in the bank account at time 0, The balance at time 1 (after making the first payment) would be 535.80*1.05-100 = 462.59 The balance at time 2 (after making the second payment) would be 462.59*1.05-200 = 285.72 The balance at time 3 (after making the third payment) would be 285.72*1.05-300 = 0.01 This shows that $535.80 would be the amount required in the account at time 0, which will be sufficient to make the required payments, with nothing left over (apart from a cent arising from rounding off). Reasonableness check: the present value of a series of payments should be less than the sum of the payments (assuming the interest earned on the account is positive). In this case $535.80 is less than $600 (i.e. less than 100 + 200 + 300) so this is reasonable.(b) PV = 1800 * 1.02 ^ -6 + 1200 * 1.02^-8 + 500 * 1.02 ^ -12 = 3016.78Check: if you had $3016.79 in the bank account at time 0, The balance at time 6 (after making the first payment) would be 3016.79*1.02^6-1800 = 1597.40 The balance at time 8 (after making the second payment) would be 1597.40*1.02^2-1200 = 461.94 The balance at time 12 (after making the third payment) would be 461.94*1.02^4 -500 = 0.02 This shows that $3016.79 would be the amount required in the account at time 0, which will be sufficient to make the required payments, with nothing left over (apart from 2 cents arising from rounding off). Reasonableness check: the present value of a series of payments should be less than the sum of the payments (assuming the interest earned on the account is positive) In this case $3016.78 is less than $3500 (i.e. less than 1800 + 1200 + 500) so this is reasonable.

Q2. Present Value of Regular Payments (a) Ann is going to put $P1 in a bank account today. It will earn interest at 5% per annum compound interest. This will accumulate to $10000 at the end of the year. What is the amount P1?(b) Ann is going to put $P2 in a bank account today. It will earn interest at 5% per annum. This will accumulate to $10000 at the end of the 2 years. What is the amount P2?(c) Ann is going to put $P3 in a bank account today. It will earn interest at 5% per annum. This will accumulate to $10000 at the end of the 3 years. What is the amount P3?(d) Ann is going to put $P4 in a bank account today. It will earn interest at 5% per annum. This will accumulate to $10000 at the end of the 4 years. What is the amount P4?(e) What is the total amount Ann should put in the bank now, in order to provide $10,000 per annum at the end of each year for the next 4 years?Solution(a) Present Value for first payment = 10,000/1.05 = $9,523.81 (b) Present Value for first payment = 10,000/1.05^2 = $9,070.29 (c) Present Value for first payment = 10,000/1.05^3 = $8,638.38 (d) Present Value for first payment = 10,000/1.05^4 = $8,227.02 (e) Total for all four payments = 10,000 / 1.05 + 10,000 / 1.05^2 + 10,000 / 1.05^3 + 10,000/ 1.05^4 = 35,459.51

Q3. Proof by induction for Present Values of a series of payments. Suppose that an account has an initial balance of P at time 0. The account will earn interest at rate i p.a. Money will be withdrawn from the account at the end of each year, and the amount withdrawn at time t is denoted (for example if $100 is withdrawn at the end of the second year then There might also be an immediate withdrawal, i.e. a withdrawal at time 0. No additional deposits are made into the account, so it only increases as a result of investment income. Note that the account balance might become negative if there is not enough money to make all the repayments.(a) Use the method of induction to show that the account balance at time n (just after any payment withdrawn at that time) will be given by

(b) Suppose that P, the money in the account at time 0, is exactly enough to make all the payments up until time n, with nothing left over. {That is, P is the present value of the payments.} Set the account balance at time n equal to 0 and solve for P in order to find an expression for the present value.{This should verify the statement made in lectures: the present value of a series of payments is the sum of the present values of each individual payment.}SolutionStep 1: show that the proposition is true for n = 1Suppose that the contribution at time 0 is C0. This will earn interest for one year. At that time an additional contribution will be made, denoted C1. So the accumulated balance at the end of the first year will be . This may be re-expressed as So the proposition is true for n = 1Step 2: show that if the proposition is true for n = k, then the proposition is true for n = k+1Let be the balance at time k (just after the payment has been added to the account). By assumption,

Let be the balance at time k+1 (just after the payment has been added to the account)The balance at time k+1 will be the balance at time k, plus one years interest, plus the additional contribution made at the end of the year. So... Now by using our assumption, and substituting for Bk,

So if the proposition is true for n = k, it is also true for n = k+1Step 3 Combining step 1 and step 2, and applying the Principle of Mathematical Induction the proposition is true for n = 1,2,3,....

Q4. Annuities payable in arrears. (a) Assuming that there are n equal payments of amount C, payable at the end of each year from year 1 to n, then the present value of these payments is given by

Use the formula for the sum of a GP to derive the present value of these payments. This is the formula for a fixed term annuity with payments payable in arrears. In actuarial notation the present value of $1 p.a. payable in arrears for a fixed term of n years is denoted (b) Use this formula to calculate that the total amount needed to pay $10,000 per annum at the end of each year for the next four years, assuming interest is earned at 5% p.a., and verify that this matches the answer to Q2 above (Anns payments). In actuarial notation this would be

(c) Assuming that the payments are increasing over time, so that the payment at the end of the tth year, denoted Ct, is C*(1+f)^(t-1). Then the present value of these payments is given by

Use the formula for the sum of a GP to derive the present value of these payments.

Solution:(a) If the payments are fixed at C per annum, and the first payment is at time t=1, then the present value is

The present value of the payments is a geometric progression with initial term constant ratio n termsSo using the formula for the sum of a GP gives the present value of the payments asMultiple by (1+i)/(1+i) to get

This gives us the standard formula for the present value of a series of payments of $C payable annually in arrears for n years. In actuarial notation this is denoted as (b) We can use this formula to find the PV of $10000 per annum payable annually in arrears for 4 years at 5% p.a

PV =

PV = 35,459.51

This matches the answer from question 2.

(c) Assuming that the payments are , so that the payment at the end of the tth year, denoted Ct, is C*(1+f)^(t-1). Then the present value of these payments is given by increasing over time fixed at C per annum, and the first payment is at time t=1, then the present value is

The present value of the payments is a geometric progression with initial term constant ratio n termsSo using the formula for the sum of a GP gives the present value of the payments asMultiple by (1+i)/(1+i) to get

Example: Suppose we have payments of 10000 at the end of the first year, 10000*1.02 in the second year, 10000*1.02^2 in the third year and so on, for 10 paymentsUsing an Excel spreadsheet and calculating the present value of each payment at 5% p,a1 $ 10,000.00 $ 9,523.81

2 $ 10,200.00 $ 9,251.70

3 $ 10,404.00 $ 8,987.37

4 $ 10,612.08 $ 8,730.58

5 $ 10,824.32 $ 8,481.14

6 $ 11,040.81 $ 8,238.82

7 $ 11,261.62 $ 8,003.43

8 $ 11,486.86 $ 7,774.76

9 $ 11,716.59 $ 7,552.62

10 $ 11,950.93 $ 7,336.83

sum $ 83,881.06

Using the formula with i = 0.05 and f = 0.02 and C =10000 and n = 10 gives:PV = 10000 * [1 - (1.02/1.05)^10] / (0.05-0.02) = 83,881.06

Q5. Annuities payable in advance. Assume that there are n equal payments of amount $ 1, payable at the start of each year (i.e. at times t = 0,1,2,...n-1). This is called a fixed term annuity payable in advance. In actuarial notation, the present value of $1 p.a. payable in advance for a fixed term of n years is denoted , where the double-dots indicate payment in advance. Show that Solution: In this case we have a series of payments of $1 per annum at times 0,1,2,3...n-1.(a) Using the result derived in a previous question, the present value of these payments is

These payments are in a geometric progression with first term 1 and constant ratio (1+i)-1, with n terms, so the present value of all payments combined is = Multiple by (1+i)/(1+i) to get = = = In general, the present value of a series of payments made annually in advance is (1+I) multiplied by the present value of the same set of payments made annually in arrears

Q6. More practice with fixed term annuities(a) Joseph intends to retire on his 65th birthday. He expects to live for exactly 20 years and would like to have an income of $40,000 per annum, payable at the end of each year for 20 years. How much does he need to have in his superannuation account at age 65, in order to meet this objective? Assume that he will be able to earn 6% p.a. (b) Now allow for inflation in the cost of living at 2% pa. Joseph intends to retire on his 65th birthday. He expects to live for exactly 20 years and would like to have withdraw of $40,000 per annum at the end of the first year, $40,000 * 1.02 at the end of the second year, $40,000 * 1.02^ 2 at the end of the third year, and so on. How much does he need to have in his superannuation account at age 65, in order to meet this objective? Assume that he will be able to earn 6% p.a. (c) Maryann intends to retire on her 60th birthday. She expects to live for exactly 30 years and would like to have an income of $50,000 per annum, payable at the start of each year for 30 years. How much does she need to have in her superannuation account at age 60, in order to meet this objective? Assume that she will be able to earn 5% p.a.(d) Now allow for inflation in the cost of living at 1% p.a. Maryann intends to retire on her 60th birthday. She expects to live for exactly 30 years and would like to withdraw $50,000 at the end of the first year, $50,000 * 1.01 at the end of the second year, $50,000 * 1.01^2 at the end of the third year, and so on. How much does she need to have in her superannuation account at age 60, in order to meet this objective? Assume that she will be able to earn 5% p.a.Solution(a) Present value of payments of $40,000 p.a. payable in arrears for 20 years at 6% =40000 * [1-1.06^-20]/0.06 = 458,796.85(b) Present value of payments of $40,000 in the first year, increasing at 2%p.a, in arrears for 20 years.PV of first payment = 40000 / 1.06PV of second payment = 40000 * 1.02 / 1.06^2PV of third payment = 40000 * 1.02^2 / 1.06^3 and so on.This is a GP where the first term is 40000/1.06 and the constant ratio is 1.02/1.06 and there are 20 termsUsing the sum of a GP formula, the present value of all the payments combined PV of all payments = 40000 / 1.06 * [(1.02/1.06)^20 - 1] / [(1.02/1.06) - 1] = 536,674.58Reasonableness check - since the payments in (b) are higher than the payments in (a), the answer to (b) should be higher than the answer to (a)(c) Present value of payments of $50,000 p.a. payable in advance for 30 years at 5% = 50,000 * [1-1.05^-30]/0.05 * 1.05 = 807,053.68(d) Present value of payments of $50,000 in the first year, increasing at 1%p.a, in arrears for 30 years, at interest rate 5% p.a.PV of first payment = 50000 / 1.05PV of second payment = 50000 * 1.01 / 1.05^2PV of third payment = 50000 * 1.01^2 / 1.05^3 and so on.This is a GP where the first term is 50000/1.05 and the constant ratio is 1.01/1.05 and there are 30 termsUsing the sum of a GP formula, the present value of all the payments combined PV of all payments = 50000 / 1.05 * [(1.01/1.05)^30 - 1] / [(1.01/1.05) - 1] = 860,172.70

Q7. Actuarial Notation and Reasonableness checks. Actuaries often have to calculate the present value of a series of regular payments, so there is a standard notation for this. at i represents the present value of payments of R per annum at the end of each year for n years, at interest rate i per annum.

(a) Use a general reasoning argument to explain why at i (i > 0) should always be less than R * n.(b) For a given set of non-negative payments: the higher the interest rate, the lower the present value. Verify this mathematically.Solution(a) If the account did not earn any interest, then we would need an amount of R*n to make all the payments.If the account does earn interest, then the investment income will cover some part of each payment. So we dont need to have the entire sum R*n in the account, we can start with a lower amunt

Hence if the interest rate is positive, then at i (i > 0) should always be less than R * n.(b) The present value is a function of the interest rate. We can show that the function is monotonic decreasing, i.e. the first derivative with respect to i is negative , when the interest rate is positive.

The first derivative with respect to is

If the values of t, Ct, and i are all positive, then the first derivative must be negative. Hence the higher the interest rate, the lower the present value.

Q8. Andrew has $300,000 as a lump sum at age 65. He intends to withdraw $R per annum at the end of each year for the next 30 years, and then die at time t = 30. He doesnt intend to have any money left over after he dies. (a) How much can he withdraw every year, assuming that the rate of interest earned is 5% p.a.?

(b) Suppose that Andrew withdraws $25,000 p.a in arrears. How long will it be until he runs out of money? [Find the first year in which his balance at the end of the year is negative]

(c) Suppose that Andrew withdraws $14,000 p.a. in arrears. How long will it be until he runs out of money?

(d) Suppose that due to a breakthrough in biomedical technology, Andrew will live forever. He will withdraw $Y at the end of every year, forever [Note that this is called a perpetuity]. What is the value of Y which will make the present value of the payments equal to $300,000? Verify your answer by general reasoning. Solution(a) Solve for the value of R such that

R = 19,515.43

(b) Solve for the value of n such that

n > 18.78 yearsThis shows that Andrew will be able to withdraw $25,000 p.a. for 18 years, but in the 19th year the amount in the account will be less than $25,000. He will be able to withdraw a reduced amount at that time.Practice: Try to do this question with the EXCEL solver just to make sure you know how to use the Solver.

(c) Trick question ! Andrew will never run out of money. The account of $300,000 earns interest at 5% p.a., which means that the interest in the first year is $15,000. Andrew is only withdrawing $14,000. At the end of the year, the balance will increase to $301,000. The balance will keep increasing each year, so Andrew will never run out of money. When he dies, his heirs will receive an inheritance of more than $300,000.Note that this system will only work if the interest rate is always 5% p.a. If the interest rate fluctuates randomly, and might have low values, then there is a chance that Andrew will run out of money.(d) The account earns $15,000 per annum. Andrew can withdraw this amount every year, and he will never run out of money. A sum of P invested at rate i per annum will provide a perpetuity of C = P * i per annum.You can deduce this mathematically by finding the present value of payments of C per annum payable forever. PV = C* (1+i) ^-1 + C * (1+i) ^ -2 + C*(1+i)^-3+......The sum to infinity of a geometric progression with first term a and constant ratio r is a/(1-r), as long as the absolute value of r is less than 1.In this case a = C * (1+i)^-1 and r = (1+i)^-1Verify that the present value of an infinite number of payments of C per annum is given byPV = C/i Hence the amount required to fund a perpetuity of C per annum is C/iAlternatively a sum of P invested at i per annum will fund a perpetuity of C = iP

Q9. Bob has $300,000 as a lump sum at age 65. Bob knows that the cost of living will increase over time, and he estimates that the inflation rate will be 2% per annum. He intends to withdraw $R at the end of the first year, $R * 1.02 at the end of the second year, $R*1.02^2 at the end of the third year, and so on for the next 30 years. He expects to die before the 31st payment falls due. He doesnt intend to have any money left over after he dies. (a) How much can he withdraw every year, assuming that the rate of interest earned is 5% p.a.?

(b) Suppose that due to a breakthrough in biomedical technology, Bob will live forever. He will withdraw $Y * (1.02)^(t-1) at the end of the tth year, forever [an increasing perpetuity]. What is the value of Y which will make the present value of the payments equal to $300,000? Verify your answer by general reasoning.

(c) Suppose that inflation is 6% p.a. Revise your answer to part (b) allowing for payments that increase at 6% p.a.

Solution:

(a) Use the formula we derived previously, for the present value of an increasing annuity, with f = 2% and i = 5% and n = 30

Set this present value equal to 300,000 and solve for R. R = 15,493.42(the cash flows are shown on the next page)(b) We want the balance to keep increasing in line with inflation. This ensures that the real value of our wealth remains constant. Hence we want to find the value of the withdrawal which satisfiesBalance at start of year * (1+i) - withdrawal = Balance at end of yearBalance at end of year = Balance at start of year * (1+f)In this case, for the first year:300,000 * 1.05 - Y = 300,000 * 1.02Y = 300,000 * (1.05-1.02) Y = 9000(the cash flows are shown on the next page)(c) There is no sensible answer for part (c). If inflation is making prices increase at 6% p.a, and your investment return is only 5% p.a, then your standard of living must decline.

(a) Cash flows for 30 years of payments

balanceinterestpaymentbalance

yearstart yearearnedend yearend year

1 $ 300,000.00 $ 15,000.00 $ 15,493.42 $ 299,506.58

2 $ 299,506.58 $ 14,975.33 $ 15,803.29 $ 298,678.62

3 $ 298,678.62 $ 14,933.93 $ 16,119.35 $ 297,493.20

4 $ 297,493.20 $ 14,874.66 $ 16,441.74 $ 295,926.12

5 $ 295,926.12 $ 14,796.31 $ 16,770.58 $ 293,951.85

6 $ 293,951.85 $ 14,697.59 $ 17,105.99 $ 291,543.45

7 $ 291,543.45 $ 14,577.17 $ 17,448.11 $ 288,672.52

8 $ 288,672.52 $ 14,433.63 $ 17,797.07 $ 285,309.08

9 $ 285,309.08 $ 14,265.45 $ 18,153.01 $ 281,421.52

10 $ 281,421.52 $ 14,071.08 $ 18,516.07 $ 276,976.53

11 $ 276,976.53 $ 13,848.83 $ 18,886.39 $ 271,938.96

12 $ 271,938.96 $ 13,596.95 $ 19,264.12 $ 266,271.79

13 $ 266,271.79 $ 13,313.59 $ 19,649.40 $ 259,935.98

14 $ 259,935.98 $ 12,996.80 $ 20,042.39 $ 252,890.38

15 $ 252,890.38 $ 12,644.52 $ 20,443.24 $ 245,091.67

16 $ 245,091.67 $ 12,254.58 $ 20,852.10 $ 236,494.15

17 $ 236,494.15 $ 11,824.71 $ 21,269.14 $ 227,049.71

18 $ 227,049.71 $ 11,352.49 $ 21,694.53 $ 216,707.67

19 $ 216,707.67 $ 10,835.38 $ 22,128.42 $ 205,414.63

20 $ 205,414.63 $ 10,270.73 $ 22,570.99 $ 193,114.38

21 $ 193,114.38 $ 9,655.72 $ 23,022.41 $ 179,747.69

22 $ 179,747.69 $ 8,987.38 $ 23,482.85 $ 165,252.22

23 $ 165,252.22 $ 8,262.61 $ 23,952.51 $ 149,562.32

24 $ 149,562.32 $ 7,478.12 $ 24,431.56 $ 132,608.87

25 $ 132,608.87 $ 6,630.44 $ 24,920.19 $ 114,319.12

26 $ 114,319.12 $ 5,715.96 $ 25,418.60 $ 94,616.48

27 $ 94,616.48 $ 4,730.82 $ 25,926.97 $ 73,420.34

28 $ 73,420.34 $ 3,671.02 $ 26,445.51 $ 50,645.85

29 $ 50,645.85 $ 2,532.29 $ 26,974.42 $ 26,203.72

30 $ 26,203.72 $ 1,310.19 $ 27,513.91 $ 0.00

(b) Cash flows for a perpetuity where payments increase at 2% p.a.Note that the balance must increase by 2% p.a. in order to fund payment which increase at 2% p.a.balanceinterestpaymentbalance

yearstart yearearnedend yearend year

1 $ 300,000.00 $ 15,000.00 $ 9,000.00 $ 306,000.00

2 $ 306,000.00 $ 15,300.00 $ 9,180.00 $ 312,120.00

3 $ 312,120.00 $ 15,606.00 $ 9,363.60 $ 318,362.40

4 $ 318,362.40 $ 15,918.12 $ 9,550.87 $ 324,729.65

5 $ 324,729.65 $ 16,236.48 $ 9,741.89 $ 331,224.24

6 $ 331,224.24 $ 16,561.21 $ 9,936.73 $ 337,848.73

7 $ 337,848.73 $ 16,892.44 $ 10,135.46 $ 344,605.70

8 $ 344,605.70 $ 17,230.29 $ 10,338.17 $ 351,497.81

9 $ 351,497.81 $ 17,574.89 $ 10,544.93 $ 358,527.77

10 $ 358,527.77 $ 17,926.39 $ 10,755.83 $ 365,698.33

11 $ 365,698.33 $ 18,284.92 $ 10,970.95 $ 373,012.29

12 $ 373,012.29 $ 18,650.61 $ 11,190.37 $ 380,472.54

13 $ 380,472.54 $ 19,023.63 $ 11,414.18 $ 388,081.99

14 $ 388,081.99 $ 19,404.10 $ 11,642.46 $ 395,843.63

15 $ 395,843.63 $ 19,792.18 $ 11,875.31 $ 403,760.50

16 $ 403,760.50 $ 20,188.03 $ 12,112.82 $ 411,835.71

17 $ 411,835.71 $ 20,591.79 $ 12,355.07 $ 420,072.43

18 $ 420,072.43 $ 21,003.62 $ 12,602.17 $ 428,473.87

19 $ 428,473.87 $ 21,423.69 $ 12,854.22 $ 437,043.35

20 $ 437,043.35 $ 21,852.17 $ 13,111.30 $ 445,784.22

21 $ 445,784.22 $ 22,289.21 $ 13,373.53 $ 454,699.90

22 $ 454,699.90 $ 22,735.00 $ 13,641.00 $ 463,793.90

23 $ 463,793.90 $ 23,189.70 $ 13,913.82 $ 473,069.78

24 $ 473,069.78 $ 23,653.49 $ 14,192.09 $ 482,531.17

25 $ 482,531.17 $ 24,126.56 $ 14,475.94 $ 492,181.80

26 $ 492,181.80 $ 24,609.09 $ 14,765.45 $ 502,025.43

27 $ 502,025.43 $ 25,101.27 $ 15,060.76 $ 512,065.94

28 $ 512,065.94 $ 25,603.30 $ 15,361.98 $ 522,307.26

29 $ 522,307.26 $ 26,115.36 $ 15,669.22 $ 532,753.41

30 $ 532,753.41 $ 26,637.67 $ 15,982.60 $ 543,408.48

Discussion Question

There have been many studies which have looked at the issue of retirement adequacy, i.e. whether the old age pension plus 9.5% compulsory superannuation savings will be sufficient to provide an adequate standard of living in retirement.Some people argue that the 9.5% contribution rate is not enough and the compulsory rate should be increased to 12%.Others would argue that the compulsory superannuation contributions should just provide a modest standard of living and then people should be left to make their own decisions about making additional voluntary contributions., if they want a better standard of living in retirement.(a) Suppose that you were just about to start a new job. Your boss asks if you would like to make voluntary contributions into your super fund, on top of the compulsory contributions. These voluntary contributions would be deducted from your pay and sent to the superannuation fund. What would you say? Give some reasons why people would decide AGAINST paying additional contributions when they are under age 30.

(b) Government policy-makers are concerned that many people fail to make voluntary contributions (until shortly before retirement) and hence will have inadequate retirement savings. As mentioned in lectures, the Financial Services Council has issued press releases about the Retirement Savings Gap (a copy of the press release is available on iLearn). There have been several suggestions for persuading people to make additional contributions.

a. The government could provide better financial incentives for savers. Look up the Australian co-contribution rules (at http://www.rest.com.au/co-cont) and see if that influences your decision.

b. Behavioural finance researchers have attempted to devise methods which will lead to increased superannuation savings. One well-known experiment is the Save More Tomorrow (SMarT) program. Watch http://befi.allianzgi.com/en/Topics/Pages/save-more-tomorrow.aspx.

c. Behavioural finance studies have led to the development of the concept of soft compulsion. Find out about this idea. Do you think this sort of system should be introduced in Australia? Some Australian organisations such as the Association of Superannuation Funds of Australia (ASFA) have advocated soft compulsion systems.

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