Society of Actuaries Ireland Financial Maths November 2015 Society of Actuaries Ireland

Financial Maths Compound Interest Present Value Valuing Annuities Page 22 Financial Maths Page 30 Compound Interest Present Value Valuing Annuities Important Formula are in your Formulae and Tables booklet, pages 22, 30, 31, 32. Page 31 Page 32 Hand out copies of the four pages of formulae. This first area – compound interest - is an extension of the compound interest studied for the junior cert. Thus the first formula on page 30 should be very familiar – regardless of whether or not the teacher has got around to covering this topic yet. Simple example (JC): Calculators out. Question is on next slide (hyperlinked): If I start with a principal amount of €100 and I have a compound interest rate of 5% p.a. then how much will I end up (final) with at the end of 5 years? F=P(1+i)^t = 100(1.05)^5 = €127.63. If you got that right you have already moved on from JC because at JC the questions were restricted to just 3 years – not 5 like in that example. At leaving cert the time periods are likely to be much longer though and not always years – could be months eg. There is a formula on page 32 which might be used to switch between years and months. (Not very useful formula really). The next area is Present Value. Note that in the previous example we started with an amount called the Principal and rolled it forward in time and ended up with an amount called the Final Amount. We could equally use this formula with the terminology Present instead of Principal and Future instead of Final. The second formula on page 30 uses this terminology. Notice how this is really the same as the first formula? – just rearranged a bit. Instead of rolling forward in time however, this formula is used to bring a payment back from the future to its present value. Take care when using either of these formulae – draw a time line if it helps and mark two dates on it - when the payment is due (normally some time in the future) and when you want to value it (normally today – present time). Knowing the present value of a financial product helps you - the consumer - to choose between them, to ‘shop around for the best price’ so to speak. Lets have a little bit of practice in using the present value formulae: Hyperlink to question on companies ABC and XYZ. Another fundamental difference between leaving cert and Junior Cert. is that there is likely to be more than one payment under discussion. In fact there is probably a whole series of payments, which brings us to the next item on the agenda for tonight. An annuity is simply a regular series of equal payments. Finding the value of an annuity is like two topics in one – it brings together what you know about value (present or future) with what you know about series – specifically a geometric series. You’ll find the formula for Geometric series on page 22. We are interested in the middle formulae which gives the sum of the series. Lets practice using this formula, Question re Lottery hyperlinked. Long Question (pension planning) – hyperlinked. CI PV VA Q

Compound Interest €100 to invest @ 5% p.a. compound for 5 years After 5 yrs = 100 (1+0.05)5 = 127.6281563 Ans: €127.63 Page 30

Present Value Investment Opportunity! Page 30 Investment Opportunity! In return for your investment, company ABC Ltd. is offering to pay you €12,700 in 5 years time. Current market interest rates are 5% p.a. compound. How much would you be willing to invest? (hint: What is the present value?) For the same investment, a second company XYZ Ltd. is offering to pay €13,000 after 66 months. Which company is offering the better value for money? Questions could be worded something like … Company A is offering 12700 in 5 years in return for an investment of €

Present Value Investment Opportunity! ABC Ltd. : 𝐹= €12,700 𝑡=5 Page 30 Investment Opportunity! ABC Ltd. : 𝐹= €12,700 𝑡=5 i = 0.05 𝑃= 𝐹 1+𝑖 𝑡 𝑃= 12700 1+0.05 5 =€9950.78 Questions could be worded something like … Company A is offering 12700 in 5 years in return for an investment of €9950. What is the compound annual rate of return (CAR) to 2 d.p.? The unknown (i) is inside the indexed bracket. So, this involves getting the 5th root on your calculator (fractional powers). Ans: 5.00%. Alternatively, Company B wants to offer €13000 in return for an investment of €10000 using a CAR of 5%. When (to the nearest month) should they return the money? The unknown (t) is the exponent. So, this involves using logs. Natural logs are easiest on the calculator. Ans: 65 months.

Present Value Investment Opportunity! XYZ Ltd. : 𝑡= 66 12 =5.5 Page 30 Investment Opportunity! XYZ Ltd. : 𝑡= 66 12 =5.5 𝑃= 13000 1+0.05 5.5 =€9940.36 The investment in XYZ Ltd. is valued slightly lower than the same investment in ABC Ltd. (€9950. 78). So, ABC Ltd. offers the better value.

Geometric Series Lottery prize Option A: €100,000 today Page 22 Lottery prize Option A: €100,000 today Option B: 10 annual payments of €11,000 each, with the first one paid today. Which of these options is the better value, given that you can invest at a rate of interest of 3% p.a. compound?

Society of Actuaries Ireland Page 30 Lottery prize – Option B? 𝑃𝑉 𝐵 = 11000 1.03 0 + 11000 1.03 1 + 11000 1.03 2 … 11000 1.03 9 =11000 1 1.03 0 + 1 1.03 1 + 1 1.03 2 +…+ 1 1.03 9 =11000 (𝑆 𝑛 ) Geometric Series Society of Actuaries Ireland

Lottery prize – Option B? Page 22 Lottery prize – Option B? 𝑎=𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚= 1 1.03 0 =1 𝑟=𝑐𝑜𝑚𝑚𝑜𝑛 𝑟𝑎𝑡𝑖𝑜= 1 1.03 1 = 1 1.03 𝑛=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑟𝑚𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑟𝑖𝑒𝑠=10 What is a Geometric Series? Here is the formula and it is worth learning.

Lottery prize – Option B? 𝑆 𝑛 = 𝑎 1− 𝑟 𝑛 1−𝑟 = 1 1− 1 1.03 10 1− 1 1.03 =8.786108922

Lottery prize – Option B? 𝑃𝑉 𝐵 =11000(𝑆 𝑛 ) =11000×8.786108922 =€96,647.20 Hence, option A (€100,000) is the better value.

Society of Actuaries Ireland Retirement Planning A young man decides he will start preparing for his retirement. He intends to save €500 every month, starting on his 25th birthday and ending one month before his 65th birthday. He can invest at a rate of 3% AER. Draw a time line, start at the 25th birthday, go to 65th birthday, indicate time is monthly. Mark in the first few payments, and the last few. Draw a second line for the interest rate. It is annual so the line should be one year long. Mark in the months to match the first time line. Society of Actuaries Ireland

Society of Actuaries Ireland Retirement Planning How many monthly payments will he make? Calculate the monthly return he can expect. Express your answer as a percentage correct to 4 decimal places. How much should he have in his retirement fund when he reaches 65? 12 a year for 40 years = 12 x 40 = 480 payments Call the monthly return i% or something. Put it on your timeline. You should be able to see that there will be 12 period where i% applies; i.e. 12 places to put i on the timeline - but only one place to write the 3%. See next slide. Question 3 gives us the date at which we want to calculate the value, its not now, but in the future. Mark this on your payments timeline. Now underneath each of your payments write the number of months left before you reach that future date. Society of Actuaries Ireland

Society of Actuaries Ireland Retirement Planning Solving for the unknown i involves taking the 12th root of both sides, or raising to the power of 1 12 and rearranging. 1+𝑖% 12 = 1+3% 1 Society of Actuaries Ireland

Society of Actuaries Ireland Retirement Planning 2) Answer = 0.2466% 1+ 𝑖 100 12 =1.03 1+ 𝑖 100 12 1 12 = 1.03 1 12 1+ 𝑖 100 = 1.03 1 12 =1.00246627 i=0.246627≈0.2466 𝑡𝑜 4 𝑑.𝑝. Society of Actuaries Ireland

Society of Actuaries Ireland Retirement Planning Apply the Future Value formula to each of the payments, using your value for i (unrounded) and these values for t. Never round before you get to the end. Society of Actuaries Ireland

Society of Actuaries Ireland Retirement Planning 3) Answer = €459,726.01 Underline importance of not rounding. If rounded i, and round at each potential part it is possible to get an answer of €464,000 ; i.e.€4274 out. Being just 1% out is a difference of €4597. 𝑎=1.00246627 𝑆 480 =919.4520245 𝑟=1.00246627 500𝑆 480 =459726.0122 Society of Actuaries Ireland

Society of Actuaries Ireland Retirement Planning He intends to use his retirement fund to provide him with a pension payable monthly in arrears every month up to and including the month he turns 85. How much should he expect to receive each month if he assumes interest rates over that time will be 4% p.a. compounded quarterly? New time lines – both for the payments and the interest. Society of Actuaries Ireland

Society of Actuaries Ireland Retirement Planning 1+𝑖% 3 = 1+1% 1 𝑖=0.3322284 Society of Actuaries Ireland

Society of Actuaries Ireland Retirement Planning Answer = €2,782.64 𝑎=𝑟= 1 1.003322284 𝑆 240 =165.2122867 Society of Actuaries Ireland

Society of Actuaries Ireland Retirement Planning 459726.01=𝐴 𝑆 240 459726.01=𝐴 165.2122867 𝐴= 459726.01 165.2122867 =2782.638139 Society of Actuaries Ireland

Society of Actuaries Ireland Retirement Planning Alternatively use the Amortisation Formula on page 31: 𝑃=459726.01 𝑖=0.003322284 𝑡=240 𝐴= 459726.01 0.003322284 1.003322284 240 1.003322284 240 −1 Students need to be able to derive that formula! Society of Actuaries Ireland