Financial Mathematics

Financial Mathematics Financial Maths The contents of this presentation is mainly for LCH but covers a lot of the LCO syllabus http://www.projectmathsbooks.com

The Syllabus -Solve problems and perform calculations on compound interest and depreciation (reducing-balance method) -Use present value when addressing problems involving loan repayments and investments -Solve problems involving finite and infinite geometric series -Use financial applications such as deriving the formula for a mortgage repayment http://www.projectmathsbooks.com

Present Value • The time value of money -value of money when factoring in a given amount of interest over a given period of time • Present Value - value on a given date of a future payment or a series of future payments http://www.projectmathsbooks.com

Present Value • Used throughout the financial mathematics material • Not always in the same format as seen in the formulae and tables but a simple manipulation usually gets us the formula we need http://www.projectmathsbooks.com

Present Value Formula http://www.projectmathsbooks.com

Watch the language used! http://www.projectmathsbooks.com

Using Present Value for Decision Making An investment opportunity arises for Andy. He will receive a payment of €10,000 for each of the next three years if he invests €25,000 now. Growth over this time period is estimated to be 5%. Use present values to assess this investment. http://www.projectmathsbooks.com

Solution To assess the investment, we need to compare like with like; therefore, it is necessary to calculate the present values of the future cash inflows http://www.projectmathsbooks.com

Solution http://www.projectmathsbooks.com

Making the Decision Net Present Value = Present Value of All Cash inflows – Present Value of All Cash outflows NPV ≤ 0 Do Not Invest in the Project NPV > 0 Invest in the Project NPV = €27,232.38 - €25,000 = €2,232.48 As the NPV is positive, Andy should invest http://www.projectmathsbooks.com

A few questions to try now http://www.projectmathsbooks.com

What to watch out for in the question! http://www.projectmathsbooks.com

Answer • Restaurant NPV = €18,175.41 Amusements NPV = €4,963.20 http://www.projectmathsbooks.com

Worked Solution http://www.projectmathsbooks.com

Interest: Loans and Investments • APR = Annual Percentage Rate (LOANS) • AER = Annual Equivalent Rate (INVESTMENTS) Points to Note: • Several different names used for AER in Ireland • AER and APR are always the “i” in the formulae we use http://www.projectmathsbooks.com

Calculating AER/APR Mark invested money in a 5.5 year bond when he started First Year. In the middle of Sixth Year the bond matures and he has earned 21% interest in total. Calculate the AER for this bond. http://www.projectmathsbooks.com

Solution • Step 1: Write down the formula. F = P(1 + i)t • Step 2: Identify the parts that we are given in the question. F Final value = Original amount + interest = 100% + 21% = 121% = 1.21 P Principal = Original amount = 100% = 1.00 t Time in years = 5.5 i Annual equivalent rate = ? [This is what we are looking for] http://www.projectmathsbooks.com

Solution • Step 3: Solve for the unknown value i. 1.21 = 1.00(1 + i)5.5 1.21 = (1 + i)5.5 5.5√1.21 = 1 + i 1.0353 = 1 + i 1.0353 – 1 = i i = 0.0353 ⇒ i = 3.53% ∴ The annual equivalent rate is 3.53%. http://www.projectmathsbooks.com

A question to try now The following advert appears on a Bank website. Can you verify that it displays the correct AER? http://www.projectmathsbooks.com

Watch your Savings Grow Online 4.5% 15month Fixed Term rate (3.58% AER fixed) http://www.projectmathsbooks.com

What to watch out for Watch your Savings Grow Online 4.5% 15month Fixed Term rate (3.58% AER fixed) http://www.projectmathsbooks.com

Depreciation (Reducing Balance) Formula: F=P (1 - i)t F is called the later value in the Formulae and Tables (page 30). In accounting, this is known as the Net Book Value (NBV) of the asset http://www.projectmathsbooks.com

Depreciation (Reducing Balance) http://www.projectmathsbooks.com

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Practical Style Questions http://www.projectmathsbooks.com

Links with other topics http://www.projectmathsbooks.com

A Question to Try A company has a policy to depreciate all computers at a reducing-balance rate of 20%. Computers owned by the firm are valued (net book value) at €150,000. An auditor recently pointed out that due to increases in technology, computers were losing value at a much quicker rate than in previous years. The auditor estimated that the value of the computers in two years’ time would only be €95,000. Does the firm have an adequate depreciation policy? Explain your answer. http://www.projectmathsbooks.com

Applications and Problemsinvolving Geometric Series Students need to be familiar with financial products that are on the market Annuities (e.g. Pensions) Perpetuities Bonds Investment schemes etc http://www.projectmathsbooks.com

A simple example A building society offers a savings account with an AER of 4%. If a customer saves €1,000 per annum starting now, how much will the customer have in five years’ time? http://www.projectmathsbooks.com

Method 1-Using the formula http://www.projectmathsbooks.com

Method 2-Write as a series A = 1,000(1.04)5 + 1,000(1.04)4 + 1,000(1.04)3 + 1,000(1.04)2+1,000(1.04)1 = 1,000[(1.04)5 + (1.04)4 + (1.04)3 + (1.04)2 + (1.04)1] This is a geometric series with a = (1.04)5 r = 1/1.04 n = 5 http://www.projectmathsbooks.com

Amortisation: Mortgages and Loans A = Annual repayment amount i = Interest rate (as decimal) P = Principal t = Time (in years) http://www.projectmathsbooks.com

A more testing example If a loan for €60,000 is taken out at an APR of 3%, how much should the annual repayments be if the loan is to be repaid in 10 equal instalments over a 10-year period? Assume the first instalment is paid one year after the loan is drawn down. Give your answer correct to the nearest euro. http://www.projectmathsbooks.com

Why both methods? A question could specify that a candidate must use a geometric series to provide a solution to the problem http://www.projectmathsbooks.com

Regular payments at Intervals other than Annually • Calculations are the same as for annual payments, but the AER or APR must be treated properly. • Option 1 . Leave time in years. . Do not change the APR/AER. . Use fractional units of time. • Option 2 . Switch to a different time period. . We must adjust the APR/AER. . Use integer units of time. http://www.projectmathsbooks.com

Example Alan borrows €10,000 at an APR of 6%. The terms of the loan state that the loan must be repaid in equal monthly instalments over 10 years. The first repayment will be one month from the date the loan is taken out. How much should the monthly repayment be? Give your answer to the nearest cent. http://www.projectmathsbooks.com

Solution http://www.projectmathsbooks.com

A question to try http://www.projectmathsbooks.com

Financial Mathematics