Grade 12 Mathematics of Finance VOCABULARY Interest – the cost of borrowing money Nominal interest rate – the quoted rate Effective interest rate – the actual rate (what happens in effect) Interest period – the constant amount of time between successive (compound) interest calculations(e.g. annually, half-yearly/semi-annually, quarterly, monthly, daily), which may or may not be the same as the amount of time between successive payments (Re)payment period – constant amount of time between successive payments towards a loan or an investment Principal (P) – the initial amount borrowed/deposited/etc. at the start of a period of time Final amount (A) – amount owing/saved/etc. at the end of a period of time Amortise – to repay Future Value (FV) – the value to which a present amount of money will grow/fall Present Value (PV) – the current value of an amount of money prior to any growth or loss Inflation – the decline in the value of money Scrap value – the value to which an asset will be deemed to have depreciated over its useful life Book value – the (depreciated) value of an asset at a given point in time Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance VOCABULARY Decay – the process of getting smaller (alt. Depreciation – the loss of value of an asset) Simple decay/depreciation – constant loss of size/value Calculated using the “Straight-Line Method” Compound decay/depreciation – increasing loss of size/value Calculated using the “Reducing Balance Method” Compounding – the process of calculating the Future Value of an amount of money (over successive interest period(s)) using the preceding amount (working forwards in time) by multiplying the PV by the interest growth factor (assuming growth) for the next period(s) Discounting – the process of calculating the Present Value of a future amount of money (working backwards in time) by dividing the FV by the interest growth factor (assuming growth) over the entire period Future Value (FV) Annuity – a series of regular payments of a fixed amount, towards a lump sum, over a period of time Sinking Fund – a particular FV annuity used by companies to replace old/scrapped assets Retirement Annuity – a particular FV annuity sold by insurance companies (e.g. Old Mutual) Present Value (PV) Annuity – a series of regular repayments of a fixed amount to amortise a loan Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance BASIC FORMULAE: where i(m) is the nominal rate for interest compounded m times a year Either side of the equation above may be thought of as the “interest growth factor” i.e. the factor by which the principal is multiplied to realise the final amount (assuming growth). This equation simply gives the relationship between the nominal interest rate for the interest period and the effective interest rate for the entire period. Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance NOTES: Annuities often start one time period from the present (but not always) Timelines are an essential tool to visualise the annuity Notation: Tn = end of an repayment/interest period T0 = the present moment Normally use calendar months and/or calendar years In some examples, the interest period does not coincide with the repayment period, which requires the nominal interest rate for the interest period to be converted to an equivalent nominal interest rate for the repayment period as follows: where i(m) is the nominal rate for interest compounded m times a year and i(n) is the nominal rate for interest compounded n times a year. Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance EXAMPLE: Interest on an investment is compounded monthly at a nominal rate of 10% p.a. Calculate the equivalent nominal rate if interest is compounded quarterly. SOLUTION: i(4) = quarterly nominal interest rate i(12) = monthly nominal interest rate Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance EXAMPLE: Calculate the Future Value (FV) of R100 in 5 years time if inflation remains constant at 10% p.a. SOLUTION: T0 T1 T2 T3 T4 T5 PV = R100 FV = R59,05 COMPOUNDING PV FV DISCOUNTING PV FV Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance EXAMPLE: Calculate the Future Value (FV) of R100 in 5 years time if it is invested at 10% p.a. compounded annually. SOLUTION: T0 T1 T2 T3 T4 T5 PV = R100 FV = R161,05 COMPOUNDING PV FV DISCOUNTING PV FV Prepared by: Mr. C. Hull

(Future Value) Future Lump Sum Grade 12 Mathematics of Finance FUTURE VALUE (FV) ANNUITY A series of regular payments of fixed amounts, over an agreed time interval or investment period, for future use (e.g. a retirement annuity) P1 P2 P3 Future Lump Sum (Future Value) Pn ………………………………………….. x x x Lump sum available at the end of an agreed time interval Each payment (P1 to Pn-1) grows in value (is compounded) over the agreed time interval (investment period) The final payment (Pn) does not have any growth – it forms part of the future lump sum The future lump sum (FV) is the sum total of the compounded value of each of the individual payments (P1 … Pn-1 ) over the investment period PLUS Pn The interest rate used is the nominal (quoted) rate Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance EXAMPLE: Joe invests R100 each month into an annuity at 12% p.a. for one year, starting one month from now. What is the Future Value of his annuity at the end of the agreed time interval (or investment period)? SOLUTION: Interest period = 1 month; Interest rate = = 1% per month T0 T1 T2 T3 T6 T11 T12 Timeline 100 100(1 + 0.01)11 100 100(1 + 0.01)10 100(1 + 0.01)9 100(1 + 0.01)1 FV sum is just the sum of a Geometric Series The 12th payment Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance In general, Future Value (FV) annuities are calculated as follows: CRITICAL 4-POINT CHECK BEFORE USING THE FV FORMULA: Are you dealing with a FV annuity? i.e. are there regular payments of a fixed amount for future use? Is the first payment made one interest period from the present? Is the final payment made at the time that the FV is calculated and paid out? – the date when an annuity is due to be paid out is called “at maturity”. Is interest compounded with the same regularity that payments are made? Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance EXAMPLE (with a slight complication): Mr. Klein opens a savings account for his son’s future education. On opening the account, he deposits R850 and then commits to making regular monthly payments of R850 at the end of each month for 10 years. The interest rate is set at 12% p.a. compounded monthly. Calculate how much Mr. Klein will have accumulated at the end of the investment period. SOLUTION: Do the “4-point check”: FV annuity? YES First payment made one interest period later? NO (so adjust formula with n +1 payments) …….. This is the complication  Final payment made at maturity? YES Compounding “synched” with payment time periods? YES Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance EXAMPLE: Tim plans to have R 3 000 000 in his bank account when he turns 50 in 20 years time. Calculate how much Tim must save each month, starting in one month’s time, if interest at 9.5% p.a. is compounded monthly. SOLUTION: 4-point check: FV annuity? YES, having to calculate monthly payments instead of knowing them at the start doesn’t change that we are dealing with building a lump sum for future use Savings (“payments”) start one interest period from present? YES, so n monthly savings Final saving made at maturity? YES, implied in question. Compounding “synched” with savings time periods? YES Prepared by: Mr. C. Hull

(Present Value) Present Lump Sum Grade 12 Mathematics of Finance PRESENT VALUE (PV) ANNUITY A series of regular payments of fixed amounts, over an agreed time interval, to repay a loan i.e. the lump sum of money is “available immediately” Present Lump Sum (Present Value) P1 P2 P3 Pn …………………………………….…… x x x x The lump sum is available now (at the present time) Each (re)payment x (P1 to Pn) made in the future has a “present value” which is calculated by discounting (opposite concept of compounding) over its relevant number of investment periods The present lump sum (PV) is the sum total of the discounted value of each of the repayments (P1 to Pn) over the loan period The interest rate used is the nominal (quoted) rate, usually an annual one, adjusted as required Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance EXAMPLE: Joe takes out a loan and agrees to repay it over one year, starting one month from now. His monthly repayment is R1000 and interest is compounded at 12% p.a. What is the Present Value of his annuity (i.e. “how much money did he borrow” or “what is the value of the loan”)? SOLUTION: Interest period = one month; Interest rate = = 1% per month T0 T12 T1 T3 T6 T2 Timeline 1000(1 + 0.01)-1 1000 1000(1 + 0.01)-2 1000 1000(1 + 0.01)-3 1000 PV sum is just the sum of a Geometric Series 1000(1 + 0.01)-12 1000 Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance SOLUTION (cont.): In general, Present Value (PV) annuities are calculated as follows: Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance CRITICAL 4-POINT CHECK BEFORE USING THE PV FORMULA: Are you dealing with a PV annuity? i.e. are there regular repayments of a fixed amount towards settling a loan amount (typically), which was made available upfront? Is the first repayment made one interest period from the present? Is the final repayment made at Tn, the end of the final interest period? Is interest compounded with the same regularity that repayments are made? Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance EXAMPLE: Mrs. Ndlovu takes out a mortgage bond of R980 000 on a property at a rate of 14% p.a. compounded monthly over 25 years. Calculate the monthly repayments required to settle the bond if Mrs. Ndlovu makes the first payment in one month’s time. SOLUTION: 4-point check: 1. Regular repayments towards a loan? ✓ 2. First payment one interest period from now? ✓ 3. Final payment at the end of last interest period? ✓ 4. Interest calculations and repayments in synch? ✓ Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance EXAMPLE: Abraham buys a car for R163 500. He pays a 10% deposit and settles the rest of his debt with quarterly payments over 5 years, starting in 3 month’s time. Calculate the value of his quarterly payments if interest on the loan is charged at 13% p.a. compounded quarterly. SOLUTION: NB! Interest period = 3 months 4-point check: 1. Regular repayments towards a loan? ✓ 2. First payment one interest period from now? ✓ 3. Final payment at the end of last interest period? ✓ 4. Interest calculations and repayments in synch? ✓ Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance NOTE: In some PV annuity examples, the first repayment is made immediately. In that case, this first payment must be subtracted from the loan and the PV formula has to be adjusted as follows: Always bear in mind that a PV annuity is most commonly thought of as an upfront loan i.e. funds are available immediately. EXAMPLE (cont.): Abraham buys a car for R163 500. He pays a 10% deposit and settles the rest of his debt with quarterly payments over 5 years, starting in 3 month’s time. Calculate how much interest Abraham pays on the loan. SOLUTION: Loan = R163 500 × 0.9 = R 147 150. 20 repayments × R10 120,81 = R 202 416,20 Thus, interest paid = R 202 416,20 – R147 150 = R55 266,20. Prepared by: Mr. C. Hull

Grade 12 Mathematics of Finance Prepared by: Mr. C. Hull