Financial Maths

FINANCIAL MATHEMATICS 1. RATE OF RETURN 2. SIMPLE INTEREST 3. COMPOUND INTEREST 4. MULTIPLE CASH FLOWS 5. ANNUITIES 6. LOAN REPAYMENT SCHEDULES Financial Math Support Materials Page 1 of 85 (1) RATE OF RETURN FINANCIAL MATHEMATICS CONCERNS THE ANALYSIS OF CASH FLOWS BETWEEN PARTIES TO A CONTRACT. IF MONEY IS BORROWED THERE IS AN INTIAL CASH INFLOW TO THE BORROWER BUT AFTERWARDS THERE WILL BE A CASH OUTFLOW IN THE FORM OF REPAYMENTS. A person borrows $100 and promises to repay the lender $60 after 1 year and $60 after 2 years. Show the resulting cash flows for the borrower and lender. Financial Math Support MaterialsPage 2 of 85 Time Now 1 End of 2 years Borrower 0 End of 1 year Lender 2 $100 is loaned out $120 is received back The extra $20 is the lenders compensation for foregoing current consumption to obtain future consumption. The lender requires compensation for: Financial Math Support Materials Page 3 of 85 THE â€Å"TIME VALUE† OF MONEY CONSIDER A CHOICE OF ? $100 NOW, OR ? $100 LATER ANY RATIONAL PERSON WOULD CHOOSE $100 NOW! BUT WHY? â€Å"MONEY HAS A TIME VALUE† Financial Math Support Materials Page 4 of 85 Time Value of Money (TVM) ? Refers to the difference between ? The concept enables ? Provides the means for valuing multiple cash lows that occur at different times The level of interest rates is the index used to determine prevailing TVM. Interest rates are determined by the level of †¦ For every type of financing transaction there is potentially a different interest rate. Interest rates are distinguished by the nature of the underlying transaction and focus on three characteristics: ? ? ? Financial Math Support Materials Page 5 of 85 An important aspect of valuation is applying the appropriate interest rate. For example, valuing a fixed-rate loan to a highly speculative company using a government bond rate is inappropriate; an adjustment must be made reflecting he relative creditworthiness of the borrower. While different TVMs may exist for every borrower and lender, it is the Most financial math formulae are a form of present value calculation; that is, these formulae identify the future cash flows of a financial instrument and then calculate the value at which these instruments could be exchanged for cash today. Financial Math Support Materials Page 6 of 85 RATE OF RETURN Suppose I purchase a watch for $200 and sell it a year later for $250. What is the dollar return and rate of return of this transaction? Financial Math Support Materials Page 7 of 85 Interest Interest a fee for borrowing money – about as old as civilisation itself Prime rate – the interest charged to the largest and most secure corporations. Interest is a cost to business, hence it is very important to understand how it is calculated and how it impacts on the business. There are two basic types of interest Simple Interest and Compound Interest Simple Interest Compound Interest Financial Math Support Materials Page 8 of 85 (2) SIMPLE INTEREST When a financial institution quotes an interest rate for a loan it can do so in different ways. For example, a quote 10% p. a. simple interest has different cash flows than a quote of 10% p. . compound interest payable quarterly. If the quote is offered as a SIMPLE INTEREST RATE, then the rate is taken as a proportion of the initial loan amount. eg 12% p. a. (SIMPLE), is equivalent to 1% per month, or 3% per quarter, or 6% semi-annually. * NOTE – The quoted rate is often referred to as the nominal rate. Financial Math Support Materials Page 9 of 85 SIMPLE INTEREST Suppose we lend $300 and quote a simple interest rate of 8% p. a. What will be the interest and repayment if the loan is made over: (a) six months, (b) one year, (c) three years. (a) 8% p. a. = Interest = Repayment = (b) Interest = Repayment = (c) 8% p. a. =Interest = Repayment = Financial Math Support Materials Page 10 of 85 Symbolically: Interest amount = I = P i t P ~ principal (or amount bor rowed = PV) i ~ rate of interest as a percentage t ~ time is the number of years, or fraction of a year, for which the loan is made The simple interest (I) charged on a loan of $800 for 2. 5 years at 8. 5% is: I = Pit = Simple interest is usually associated with short-term loans, that is, less than 12 months. In the formula time (t) is expressed in years, or fraction of a year. Example: $800 for 9 months at 8. 5% is: I= Financial Math Support Materials Page 11 of 85 Example: $800 for 88 days at 8. % is: I = Pit At the end of the period the amount repaid is: FV = PV(1 + t i) Where t represents the fraction of a year during which the money is borrowed. Financial Math Support Materials Page 12 of 85 SIMPLE INTEREST In general, the amount repayable, or Future Value (FV) of a loan quoted as simple interest is given by: ? ? ? ? f ? i? FV ? PV 1 ? 365 ? ? ? ? ? ? Where: FV – is the future value (amount repayable) PV – is the present value (Principle) f – is the numbe r of days i – is the annual simple interest rate PV = EQUIVALENTLY, Financial Math Support Materials FV ?f? 1+ ? ?i ? 365 ? Page 13 of 85 SIMPLE INTEREST Question 4(a) from 2001, 2nd semester final exam) Leanne buys a watch for $80 and sells it a month later for $85. What nominal annual interest rate of return does she earn? Rate of return in one month = Annual nominal rate = Financial Math Support Materials Page 14 of 85 Principal unknown A borrower can pay an interest amount of $120 at the end of 6 months. If the current interest rate for personal loans is 9% what is the maximum that can be borrowed, that is, what is PV? ? f ? i I ? PV ? ? ? 365 ? I PV ? ?f? ? ?i ? 365 ? Note: Financial Math Support Materials Page 15 of 85 Interest rate unknown A loan of $18,000 for 8 months had an nterest charge of $888. What was the annual rate of interest rate? ? f ? i I ? PV ? ? ? 365 ? I i? ?f? ? ? PV ? 365 ? Financial Math Support Materials Page 16 of 85 Rayleen’s birthday was on the 14th August last year. On this date she received a gift of $4,800 from her family which she placed in an interest earning account at a nominal rate of 5. 75% per annum. If she withdraws all funds in the account on the 8th April this year, how much will she receive? How much interest is earned? August September October November December January February March April 17 30 31 30 31 31 29 31 8 Total number of days = 237 ? ? f FV ?PV ? 1 ? ? ?i ? ? ? 365 ? ? FV = I= Financial Math Support Materials Page 17 of 85 Barns & Co Ltd. currently has a non tradable bank note with a face value of $500,000 that will mature in 85 days. Barns & Co has negotiated with its lender to obtain a loan using the note as security. The lender requires an establishment fee of $440 and charges simple interest of 9% pa. How much will Barns & Co receive, and what is the total cost of the funds? ? ? f FV ? PV ? 1 ? ? ?i ? ? ? 365 ? ? ent ? Establishm ? FV PV ? ? f? ? fee ? ? 1? ? ?i ? 365 ? Cost of funds Cost in simple interest terms Financial Math Support Materials $500,000 – $489,295. 68 = $10,704. 32 = Page 18 of 85 A bill with a face value of $500,000 and term to maturity of 180 days is sold at a yield of 8% p. a. What are the proceeds of the sale? Proceeds = PV ? PV ? FV ?f? 1? ? ?i ? 365 ? $500,000 ? 180 ? 1? ? ? 0. 08 ? 365 ? PV ? $481, 022. 67 Calculate the effective annualised return for a $100,000 investment which earned: ? 6. 5% p. a. for 90 days, then ? 7. 5% p. a. for 60 days, then ? 6. 2% p. a. for 45 days Value of investment after 90 days: 90 ? $100,000 ? 1+ ? 0. 065 = $101,602. 70 ? ? 365 ? Financial Math Support Materials Page 19 of 85 Value of investment after 90 + 60 days:Value of investment after 90 + 60 + 45 days: Value after 195 days = $103,641. 60 Annualised return = Financial Math Support Materials Page 20 of 85 APPLICATIONS OF SIMPLE INTEREST ? TREASURY NOTES ? BILLS OF EXCHANGE ? PROMISSORY NOTES – WHEN CREATED (ISSUED) – WHEN TRADED LA TER We cover these applications in greater detail in a later topic. Financial Math Support Materials Page 21 of 85 (3) COMPOUND INTEREST THE BASIC IDEA: ? PRINCIPAL GENERATES INTEREST ? RE-INVEST INTEREST TO GENERATE STILL MORE INTEREST ? RE-INVEST AGAIN TO GENERATE EVEN MORE INTEREST . . .etc Financial Math Support Materials Page 22 of 85 COMPOUND INTERESTSuppose we invest $100 000 at 10% p. a. with interest payable annually. What annual cash flows result from this investment? $100,000 Invested at 10% Compound Interest $800,000 Amount $700,000 $600,000 $500,000 $400,000 $300,000 $200,000 $100,000 $0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time in years Financial Math Support Materials Page 23 of 85 A LGEBRAICALLY ? ? ? ? ? ? Balance at end of year ? ? ? ? ? ? ? Balance after ? ? ? ? ? n FV = Financial Math Support Materials Balance at start of year ? ? ? ? ? ? (1? i) years is: PV(1 + i) n Page 24 of 85 G ENERALISING Suppose we invest $100 000 at 10% p. a. with nterest payable annually. What is the future value of this investment after 4 years? FV = $ Financial Math Support Materials Page 25 of 85 T HE POWER OF COMPOUNDING WITH COMPOUND INTEREST, â€Å"SMALL† SUMS NOW BECOME â€Å"LARGE† SUMS LATER (a) $1000 AT 13% pa FOR 50 YEARS : FV = $ (b) $1000 AT 14% pa FOR 50 YEARS : FV = $ Financial Math Support Materials Page 26 of 85 PRESENT VALUE : REARRANGING THE COMPOUND INTEREST FORMULA: PV ? FV n (1 ? i) COMPOUNDING NOW SHOWS THAT â€Å"LARGE† SUMS TO BE PAID LATER ARE WORTH ONLY â€Å"SMALL† SUMS NOW What is the present value of $1 million to be paid in 100 years’ time if the interest rate is 15% pa?Financial Math Support Materials Page 27 of 85 PRESENT VALUE : (Question 5 from 2001 2nd semester final exam) Tran Van Ng is to receive from his parents $1,000, $1,500 and $2,500 in 1 year, 2 years and 3 years respectively if he passes all subjects in his university degree each year. (a) What is the present value of th ese cash flows assuming a discount rate of 9% over the three years? (b) What is the present value of this these cash flows assuming a discount rate of 9% in the first year, 8% in the second year and 6% in the third year? Financial Math Support Materials Page 28 of 85 Present Value (a) PV ? $1, 000 ?1. 09 ? 1 ? $1,500 ?1. 9 ? 2 ? $2,500 ?1. 09 ? 3 ? $4,110. 41 (b) The value today of $2,500 received in 3 years time? 0 1 2 3 $2,500 Financial Math Support Materials Page 29 of 85 Measuring Average Growth Rates COMPOUND INTEREST IS A SPECIAL CASE OF COMPOUND GROWTH WHERE THE GROWTH RATE IS THE SAME EACH PERIOD IN COMPOUND GROWTH GENERALLY, THE GROWTH RATE MAY CHANGE EACH PERIOD IN PRACTICE, GROWTH RATES CHANGE FROM YEAR TO YEAR. WE NEED TO BE ABLE TO CALCULATE THE FUTURE VALUE AND PRESENT WHERE VALUE RATES THROUGHOUT OF OF THE AN INVESTMENT RETURN LIFE CHANGE OF THE INVESTMENT. Financial Math Support Materials Page 30 of 85 Measuring Average Growth RatesSUPPOSE YOU INITIALLY INVEST $1,000 IN AN ASSET WHOSE VALUE CHANGED YEAR BY YEAR, AS FOLLOWS: YEAR GROWTH RATE %pa 1 2 3 4 5 35 15 9 What is the future value of this investment? What is the average annual growth rate of this investment? After 4 years, the value of the asset is : Financial Math Support Materials Page 31 of 85 Measuring Average Growth Rates AVERAGE ANNUAL GROWTH RATE (g) NOTE THAT THE ANSWER IS NOT: Financial Math Support Materials Page 32 of 85 Average Growth Rate Suppose we invest $1million in an asset whose value changes as follows, year 1 growth rate 20% 2 -8% 3 -15% 4 3% What is the future value of this nvestment? What is the average annual growth rate of this investment? Financial Math Support Materials Page 33 of 85 Average Growth Rate The average annual growth rate is : Financial Math Support Materials Page 34 of 85 Average Growth Rate (Question 6(a),(b) from 2001 2nd semester final exam) House prices in Melbourne have soared in the past four years. The median price of a house in Clayton at the end of each year is as follows: 1997 – $122,000 1998 – $135,000 1999 – $147,000 2000 – $185,000 (a) What is the annual compounding growth rate for housing prices calculated at the end of each year, that is 1998, 1999 and 2000? b) What is the average annual compound growth rate for housing prices over this period? Financial Math Support Materials Page 35 of 85 Average Growth Rate (a) 1998 – 1999 – 2000 – (b) ?1 ? r ? ? 3 Financial Math Support Materials Page 36 of 85 Calculating Average Growth rate – continued g = average growth rate The average rate of growth per period over n time periods is: n ? ? ? Value at end – Value at beginning ? 1 + g ? = ? 1 + ? ? Value at beginning ? ? ? ? Solving for g, 1 ? Value at end – Value at beginning ? n g = ? 1 + ? -1 Value at beginning ? ? 1 $185,000 – $122,000 ? 3 ? g = ? 1 + ? -1 $122,000 ? ? 1 ?3 ? g = ? 1. 16793 ? ? ? – 1 = 0. 148869 g = 14. 89% Financial Mat h Support Materials Page 37 of 85 Real (after Inflation) Interest Inflation reduces the purchasing power of money. We require a methodology to adjust rates of return for the impact of inflation. TODAY 1 box of biscuits costs $2. 00 I have $200 I can consume 100 boxes of biscuits IN ONE YEAR Inflation rate (10%) 1 box costs $2. 20 To consume the same quantity of biscuits I will require To have a real return of, say, 4% pa, I need to be able to purchase 104 boxes. Financial Math Support Materials Page 38 of 85 Real (after Inflation) Interest Real increase in consumption of 4%Financial Math Support Materials Page 39 of 85 Real (after Inflation) Interest FORMULA : (1 + q) = (1 + r)(1 + p) where : q is the quoted interest rate r is the real interest rate p is the inflation rate A lender quotes an interest rate of 18% pa for an investment. If the inflation rate is currently at 4% pa, what is the real interest rate earned by the investor ? Rearranging: Financial Math Support Materials (1 + q) = (1 + r)(1 + p) Page 40 of 85 EFFECTIVE and NOMINAL (QUOTED) Interest Rates A BANK LENDS $1,000 AND QUOTES AN INTEREST RATE OF: (a) 12% pa, payable quarterly (that is, 3% each quarter) (b) 12% pa, payable semi-annually that is, 6% each half year) (c) 12% pa, payable annually (that is, 12% at the end of the year) How much interest does the bank earn at the end of one year under each of these three scenarios? Financial Math Support Materials Page 41 of 85 EFFECTIVE and NOMINAL (QUOTED) Interest Rates ? interest rate of 12% pa, payable quarterly REPAYMENTS $30 $30 1 2 $30 3 $30 4 Quarter The value at the end of the year of the interest payment in the The bank has effectively earned : Financial Math Support Materials Page 42 of 85 EFFECTIVE and NOMINAL (QUOTED) Interest Rates ? INTEREST RATE OF 12% pa, PAYABLE SEMI ANNUALLY REPAYMENTS $60 1 $60 2 Half YearThe value at the end of the year of the interest payment in the The bank has effectively earned : Financial Math Support Materia ls Page 43 of 85 EFFECTIVE and NOMINAL (QUOTED) Interest Rates ? INTEREST RATE OF 12% pa, PAYABLE ANNUALLY REPAYMENTS $120 1 YEAR The value at the end of the year of the interest payment is $120 The bank has effectively earned : Financial Math Support Materials Page 44 of 85 EFFECTIVE and NOMINAL (QUOTED) Interest Rates So a quoted (Nominal) interest rate of, 12% pa payable = 12. 55% payable annually. quarterly 12% pa payable = 12. 36% payable annually semi annually 12% pa payable = 12. 00% payable annually nnually To compare interest rate quotations (the nominal interest rate) we refer to an effective interest rate, that is, the interest rate that we would receive if interest were paid once at the end of the year. In the above example: Nominal (Quoted Rate) 12% pa payable quarterly 12% pa payable semi annually 12% pa payable annually Financial Math Support Materials Effective Rate 12. 55% pa 12. 36% pa 12. 00% pa Page 45 of 85 Formula Development If the nominal rate is j percent pa , compounding m times pa, Then after one year the principal, P, becomes: m j? ? S = P ? 1 + ? m? ? (C1) The effective annual interest rate, i, is therefore: = S-P S = -1 P P (C2) Replacing S in (C2) with equation (C1) produces: j? ? P ? 1 + ? m? ? i= P m -1 m j? ? i = ? 1 + ? m? ? Financial Math Support Materials ?1 (C3) Page 46 of 85 Effective and Nominal Interest Rates (a) NOMINAL TO EFFECTIVE If the nominal rate is 15% p. a. payable monthly, then the effective rate is : (b) EFFECTIVE TO NOMINAL If the effective rate is 15% pa then the nominal pa, with monthly payments, is : Financial Math Support Materials Page 47 of 85 Effective and Nominal Interest Rates (Question from 2002 mid semester exam) Abdul Hafahed purchases a car for $5,000 and sells it four months later for $6,000. a) What nominal annual rate of return did Abdul receive? (b) What effective annual rate of return did Abdul receive? (c) If inflation is at 2% pa, what real annual effective rate of return did Abdul receive ? Show your calculations. Financial Math Support Materials Page 48 of 85 Effective and Nominal Interest Rates (a) Four month return Annual nominal return = (b) Effective rate (c) Real annual effective rate : (1 + q) = (1 + r)(1 + p) Financial Math Support Materials Page 49 of 85 Compound Interest Formula j? ? FV = PV ? 1 + ? m? ? n Where: FV = future value PV = principal (present value) j = interest rate per annum as a percentage = mT = total number of periods over which investment is held m = number of interest payments per annum Solving for other terms by rearranging variables: PV = FV j? ? 1+ ? ? m? ? n 1 ? ? ? FV ? n ? j = ? – 1? m PV ? ? ? Microsoft Excel functions: Future value: FV(rate, nper, pmt, pv, type) Present value: PV(rate, nper, pmt, pv, type) Financial Math Support Materials Page 50 of 85 CONTINUOUS COMPOUNDING Nominal interest rate We know j? ? FV ? PV ? 1 ? ? ? m? mT Number of years Number of compounding periods per year What if compounding takes place at every moment, that is ? m ? ? ? . It can be shown that as ? m ? ? then: m j? ? j ?1 ? ? ? e lim ? m ? m where e is the base of natural logarithms (e ? 2. 71828) The Future value formula then reduces to: FV ? PVe jT or, FV ? jT PV ? jT ? FVe e Financial Math Support Materials Page 51 of 85 COMPOUNDING FREQUENCY $1,000 invested for 1 year at 12%: Compounding frequency Payment at end of year Annual Semi-Annual Quarterly Monthly Daily Continuously ? As the compounding frequency increases for a given nominal interest rate, the higher the interest repayments. However the interest repayment reaches a maximum with continuous compounding. Financial Math Support Materials Page 52 of 85Continuously Compounded Returns Nominal interest rate Recall FV ? PVe Using the notation and r where r ? jT . pt ? 1 ? PV pt ? FV pt ? pt ? 1e then Number of years rt pt e? pt ? 1 rt and rearranging we have; ? pt ? ln? e ? ? ln? ?p ? ? ? t ? 1 ? rt and ? pt ? ? rt ? ln ? ?p ? ? ? t ? 1 ? rt is the continuously c ompounding return from time period t-1 to t. Financial Math Support Materials Page 53 of 85 Continuously Compounded Returns pt Note: the term pt ? 1 is referred as the Price Relative. It is the proportional price change from time t-1 to t. The logarithm of the price relative is the continuously compounding return.Continuously compounding returns are often easier to work with. Two important properties: (1) Continuously compounding returns over a period can be added up to give a total continuously compounding return. (2) The average continuously compounding return over a period is the arithmetic average of each individual continuously compounding return. Financial Math Support Materials Page 54 of 85 Continuously Compounded Returns A stock price has a closing price of $3. 00, $3. 25 and $2. 90 over 3 days. What is the continuously compounding return on each day? What is the total and average continuously compounding return? Time 0 1 Price 3. 00 . 25 2 Return 2. 90 Total return from ti me 0 to 2 = Financial Math Support Materials Page 55 of 85 Continuously Compounded Returns An investor is given a choice of: (a) investing at 16. 5% p. a. , (b) investing at 4% per quarter, for 1 year (c) investing at 16. 3% p. a. and compounded daily. (d) 16. 3% p. a. continuously compounding. Which investment is chosen? Financial Math Support Materials Page 56 of 85 Calculate the effective rate in each case. (a) 16. 5% pa (b) (c) (d) Financial Math Support Materials Page 57 of 85 (4) MUTIPLE CASH FLOWS Cash Flow Stream : Future Value 0 1 2 3 $200 3. 5 $450 4 5 6 $800 1 2 3 If interest rate 9%pa = $1,712. 50 1 :$ 2 :$ 3 :$ Stream Future Value Financial Math Support Materials Page 58 of 85 Cash Flow Stream : Present Value 0 1 2 3 $200 3. 5 $450 4 5 6 $800 1 2 3 If interest rate = 9%pa 1 2 3 Stream Present Value = $ Financial Math Support Materials Page 59 of 85 Net Present Value – NPV The present value of the following stream of cash flows, using a discount rate of 7. 5%, is: 0 Cash flows: 1 2 3 $880 $560 $420 4 $980 PV’s =$ Suppose it cost the investor $2,000 to purchase this stream of cash flows, the net present value of this stream is: NPV = -$2,000 + $ =$ outflow Investment projects where NPV ? 0 are viable.Financial Math Support Materials Page 60 of 85 Internal Rate of Return – IRR One period: YEAR $ 0 -1000 1 +1120 Dollar return = $ Equivalently, solve for r : What value of r will produce an NPV = 0 ? PV of $1,120 Using discount Rate of r for 1 period Financial Math Support Materials No discounting required since $1,000 occurs â€Å"now† Page 61 of 85 Internal Rate of Return – IRR Two periods YEAR $ 0 -1000 1 +1120 2 + 25 Clearly, IRR > 12% pa but IRR < 14. 5% pa Why? Because this would be the rate of return if the additional $25 was received in year 1. That is, Thus 12% < IRR < 14. 5% But where in this range is the IRR ?