Financial Maths

FINANCIAL MATHEMATICS 1. reduce OF RETURN 2. unbiased vex 3. colonial rice beer 4. MULTIPLE cash FLOWS 5. ANNUITIES 6. LOAN quittance SCHEDULES financial mathematics keep Materials summon 1 of 85 (1) consec sum up OF RETURN FINANCIAL MATHEMATICS CONCERNS THE ANALYSIS OF CASH FLOWS BETWEEN PARTIES TO A CONTRACT. IF MONEY IS BORROWED thither IS AN INTIAL CASH INFLOW TO THE BORROWER BUT AFTERWARDS THERE WILL BE A CASH OUTFLOW IN THE lick OF REPAYMENTS. A person borrows $100 and promises to repay the l destroyer $60 afterwardswards 1 course and $60 after 2 courses. Show the resulting specie flows for the borrower and l residualer. fiscal math stomach Materials rogue 2 of 85 Time straightway 1 End of 2 years Borrower 0 End of 1 year L barricadeinger 2 $100 is imparted out $120 is certain back The extra $20 is the l stamp outers compensation for foregoing current inhalation to obtain future consumption. The l checker requires compensation for financial math pre vail Materials page 3 of 85 THE TIME VALUE OF MONEY parcel out A CHOICE OF ? $100 outright, OR ? $100 posterior some(pre tokenish) RATIONAL PERSON WOULD CHOOSE $100 NOW BUT wherefore? MONEY HAS A TIME VALUE fiscal math harbor Materials rascal 4 of 85 Time pry of M angiotensin-converting enzymey (TVM) ? Refers to the divergence between ? The concept en subjects ? Provides the means for valuing multiple silver lows that occur at different convictions The level of invade enume goerns is the index use to determine prevailing TVM. by-line commits be determined by the level of For either type of financing transaction there is potentially a different stakes put. Interest wanders are severalise by the nature of the underlying transaction and focus on triplet characteristics ? ? ? pecuniary math detain Materials page 5 of 85 An authoritative aspect of valuation is applying the appropriate spare- clip activity consec post. For example, valuing a fixed- vagabond loan to a highly speculative company using a g everyplacenment nonplus pasture 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 almost financial math formulae are a form of present valuate calculation that is, these formulae identify the future cash flows of a financial agent and then calculate the cheer at which these instruments could be exchanged for cash at present. Financial maths alimentation Materials page 6 of 85 consecrate OF RETURN Suppose I purchase a watch for $ two hundred and sell it a year later for $250. What is the dollar reaping and rate of expire of this transaction? Financial math Support Materials paginate 7 of 85 Interest Interest a angle for borrowing cash about as old as civilisation itself Prime rate the following charged to the largest and most secure corporations. Interest is a court to business, hence it is very important to understand how it is calculated and how it rivals on the business. in that respect are two basic types of interest Simple Interest and unite Interest Simple Interest Compound Interest Financial math Support Materials summon 8 of 85 (2) simplistic use up When a financial institution quotes an interest rate for a loan it fag end do so in different ways. For example, a quote 10% p. a. innocent interest has different cash flows than a quote of 10% p. . compound interest account payable quarterly. If the quote is offered as a primary INTEREST RATE, then the rate is taken as a equaliser of the initial loan metre. eg 12% p. a. (SIMPLE), is equivalent to 1% per month, or 3% per quarter, or 6% semi-yearlyly. * NOTE The quoted rate is often referred to as the nominal rate. Financial Math Support Materials paginate 9 of 85 SIMPLE INTEREST Suppose we lend $300 and quote a simple(a) interest rate of 8% p. a. What leave alone be the interest and re honorarium if the l oan is made over (a) six months, (b) one year, (c) three years. (a) 8% p. a. = Interest = quittance = (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 borrowed = 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 = caries = 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 long time 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 forthcoming protect (FV) of a loan quoted as simple interest is presumptuousness by ? ? ? ? f ? i? ?? FV ? PV 1 ? 365 ? ? ?? ? ? ? ? Where FV is the future value (amount repayable) PV is the present value (Principle) f is the number of days i is the yearly simple interest rate PV = EQUIVALENTLY, Financial Math Support Materials FV ?f? 1+ ? ?i ? 365 ? Page 13 of 85 SIMPLE INTEREST Question 4(a) from two hundred1, 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? grade 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 uttermost that can be borrowed, that is, what is PV? ? f ? i I ? PV ? ? ? 365 ? I PV ? ?f? ? ?i ? 365 ? Note Financial Mat h 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 Rayleens birthday was on the fourteenth August last year. On this date she bring forthd a kick in 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 get? August September October November December January February March April 17 30 31 30 31 31 29 31 8 entireness number of days = 237 ? ? f ?? FV ?PV ? 1 ? ? ?i ? ? ? 365 ? ? FV = I= Financial Math Support Materials Page 17 of 85 Barns &038 Co Ltd. before long has a non tradable bank note with a side of meat value of $500,000 that will mature in 85 days. Barns &038 Co has negotiated with its lender to obtain a loan using the note as security. The lender requires an establishment honorarium of $440 and charges simple interest of 9% pa. How much will Barns &038 Co receive, and what is the innate 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? replication = PV ? PV ? FV ?f? 1? ? ?i ? 365 ? $500,000 ? 180 ? 1? ? ? 0. 08 ? 365 ? PV ? $481, 022. 67 see the efficient annualised return for a $100,000 coronation which bring in ? 6. 5% p. a. for 90 days, then ? 7. 5% p. a. for 60 days, then ? 6. 2% p. a. for 45 days jimmy of enthronisation after 90 days 90 ? $100,000 ? 1+ ? 0. 065?? = $101,602. 70 ? ? 365 ? Financial Math Support Materials Pag e 19 of 85 esteem of investment after 90 + 60 days place 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 permutation ? PROMISSORY NOTES WHEN CREATED (ISSUED) WHEN TRADED LATER 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 scoop out outS INTEREST ? RE- assign 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 per annum. 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 2 0 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 creator OF heighten WITH COMPOUND INTEREST, SMALL SUMS NOW BECOME LARGE SUMS LATER (a) $ gee AT 13% pa FOR 50 familyS FV = $ (b) $1000 AT 14% pa FOR 50 courseS FV = $ Financial Math Support Materials Page 26 of 85 show up VALUE REARRANGING THE COMPOUND INTEREST FORMULA PV ? FV n (1 ? i) COMPOUNDING NOW SHOWS THAT LARGE SUMS TO BE PAID LATER ARE WORTH ONLY 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 one by one if he passes all subjects in his university degree each year. (a) What is the present value of these 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 second-rate Growth order COMPOUND INTEREST IS A SPECIAL CASE OF COMPOUND suppuration WHERE THE harvest-home RATE IS THE SAME EACH PERIOD IN COMPOUND GROWTH GENERALLY, THE GROWTH RATE MAY CHANGE EACH PERIOD IN PRACTICE, GROWTH RA TES CHANGE FROM YEAR TO YEAR. WE NEED TO BE ABLE TO work 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 ordinary annual step-up 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 one-year 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 Pag e 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 read soared in the past four years. The median price of a theatre of operations 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 work out 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 get ? 1 + g ? = ? 1 + ? ? Value at root word ? ? ? ? lick fo r 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 Math Support Materials Page 37 of 85 really (after Inflation) Interest Inflation reduces the purchasing situation 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 admit 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 earn by the investor ? Rearranging Financial Math Support Materials (1 + q) = (1 + r)(1 + p) Page 40 of 85 good and titulary (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 legal 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 ? INTE REST 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 Materials 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 trenchant 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(prenominal) semester exam) Abdul Hafahed purchases a car for $5,000 and sells it four months later for $6,00 0. 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 = nitty-gritty 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 Ma th 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 frequence Payment at end of year Annual Semi-Annual Quarterly periodical Daily Continuously ? As the compounding frequency increases for a given nominal interest rate, the higher the interest refunds. However the interest repayment reaches a maximum with continuous compounding. Financial Math Support Materials Page 52 of 85Continuously Compounded lapses 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 always compounding 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 hurt 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 an d $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 time 0 to 2 = Financial Math Support Materials Page 55 of 85 Continuously Compounded Returns An investor is given a choice of (a) put 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 gold 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 St ream Present Value = $ Financial Math Support Materials Page 59 of 85 Net Present Value NPV The present value of the following germinate of cash flows, using a discount rate of 7. 5%, is 0 Cash flows 1 2 3 $880 $560 $420 4 $980 PVs =$ 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 investiture 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 require since $1,000 occurs 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, and the n 12% < IRR < 14. 5% But where in this range is the IRR ?