Tài chính doanh nghiệp - Chapter 8: Mathematics of finance: An introduction to basic concepts and calculations

Chapter 8Mathematics of Finance: An Introduction to Basic Concepts and CalculationsCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye WatsonLearning ObjectivesDifferentiate between simple and compound interest rate calculationsDifferentiate between nominal and effective interest rate calculationsCalculate present and future values of cash flowsCalculate the yield of a securityCalculate the present value of an annuityCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye WatsonChapter Organisation8.1 Introduction8.2 Simple InterestSimple interest accumulationPresent valueYieldsHolding period yieldCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye WatsonChapter Organisation (cont.)8.3 Compound Interest Compound interest accumulation (future value)Present valuePresent value of an annuityAccumulated value of an annuity (future value)Effective rates of interest8.4 Summary Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.1 IntroductionFocus is on the mathematical techniques for calculating the cost of borrowing and the return earned on an investmentTable 8.1 defines the symbols of various formulaeAlthough symbols vary between textbooks, formulae are consistentCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.1 Introduction (cont.)Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye WatsonChapter Organisation8.1 Introduction8.2 Simple InterestSimple interest accumulationPresent valueYieldsHolding period yieldCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple InterestSimple interest is interest paid on the original principal amount borrowed or investedThe principal is the initial, or outstanding, amount borrowed or investedWith simple interest, interest is not paid on previous interestCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Simple interest accumulationThe amount of interest paid on debt, or earned on a deposit is I = A  n  i (8.1) whereA is the principaln is the duration of the loan, expressed as the number of interest payment periods (usually one year)i is the interest rate, expressed as a decimalCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Simple interest accumulation (cont.)Example 1: If $10,000 is borrowed for one year, and simple interest of 8.00 per cent per annum is charged, then the total amount of interest paid on the loan would be:I = A  n  I = 10 000  1  0.08 = $800Example 2: Had the same loan been for two years then the total amount of interest paid would be:I = 10 000  2  0.08 = $1600Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Simple interest accumulation (cont.)The market convention (common practice occurring in a particular financial market) for the number of days in the year is 365 in Australia and 360 in the USA and EuromarketsExample 3: If the amount is borrowed, at the same rate of interest, but for a ninety day term, then the total amount of interest paid would be:I = 10 000  90/365  0.08 = $197.26Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Simple interest accumulation (cont.)The final amount payable (S) on the borrowing is the sum of the principal plus the interest amountAlternatively, the final amount payable can be calculated in a single equation S = A + I = A + (A  n  i)S = A[1 + (n  i)] (8.2)Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Simple interest accumulation (cont.)The final amounts payable in the previous examples areExample 1a:S = 10 000 [1 + ( 1  0.08)] = 10 800Example 2a:S = 10 000 [1 + ( 2  0.08)] = 11 600Example 3a:S = 10 000 [1 + ( 90/365  0.08)] = 10 197.26Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Present value with simple interestThe present value is the current value of a future cash flow, or series of cash flows, discounted by the required rate of returnAlternatively, the present value of an amount of money is the necessary amount invested today to yield a particular value in the futureThe yield is the effective rate of return receivedCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Present value with simple interest (cont.)Equation for calculating the present value of a future amount A = S/[1 + (n  i)] (8.3)Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Example 5: A company discounts (sells) a commercial bill with a face value of $500,000, a term to maturity of 180 days, and a yield of 8.75 per cent per annum. How much will the company raise on the issue? (Commercial bills are discussed in Chapter 9.) Briefly, a bill is a security issued by a company to raise funds. A bill is a discount security, that is, it is issued with a face value payable at a date in the future, but in order to raise the funds today the company sells the bill today for less than the face value. The investor who buys the bill will get back the face value at the maturity date. The price of the bill will be:Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Example 5 (cont.):Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Present value with simple interest (cont.)Equation 8.3 may be rewritten to facilitate its application to calculating the price (i.e. present value) of another discount security, the Treasury note (T-note)Price = 365  face value (8.4) 365 + (yield/100 days to maturity)Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Present value with simple interest (cont.)Example 6: What price per $100 of face value would a funds manager be prepared to pay to purchase 180-day T-notes if the current yield on these instruments was 7.35 per cent per annum?Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Calculations of yieldsIn the previous examples the return on the instrument or yield was givenHowever, in other situations, it is necessary to calculate the yield on an instrument (or cost of borrowing) i = 365/n  I/A (8.5)Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Calculations of yields (cont.)Example 7: What is the yield (rate of return) earned on a deposit of $50,000 with a maturity value of $50,975 in 93 days? That is, this potential investment has a principal (A) of $50,000, interest (I) of $975 and an interest period (n) of 93 days.Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Holding period yield (HPY)HPY is the yield on securities sold in the secondary market prior to maturityShort-term money market securities (e.g. T-notes) may be sold prior becauseIntended short-term management of surplus cash held by investorThe investor’s cash flow position has unexpectedly changed and cash is neededA better rate of return can be earned in an alternative investmentCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Holding period yield (HPY) (cont.)The yield to maturity is the yield obtained by holding the security to maturityThe HPY is likely to be different from the yield to maturityThis is illustrated in Example 9 of the textbook with a discount securityA discount security pays no interest, but is sold today for less than its face value which is payable at maturity e.g. T-noteCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.2 Simple Interest (cont.)Holding period yield (HPY) (cont.)The HPY will beGreater than the yield to maturity when the market yield declines from the yield at purchase i.e. interest rates have declined and the price of the security increasesLess than the yield to maturity when the market yield increases from the yield at purchase i.e. interest rates have increased and the price of the security decreasesCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye WatsonChapter Organisation (cont.)8.3 Compound Interest Compound interest accumulation (future value)Present valuePresent value of an annuityAccumulated value of an annuity (future value)Effective rates of interest8.4 Summary Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound InterestCompound interest is paid on the initial principal plus accumulated previous interest entitlementsAccumulation (future value)When an amount is invested for only a small number of periods it is possible to calculate the compound interest payable in a relatively cumbersome way as illustrated in Example 10 in the textbookCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Accumulation (future value) (cont.)The cumbersome method in Example 10 can be simplified using the general form of the compounding interest formula S = A(1 + i)n (8.6)Applying equation (8.6) to Example 10S = 5000 (1 + 0.15)3 = $7604.38 Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Accumulation (future value) (cont.)On many investments and loans, interest will accumulate more frequently than once a year e.g. daily, monthly, quarterly etc.Thus, it is necessary to recognise the effect of the compounding frequency on the inputs i and n in equation 8.6If interest had accumulated monthly on the previous loan, then i = 0.15/12 = 0.0125 and n = 3  12 = 36Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Accumulation (future value) (cont.)Example 11a: The effect of compounding can be further understood by considering a similar deposit of $8000 paying 12 per cent per annum, but where interest accumulates half-yearly for four years:I = 12.00 % p.a. / 2 = 0.06and:n = 4  2 = 8 periodsso:S = 8000(1 + 0.06)8 = 8000(1.593848) = $12 750.78Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Present valueThe present value of a future amount is the future value divided by the interest factor (referred to as the discount factor) and is expressed in equation form as A = S/(1 + i)n (8.7a)A = S(1 + i)-n (8.7b)Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Example 12: What is the present value of $18 500 received at the end of three years, if funds could presently be invested at 7.25 per cent per annum, compounded annually? Using Equation 8.7a:Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Present value of an annuityAn annuity is a series of periodic cash flows of the same amountOrdinary annuity—series of periodic cash flows occur at end of each period (equation 8.8)(8.8)Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Present value of an annuity (cont.)Example 14: The present value of an annuity of $200, received at the end of every three months for ten years, where the required rate of return is 6.00 per cent per annum, compounded quarterly, would be:Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Example 14 (cont):Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Present value of an annuity dueAnnuity due—cash flows occur at the beginning of each period (equation 8.9)(8.9)Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Present value of an annuity due (cont.)Example 15: The present value of an annuity of $200, received at the end of every three months for ten years, where the required rate of return is 6.00 per cent per annum, compounded quarterly, would be:Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Example 15 (cont):Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Present value of a Treasury bondEquation 8.10 is used to calculate the price (or present value) of a Treasury bondExample 16 in the textbook illustrates the application of equation 8.10(8.10)Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Accumulated value of an annuity (future value)(8.11)Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Accumulated value of an annuity (future value) (cont.)Example 17: A university student is planning to invest the sum of $200 per month for the next three years in order to accumulate sufficient funds to pay for a trip overseas once she has graduated. Current rates of return are 6 per cent per annum, compounding monthly. How much will the student have available when she graduates?Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Example 17 (cont):Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Effective rates of interestThe nominal rate of interest is the annual rate of interest, which does not take into account the frequency of compoundingThe effective rate of interest is the rate of interest after taking into account the frequency of compoundingCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Effective rates of interest (cont.)Example 18a: A deposit of $8000 is made for four years, and will earn 12 per cent per annum, with interest compounding semi-annually. What will be the value of the deposit at maturity?Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Example 18a (cont):Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Effective rates of interest (cont.)Example 18b: What would the maturity value of the same deposit be if interest was compounded annually, rather than semi-annually as in Example 18a?Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Effective rates of interest (cont.)The formula for converting a nominal rate into an effective rate is(8.12)Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)Example 19a: What is the effective rate of interest if you are quoted:a 10 per cent per annum, compounded annually?b 10 per cent per annum, compounded semi-annually?c 10 per cent per annum, compounded monthly?aCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.3 Compound Interest (cont.)b cCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye WatsonChapter Organisation (cont.)8.3 Compound Interest Compound interest accumulation (future value)Present valuePresent value of an annuityAccumulated value of an annuity (future value)Effective rates of interest8.4 Summary Copyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.4 SummarySimple interest is interest paid on the original principal amount borrowed or investedCompound interest is paid on the initial principal plus accumulated previous interest entitlementsThe present value and future value of an investment or loan can be calculated using either simple or compound interestCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson8.4 Summary (cont.)An annuity is a series of periodic cash flows of the same amount, of which both the present value and the future value can be calculatedUnlike the nominal rate of interest, which ignores the frequency of compounding, the effective rate of interest takes into account the frequency of compoundingCopyright  2003 McGraw-Hill Australia Pty Ltd PPTs t/a Financial Accounting by WillisSlides prepared by Kaye Watson