Tài chính kế toán - Chapter 8: Mathematics of finance: an introduction to basic concepts and calculations

Chapter 8Mathematics of finance: An introduction to basic concepts and calculationsLearning objectivesUnderstand and carry out simple interest calculations to determine:accumulated amountpresent valueyieldsholding period yieldUnderstand and carry out compound interest calculations to determine:accumulated amount (future value)present valuepresent value of an annuityaccumulated value of an annuity (future value)effective rate of interestChapter organisation8.1 Simple interestSimple interest accumulationPresent valueYieldsHolding period yield8.2 Compound interest Compound interest accumulation (future value)Present valuePresent value of an annuityAccumulated value of an annuity (future value)Effective rates of interest8.3 Summary8.1 Simple interestIntroductionFocus 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 consistent(cont.)8.1 Simple interest (cont.)(cont.)8.1 Simple interest (cont.)Simple 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 interestThe amount of interest paid on debt, or earned on a deposit is: where:A is the principald 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 decimal(cont.)Simple interest accumulationExample 1: If $10 000 is borrowed for one year, and simple interest of 8% per annum is charged, the total amount of interest paid on the loan would be:I = A  d/365  i = 10 000  365/365  0.08 = $800Example 2: Had the same loan been for two years the total amount of interest paid would be:I = 10 000  730/365  0.08 = $1600(cont.)Simple interest accumulation (cont.)The market convention (common practice occurring in a particular financial market) is for the number of days in the year to be 365 in Australia and 360 in the US and the euromarketsExample 3: If the amount is borrowed at the same rate of interest but for a 90-day term, the total amount of interest paid would be:I = 10 000  90/365  0.08 = $197.26(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)] (cont.)Simple interest accumulation (cont.)The final amounts payable in the three previous examples are:Example 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)] = 10 197.26Simple interest accumulation (cont.)The 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 received(cont.)Present value with simple interestPresent value with simple interest (cont.)Equation for calculating the present value of a future amount is a re-arrangement of Equation 8.2(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 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, i.e. 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 receive face value at the maturity date. The price of the bill will be:(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)(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 5.82% per annum?Present value with simple interest (cont.)In 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 x I d A(cont.)Calculation of yieldsExample 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 (d) of 93 days.i = 365 x $975 93 $50 000 = 0.07653 = 7.65%Calculation of yields (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 to maturity because:investment was intended as 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 investment(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 security using Equation 8.3 (8.4 can also be used) A discount security pays no interest but is sold today for less than its face value, which is payable at maturity, e.g. T-note(cont.)Holding period yield (HPY) (cont.)The HPY will be:greater than the yield to maturity when the market yield declines from the yield at purchase, i.e. interest rates have decreased 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 decreasesChapter organisation8.1 Simple interestSimple interest accumulationPresent valueYieldsHolding period yield8.2 Compound interest Compound interest accumulation (future value)Present valuePresent value of an annuityAccumulated value of an annuity (future value)Effective rates of interest8.3 Summary8.2 Compound interestCompound interest (unlike simple interest) is paid on both:the initial principal; andthe accumulated previous interest entitlementsCompound interest accumulation (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 (illustrated in Example 10 in the textbook)This method can be simplified using the general form of the compounding interest formula S = A(1 + i)n Applying Equation 8.6 to Example 10S = 5000 (1 + 0.15)3 = $7604.38(cont.)On many investments and loans, interest will accumulate more frequently than once a year; e.g. daily, monthly, quarterlyThus, 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 = 36(cont.)Compound interest accumulation (future value) (cont.)Example 11a: The effect of compounding can be further understood by considering a similar deposit of $8000 paying 12% per annum, but where interest accumulates quarterly for four years:I = 12.00 % p.a. / 4 = 0.03and:n = 4  4 = 16 periodsso:S = 8000(1 + 0.03)16 = 8000(1.604706) = $12 837.65Compound interest accumulation (future value) (cont.)Present value with compound interestThe 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)nA = S(1 + i)-n (cont.)Example 12: What is the present value of $18 500 received at the end of three years, if funds could currently be invested at 7.25% per annum, compounded annually? Using Equation 8.7a: A = S (1 + i)n = $18 500 (1 + 0.0725)3 = $18 500 = $14 996.15 1.233650Present value with compound interest (cont.)Present value of an ordinary 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) A = C [ 1 – (1 + i)-n ] i(cont.)Present value of an ordinary annuity (cont.)Example 14: The present value of an annuity of $200, received at the end of every three months for 10 years, where the required rate of return is 6.00% per annum, compounded quarterly, would be: C = $200 i = 6.00%/4 = 1.50% or 0.015 n = 4 x 10 = 40 Therefore:A = $200 [ 1 – (1 + 0.015)-40 ] 0.015 = $200 [ 29.915 845 2 ] = $5983.17Annuity due—cash flows occur at the beginning of each period (Equation 8.9)A = C [ 1 – (1 + i)-n ] (1 + i) i(cont.)Present value of an annuity dueExample 15: The present value of an annuity of $200, received at the beginning of every three months for ten years, where the required rate of return is 6.00% per annum, compounded quarterly, would be: C = $200 i = 6.00%/4 = 1.50% or 0.015 n = 4 x 10 = 40 Therefore:A = $200 [ 1 – (1 + 0.015)-40 ] 0.015 = $200 [ 29.915 845 2 ](1.015) = $6072.92Present value of an annuity due (cont.)Equation 8.10 is used to calculate the price (or present value) of a corporate bond A = C [ 1 – (1 + i)-n ] + S(1 + i)-n iExample 16 in the textbook illustrates the application of Equation 8.10Present value of a Treasury bondThe accumulated (or future) value of an annuity is given by Equation 8.11 S = C [ (1 + i)n - 1 ] i(cont.)Accumulated value of an annuity (future value)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 annum, compounding monthly. How much will the student have available when she graduates? C = $200 i = 6.00%/12 = 0.50% or 0.005 n = 3 x 12 = 36 Therefore:S = $200 [ (1 + 0.005)36 - 1 ] 0.005 = $200 [ 39.3361 ] = $7867.22Accumulated value of an annuity (future value) (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 compounding(cont.)Effective rates of interest (cont.)Example 18a: A deposit of $8000 is made for four years and will earn 12% per annum, with interest compounding semi-annually. What will be the value of the deposit at maturity? A = $8000 i = 12%/2 = 6% or 0.06 n = 4 x 2 = 8 Therefore:S = $8000 (1 + 0.06)8 = $8000 (1.06)8 = $12 750.78(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?S = $8000 (1.12)4 = $12 588.15(cont.)Effective rates of interest (cont.)The formula for converting a nominal rate into an effective rate is: ie = (1 + i/m )m – 1 (cont.)Effective rates of interest (cont.)Example 19: What is the effective rate of interest if you are quoted: (a) 10% per annum, compounded annually? (b) 10% per annum, compounded semi-annually? (c) 10% per annum, compounded monthly? (a) ie = (1 + 0.10/1)1 – 1 = (1.10)1 – 1 = 0.10 or 10% (b) ie = (1 + 0.10/2)2 – 1 = (1.05)2 – 1 = 0.1025 or 10.25% (c) ie = (1 + 0.10/12)12 – 1 = (1.008333)12 – 1 = 0.1047 or 10.47%Chapter organisation8.1 Simple interestSimple interest accumulationPresent valueYieldsHolding period yield8.2 Compound interest Compound interest accumulation (future value)Present valuePresent value of an annuityAccumulated value of an annuity (future value)Effective rates of interest8.3 Summary8.3 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 interest(cont.)8.3 Summary (cont.)An annuity (ordinary or due) 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 compounding