Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-1 Chapter 8 Mathematics of Finance: An Introduction to Basic Concepts and Calculations

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-2 Learning Objectives Differentiate between simple and compound interest rate calculations Differentiate between nominal and effective interest rate calculations Calculate present and future values of cash flows Calculate the yield of a security Calculate the present value of an annuity

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-3 Chapter Organisation 8.1 Simple Interest –Simple interest accumulation –Present value –Yields –Holding period yield 8.2 Compound Interest –Compound interest accumulation (future value) –Present value –Present value of an annuity –Accumulated value of an annuity (future value) –Effective rates of interest 8.3 Summary

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger Simple Interest Introduction –Focus is on the mathematical techniques for calculating the cost of borrowing and the return earned on an investment –Table 8.1 defines the symbols of various formulae –Although symbols vary between textbooks, formulae are consistent

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger Simple Interest (cont.)

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger Simple Interest (cont.) Simple interest is interest paid on the original principal amount borrowed or invested –The principal is the initial, or outstanding, amount borrowed or invested –With simple interest, interest is not paid on previous interest

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-7 Simple interest accumulation The amount of interest paid on debt, or earned on a deposit is (8.1) where Ais the principal dis the duration of the loan, expressed as the number of interest payment periods (usually one year) iis the interest rate, expressed as a decimal

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-8 Simple interest accumulation (cont.) –Example 1: If $ 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 =  365/365  0.08 = $800 –Example 2: Had the same loan been for two years then the total amount of interest paid would be: I =  730/365  0.08 = $1600

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-9 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 US and the euromarkets –Example 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 =  90/365  0.08 = $197.26

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-10 Simple interest accumulation (cont.) –The final amount payable (S) on the borrowing is the sum of the principal plus the interest amount –Alternatively, 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  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-11 Simple interest accumulation (cont.) The final amounts payable in the three previous examples are: –Example 1a: S = [1 + ( 1  0.08)] = –Example 2a: S = [1 + ( 2  0.08)] = –Example 3a: S = [1 + ( 90/365  0.08)] =

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-12 Present value with simple interest The present value is the current value of a future cash flow, or series of cash flows, discounted by the required rate of return Alternatively, the present value of an amount of money is the necessary amount invested today to yield a particular value in the future –The yield is the effective rate of return received

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-13 Present value with simple interest (cont.) Equation for calculating the present value of a future amount is a re-arrangement of Equation

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-14 Present value with simple interest (cont.) –Example 5: A company discounts (sells) a commercial bill with a face value of $ , 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 get back the face value at the maturity date. The price of the bill will be:

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-15 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) 8.4

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-16 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 annum?

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-17 Calculations of yields –In the previous examples the return on the instrument or yield was given –However, in other situations it is necessary to calculate the yield on an instrument (or cost of borrowing) i = 365 x I (8.5) d A

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-18 Calculations of yields (cont.) –Example 7: What is the yield (rate of return) earned on a deposit of $ with a maturity value of $ 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 $975 93$ = = 7.65%

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-19 Holding period yield (HPY) HPY is the yield on securities sold in the secondary market prior to maturity –Short-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 investor  The investor’s cash flow position has unexpectedly changed and cash is needed  A better rate of return can be earned in an alternative investment

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-20 Holding period yield (HPY) (cont.) –The yield to maturity is the yield obtained by holding the security to maturity –The HPY is likely to be different from the yield to maturity –This 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

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-21 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 increases –Less 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 decreases

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-22 Chapter Organisation 8.1 Simple Interest –Simple interest accumulation –Present value –Yields –Holding period yield 8.2 Compound Interest –Compound interest accumulation (future value) –Present value –Present value of an annuity –Accumulated value of an annuity (future value) –Effective rates of interest 8.3 Summary

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger Compound Interest Compound interest (unlike simple interest) is paid on –The initial principal –Accumulated previous interest entitlements

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-24 Compound interest accumulation (future value) (cont.) 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 (8.6) Applying Equation 8.6 to Example 10 S = 5000 ( ) 3 = $

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-25 Compound interest 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 8.6 –If interest had accumulated monthly on the previous loan, then i = 0.15/12 = and n = 3  12 = 36

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-26 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 half-yearly for four years: I = % p.a. / 2 = 0.06 and: n = 4  2 = 8 periods so: S = 8000( ) 8 = 8000( ) = $

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-27 Present value with compound interest –The 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(8.7a) (1 + i) n A = S(1 + i) -n (8.7b)

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-28 Present value with compound interest (cont.) –Example 12: What is the present value of $ received at the end of three years if funds could presently be invested at 7.25% per annum, compounded annually? Using Equation 8.7a: A = S (1 + i) n = $ ( ) 3 = $18 500= $

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-29 Present value of an ordinary annuity An annuity is a series of periodic cash flows of the same amount –Ordinary annuity—series of periodic cash flows occur at end of each period (equation 8.8) A = C [ 1 – (1 + i) -n ](8.8) i

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-30 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 ten years, where the required rate of return is 6.00 per cent per annum, compounded quarterly, would be: C = $200 i = 6.00%/4 = 1.50% or n = 4 x 10 = 40 Therefore A = $200 [ 1 – ( ) -40 ] = $200 [ ] = $

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-31 Present value of an annuity due Annuity due—cash flows occur at the beginning of each period (Equation 8.9) A = C [ 1 – (1 + i) -n ] (1 + i)(8.9) i

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-32 Present value of an annuity due (cont.) –Example 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 cent per annum, compounded quarterly, would be: C = $200 i = 6.00%/4 = 1.50% or n = 4 x 10 = 40 Therefore A = $200 [ 1 – ( ) -40 ] = $200 [ ](1.015) = $

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-33 Present value of a Treasury bond –Equation 8.10 is used to calculate the price (or present value) of a Treasury bond A = C [ 1 – (1 + i) -n ] + S(1 + i) -n (8.10) i –Example 16 in the textbook illustrates the application of Equation 8.10

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-34 Accumulated value of an annuity (future value) The accumulated (or future) value of an annuity is given by Equation 8.11 S = C [ (1 + i) n - 1 ] (8.11) i

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-35 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? C = $200 i = 6.00%/12 = 0.50% or n = 3 x 12 = 36 Therefore S = $200 [ ( ) ] = $200 [ ] = $

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-36 Effective rates of interest The nominal rate of interest is the annual rate of interest, which does not take into account the frequency of compounding The effective rate of interest is the rate of interest after taking into account the frequency of compounding

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-37 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 ( ) 8 = $8000 (1.06) 8 = $

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-38 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 = $

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-39 Effective rates of interest (cont.) The formula for converting a nominal rate into an effective rate is i e = (1 + i/m ) m – 1

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-40 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) i e = ( /1) 1 – 1 = (1.10) 1 – 1 = 0.10 or 10% (b) i e = ( /2) 2 – 1 = (1.05) 2 – 1 = or 10.25% (c) i e = ( /12) 12 – 1 = ( ) 12 – 1 = or 10.47%

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 8-41 Chapter Organisation 8.1 Simple Interest –Simple interest accumulation –Present value –Yields –Holding period yield 8.2 Compound Interest –Compound interest accumulation (future value) –Present value –Present value of an annuity –Accumulated value of an annuity (future value) –Effective rates of interest 8.3 Summary

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger Summary Simple interest is interest paid on the original principal amount borrowed or invested Compound interest is paid on the initial principal plus accumulated previous interest entitlements The present value and future value of an investment or loan can be calculated using either simple or compound interest

Copyright  2007 McGraw-Hill Australia Pty Ltd PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney Slides prepared by Anthony Stanger 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 calculated Unlike the nominal rate of interest, which ignores the frequency of compounding, the effective rate of interest takes into account the frequency of compounding