Annuities and Amortization Finite Mathematics 111 Dick Schwanke Session #3 1 In the Previous Two Sessions Calculating Simple Interest Finding the Amount Owed Computing Discounted Loans Quick Review of Exponents and Logarithms Calculating Compound Interest Comparing different Compounding Periods Computing the Present Value of A Dollars About Zero-Coupon Bonds 2 Time Value of Money Problems Exercise 6.2, pp 311-315 Recall P = A(1+(r/n)) -nt or A = P(1+(r/n)) nt Present value is the P dollars (at the start) Future value is the A dollars (at the end) Find the present value: #20, 22, 26 Find the effective rate of interest: #34, 36 Find investment amounts: #48, 50 Find value of zero-coupon bond: #86, 88 Discuss the approach to question #97 with four banks, summary data on p 316 3 End Sections of Chapter 6; Using Excel for Financial Calculations Page 1 of 10
Annuities Definition: Annuity is a sequence of equal periodic deposits Investing money in small amounts at periodic intervals is typical method Annual life insurance premiums (whole life) Monthly savings deposits at the bank perhaps for a college fund or house down payment Installment loan payments perhaps for a car 401(k) dollar averaging in the stock market Funding for a sinking fund payback Amount of annuity is the sum of all deposits plus all interest accumulated 4 Finding the amount of an Annuity Using compound interest A = P(1+(r/n)) nt Where t is the number of years P is the principal invested r is the annual interest rate n is the times per year it is compounded First deposit is worth A 1 = P(1+i) n Second deposit worth A 2 = P(1+i) (n-1) Third deposit worth A 3 = P(1+i) (n-2) until final deposit is P without interest 5 Finding the amount of an Annuity Add all those deposits and interest Total A = A 1 + A 2 + A 3 +... + A (n-1) + A n A= P(1+i) n + P(1+i) (n-1) + P(1+i) (n-2) + P(1+i) (n-3) +... + P(1+i) + P Factor out the P and reverse order Total A = P[ 1+ (1+i) +... + (1+i) (n-1) ] The (1+i) terms geometric sequence = [1-(1+i) n ] / [1-(1+i)] (from Appendix 4) 6 End Sections of Chapter 6; Using Excel for Financial Calculations Page 2 of 10
Finding the amount of an Annuity Simply the sequence to get the value of the annuity after n deposits is A = P [ ((1+i) n -1)/ i ] Note: when the n th deposit is made the first deposit has earned interest for n-1 compounding periods (i.e. be careful with your arithmetic) 7 Working a few examples: page 326 Saving for a house (or one year at Harvard) #28 Funding a Keogh Plan for retirement #30 Sinking Fund to pay off bonds #32 (also see using Excel in examples #10-11, on pages 320-323; website) Value of an IRAs #36 So you want to be a millionaire #40 8 Finding Present Value of an Annuity (recall) Present Value is the amount of money needed now, to obtain an amount A in the future. Present Value of an annuity is the sum of present values of the withdraws. Alternately: Present Value of an annuity is the money need now, so that if invested at i percent, n equal dollar amounts can be withdrawn to zero left 9 End Sections of Chapter 6; Using Excel for Financial Calculations Page 3 of 10
Finding Present Value of an Annuity Similar to earlier method Recall we defined a -n = 1/ a n if a 0 Sum the present values (V n n) for all of the planned withdraws First withdraw is worth V 1 = P(1+i) -1 Second withdraw is worth V 2 = P(1+i) -2 until final (n th ) withdraw is V 2 = P(1+i) -n 10 Finding Present Value of an Annuity Add Present Values for all withdraws Total V = V 1 + V 2 + V 3 +... + V (n-1) + V n V = P(1+i) -1 + P(1+i) -2 +...+ P(1+i) -n Factor out the P(1+i) -n and substitute the geometric sequence (as before) to get Present Value of the Annuity is V = P[1-(1+i) -n ]/ i] 11 Work some examples Retirement Account, page 342, #16 How much to invest today? At age 65, can expect to live 25 years Invested at 10% PA, compound monthly Need to guarantee $300 / month Corporate Leasing page 342, #46 Which piece is the better investment? Model A costs $50K, saves $12K/year in labor, has a useful life of 10 years Model B costs $42K, saves $10K/year in labor costs, has useful life of 8 years Time value of money is 10% per annum 12 End Sections of Chapter 6; Using Excel for Financial Calculations Page 4 of 10
Amortization, or who is Mort? A fixed interest rate loan is said to be amortized, if both principal and interest are paid by sequence of equal payments made over equal periods of time Take Present Value of the Annuity V = P[1-(1+i) -n ] / i], solve for Payment Payment required to pay off a loan of V dollars, borrowed for n payment periods, at i interest rate per period is P = V[ i / (1-(1+i) -n ] 13 Work an example Mortgage Payments, page 343 #32 What will the monthly payment be? Summer home will cost $180,000 $60,000 in present home equity for down payment Finance 25 years at 10.2% compounded monthly BTY, the Refinancing Mortgage #30 would make a great test question Note: we will return to mortgage payments with Excel later 14 Pricing Bonds Definition: Face Amount (or Face Value or Par Value or Denomination) of a bond is the amount paid to the bond holder at maturity. Note: often this is the amount paid by the bondholder at original issue Nominal Interest (or Coupon Rate) is the contractual interest paid on the bond Note: normally quoted as annual percentage rate of the face amount Note: conventionally paid semi-annually 15 End Sections of Chapter 6; Using Excel for Financial Calculations Page 5 of 10
More About Bond Pricing Prices of Corporate Bonds fluctuate Reasons might include Trading at a Premium means price is higher than face amount Trading at a Discount means price is lower than face amount 16 Still More About Bond Pricing True Yield means the combination of trading price and interest rate To calculate the true interest rate = sum an annuity of semiannual interest payments (for now until maturity date) plus the present value of a single future payment at maturity Work example, page 344, #48 (also see using Excel in examples #11-12, on pages 337-339; website) 17 Quiz on A-3, 6.1 and 6.2 Individual Effort Calculator use is okay Open book Open notes 20 minute limit 18 End Sections of Chapter 6; Using Excel for Financial Calculations Page 6 of 10
Using Technology to Assist in Financial Calculations Calculators: TI-83 and HP-12C Software: Microsoft Excel 2007/2010 Session #3 of Finite Mathematics 19 TI-83 / 84 Graphing Calculator Section 5.5 of textbook discusses recursive sequences TI-83 can do these: see PDF file of 6.5 linked from course lecture page http://www.harford.edu/faculty/dschwank/finite111 /Data_sets/ch06_TI-83.pdf This material is not going to be tested 20 TI-83 / 84 Graphing Calculator Details of more financial at TI website, see chapter 14 of guidebook http://education.ti.com/downloads/guidebooks/gra phing/83p/83m$book-eng.pdf p gp Functions of interest APPS ENTER select FINANCE from menu select TVM (time value of money) enter, input all known values, with (-) for PMT, move cursor to unknown value press ALPHA SOLVE 21 End Sections of Chapter 6; Using Excel for Financial Calculations Page 7 of 10
HP-12c Calculator Storage registers specifically for n, i, PV, PMT, and FV Know any couple, can get all others Functions of multiply by 12 and divide by 12 direct to register contents HP website has detailed instructions, e.g.: http://h10025.www1.hp.com/ewfrf/wc/document?docname=c01800821&cc=us&lc=en&dlc=en& product=81575 22 About Microsoft Excel Is a spreadsheet with multiple pages Columns by letter Rows by number Cells by letter-number location Helps keep numbers organized Power is in the built-in functions To use a function Select a cell for the answer Type function / formula, beginning with = Can use numbers or cell locations 23 About Microsoft Excel Useful functions (for chapter 5) begin at Formulas then select Financial See Function Description handout Eight functions of particular use to us PV Returns the present value of an investment RATE Returns the interest rate per period of an annuity 24 End Sections of Chapter 6; Using Excel for Financial Calculations Page 8 of 10
Functions in Microsoft Excel Eight functions of particular use FV Returns the future value of an investment based on periodic constant payments and a constant interest rate IPMT Returns the interest payment for an investment for a given period NOMINAL Returns the annual nominal interest rate 25 Functions in Microsoft Excel Eight functions of particular use NPER Returns the number of periods for an investment PMT Returns the periodic payment for an annuity PPMT Returns the payment on the principal for an investment for a given period 26 Examples of Microsoft Excel Funding a loan: Example 11, page 323 Start with textbook provided data Use PV function Use FV function Expand to full payment schedule Home Mortgage Payments: #7, p. 335 Pricing a bond: Example 12, page 338 Start with textbook provided data Use PV function Expand to full payment schedule 27 End Sections of Chapter 6; Using Excel for Financial Calculations Page 9 of 10
Problems using Microsoft Excel Student technology project #1: Page 327, #31, sinking fund payment Page 344, #49, corporate bond Page P 351, #29, house mortgage, make a 25 year / 300 month schedule showing: Date of each monthly payment Interest paid that month Interest paid year to date (reset in Jan.) Current equity in home each month 28 Concluding Chapter 6 Review on bottom of page 349 Lists types of problems to solve Gives number of related review exercise Important formulas top of page 350 Root of all formulas (p. 295) I = Prt Amount (p. 296) A = P + I = P(1+rt) Five other formulas In class quiz: open book / open notes 29 Notes of the Day 1. Excel examples to be posted Page 322, #10-11 Find Annuity Amount Page 339, #11-12 Find Present Value Page 333, #6-7: Home Mortgage 2. Problem numbers for Excel Technology Assignment #1 are: Page 327, #31 Sinking Fund Payment Page 344, #49 Corporate Bonds Page 351, #29 plus Home Mortgage 3. Excel project is due at class #5 30 End Sections of Chapter 6; Using Excel for Financial Calculations Page 10 of 10