Dick Schwanke Finite Math 111 Harford Community College Fall 2013

Transcription

1 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 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 End Sections of Chapter 6; Using Excel for Financial Calculations Page 1 of 6

2 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 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 the 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 6

3 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 Note: be careful with your arithmetic, e.g.: n deposits using (n-1) periods e.g. : multiple nested parenthesis 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 ; 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 6

4 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 ) 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 V (n-1) + V n V = P(1+i) -1 + P(1+i) 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 4% 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 6

5 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 5.1% 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 6

6 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 ; website) 17 Notes of the Day Session #3 - got to slide #8 problems Resume this same set of session #3 slides at session #4 Session #4 do slide #12 problems Session #4 discuss discount and premium pricing for bonds 18 End Sections of Chapter 6; Using Excel for Financial Calculations Page 6 of 6