# Topics. Chapter 5. Future Value. Future Value - Compounding. Time Value of Money. 0 r = 5% 1

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1 Chapter 5 Time Value of Money Topics 1. Future Value of a Lump Sum 2. Present Value of a Lump Sum 3. Future Value of Cash Flow Streams 4. Present Value of Cash Flow Streams 5. Perpetuities 6. Uneven Series of Cash Flows 7. Other Compounding Periods 8. Effective Annual Rates (EAR) 9. Amortized Loans Future Value - Compounding Future Value - What is something worth in the future at a rate of interest. ex. \$1,000 deposited into a bank today and left in the account for 1 year earning interest at 5% Future Value 0 r = 5% 1 PV 0 = \$1,000 FV 1 = FV1 = PV + Interest = \$1,000 + Interest ( r = \$1,000 *.05) = \$1,000 + \$50 = \$1050 1

2 How about in 2 years? Compounding Compounding = interest on interest 0 r = 5% 1 2 PV = \$1,000 \$1,050 FV2 = \$52.50 \$1, \$1, FV 1 = PV + ( PV X r ) = PV ( 1 + r ) FV 2 = PV ( 1 + r ) ( 1 + r ) = PV ( 1 + r ) 2 General form of equation = FV n = PV ( 1 + r ) n Future Values: General Formula FV = PV(1 + r) n FV Practice What is the future value (FV) of \$100 if it is invested 1 year at 10%? FV = future value PV = present value r = period interest rate, expressed as a decimal n = number of periods FV 1 = PV ( 1 + r ) 1 = \$100 (1.10) 1 = \$110 2

3 FV Practice FV of \$100 at 10% What is the future value (FV) of \$100 if it is invested 2 years at 10%? FV 2 = PV ( 1 + r ) 2 = \$100 (1.10) 2 = \$121 Solving for FV What is the FV of \$100 if invested for 3 years earning 10% interest? Solve with the equation FV n = PV ( 1 + r ) n = \$100 ( 1.10 ) 3 = \$ Solving for FV Solve with the calculator PV = 100 N = 3 I/Y = 10% FV =

4 Calculator Rules 1. Clear Register 2. Set to 1 period per year 3. Set proper mode (Begin or End) Calculator Keys Texas Instruments BA-II Plus FV = future value PV = present value I/Y = period interest rate N = number of periods Texas Instruments BA-II Plus Using The TI BAII+ Calculator The highlighted row of keys is the TVM register. Solve for FV, PV, PMT, N, or I/Y Focus on 3rd Row of keys N I/Y PV PMT FV Number of periods Interest rate per period Present value Payment per period Future value CLR TVM: Clears all keys in the TVM register 4

5 Hewlett-Packard 10B Calculator Keys FV = future value PV = present value I/YR = period interest rate N = number of periods Practice What is the future value (FV) of \$1,000 if it is invested 20 years at 5%? PV = \$1,000 I/Y = 5% N = 20 FV 20 = \$2, Practice Present Value - Discounting What is the future value (FV) of \$1,000 if it is invested 20 years at 10%? PV = \$1,000 I/Y = 10% N = 20 0 r = 10% 1 PV 0 = FV 1 = \$1,320 FV 20 = \$6,

6 Solve with the equation Solve with the calculator PV 0 = FV n / ( 1 + r ) n = \$1,320 / (1.10) 1 = \$1,200 PV = \$1,320 I/Y = 10% N = 1 FV 1 = \$1, Problem Answer Oscar Wright of Carson City, Nevada, while combing through his great-grandmothers trunk of remembrances, found a State of Nevada Bond that was issued the day Nevada became the 36th state on October 31, This State of Nevada bond has a face value of \$100 and a stated interest rate of 22%. Why do you think the interest rate is so high? If the State of Nevada agrees to pay Oscar the face value of this bond plus interest through October 31 of last year, how much will he receive? PV = \$100 I/Y = 22% N = 147 FV 147 = \$4.953 x

7 Doubling Money How long will it take a \$1,000 investment to double if it is invested in an account paying 8% interest? PV = -\$1,000 FV = +\$2,000 I/Y = 8% N = yrs Doubling Money The Rule of 72. A method to approximate when solving for n or r. n = 72 / r or r = 72 / n Problem A recent advertisement in the financial section of a magazine carried the following claim: "Invest your money with us at 6 percent, compounded annually, and we guarantee to double your money sooner than you imagine." Ignoring taxes, approximately how long would it take to double your money at a nominal rate of 6 percent, compounded annually? How long if the rate is 3%? How much if the rate is 18%? PV = -\$1 FV = +\$2 I/Y = 6% N = yrs 7

8 PV = -\$1 FV = +\$2 I/Y = 3% N = yrs PV = -\$1 FV = +\$2 I/Y = 18% N = yrs Problem Answer If you had invested \$10,000 in Fidelity's Magellan fund when Peter Lynch became the fund manager on 5/31/77 your money would have grown to \$233, by the time Lynch retired on 5/30/92. What is the annual rate of return on your investment? PV = -\$10,000 FV = +\$233, N = 15 I/Y = 23.36% 8

9 Annuities An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods. Examples of Annuities Include: Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings Types of Annuities An Annuity represents a series of equal payments (or receipts) occurring over a finite period of time. Ordinary Annuity: Payments or receipts occur at the end of each period. Annuity Due: Payments or receipts occur at the beginning of each period. Future Value of an Ordinary Annuity A payment of \$100/year for 3 years at 10%, with the first payment one year from today \$100 \$100 \$100 FV 3 = FV of an Ordinary Annuity Find the FV 3 of each \$100 at 10% \$100 \$100 \$100 \$110 \$121 \$331 9

10 Equation FV of an Ordinary Annuity FV = PMT * ( 1 + r )n - 1 r = \$100 * ( 1.10 ) = \$100 * = \$ FV of an Ordinary Annuity Calculator PMT=100 N=3 I/Y=10% FV = \$ Future Value of an Annuity Due A payment of \$100/year for 3 years at 10%, with the first payment today \$100 \$100 \$100 FV 3 = FV of an Annuity Due Find the FV 3 of each \$100 at 10% \$100 \$100 \$100 \$ \$ \$ \$

11 FV of an Annuity Due FV of an Annuity Due Equation FV = PMT X ( 1 + r )n - 1 X ( 1 + r ) r = \$100 X ( 1.10 ) 3-1X (1.10).10 = \$100 X = \$ Calculator PMT=100 N=3 I/Y=10% Beg Mode FV = \$ Present Value of an Ordinary Annuity A payment of \$100/year for 3 years at 10%, with the first payment one year from today \$100 \$100 \$100 PV 0 = PV of an Ordinary Annuity Find the PV 0 of each \$100 at 10% \$100 \$100 \$100 \$ \$ \$ \$

12 PV of an Ordinary Annuity Equation PV = PMT x [ 1 1 / ( 1 + r ) n ] r = PMT x [ 1 1 / ( 1.10 ) 3 ].10 = \$100 x = \$ PV of an Ordinary Annuity Calculator PMT=100 N=3 I/Y=10% PV = \$ Present Value of an Annuity Due A payment of \$100/year for 3 years at 10%, with the first payment today \$100 \$100 \$100 PV 0 = PV of an Annuity Due Find the PV 0 of each \$100 at 10% \$100 \$100 \$100 \$ \$ \$

13 PV of an Annuity Due PV of an Annuity Due Equation PV = PMT x [ 1 1 / ( 1 + r ) n ] x ( 1 + r ) r = 100 x [ 1 1 / ( 1.10 ) 3 ] x (1.10).10 = \$100 x x 1.10 = \$ Calculator BGN PMT=100 N=3 I/Y=10% PV = \$ Problem Problem Assume you start investing for your retirement by opening a Roth IRA and depositing money into a mutual fund on a yearly basis. These deposits consist of \$3,000 per year and are in the form of a 40-year annuity due. Assume this fund earns 11% per year over these 40 years. How much money will be in your retirement account the day you retire? When you retire in year 40 you move your IRA nest egg into a safer account earning 5% per year. Assume you wish to withdraw an equal annual amount for 25 years as an ordinary annuity until all the money is gone. How much can you withdrawal every year? You decide that when you retire 50 years from now, you will need \$200,000 a year to live comfortably for the next 20 years (you receive these payments starting 51 years from now). If money is deposited into an account with a contract rate of interest of 10 percent, how much will you need to save every year (deposit in an account) for the next 50 years (assume you make 50 payments). Assume the first savings deposit starts today and money remains in this account paying the same interest rate until all funds are paid out. 13

14 Perpetuities Infinite series of payments An annuity that goes on forever PV = PMT r Example A bond is contracted to make a \$90 payment per year, the first payment is one year from today, there is no maturity date, and the market rate of interest is 10%. What is the PV? PV = PMT = \$90 = \$900 r.10 Example , \$ : 0.77 : \$ Present Value of a Growing Perpetuity PV = CF r g 14

15 Problem The Andre Aggasi Foundation wishes to endow the Las Vegas Animal Shelter with a annual gift that will grow with inflation. The amount given the first year is \$10,000 and inflation is expected to be 3% per year. If the discount rate is 10%, what is the present value of this endowment? Problem If instead inflation is estimated to be 5% per year and the discount rate remains at 10%, what is the present value of this endowment? Uneven Series of Cash Flows r = 10% \$100 \$500 \$500 \$500 \$500 \$750 Uneven Series of Cash Flows r = 10% \$100 PV 0 =? PV 0 =\$

16 Uneven Series of Cash Flows r = 10% \$500 \$500 \$500 \$500 PV 1 =\$1, Uneven Series of Cash Flows r = 10% \$750 PV 0 = PV 0 =\$1, Uneven Series of Cash Flows Complex TVM Total PV 0 = \$ \$1, \$ = \$1, Assuming a discount rate of 9%, what is the value of these cash flows in year 6? Year 7?, Year 10?, Year 11? \$6M \$6M \$6M \$6M 16

17 Value in Year װ \$6M \$6M \$6M \$6M END PMT=\$6M r = 9% n = 4 PV 6 = \$19.44M Value in Year װ \$6M \$6M \$6M \$6M BGN PMT=\$6M r = 9% n = 4 PV 7 = \$21.19M Value in Year װ \$6M \$6M \$6M \$6M END PMT=\$6M r = 9% n = 4 FV 10 = \$27.44M Value in Year װ \$6M \$6M \$6M \$6M BGN PMT=\$6M r = 9% n = 4 FV 11 = \$29.91M 17

18 What is the value today? 1. FV 6 =\$19.44M; r=9%; n=6; PV 0 =\$ FV 7 =\$21.19M; r=9%; n=7; PV 0 =\$ FV 10 =\$27.44M; r=9%; n=10; PV 0 =\$ FV 11 =\$29.91M; r=9%; n=11; PV 0 =\$11.59 Compounding Other then Yearly 10% Annual Compounding 0 1 \$100 \$100(1.10) \$110 Compounding Other then Yearly 10% Semi-Annual Compounding 0 6 mn 1 \$100 \$100(1.05) \$105(1.05) \$105 \$ How to Solve? Formula FV n = PV x ( 1 + r/m ) mxn FV 1 = \$100 x ( /2 ) 2x1 FV 1 = \$

19 How to Solve? Calculator Note: r = ( r / m ), n = (n * m) PV = 100 r = 10% / 2 = 5% n = 1 * 2 = 2 FV = \$ Problem If you deposit \$10,000 in this account and earn 11% compounded quarterly, what is the future value in 4½ years? PV 0 =\$10,000 r=11%/4 = 2.75% N=4.5x4=18 FV 4.5 = \$16, Effective Annual Rates (EAR) EAR = ( 1 + r/m ) m -1 r = Annual Percentage Rate (APR) m = number of periods in a year Example If the APR is 10% what is the EAR for these different compounding periods? annual (m=1) = 10.00% semi-annual (m=2) = 10.25% quarterly (m=4) = % monthly (m=12) = % daily (m=365) = % 19

20 Problem You see an advertisement in Money magazine from an investment company that offers an account paying a nominal interest rate of 7%. What is the effective annual rate (EAR) of 7% compounded semi-annually? What is the EAR of 7% compounded monthly? Amortized Loans Loan that is paid off in equal payments over a set period of time. Define: PV = loan amount PMT 1 = PMT 2 =... = PMT t n = number of payments r = given FV t = 0 Example Amortization Table Home mortgage: 30 year, \$300,000, 6%, paid monthly Mn 1 Beg. Balance \$300, Payment (P&I) \$1, Principal \$ Interest \$1, End Balance \$299, What is the monthly payment? \$1, װ \$1, װ \$ װ װ 2 װ \$299, \$1, \$299, PV 0 = \$300,000; n = 30x12; r = 6%/ \$3, \$1, \$1, \$1, \$1, \$1, \$17.85 \$8.95 \$1, PMT = \$1,

21 30-year vs. 15-year Mortgage PMT 30 = \$1, month 1 I = 1, P = month 2 I = 1, P = PMT 15 = \$2, month 1 I = 1, P = 1, month 2 I = 1, P = 1, Mortgage Balance 30-year vs. 15-year Mortgage Balance in 5 years Bal 30 =\$279, Bal 15 =\$228, Balance in 10 years Bal 30 =\$251, Bal 15 =\$130, Balance in 15 years Bal 30 =\$213, Bal 15 =\$0 Does this make sense? Difference in balances in 5 years \$279, \$228, = \$51, Differences in payment \$1, \$2, = - \$ n = 5, r = 6, PMT = , m = 12 FV = \$51, Problem You purchased your house 6 years ago using a 30-year, \$200,000 mortgage with a contractual rate of 7% that calls for monthly payments. What is your monthly payment? What is the loan balance today after making 6 years worth of monthly payments? How much equity do you have in your house if appreciation has averaged 3% per year? 21

22 Amortization table 5-year, \$100,000 loan, 8% rate, annual payments Amortization table 5-year, \$100,000 loan, 8% rate, annual payments Year 1 Beg Balance Payment Principal Interest End Balance Year 1 Beg Balance \$100,000 Payment 25, Principal 17, Interest 8, End Balance 82, , , , , , , , , , , , , , , , , , , , \$0 22

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