Basic Finance The Time Value of Money 7 An introduction to financial institutions, investments & Management Eleventh Edition

Time Value What is Time Value of Money?

Time Value Process of expressing the present in the future (compounding) the future in the present (discounting)

Time Value Compounding Future value of a dollar Process by which interest is paid on interest that was previously earned. Future value of a dollar Amount to which a single payment will grow at some rate of interest.

Time Value Payments are either a single payment a series of equal payments (an annuity)

Time Value Time value of money problems may be solved by using Interest tables Financial calculators Software

Variables for Time Value of Money Problems PV = present value FV = future value PMT = annual payment N = number of time periods I = interest rate per period

Financial Calculators Express the cash inputs (PV, FV, and PMT) as cash inflows and cash outflows At least one of the cash variables must be an inflow (+) an outflow (-)

Future Value P0(1 + i)n = Pn Future value of $1 takes a single payment in the present into the future General equation for the future value of $1: P0(1 + i)n = Pn

Future Value Illustrated PV = -100 I = 5 N = 20 PMT = 0 FV = ? = 265.33

Greater Terminal Values Higher interest rates Longer time periods Result in greater terminal values

Greater Terminal Values

Present Value Present value of $1 brings a single payment in the future back to the present General equation for the present value of $1: P0 = Pn (1+i)n

Present Value Present Value Discounting Current value of a dollar to be received in the future. Discounting Process of determining present value.

Present Value Illustrated FV = 100 I = 6 N = 5 PMT = 0 PV = ? = -74.73

Lower Present Values Higher interest rates Longer time periods Result in lower present values

Lower Present Values

Annuities What is an Annuity?

Annuities - Future Sum Annuity: Series of equal, annual payments. Future value of an annuity: Amount to which a series of equal payments will grow at some rate of interest. Annuity Due: Annuity in which the payments are made at the beginning of the time period. Ordinary Annuity: Annuity in which the payments are made at the end of the time period.

Annuities - Future Sum Future sum of annuity takes a series of payments into the future Payments may be made at the end of each time period (ordinary annuity) at the beginning of each time period (annuity due)

FV Time Lines Ordinary annuity Year Payment - $100 100 100 0 1 2 3

FV Time Lines Annuity due Year Payment $100 100 100 - 0 1 2 3

Future Value of an Ordinary Annuity Illustrated PV = 0 PMT = -100 I = 5 N = 3 FV = ? = 315.25

Greater Terminal Values Higher interest rates Longer time periods Result in greater terminal values

Greater Terminal Values

Present Value of an Annuity The present value of an annuity brings a series of payments in the future back to the present.

Present Value of an Ordinary Annuity Illustrated FV = 0 PMT = 100 I = 6 N = 3 PV = ? = -267.30

Annuities - Present Value Higher interest rates result in lower present values But longer time periods increases the present value (because more payments are received)

Annuities - Present Value

Time Value Illustrations The following is a series of problems or questions that use the time value of money.

Illustration 1 You deposit $1,000 in an account at the end of each year for twenty years. What is the total amount in the account if you earn 6 percent annually?

Future Value of an Ordinary Annuity The unknown: FV The givens: PV = 0 PMT = -1,000 N = 20 I = 6 The answer: 36,786 ($36,786)

Interpretation For an annual cash payment of $1,000, you will have $36,786 after twenty years Of the $36,786 $20,000 is the total cash outflow $16,786 is the earned interest

Illustration 2 What is the present value of (or required cash outflow to purchase) an ordinary annuity of $1,000 for twenty years, if the rate of interest is 6 percent?

Present Value of an Annuity The unknown: PV The givens: FV = 0 PMT = 1,000 N = 20 I = 6 The answer: -11,470 ($11,470)

Interpretation For a present payment of $11,470, the individual will annually receive $1,000 for the next twenty years The $11,470 is an immediate cash outflow The $1,000 annual payment to be received is a cash inflow

Illustration 3 You buy a stock for $10 and expect the price to increase 9 percent annually. After 10 years, what is the anticipated price of the stock?

Future Value of $1 The unknown: FV The givens: PV = -10 PMT = 0 N = 10 The answer: 23.67 ($23.67)

Interpretation A $10 stock will be worth $23.67 after 10 years if its price grows 9% annually.

Illustration 4 What is the cost of a stock that was sold for $23.67, held for 10 years and whose value appreciated 9 percent annually?

Present Value of $1 The unknown: PV The givens: FV = 23.67 PMT = 0 The answer: -10($10)

Interpretation $23.67 received after ten years is worth $10 today if the rate of return is 9 percent.

Interpretation of Future and Present Values These two problems are the same: In the first case the $10 is compounded into its future value ($23.67) In the second case the future value ($23.67) is discounted back to its present value ($10)

Illustration 5 A stock was purchased for $10 and sold for $23.67 after 10 years. What was the return?

Future Determination of the Interest Rate The unknown: I The givens: PV = -10 PMT = 0 N = 0 FV = 23.67 The answer: 9(9%)

Interpretation The yield on a $10 investment that was sold after 10 years for $23.67 is 9%.

Illustration 6 If an investment pay $50 a year for 10 years and repays $1,000 after 10 years, what is this investment worth today if you can earn 6 percent?

Determination of Present Value The unknown: PV The givens: FV = 1,000 PMT = 50 I = 6 N = 10 The answer: -926($926)

Interpretation If you collect $50 a year for 10 years and receive $1,000 after 10 years, those cash inflows are currently worth $926 at 6 percent.

Illustration 7 Time value is used to determine a loan repayment schedule such as a mortgage.

Loan Repayment Schedule Amount borrowed (PV) = 80,000 Interest rate (I) = 8(8%) Term of the loan (N) = 25 years No future value since loan is repaid Amount of the annual payment = 7,494.30 ($7,494.30)

Loan Repayment Schedule Principal Balance Pmnt Interest Repayment Owed 1 $6,400.00 $1,094.15 $78,905.85 2 6,312.47 1,181.68 77,724.17 . 25 555.13 6,939.17 .00

Illustration 8 You have $470,000 and spend $96,000 a year. If you earn 8% annually, how long will your funds last?

Determination of Number of Years The unknown: N The givens: PV = 470,000 I = 8 FV = 0 PMT = -96,000 The answer: 6.46 (6.5 years)

Interpretation If you have $470,000 and earn 8 percent annually, you can spend $96,000 per year for approximately 6 years and 6 months.

Non-annual Compounding More than one interest payment a year More frequent compounding

Non-annual Compounding Multiply number of years by frequency of compounding Divide interest rate by frequency of compounding

Non-annual Compounding Illustration What is the future value of $100 that pays 8 percent compounded quarterly for five years?

Determination of Future Value FV = $100 (1+0.08/4) 5x4 = $100 (1+0.02)20 = $148.59

Determination of Future Value PV = -100 I = 8/4 = 2 N = 5x4 = 20 PMT = 0 FV = The answer: 148.59 ($148.59)

Interpretation $100 grows to $148.59 at 8 percent in five years with interest compounded quarterly.

Periods less than One Year Same variables as in all time value problems except N < 1.

Illustration What is the return on an investment that costs $98,543 and pays $100,000 after 45 days?

Determination of Return The unknown: I The givens: PV = -98,543 N = 0.1233 FV = 100,000 PMT = 0 The answer: 12.64 (12.64%)

$98,543 invested for 45 days grows to $100,000 at 12.64 percent. Interpretation $98,543 invested for 45 days grows to $100,000 at 12.64 percent.