Time Value of Money. Outline Meaning of Time Value Concept of Future Value and Compounding (FV)...
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Transcript of Time Value of Money. Outline Meaning of Time Value Concept of Future Value and Compounding (FV)...
Time Value of Money
Outline
Meaning of Time Value Concept of Future Value and Compounding (FV) Concept of Present Value and Discounting (PV) Frequency of Compounding Present Value versus Future Value Determining the Interest rate (r) Determining the Time Period (n) Future Value and Present Value of Multiple Cash Flows Annuities and Perpetuities
Time Value of Money
Basic Problem:– How to determine value today of cash flows that are expected in the
future? Time value of money refers to the fact that a dollar in hand today
is worth more than a dollar promised at some time in the future Which would you rather have -- $1,000 today $1,000 today or $1,000 in 5 $1,000 in 5
years?years? Obviously, $1,000 today$1,000 today. Money received sooner rather than later allows one to use the
funds for investment or consumption purposes. This concept is referred to as the TIME VALUE OF MONEYTIME VALUE OF MONEY!!
TIMETIME allows one the opportunity to postpone consumption and earn INTERESTINTEREST.
Future Value and Compounding
Future value refers to the amount of money an investment will grow to over some length of time at some given interest rate
To determine the future value of a single cash flows, we need: present value of the cash flow (PV) interest rate (r), and time period (n)
FVn = PV0 × (1 + r)n
Future Value Interest Factor at ‘r’ rate of interest for ‘n’ time periods
Examples on computation of future value of a single cash flow
If you invested $2,000 today in an account that $2,000 today in an account that pays 6pays 6% interest, with interest compounded annually, how much will be in the account at the end of two years if there are no withdrawals?
Future Value (Graphic)Future Value (Graphic)
0 1 2
$2,000$2,000
FVFV
6%
FVFV11 = PVPV (1+r)n
= $2,000$2,000 (1.06)2
= $2,247.20$2,247.20
Future Value (Formula)Future Value (Formula)
FV = future value, a value at some future point in timePV = present value, a value today which is usually designated as time 0r = rate of interest per compounding period n = number of compounding periods
Calculator Keystrokes: 1.06 (2nd yx) 2 x 2000 =
John wants to know how large his $5,000$5,000 deposit will become at an annual compound interest rate of 8% at the end of 5 years5 years.
Future Value (Example)Future Value (Example)
0 1 2 3 4 55
$5,000$5,000
FVFV55
8%
Future Value SolutionFuture Value Solution
Calculation based on general formula: FVFVnn = PV (1+r)n
FVFV55 = $5,000 (1+ 0.08)5
= $7,346.64$7,346.64
Calculator keystrokes: 1.08 2nd yx x 5000 =
Present Value and Discounting
The current value of future cash flows discounted at the appropriate discount rate over some length of time period
Discounting is the process of translating a future value or a set of future cash flows into a present value.
To compute present value of a single cash flow, we need: Future value of the cash flow (FV) Interest rate (r) and Time Period (n)
PV0 = FVn / (1 + r)n
PVIF (r,n) Examples
Assume that you need to have exactly $4,000 saved 10 years from now. How much must you deposit today in an account that pays 6% interest, compounded annually, so that you reach your goal of $4,000?
0 5 5 10
$4,000$4,000
6%
PVPV00
Present Value (Graphic)Present Value (Graphic)
PV0 = FV / (1+r)10
= $4,000 / (1.06)10
= $2,233.58
Present Value (Formula)Present Value (Formula)
0 5 5 10
$4,000$4,000
6%
PVPV00
Joann needs to know how large of a deposit to make today so that the money will grow to $2,500 in 5 years. Assume today’s deposit will grow at a compound rate of 4% annually.
Present Value ExamplePresent Value Example
0 1 2 3 4 55
$2,500$2,500PVPV00
4%
Calculation based on general formula: PVPV00 = FVFVnn / (1+r)n
PVPV00 = $2,500/(1.04)$2,500/(1.04)55
= $2,054.81
Calculator keystrokes: 1.04 2nd yx 5 = 2nd 1/x X 2500 =
Present Value SolutionPresent Value Solution
General Formula:
FVn = PVPV00(1 + [r/m])mn
n: Number of Years
m: Compounding Periods per Year
r: Annual Interest Rate
FVn,m: FV at the end of Year n
PVPV00: PV of the Cash Flow today
Frequency of CompoundingFrequency of Compounding
Frequency of Compounding Example
Suppose you deposit $1,000 in an account that pays 12% interest, compounded quarterly. How much will be in the account after eight years if there are no withdrawals?
PV = $1,000
r = 12%/4 = 3% per quarter
n = 8 x 4 = 32 quarters
Solution based on formula:
FV= PV (1 + r)n
= 1,000(1.03)32
= 2,575.10
Calculator Keystrokes:
1.03 2nd yx 32 X 1000 =
Present Value versus Future Value
Present value factors are reciprocals of future value factors
Interest rates and future value are positively related Interest rates and present value are negatively related Time period and future value are positively related Time period and present value are negatively related
Determining the Interest Rate (r)
At what rate of interest should we invest our money today to get a desired amount of money after a certain number of years?
Essentially, we are trying to determine the interest rate given present value (PV), future value (FV), and time period (n)
Examples The rate which money can be doubled/tripled
Determining the Time Period (n)
For how long should we invest money today to get a desired amount of money in the future at a given rate of interest
Determining the time period (n) for which a current amount (PV) needs to be invested to get a certain future value (FV) given a rate of interest (r).
Examples The time period needed to double/triple our current
investment
Future Value of Multiple Uneven Cash Flows
Compute the future value of each single cash flow using future value formula and add them up over all the cash flows
Example
Present Value of Multiple Uneven Cash Flows
Compute the present value of each single cash flow using present value formula and add them over all the cash flows
Examples
Annuities
A series of level/even/equal sized cash flows that occur at the end of each time period for a fixed time period
Examples of Annuities: Car Loans House Mortgages Insurance Policies Some Lotteries Retirement Money
Present Value of an Annuity– Examples
Computing Cash Flow per period in annuity– Examples
Perpetuities
A series of level/even/equal sized cash flows that occur at the end of each period for an infinite time period
Examples of Perpetuities: Consoles issued by British Government Preferred Stock
Present Value of a Perpetuity
Effective Annual Rate
Compounding other than annual