Variable key
Where: |
FV |
= Future value of an investment |
PV |
= Present value of an investment (the lump sum) |
r |
= Return or interest rate per period (typically 1 year) |
n |
= Number of periods (typically years) that the lump sum is invested |
PMT |
= Payment amount |
CFn |
= Cash flow steam number |
m |
= # of times per year r compounds |
Equation guide
Future value of a lump sum: |
FV = PV x (1 + r)n |
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Future-value factor (FVF) table |
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Excel future value formula FV= |
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Compound interest. Formula for simple interest is PV + (n x (PV x r)) |
Future Value of an Ordinary Annuity |
FV = PMT x { [ ( 1 + r )n - 1 ] / r} |
Future Value of an Annuity Due |
FV (annuity due) = PMT x { [ ( 1 + r)n -1 ] / r } x (1 + r) |
Future Value of Cash Flow Streams |
FV = CF1 x (1 +r)n-1 + CF2 x (1 + r)n-2 + ... + CFn x (1 + r)n-n |
Present value of a lump sum in future |
PV = FV / (1 + r)n = FV x [ 1 / (1+ r)n ] |
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Present-value factor (FVF) table |
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Excel present value formula PV= |
Present Value of a Mixed Stream |
PV = [CF1 x 1 / (1 + r)1] + [CF2 x 1 / (1 + r)1] + ... + [CFn x 1 / (1 + r)1] |
Present Value of an Ordinary Annuity |
PV = PMT/r x [1 - 1 / (1 + r)n] |
Present Value of Annuity Due |
PV (annuity due) = PMT/r x [1 - 1 / (1 + r)n] x (1 + r) |
Lump sum future value in excel
Present Value of a Growing Perpetuity
Most cash flows grow over time |
This formula adjusts the present value of a perpetuity formula to account for expected growth in future cash flows |
Calculate present value (PV) of a stream of cash flows growing forever (n = ∞) at the constant annual rate g |
Loan Amortization
A borrower makes equal periodic payments over time to fully repay a loan |
E.g. home loan |
Uses |
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Total $ of loan |
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Term of loan |
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Frequency of payments |
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Interest rate |
Finding a level stream of payments (over the term of the loan) with a present value calculated at the loan interest rate equal to the amount borrowed |
Loan amortization schedule Used to determine loan amortisation payments and the allocation of each payment to interest and principal |
Portion of payment representing interest declines over the repayment period, and the portion going to principal repayment increases |
PMT = PV / {1 / r x [ 1 - 1 / (1 + r)n ] }
Deposits Needed to Accumulate a Future Sum
Determine the annual deposit necessary to accumulate a certain amount of money at some point in the future |
E.g. house deposit |
Can be derived from the equation for fi nding the future value of an ordinary annuity |
Can also be used to calc required deposit |
PMT = FV {[( 1 + r)n - 1 ] / r}
Once this is done substitute the known values of FV, r, and n into the righthand
side of the equation to find the annual deposit required.
Stated Versus Effective Annual Interest Rates
Make objective comparisons of loan costs or investment returns over different compounding periods |
Stated annual rate is the contractual annual rate charged by a lender or promised by a borrower |
Effective annual rate (EAR) AKA the true annual return, is the annual rate of interest actually paid or earned |
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Reflects the effect of compounding frequency |
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Stated annual rate does not |
Maximum effective annual rate for a stated annual rate occurs when interest compounds continuously |
EAR = ( 1 + r/m )m - 1
Compounding continuously: EAR (continuous compounding) = er - 1
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Concept of future value
Apply simple interest, or compound interest to a sum over a specified period of time. |
Interest might compound: annually, semiannual, quarterly, and even continuous compounding periods |
Future value value of an investment made today measured at a specific future date using compound interest. |
Compound interest is earned both on principal amount and on interest earned |
Principal refers to amount of money on which interest is paid. |
Important to understand
After 30 years @ 5% a $100 principle account has:
- Simple Interest: balance of $250.
- Compound interest: balance of $432.19
FV = PV x (1 + r)n
The Power of Compound Interest
Future Value of One Dollar
Present value
Used to determine what an investor is willing to pay today to receive a given cash flow at some point in future. |
Calculating present value of a single future cash payment |
Depends largely on investment opportunities of recipient and timing of future cash flow |
Discounting describes process of calculating present values |
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Determines present value of a future amount, assuming an opportunity to earn a return (r) |
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Determine PV that must be invested at r today to have FV, n from now |
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Determines present value of a future amount, assuming an opportunity to earn a given return (r) on money. |
We lose opportunity to earn interest on money until we receive it |
To solve, inverse of compounding interest |
PV of future cash payment declines longer investors wait to receive |
Present value declines as the return (discount) rises. |
E.g. value now of $100 cash flow that will come at some future date is less than $100
PV = FV / (1 + r)n = FV x [ 1 / (1+ r)n ]
The Power of Discounting
Special applications of time value
Use the formulas to solve for other variables |
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Cash flow |
CF or PMT |
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Interest / Discount rate |
r |
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Number of periods |
n |
Common applications and refinements |
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Compounding more frequently than annually |
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Stated versus effective annual interest rates |
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Calculation of deposits needed to accumulate a future sum |
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Loan amortisation |
Compounding More Frequently Than Annually
Financial institutions compound interest semiannually, quarterly, monthly, weekly, daily, or even continuously. |
The more frequently interest compounds, the greater the amount of money that accumulates |
Semiannual compounding |
Compounds twice per year |
Quarterly compounding |
Compounds 4 times per year |
m values: |
Semiannual |
2 |
Quarterly |
4 |
Monthly |
12 |
Weekly |
52 |
Daily |
365 |
Continuous Compounding |
m = infinity |
e = irrational number ~2.7183.13 |
General equation: FV = PV x (1 + r / m)mxn
Continuous equation: FV (continuous compounding) = PV x ( erxn )
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Future Value of Cash Flow Streams
Evaluate streams of cash flows in future periods. |
Two types: |
Mixed stream = a series of unequal cash flows reflecting no particular pattern |
Annuity = A stream of equal periodic cash flows |
More complicated than calc future or present value of a single cash flow, same basic technique. |
Shortcuts available to eval an annuity |
AKA terminal value |
FV of any stream of cash flows at EOY = sum of FV of individual cash flows in that stream, at EOY |
Each cash flow earns interest, so future value of stream is greater than a simple sum of its cash flows |
FV = CF1 x (1 +r)n-1 + CF2 x (1 + r)n-2 + ... + CFn x (1 + r)n-n
Future Value of an Ordinary Annuity
Two basic types of annuity: |
Ordinary annuity = payments made into it at end of each period |
Annuity due = payments made into it at the beginning of each period (arrives 1 year sooner) |
So, future value of an annuity due always greater than ordinary annuity |
Future value of an ordinary annuity can be calculated using same method as a mixed stream |
FV = PMT x { [ ( 1 + r )n - 1 ] / r}
Finding the Future Value of an Annuity Due
Slight change to those for an ordinary annuity |
Payment made at beginning of period, instead of end |
Earns interest for 1 period longer |
Earns more money over the life of the investment |
FV (annuity due) = PMT x { [ ( 1 + r)n -1 ] / r } x (1 + r)
Present Value of Cash Flow Streams
Present values of cash flow streams that occur over several years |
Might be used to: |
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Value a company as a going concern |
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Value a share of stock with no definite maturity date |
= sum of the present values of CFn |
Perpetuity: A level or growing cash flow stream that continues forever |
Same technique as a lump sum |
Present Value of a Mixed Stream = Sum of present values of individual cash flows |
Mixed stream:
PV = [CF1 x 1 / (1 + r)1] + [CF2 x 1 / (1 + r)1] + ... + [CFn x 1 / (1 + r)1]
Present value of an ordinary annuity
Present Value of an Ordinary Annuity
Similar to mixed stream |
Discount each payment and then add up each term |
PV = PMT/r x [1 - 1 / (1 + r)n]
Present Value of Annuity Due
Similar to mixed stream / ordinary annuity |
Discount each payment and then add up each term |
Cash flow realised 1 period earlier |
Annuity due has a larger present value than ordinary annuity |
PV (annuity due) = PMT/r x [1 - 1 / (1 + r)n] x (1 + r)
Present Value of a Perpetuity
Level or growing cash fl ow stream that continues forever |
Level = infinite life |
Simplest modern example = prefered stock |
Preferred shares promise investors a constant annual (or quarterly) dividend payment forever |
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express the lifetime (n) of this security as infi nity (∞) |
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