Variable keyWhere:  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 guideFuture value of a lump sum:  FV = PV x (1 + r)^{n}    Futurevalue factor (FVF) table    Excel future value formula FV=    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)^{n1} + CF2 x (1 + r)^{n2} + ... + CFn x (1 + r)^{nn}  Present value of a lump sum in future  PV = FV / (1 + r)^{n} = FV x [ 1 / (1+ r)^{n} ]    Presentvalue factor (FVF) table    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 excelPresent Value of a Growing PerpetuityMost 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 
Stated Versus Effective Annual Interest RatesMake 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    Reflects the effect of compounding frequency    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) = e^{r}  1 Deposits Needed to Accumulate a Future SumDetermine 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. Loan AmortizationA borrower makes equal periodic payments over time to fully repay a loan  E.g. home loan  Uses    Total $ of loan    Term of loan    Frequency of payments    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} ] }   Concept of future valueApply 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 InterestFuture Value of One Dollar Present valueUsed 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    Determines present value of a future amount, assuming an opportunity to earn a return (r)    Determine PV that must be invested at r today to have FV, n from now    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 DiscountingSpecial applications of time valueUse the formulas to solve for other variables    Cash flow  CF or PMT    Interest / Discount rate  r    Number of periods  n  Common applications and refinements    Compounding more frequently than annually    Stated versus effective annual interest rates    Calculation of deposits needed to accumulate a future sum    Loan amortisation 
Compounding More Frequently Than AnnuallyFinancial 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 ( e^{rxn} )   Future Value of Cash Flow StreamsEvaluate 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)^{n1} + CF2 x (1 + r)^{n2} + ... + CFn x (1 + r)^{nn} Future Value of an Ordinary AnnuityTwo 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 DueSlight 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 StreamsPresent values of cash flow streams that occur over several years  Might be used to:    Value a company as a going concern    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 AnnuitySimilar 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 DueSimilar 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 PerpetuityLevel 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    express the lifetime (n) of this security as infi nity (∞) 

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