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} 
 
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 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 
 
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} ] }
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 
 
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


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 
 
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 Discounting
Special applications of time value
Use 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 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 ( e^{rxn} )


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)^{n1} + CF2 x (1 + r)^{n2} + ... + CFn x (1 + r)^{nn}
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: 
 
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 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 
 
express the lifetime (n) of this security as infi nity (∞) 

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Huy Nguyen, 16:00 7 Aug 20
it's helpful for memorial. Thanks
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