Cheatography

# Time value of money Cheat Sheet by NatalieMoore

### 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 - Future­­-value 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)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 ] - Presen­t-value 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)

### 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
PV = CF1 / r - g r > 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) = er - 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

### 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 ]

### 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 ( erxn )

### 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: - 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 (∞)
PV = PMT x 1/r = PMT/r

Very Nice .thanks

Very Nice .thanks