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Time value of money Cheat Sheet by

time     value     future     interest

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)

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
PV = CF1 / r - g r > g

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

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.

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 ] }
 

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 ( 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

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