Time value of money Cheat Sheet (2024)

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

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)

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

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

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

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

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.

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.

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

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

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

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

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

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

Similar to mixed stream

Discount each payment and then add up each term

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

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 (∞)