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Where:
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FV
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= Future value of an investment
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PV
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= Present value of an investment (the lump sum)
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r
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= Return or interest rate per period (typically 1 year)
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n
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= Number of periods (typically years) that the lump sum is invested
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PMT
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= Payment amount
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CFn
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= Cash flow steam number
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m
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= # of times per year r compounds
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Future value of a lump sum:
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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))
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Future Value of an Ordinary Annuity
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FV = PMT x { [ ( 1 + r )n - 1 ] / r}
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Future Value of an Annuity Due
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FV (annuity due) = PMT x { [ ( 1 + r)n -1 ] / r } x (1 + r)
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Future Value of Cash Flow Streams
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FV = CF1 x (1 +r)n-1 + CF2 x (1 + r)n-2 + ... + CFn x (1 + r)n-n
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Present value of a lump sum in future
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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=
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Present Value of a Mixed Stream
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PV = [CF1 x 1 / (1 + r)1] + [CF2 x 1 / (1 + r)1] + ... + [CFn x 1 / (1 + r)1]
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Present Value of an Ordinary Annuity
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PV = PMT/r x [1 - 1 / (1 + r)n]
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Present Value of Annuity Due
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PV (annuity due) = PMT/r x [1 - 1 / (1 + r)n] x (1 + r)
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Most cash flows grow over time
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This formula adjusts the present value of a perpetuity formula to account for expected growth in future cash flows
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Calculate present value (PV) of a stream of cash flows growing forever (n = ∞) at the constant annual rate g
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Make objective comparisons of loan costs or investment returns over different compounding periods
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Stated annual rate is the contractual annual rate charged by a lender or promised by a borrower
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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
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Maximum effective annual rate for a stated annual rate occurs when interest compounds continuously
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Determine the annual deposit necessary to accumulate a certain amount of money at some point in the future
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E.g. house deposit
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Can be derived from the equation for fi nding the future value of an ordinary annuity
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Can also be used to calc required deposit
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A borrower makes equal periodic payments over time to fully repay a loan
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E.g. home loan
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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
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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
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Loan amortization schedule Used to determine loan amortisation payments and the allocation of each payment to interest and principal
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Portion of payment representing interest declines over the repayment period, and the portion going to principal repayment increases
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Apply simple interest, or compound interest to a sum over a specified period of time.
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Interest might compound: annually, semiannual, quarterly, and even continuous compounding periods
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Future value value of an investment made today measured at a specific future date using compound interest.
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Compound interest is earned both on principal amount and on interest earned
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Principal refers to amount of money on which interest is paid.
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Used to determine what an investor is willing to pay today to receive a given cash flow at some point in future.
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Calculating present value of a single future cash payment
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Depends largely on investment opportunities of recipient and timing of future cash flow
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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.
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We lose opportunity to earn interest on money until we receive it
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To solve, inverse of compounding interest
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PV of future cash payment declines longer investors wait to receive
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Present value declines as the return (discount) rises.
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Use the formulas to solve for other variables
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Cash flow
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CF or PMT
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Interest / Discount rate
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r
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Number of periods
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n
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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
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Financial institutions compound interest semiannually, quarterly, monthly, weekly, daily, or even continuously.
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The more frequently interest compounds, the greater the amount of money that accumulates
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Semiannual compounding
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Compounds twice per year
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Quarterly compounding
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Compounds 4 times per year
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m values:
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Semiannual
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2
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Quarterly
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4
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Monthly
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12
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Weekly
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52
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Daily
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365
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Continuous Compounding
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m = infinity
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e = irrational number ~2.7183.13
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Evaluate streams of cash flows in future periods.
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Two types:
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Mixed stream = a series of unequal cash flows reflecting no particular pattern
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Annuity = A stream of equal periodic cash flows
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More complicated than calc future or present value of a single cash flow, same basic technique.
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Shortcuts available to eval an annuity
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AKA terminal value
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FV of any stream of cash flows at EOY = sum of FV of individual cash flows in that stream, at EOY
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Each cash flow earns interest, so future value of stream is greater than a simple sum of its cash flows
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Two basic types of annuity:
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Ordinary annuity = payments made into it at end of each period
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Annuity due = payments made into it at the beginning of each period (arrives 1 year sooner)
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So, future value of an annuity due always greater than ordinary annuity
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Future value of an ordinary annuity can be calculated using same method as a mixed stream
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Slight change to those for an ordinary annuity
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Payment made at beginning of period, instead of end
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Earns interest for 1 period longer
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Earns more money over the life of the investment
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Present values of cash flow streams that occur over several years
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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
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= sum of the present values of CFn
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Perpetuity: A level or growing cash flow stream that continues forever
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Same technique as a lump sum
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Present Value of a Mixed Stream = Sum of present values of individual cash flows
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Similar to mixed stream
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Discount each payment and then add up each term
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Similar to mixed stream / ordinary annuity
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Discount each payment and then add up each term
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Cash flow realised 1 period earlier
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Annuity due has a larger present value than ordinary annuity
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Level or growing cash fl ow stream that continues forever
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Level = infinite life
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Simplest modern example = prefered stock
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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 (∞)
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Cheatography n. The art of mapping complex subjects into clear, instantly navigable reference guides.
