Ch’s Time Value of Money Formula Sheet

Financial Management Ch's 4-6: Time Value of Money Formula Sheet, p.1

Prof. Durham

CALCULATION

MATH EQUATION

EXCEL FORMULA

[In the following three equations, you need to be consistent with your r and the N (i.e., the exponent). If compounding is annual, you need a rate per year and an N in years. If compounding is semi-annual, you need a rate per half-year and an N in half-years. If compounding is quarterly, you need a rate per quarter and an N in quarters. And so on. You do not need any new equations.]

"Earlier" Value of a Single Cash Flow:

"PV" = C / (1+r)1

= -PV( rate , 1 , , fv )**

gives a value one period before the single cash flow to get a value two periods before the single cash flow, change the 1s to 2s **note that, unlike in the annuity equations that follow, the cash flow is "fv" (not "pmt")

Present Value of a Single

PV = CN / (1+r)N

= -PV( rate , nper , , fv )**

Cash Flow at Time N:

gives a value at time zero of the single cash flow at time N

**again, note that, unlike in the annuity equations that follow, the cash flow is "fv" (not "pmt")

"Later" Value of a Single Cash Flow:

FV = C (1+r)1

= -FV( rate , 1 , , pv )**

gives a value one period after the single cash flow to get a value two periods after the single cash flow, change the 1s to 2s

to get a value three periods after the single cash flow, change the 1s to 3s, etc.

**note that, unlike in the annuity equations that follow, the cash flow is "pv" (not "pmt")

[Now, we move on to equations for annuities and perpetuities. Please note: These equations are the only ones that you need. As explained in other course materials, if the cash flows are annual, you need a rate per year and an N in years. If the cash flows are semi-annual, you need a rate per half-year and an N in half-years. If the cash flows are quarterly, you need a rate per quarter and an N in quarters. And so on. But you do not need a new equation.]

Present Value of an Ordinary Annuity: PV = C / r ( 1 ? 1/(1+r)N )

= -PV( rate , nper , pmt )

gives a value one period before the first cash flow in the annuity

if the first cash flow in the annuity is at t=1, this equation gives a value at t=0; if the first cash flow in the annuity is at t=2, this equation gives a value at t=1;

if the first cash flow in the annuity is at t=6, this equation gives a value at t=5, etc.

Using some made-up numbers, notice that the math isn't so bad:

PV = $3000 / 0.08 ( 1 ? 1/1.0820 )

= -PV( 0.08 , 20 , 3000 )

Future Value of an Ordinary Annuity:

FV = C / r ( (1+r)N ? 1 )

= -FV( rate , nper , pmt )

gives a value at the same instant as the final cash flow in the annuity

Again, using some made-up numbers:

FV = $3000 / 0.08 ( 1.0820 ? 1 )

= -FV( 0.08 , 20 , 3000 )

Present Value of an Annuity Due:

PV = C / r ( 1 ? 1/(1+r)N ) (1+r) = -PV( rate , nper , pmt , , 1 )

gives a value at the same instant as the first cash flow in the annuity

Again, using some made-up numbers:

PV = $3000 / 0.08 ( 1 ? 1/1.0820 ) 1.08

= -PV( 0.08 , 20 , 3000 , , 1 )

Future Value of an Annuity Due:

FV = C / r ( (1+r)N ? 1 ) (1+r) = -FV( rate , nper , pmt , , 1 )

gives a value one period after the final cash flow in the annuity

Using the same made-up numbers:

FV = $3000 / 0.08 ( 1.0820 ? 1 ) 1.08

= -FV( 0.08 , 20 , 3000 , , 1 )

Financial Management Ch's 4-6: Time Value of Money Formula Sheet, p.2

Prof. Durham

CALCULATION

MATH EQUATION

EXCEL FORMULA

Present Value of a (Level) Perpetuity:

"PV" = C / r

= C / r

gives a value one period before the first cash flow in the (level) perpetuity if the first cash flow in the level perpetuity is at t=1, this equation gives a value at t=0; if the first cash flow in the level perp. is at t=2, this eq'n gives a value at t=1; if the first cash flow in the level perp. is at t=15, this eq'n gives a value at t=14, etc.

Future Value of a (Level) Perpetuity: such an equation does not exist ... because you can never get to (or past) the

final cash flow in any perpetuity because, by definition, a perpetuity's cash flows are perpetual.

Present Value of a Growing Perpetuity:

PV = C / ( r - g )

= C / ( r - g )

gives a value one period before the first cash flow in the growing perpetuity if the first cash flow in the growing perp. is at t=1, this equation gives a value at t=0; if the first cash flow in the growing perp. is at t=2, this eq'n gives a value at t=1; if the first cash flow in the growing perp. is at t=23, this eq'n gives a value at t=22, etc.

Future Value of a Growing Perpetuity: such an equation does not exist ... because you can never get to (or past)

the final cash flow in any perpetuity because, by definition, a perpetuity's cash flows are perpetual.

CALCULATION

MATH EQUATION

Effective Annual Rate:

EAR = ( 1 + periodic rate )# of compounding periods in a year ? 1,

where the "periodic rate" is the rate per compounding period

if interest is compounded monthly, then periodic rate equals annual rate ? 12 and exponent = 12

if interest is compounded quarterly, then periodic rate equals annual rate ? 4 and exponent = 4

if interest is compounded semi-annually, then periodic rate equals annual rate ? 2 and exponent = 2, etc.

Excel equation: =EFFECT( annual rate , # of compounding periods in a year )

Effective Half-Year Rate:

EHYR = ( 1 + periodic rate )# of compounding periods in a half-year ? 1,

NOT COMMON; STILL USEFUL

where the "periodic rate" is the rate per compounding period

if interest is compounded monthly, then periodic rate equals annual rate ? 12 and exponent = 6

if interest is compounded quarterly, then periodic rate equals annual rate ? 4 and exponent = 2

if interest is compounded semi-annually, then periodic rate equals annual rate ? 2 and exponent = 1, etc.

Excel formula: does not really exist

Effective Quarterly Rate:

EQR = ( 1 + periodic rate )# of compounding periods in a quarter ? 1,

NOT COMMON; STILL USEFUL

where the "periodic rate" is the rate per compounding period

if interest is compounded daily, then periodic rate equals annual rate ? 365 and exponent = 91.25

if interest is compounded monthly, then periodic rate equals annual rate ? 12 and exponent = 3

if interest is compounded quarterly, then periodic rate equals annual rate ? 4 and exponent = 1, etc.

Excel formula: does not really exist

Financial Management

Ch's 4-6: Time Value of Money Formula Sheet, p.3

Prof. Durham

CALCULATION

MATH EQUATION

EXCEL FORMULA

[FROM CHAPTER 5: The equation for valuing a bond consists of nothing more than a combination of the equation for

present value of an ordinary annuity and the equation for present value a single cash flow at time N.]

Price of a Bond:

PV = C / r ( 1 ? 1/(1+r)N ) + Face / (1+r)N = -PV( rate , nper , pmt , fv )

gives a value at time zero Math's r is Excel's rate, N is nper, C is pmt, and Face is fv if the coupon pmts. in the annuity are semi-annual, the rate must be per half-year and the N must be in half-years. if the coupon pmts. in the annuity are annual, the rate must be per year and the N must be in years.

Yield to Maturity of a Bond: Price = C / y ( 1 ? 1/(1+y)N ) + Face / (1+y)N = RATE( nper , pv , pmt , fv )

Yield to maturity is the "y" that satisfies this equality and can be solved only by iterative trial-and-error Math's Price is the negative of Excel's PV, N is nper, C is pmt, and Face is fv NOTE: If using Excel, the Price must be entered as a negative number for pv

if the coupon pmts. in the annuity are semi-annual, the rate must be per half-year and the N must be in half-years. if the coupon pmts. in the annuity are annual, the rate must be per year and the N must be in years.

[FROM CHAPTER 6: A stock is typically valued using any combination of equations from Chapter 4. The price could

be simply the present value of a bunch of single cash flows. Or, the future cash flows could be any combination of things that we've learned so far: single sums, annuities, level perpetuities, growing perpetuities. I thus cannot really give you a couple of nifty equations like I'm able to for Chapters 4 and 5.]

Dividend-Growth Valuation: P0 = Div1 / ( r - g ), where dividends are simply (and conveniently) predicted to grow by the same growth rate, g, each year forever Excel formula: does not really exist

Finite-Horizon Valuation: P0 = Div1 / (1+r) + Div2 / (1+r)2 + Div3 / (1+r)3 + + DivN / (1+r)N + PN / (1+r)N, where Div1 through DivN are the dividends across the horizon and PN is the end-of-horizon selling price; sometimes PN is given and other times PN = DivN+1 / ( r - g ), where DivN+1 is the first dividend in a growing perpetuity. Excel formula: does not really exist

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