Calculating PV and FV

FV factor formula

(1+I/Y)^N represented the compounding rate of an investment, this factor or interest factor

Ordinary Annuity

An annuity that pays at the end of each compounding period. Typical CF pattern of man investment and business finance applications

FV of an Annuity due alternative methods

Another way to calculate the FV of an ordinary annuity, and simply multiply the resulting FV by [1 + periodic compounding rate (I/Y)]. Symbolically, this can be expressed as:

Solve: Assume the Kodon preferred stock in the preceding examples is scheduled to pay its first dividend in four years, and is non-cumulative (i.e., does not pay any dividends for the first three years). Given an 8% required rate of return, what is the value of Kodon's preferred stock today?

As in the previous example, PVperpetuity=4.500.08=$56.25,PVperpetuity=4.500.08=$56.25, but because the first dividend is paid at t = 4, this PV is the value at t = 3. To get the value of the preferred stock today, we must discount this value for three periods: 56.25(1.08)3 =$44.65.

FV of a single cash flow equation

FV = PV(1+I/Y)^N PV = amount of money invested today I/Y = rate of return per compounding period N = total number of compounding periods

Solve: PV of a perpetuity Kodon Corporation issues preferred stock that will pay $4.50 per year in annual dividends beginning next year and plans to follow this dividend policy forever. Given an 8% rate of return, what is the value of Kodon's preferred stock today?

Given that the value of the stock is the PV of all future dividends, we have: PVperpetuity=4.500.08=$56.25PVperpetuity=4.500.08=$56.25 Thus, if an investor requires an 8% rate of return, the investor should be willing to pay $56.25 for each share of Kodon's preferred stock. Note that the PV of a perpetuity is its value one period before its next payment.

Solve:What is the future value of an annuity that pays $200 per year at the beginning of each of the next three years, commencing today, if the cash flows can be invested at an annual rate of 10%?

Note in the time line in the following figure that the FV is computed as of the end of the last year in the life of the annuity, Year 3, even though the final payment occurs at the beginning of Year 3 (end of Year 2). To solve this problem, put your calculator in the BGN mode ([2nd] [BGN] [2nd] [SET] [2nd] [QUIT] on the TI or [g] [BEG] on the HP), then input the relevant data and compute FV. N = 3; I/Y = 10; PMT = -200; CPT → FV = $728.20. lternatively, we could calculate the FV for an ordinary annuity and multiply it by (1 + I/Y). Leaving your calculator in the END mode, enter the following inputs: N = 3; I/Y = 10; PMT = -200; CPT → FVAO = $662.00 FVAD = FVAO × (1 + I/Y) = 662 × 1.10 = $728.20

PV of a perpetuity

PMT / I/Y fixed periodic cash flow divided by the appropriate periodic rate of return. As with other TVM applications, it is possible to solve for unknown variables in the PVperpetuity equation. In fact, you can solve for any one of the three relevant variables, given the values for the other two.

PV of a single sum

PV = FV/(1+i)^n

Solve: PV of an ordinary annuity beginning later than t = 1 What is the present value of four $100 end-of-year payments if the first payment is to be received three years from today and the appropriate rate of return is 9%?

Step 1:Find the present value of the annuity as of the end of year 2 (PV2). Input the relevant data and solve for PV2. N = 4; I/Y = 9; PMT = -100; FV = 0; CPT → PV = PV2 = $323.97 Step 2:Find the present value of PV2. Input the relevant data and solve for PV0. N = 2; I/Y = 9; PMT = 0; FV = -323.97; CPT → PV = PV0 = $272.68 In this solution, the annuity was treated as an ordinary annuity. The PV was computed one period before the first payment, and we discounted PV2 = $323.97 over two years. We need to stress this important point. The PV annuity function on your calculator set in "END" mode gives you the value one period before the annuity begins. Although the annuity begins at t = 3, we discounted the result for only two periods to get the present (t = 0) value.

Solve: PV of an ordinary annuity What is the PV of an annuity that pays $200 per year at the end of each of the next three years, given a 10% discount rate?

The payments occur at the end of the year, so this annuity is an ordinary annuity. To solve this problem, enter the relevant information and compute PV. N = 3; I/Y = 10; PMT = -200; FV = 0; CPT → PV = $497.37. The $497.37 computed here represents the amount of money that an investor would need to invest today at a 10% rate of return to generate three end-of-year

Discounting

The process of finding the present value of a cash flow or a series of cash flows; discounting is the reverse of compounding. Rate referred to as; discount rate, opportunity cost, required rate of return and cost of capital

Solve: PV of a bond's cash flows A bond will make coupon interest payments of 70 euros (7% of its face value) at the end of each year and will also pay its face value of 1,000 euros at maturity in six years. If the appropriate discount rate is 8%, what is the present value of the bond's promised cash flows?

The six annual coupon payments of 70 euros each can be viewed as an ordinary annuity. The maturity value of 1,000 euros is the future value of the bond at the time the last coupon payment is made. On a time line, the promised payment stream is as shown below. he PV of the bond's cash flows can be broken down into the PV of a 6-payment ordinary annuity, plus the PV of a 1,000 euro lump sum to be received six years from now. The calculator solution is: N = 6; PMT = 70; I/Y = 8; FV = 1,000; CPT PV = -953.77 With a yield to maturity of 8%, the value of the bond is 953.77 euros. Note that the PMT and FV must have the same sign, since both are cash flows paid to the investor (paid by the bond issuer). The calculated PV will have the opposite sign from PMT and FV.

Solve: FV of a single sum Calculate the FV of a $200 investment at the end of two years if it earns an annually compounded rate of return of 10%.

To solve this problem with your calculator, input the relevant data and compute FV. N = 2; I/Y = 10; PV = -200; CPT → FV = $242

Solve: PV of a single sum Given a discount rate of 10%, calculate the PV of a $200 cash flow that will be received in two years.

To solve this problem, input the relevant data and compute PV. N = 2; I/Y = 10; FV = 200; CPT → PV = −$165.29 (ignore the sign)

Annuities Due

an annuity where the payments are received at the beginning of the period (first payment is today at t=0)

Solve: FV of an ordinary annuity What is the future value of an ordinary annuity that pays $200 per year at the end of each of the next three years, given the investment is expected to earn a 10% rate of return?

his problem can be solved by entering the relevant data and computing FV. N = 3; I/Y = 10; PMT = -200; CPT → FV = $662.00 Implicit here is that PV = 0; clearing the TVM functions sets both PV and FV to zero. As indicated here, the sum of the compounded values of the individual cash flows in this three-year ordinary annuity is $662. Note that the annuity payments themselves amounted to $600, and the balance is the interest earned at the rate of 10% per year.

Perpetuity

is a financial instrument that pays a fixed amount of money at set intervals over an infinite period of time

Begin Mode (BGN)

press [2nd] [BGN] [2nd] [SET] --> press [2nd] [QUIT]. You will normally want your calculator to be in the ordinary annuity (END) mode, so remember to switch out of BGN mode after working annuity due problems

PV of a single sum definition

single sum is todays value of a cash flow that is to be received at some point in the future. The amount that must be invested today, at a given rate of return over a given period of time in order to end with a specified FV.

Annuities

stream of equal cash flows that occurs at equal intervals over a given period. need 4/5 of relevant variables

Solve: Given a discount rate of 10%, what is the present value of an annuity that makes $200 payments at the beginning of each of the next three years, starting today?

the time line for this problem is shown in the following figure. First, let's solve this problem using the calculator's BGN mode. Set your calculator to the BGN mode ([2nd] [BGN] [2nd] [SET] [2nd] [QUIT] on the TI or [g] [BEG] on the HP), enter the relevant data, and compute PV. N = 3; I/Y = 10; PMT = -200; CPT → PVAD = $547.11 lternatively, this problem can be solved by leaving your calculator in the END mode. First, compute the PV of an ordinary 3-year annuity. Then multiply this PV by (1 + I/Y). To use this approach, enter the relevant inputs and compute PV. N = 3; I/Y = 10; PMT = -200; CPT → PVAO = $497.37 PVAD = PVAO × (1 + I/Y) = $497.37 × 1.10 = $547.11

PV of an Annuity Due (PVAd)

there is ONE LESS discounting period since cash flows occur at t=0, this already in PV. Implies that all else equal, the PV of an annuity due will be greater than the PV of an Ordinary Annuity. PVAD = PVAO × (1 + I/Y) The advantage of this second method is that you leave your calculator in the END mode and won't run the risk of forgetting to reset it. Regardless of the procedure used, the computed PV is given as of the beginning of the first period, t = 0.

To find PV of an Ordinary Annuity

to find the PV of an ordinary annuity, we use the future cash flow stream, PMT, that we used with FV annuity problems, but we discount the cash flows back to the present (time = 0) rather than compounding them forward to the terminal date of the annuity. Here again, the PMT variable is a single periodic payment, not the total of all the payments (or deposits) in the annuity. The PVAO measures the collective PV of a stream of equal cash flows received at the end of each compounding period over a stated number of periods, N, given a specified rate of return, I/Y. The following examples illustrate how to determine the PV of an ordinary annuity using a financial calculator.