Time Value of Money - College of Businesscob.ualr.edu/lcholland/finc3310/Impatica_Files/TVM3/TV3.pdf · Time Value of Money 3. Valuing Stocks and Bonds Welcome back to the major topic

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Copyright 2004 by Larry C. Holland Slide 1

BUSINESS FINANCE

Time Value of Money

3. Valuing Stocks and Bonds

Welcome back to the major topic of the Time Value of Money. This is the third module and covers the process of valuing stocks and bonds.

The idea behind finding the value of a financial asset is to first determine the future cash flows that will occur as a result of owning the asset. Then you can find the present value of those future cash flows to determine the value (or the price) of that asset today.

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Stocks and Bonds

Before we go through the calculation of finding the current value of stocks and bonds, let’s spend a few moments discussing what stocks and bonds really represent.

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Stocks represent an ownership share of the future profits. Future cash flows are the future dividend payments.

For example, when a company issues stock, it is really selling an ownership share of the company’s future profits. Each year, as a company makes a profit, a decision is made on the portion of profits that should be paid immediately as a dividend to the stockholders and how much should be reinvested to grow the firm so that larger dividends can be paid in the future. So ultimately, if you hold the stock forever, the future cash flows are the future dividend payments.

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Bonds represent a loan from the investor to a company or government entity. Future cash flows are interest and principal payments.

On the other hand, bonds represent a loan from the investor to a company or government entity.

For example, to provide more capital for expansion or growth, a company might issue 30-year bonds to investors. In exchange for the cash received from the sale of the bonds, the company promises to make periodic interest payments over the next 30 years and ultimately repay the entire principal amount at the end of the 30-year period. The future interest and principal payments are supported by the future cash earnings of the company.

Likewise, a government entity might sell bonds to provide additional cash if current expenditures are higher than tax revenues. The future interest and principal payments are then provided from future tax revenues.

In either case, from the investor’s point of view, the future cash flows from a bond are the interest and principal payments.

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1. Value of Bonds2. Value of Stocks

With this short introduction, let’s first develop the calculations for finding the current value of bonds. Later in the module, we’ll work through the calculations for finding the value of stocks.

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Current Value of a Bond

The process for finding the current value of a bond is to first identify the future cash flows, and then find the present value of those case flows.

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Future cash flows are interest and principal.

The future cash flows for a bond are the periodic interest payments and the return of the principal at the end. Let’s do a quick example problem to illustrate how to value a bond.

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Suppose you have a $1000 bond which pays $80 interest each year with the principal due in 3 years. What is the value of this bond today?

0 1 2 3

10808080

Suppose you have a $1000 bond which pays $80 interest each year with the principal due in three years. This time line shows the sequence of these future cash flows. If you also know that similar bonds offer an overall return of 6% per year, then you could consider the required rate of return to be 6% on this bond as well. This is what you would use as a discount rate to find present values. You can then find the present value of each of the individual future cash flows. The sum of these present values will then be the present value of all future cash flows.

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0 1 2 3

108080801 x1.06 75.47

For example, the present value of $80 one year from now is 80 divided by 1.06, which equals $75.47

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0 1 2 3

108080801 x1.06 75.47 1 x1.06271.20

The present value of $80 two years from now is 80 divided by 1.06 squared, or $71.20.

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0 1 2 3

108080801 x1.06 75.47 1 x1.06271.20 1 x1.063

906.79

And finally, the present value of $1080 three years from now is 1080 divided by 1.06 raised to the third power, or $906.79.

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0 1 2 3

108080801 x1.06 75.47 1 x1.06271.20 1 x1.063

906.79

1053.46

The sum of these three present values represents the present value of all future cash flows. Thus, the current value of this bond is $1053.46.

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Financial Calculator

Input OutputData Data

Number of PeriodsInterest RatePresent ValuePaymentFuture Value

Valuing a Bond

You can also use a financial calculator to calculate the current value of a bond. This slide shows the Financial Calculator spreadsheet, which supports the Time Value of Money modules. Let’s use the financial calculator to solve the example problem we just completed.

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Number of Periods 3Interest RatePresent ValuePaymentFuture Value

Valuing a Bond

To find the current value of a bond, you need to enter the input data from the example problem. Since the bond is a 3-year bond, enter a 3 in the cell for the number of periods.

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Number of Periods 3Interest Rate 6.00%Present ValuePaymentFuture Value

Valuing a Bond

Since the discount rate is 6%, enter 6% in the interest rate cell.

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Number of Periods 3Interest Rate 6.00%Present Value -213.84Payment 80.00Future Value -254.69

Valuing a Bond

And enter the annual interest payment in the payment cell. When you do this, two numbers are shown in the output data cells. We are only interested in the Present Value for this problem. This spreadsheet indicates that the Present Value of the interest payments is $213.84.

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Number of Periods 3Interest Rate 6.00%Present Value -839.62Payment -314.11Future Value 1,000.00

Valuing a Bond

If you delete the $80 payment cell and input the principal of $1,000 in the Future Value cell, there are again two numbers shown in the output data cells. We are only interested in the Present Value. This spreadsheet indicates that the Present Value of the $1,000 principal payment is equal to $839.62.

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PV of interest = 213.84PV of principal = 839.62

Total = 1,053.46

What we really need is the sum of the Present Value of the interest payments (which is 213.84) and the Present Value of the principal payment (which is 839.62). In this problem, the sum of these Present Values is $1,053.46.

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Number of Periods 3Interest Rate 6.00%Present Value -1,053.46Payment 80.00Future Value 1,000.00

Valuing a Bond

Fortunately, the Financial Calculator can sum these Present Values in one calculation if you enter both the annual payment of $80 per year and the future principal payment of $1,000. In this case, there is only one output data cell, which is the sum of the Present Value of the interest payments and the Present Value of the principal payment. The current value of this bond is therefore $1,053.46. Notice that the output data cell has a negative sign. This indicates that the cash flow is in the opposite direction (that is, you have to pay this amount of money to buy the bond). We could restate the problem to illustrate the different direction in cash flows. For example, how much would you have to pay for a bond today (this is a negative cash flow) in order to receive $80 per year and $1,000 at the maturity of the bond (these are positive cash flows). Soa negative sign means you have to pay that amount of money and a positive sign means you receive that amount of money.

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Definitions

At this point, it would be useful to define some terms related to bonds.

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Face Value = PrincipalPrice > Face Value = PremiumPrice < Face Value = Discount

First of all, the Face Value of a bond is the principal that will be paid at the end of the life of the bond. The face value of most bonds is $1,000.

A Premium bond is a bond which is currently priced higher than the face value. A discount bond is a bond which is currently priced below the face value.

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Maturity = Years to Payment of Principal

The maturity of a bond is the number of years to the end of the life of the bond, when the principal is paid.

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Coupon Rate =Interest per Year

Face Value

The Coupon Rate of a bond is the interest paid per year divided by the face value of the bond. The coupon rate also identifies the amount of interest paid per year. For example, a 6% coupon rate implies 6% times $1,000, or $60 interest per year.

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Current Yield =Interest per Year

Current Price

The Current Yield of a bond is the interest paid per year divided by the current price of a bond. When the current yield is less than the coupon rate, the bond is a premium bond. When the current yield is more than the coupon rate, the bond is a discount bond.

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The yield-to-maturity is the discount rate for which the PV of the interest payments plus the PV of the principal equals the current price.

The Yield-to-Maturity of a bond is the discount rate for which the Present Value of the interest payments plus the Present Value of the principal payment equals the current price.

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Interest payments are normally paid semi-annually.

A final aspect about bonds is that interest payments are normally paid semi-annually rather than every year.

Let’s work another example problem for a bond that pays interest semi-annually.

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Coupon Rate = 7%Maturity = 30 yearsFace Value = $1,000YTM = 6.2% per yearCurrent Price = ?

Suppose a bond has a coupon rate of 7%, a maturity of 30 years, and a face value of $1,000. This bond pays interest semi-annually and has a yield to maturity of 6.2% per year. What is the current value of this bond?

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Number of PeriodsInterest RatePresent ValuePaymentFuture Value

Valuing a Bond

To solve this problem on a Financial Calculator, all the input data must be converted to a semi-annual basis.

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Number of Periods 60Interest RatePresent ValuePaymentFuture Value

Valuing a Bond

For example, the number of periods equals 30 times 2, or 60 6-month periods.

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Number of Periods 60Interest Rate 3.10%Present ValuePaymentFuture Value

Valuing a Bond

The interest rate every 6-months is 6.2% divided by 2, or 3.1%.

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Number of Periods 60Interest Rate 3.10%Present Value -948.24Payment 35.00Future Value -5,921.57

Valuing a Bond

From the coupon rate, the interest payment every year is 7% times a face value of $1,000, or $70 per year. However, the interest payment every 6 months is $70 per year divided by 2, or $35.

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Valuing a Bond

Finally, the Future Value of the bond is $1,000. The output data cell indicates that the current value of this bond is $1108.37. This is a premium bond because it is priced above the face value of $1,000.

Notice that four pieces of input data are necessary to calculate the fifth output data value. Let’s set up a new problem like this current example, but one in which we are given the current price and are asked to calculate the annual yield to maturity.

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Current Price = $1,108.37Coupon Rate = 7%Maturity = 30 yearsFace Value = $1,000Semi-annual interest paymentsYTM = ?

Suppose a bond with a current price of $1108.37 has a coupon rate of 7%, a maturity of 30 years, and a face value of $1,000. What is the annual yield to maturity of this bond?

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Valuing a Bond

Since this is very similar to the example problem we just completed, we can use the same Financial Calculator spreadsheet with two changes.

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Number of Periods 60Interest Rate #NUM!Present ValuePayment 35.00Future Value 1,000.00

Valuing a Bond

First, we must delete the interest rate from the input data cell.

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Valuing a Bond

Then we need to enter the current price as the Present Value of –1108.37. Recall that the Present Value must be entered as a negative number because that is how much you have to payfor the bond in order to receive the interest payments and the final principal.

Also notice that the interest rate answer is 3.1%. This is still on a 6-month basis, but the problem asks for the annual yield to maturity. We still need to take the interest output answer and multiply by 2 to convert it to an annual basis. Thus, 3.1% times 2 equals a yield to maturity of 6.2% per year.

This completes the Bond portion of this module.

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Value stocks by finding the PV of all future cash flows.

The last part of this module covers how to value stocks. You use the same basic approach of finding the Present Value of all future cash flows.

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Value stocks by finding the PV of all future cash flows.

Assume the future cash flows are projected dividends.

The main difference is in identifying the future cash flows. For stocks, the future cash flows are the projected dividends. Let’s define some notation, and then we can calculate an estimate of the current stock value.

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D0 = Current dividend paid

Let D0 equal the current dividend that has been paid. This dividend has already been paid. All the other dividends that we project are estimates of future cash flows.

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D0 = Current dividend paidD1 = Dividend in 1 year

Then D1 equals the expected dividend one year from now. In reality, dividends are usually paid quarterly. But for valuation purposes, we assume that the dividend occurs once per year at the end of the year.

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D0 = Current dividend paidD1 = Dividend in 1 yearD2 = Dividend in 2 years

D2 equals the expected dividend in two years.

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D0 = Current dividend paidD1 = Dividend in 1 yearD2 = Dividend in 2 yearsD3 = Dividend in 3 years

And D3 equals the expected dividend in three years.

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D1 D2 D3 D4P0 = + + +(1+r) (1+r)2 (1+r)3 (1+r)4

D5 D6 D7+ + + . . . . .(1+r)5 (1+r)6 (1+r)7

To find the value of a stock today, we need to calculate the present value of all future cash flows. A straightforward way to do this is to find the present value of the projected dividends indefinitely into the future. To find the Present Value of the first dividend one year from now, we divide D1 by (1+r). The Present Value of the second dividend is D2 divided by (1+r) squared. The rest of the Present Values are calculated in a similar manner. The most difficult problem in this case is to project dividends for 50 to 100 years into the future (present values for cash flows beyond 50 years or so are usually small enough that they can be ignored). We usually make a few assumptions to simplify this calculation.

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D1P0 = r – g

D1 = Dividend in 1 yearr = Required rate of returng = Constant growth rate

If we are willing to assume that the dividend will growth at a constant rate indefinitely into the future, the equation to find the Present Value of all future cash flows simplifies into this simple equation: the next dividend divided by R, the required rate of return, minus g, the assumed constant growth rate indefinitely into the future. This equation is called the constant dividend growth model. Let’s use this formula in an example problem to illustrate the use of this equation.

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Suppose you project that a company will pay a dividend of $1.50 per share over the next year, and you anticipate the dividend will grow at a constant rate of 8% per year indefinitely into the future. If the required rate of return is 14%, what would be a fair price for one share of the stock today?

Suppose you project that a company will pay a dividend of $1.50 per share over the next year, and you anticipate the dividend will grow at a constant rate of 8% per year indefinitely into the future. If the required rate of return is 14%, what would be a fair price for one share of the stock today? The first thing to notice in this problem is that we can assume a constant growth rate for dividends, indefinitely into the future. Therefore, we can use the simplified Constant Dividend Growth model to value the stock. In the problem, we are given the first dividend, or D1, as $1.50 per share. We are also given a constant growth rate of 8% per year and a required rate of return of 14% per year. This is sufficient data to apply the constant dividend growth model.

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D1P0 = r – g

1.50 P0 = = 25.00

.14 – .08

This slide shows the calculation of a fair price for the stock today using the constant dividend growth model. $1.50 divided by (.14 - .08) equals a current stock price of $25.00 per share.

Let’s work another similar problem.

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Suppose you project that a company just paid a dividend of $4.00 per share, and you anticipate the dividend will grow at a constant rate of 6% per year indefinitely into the future. If the required rate of return is 16%, what would be a fair price for one share of the stock today?

Suppose you project that a company just paid a dividend of $4.00 per share, and you anticipate the dividend will grow at a constant rate of 6% per year indefinitely into the future. If the required rate of return is 16%, what would be a fair price for one share of the stock today?

Again, the constant dividend growth model can be applied because we willing to assume a constant growth in dividend indefinitely into the future. We are given a constant growth of 6% per year and a required rate of return of 16% per year. However, this time we are given the dividend that a company has just paid. This dividend represents D0, but we need D1 in order to apply the constant dividend growth model.

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D1 = D0 (1+g)

= 4.00 (1.06)

= 4.24

So we need to calculate D1 from the information given. Since we know that the dividend is growing at a constant rate, if the last dividend was $4.00 per share, the dividend one year from now must be 6% larger. Therefore, D1 equals D0, or $4.00, times 1.06, which equals $4.24.

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D1P0 = r – g

4.24 P0 = = 42.40

.16 – .06

So a fair price for the stock today would be $4.24 divided by (.16-.06), which equals $42.40.

The constant dividend growth model provides a simple way to estimate the value of a stock today if we can assume a constant growth in dividends from today forward, indefinitely into the future. Sometimes, however, we know that a company can grow faster than normal for a short period of time, and then slow down to a normal growth rate over the long run. This is a two-stage dividend growth model. We can construct such a model by changing the time frame for the simple constant dividend growth model.

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D1P0 = r – g

D2P1 = r – g

For example, if the simple constant dividend growth model works this year, it should also work next year as well. Of course, P0 next year would be P1 from the point of view of this year, and D1next year would be D2 from the point of view of this year. Therefore, from the point of view of this year, P1 must also be equal to D2 divided by (r –g).

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D1P0 = r – g

Dt+1Pt = r – g

Of course, you could use this same logic and arrive at a more generic equation: P at any year t is equal to the Dividend for year t+1 divided by (r – g). In words, the price of a stock at any point in time is equal to the next dividend divided by (r –g).

If we project a constant growth in dividends to be some time t years in the future, we then have a two-stage dividend growth model.

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D1 D2 D3 D4P0 = + + + (1+r) (1+r)2 (1+r)3 (1+r)4

Dt Pt. . . + + . . .(1+r)t (1+r)t

This slide shows such a two-stage model. In the first stage, we can identify specific dividend payments. Suppose at time t years in the future you decide to sell the stock at price Pt, just after receiving the dividend in year t, which is Dt. Of course, after selling the stock, no more dividends would occur. Therefore, the present value of all future cash flows would be the Present Value of all dividends up to the sale of the stock plus the Present Value from the sale of the stock. Notice that the last dividend and the sale of the stock both occur at the end of year t, so the denominator for finding the present value is (1+r) raised to the t power for both cash flows.

Now, if we assume a constant growth in dividends after time t, the price of the stock at that point in time would be equal to the next dividend divided by (r – g). We can substitute this equation for Pt as long as we are willing to assume a constant growth rate beyond time t.

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D1 D2 D3 D4P0 = + + + (1+r) (1+r)2 (1+r)3 (1+r)4

Dt Dt+1/(r-g) . . . + + . . .

(1+r)t (1+r)t

This slide shows such a two-stage model. Probably the easiest way to understand how to apply this two-stage model is with a couple of example problems.

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Suppose a company just paid a dividend of $2.00 per share, and you anticipate the dividend will grow 10% per year for the next two years and then at a constant rate of 8% per year indefinitely into the future. If the required rate of return is 12% per year, what would be a fair price for this stock today?

Suppose a company just paid a dividend of $2.00 per share, and you anticipate the dividend will grow 10% per year for the next two years and then at a constant rate of 8% per year indefinitely into the future. If the required rate of return is 12% per year, what would be a fair price for this stock today?

The best way to solve this type of problem is to logically set up the problem and then work the calculations.

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0 1 2 3

In this slide, we first draw a time line so that the cash flows and growth rates can be identified.

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0 1 2 3

10%

The problem indicates a growth of 10% per year for the first year,

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0 1 2 3

10%10%

and 10% growth for the second year.

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0 1 2 3

8%10%10%

Thereafter, the growth rate is 8% per year, indefinitely into the future. The key to solving this type of problem is to identify the point in time in which you can say from this time forward, the dividend will grow at a constant rate. On the time line, it is clear that this point in time is at the end of two years. Once you have identified this point in time, write down the price P with a subscript equal to that time –

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0 1 2 3

8%10%10%

P2

in this case, P2. At this point, we can use the generic form of the dividend growth model because there is constant growth thereafter.

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0 1 2 3

8%10%10%

D3P2 = r – g

Therefore, the price is equal to the next dividend divided by (r-g), or more specifically, P2 is equal to D3 divided by (r-g). From this equation, we need to find D3.

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0 1 2 3

8%10%10%

D3P2 = r – g

D0 = 2.00

The problem stated that the company just paid a dividend of $2.00. Therefore, D0 is equal to $2.00.

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0 1 2 3

8%10%10%

D3P2 = r – g

D0 = 2.00D1 = 2.00 (1.10) = 2.20

D1 grows by 10% and is equal to $2.00 times 1.10, or $2.20.

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0 1 2 3

8%10%10%

D3P2 = r – g

D0 = 2.00D1 = 2.00 (1.10) = 2.20D2 = 2.20 (1.10) = 2.42

D2 also grows by 10% and is equal to $2.20 times 1.10, or $2.42.

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0 1 2 3

8%10%10%

D3P2 = r – g

D0 = 2.00D1 = 2.00 (1.10) = 2.20D2 = 2.20 (1.10) = 2.42D3 = 2.42 (1.08) = 2.6136

D3 grows by 8% and is equal to $2.42 times 1.08, or $2.61.

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0 1 2 3

8%10%10%

D3 2.6136P2 = = = 65.34

r – g .12 - .08

D0 = 2.00D1 = 2.00 (1.10) = 2.20D2 = 2.20 (1.10) = 2.42D3 = 2.42 (1.08) = 2.6136

Now that we have D3 equal to $2.61, we can calculate P2, which is equal to $2.61 divided by (.12 - .08), or $65.34.

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0 1 2 3

2.2065.342.42

The last step is to identify all the future cash flows. The future cash flows are D1 of $2.20, D2of $2.42, and P2 of $65.34.

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0 1 2 3

2.2065.342.42

D1 D2 P2P0 = + + (1+r) (1+r)2 (1+r)2

2.20 2.42 65.34P0 = + +

(1.12) (1.12)2 (1.12)2

P0 = 1.964 + 1.929 + 52.09 = 55.98

The present value of these three cash flows is equal to $55.98, which is the current fair value of this stock. Notice in the calculation that D2 and P2occur at the same time and therefore have the same denominator in the present value calculation.

Let’s work one more problem.

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Suppose you project that a company will pay a dividend of $2.00 next year, $3.00 two years from now, and $4.00 three years from now. After that, you anticipate a constant growth of 7% per year indefinitely into the future. If the required rate of return is 13%, what would be a fair price for this stock today?

Suppose you project that a company will pay a dividend of $2.00 per share next year, $3.00 per share two years from now, and $4.00 per share three years from now. After that, you anticipate a constant growth in dividends of 7% per year indefinitely into the future. If the required rate of return is 13%, what would be a fair price for this stock today?

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4 years0

2.00 3.00 4.007 %

1 2 3

To solve this problem, we first set up a time line and identify the cash flows and growth rates in the problem. We know that D1 is equal to $2.00, D2 is equal to $3.00, D3 is equal to $4.00, and the dividend will grow at 7% per year thereafter. From the timeline, it is clear that at the end of year 3, the dividend will grow at a constant rate thereafter.

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4 years0

2.00 3.00 4.007 %

1 2 3

P3

So we immediately write down P3,

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4 years0

2.00 3.00 4.007 %

1 2 3

D4P3 = r – g

which we know equals D4 over (r – g).

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4 years0

2.00 3.00 4.007 %

1 2 3

D4P3 = r – g

D4 = 4.00 (1.07) = 4.28

Since we need D4 for this calculation, we calculate D4 as equal to D3 of $4.00 times 1.07, or $4.28.

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4 years0

2.00 3.00 4.0071.33

7 %

1 2 3

D4 4.28P3 = = = 71.33

r – g .13 - .07

D4 = 4.00 (1.07) = 4.28

This allows us to calculate P3 as $4.28 divided by (.13 - .07), which equals $71.33.

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4 years0

2.00 3.00 4.0071.33

7 %

1 2 3

D1 D2 D3 P3P0 = + + +(1+r) (1+r)2 (1+r)3 (1+r)3

2.00 3.00 4.00 71.33P0 = + + +

(1.13) (1.13)2 (1.13)3 (1.13)3

P0 = 1.770 + 2.349 + 2.772 + 49.435 = 56.33

Now that we have all the future cash flows, we can find the Present Values. The present value of D1 of $2.00 over 1.13, plus the present value of D2of $3.00 over 1.13 squared, plus the present value of D3 of $4.00 over 1.13 raised to the third power, plus the present value of P3 of $71.33 over 1.13 raised to the third power equals $56.33. This is the current fair value of this stock.

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Summary

Let’s summarize the material covered in this module.

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Summary• Valued a bond as the present value of

interest and principal payments

First, we found the fair value of a bond by finding the present value of future interest payments and the final principal payment.

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interest and principal payments• Included semi-annual interest payments

Then we extended this process by including semi-annual interest payments.

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interest and principal payments• Included semi-annual interest payments• Valued a stock as the present value of future

dividends

Then, we found the fair value of a stock by finding the present value of future dividends.

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dividends• Used the simple dividend growth model

We then used the simple constant dividend growth model,

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dividends• Used the simple dividend growth model• Used the two-stage dividend growth model

and extended this approach with a two-stage dividend growth model to find the fair value for a stock.

This concludes the module on valuing stocks and bonds.

Documents

Time Value of Money - College of Businesscob.ualr.edu/lcholland/finc3310/Impatica_Files/TVM3/TV3.pdf · Time Value of Money 3. Valuing Stocks and Bonds Welcome back to the major topic