Time Value of Money



CHAPTER 5 : How to Value Bonds and Stocks

1. RETURN OF STOCK

What will be the return of the stock if I am going to receive a dividend at the end of one year and then sell the stock?

Return =Dividend Yield + Capital Gain.

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2. VALUATION OF STOCK

How much should I pay today for this stock?

From the previous equation, we can get

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How much should the person who buys it from me pay for the stock in a year (ie. P1) if he is going to receive a dividend after one year and then he is going to sell it?

Similar to the previous logic,

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Price of the stock is the present value of the cash flows (ie, dividend) received by the investor.

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By extending the period to T, we can get

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If T is really big #, the stock price can be expressed

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Now the price of the stock is obviously independent of the time horizon, T.

As shown above, the present value of the terminal price becomes less important.

3. IMPLICATIONS

By considering how much a buyer will pay for the stock when it is repeatedly sold,

we find that the stock price is the PV of all future dividends.

4. SPECIAL CASE [1]. NO GROWTH IN DIVIDEND.

- DIV1 = DIV2 = ….=DIV

- Similar to preferred stock

- Good approximation for many utility stocks

- Ordinary perpetuity

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5. SPECIAL CASE [2]:CONSTANT GROWTH RATE IN DIVIDEND

- DIV2 = DIV1(1+g)

- DIV3 = DIV2(1+g) = DIV1(1+g)2

- And so on …..

- If dividends are expected to grow at a constant rate (ie, g), the value of the stock is

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From the previous Equation, we can get

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6. SPECIAL CASE [3]:SUPERNORMAL GROWTH RATE IN DIVIDEND

: Firms typically go through life cycles. During the early part of their lives, their growth is much faster than that of the economy as a whole; then they match the economy’s growth; and finally their growth is slower than that of the economy.

Example) UH Corporation has been growing at a rate of 20% per year in recent years. This same growth rate is expected to last another 2 years, then it will grow at a constant rate of 6% thereafter. What is UHC’s stock worth today, given D0=$1.60 and r=10% ? [$54.11]

Solve all the Questions and Problems in Text book [Pg 121 - 124, and Pg.133]

- Except 5.31 on Pg. 124

Exercise For Time Value of Money

Now 1 2 3 4 5

Option A $60 $60 $60 $60 $60+$1000

Option B ???

Q1) Given that interest rate is 2%, how much of the money should I give to you NOW so that you are indifferent between two options? (ANSWER:$1188.54)

Q2) Given that interest rate is 15%, how much of the money should I give to you NOW so that you are indifferent between two options? (ANSWER:$698.31)

Now 1 2 3 4 5

Option A $127.63

Option B ???

Q3) Given that interest rate is 5%, how much of the money should I give to you NOW so that you are indifferent between two options? (ANSWER:$100)

Now 1 2 3 4 5

Option A ???

Option B $100

Q4) Given that interest rate is 5%, how much of the money should I give to you AT THE END OF YEAR_5 so that you are indifferent between two options? (ANSWER:$127.63)

Now 1 2 3 4 5

Option A ? ? ? ? ?

Option B $100

Q5) Given that interest rate is 5%, how much of the fixed amount of money should I give to you AT THE END OF YEAR_5 so that you are indifferent between two options? (ANSWER:$23.10)

Q6) Suppose you can buy a certificate for $78.35 which will pay you $100 after 5 years. What is the interest rate you will earn on your investment? (ANSWER: 5%)

Q7) Suppose you know that a bond will provide a return of 5% per year, that it costs you $783.5 now, and that you will receive $1,000 at the maturity. But you don’t know when the bond matures. (ANSWER : n=5)

Now 1 2 3

Option A $100 $100 $100

Option B ?

Q8) Given that interest rate is 5%, how much of the money should I give to you AT THE END OF YEAR 3 so that you are indifferent between two options? (ANSWER:$315.25)

Now 1 2

Option A $100 $100 $100

Option B ?

Q9) Given that interest rate is 5%, how much of the money should I give to you AT THE END OF YEAR 2 so that you are indifferent between two options? (ANSWER:$315.25)

Now 1 2

Option A $100 $100 $100

Option B ?

Q10) Given that interest rate is 5%, how much of the money should I give to you NOW so that you are indifferent between two options? (ANSWER:$285.94)

Exercise For Time Value of Money (Continued)

Now 1 2 3 4 5

Option A 100 200 200 0 300

Option B ?

Q11) Given that interest rate is 5%, how much of the money should I give to you NOW so that you are indifferent between two options? (ANSWER:$684.47)

Now 1 2 3 4 5

Option A $100 100 200 200 0 300

Option B ? ?

Q12) Given that interest rate is 5%, how much of the money should I give to you NOW so that you are indifferent between two options? (ANSWER:$784.47)

Now 1 2 3 4 5

Option A 100 200 200 0 300

Option B ?

Q13) Given that interest rate is 5%, how much of the money should I give to you AT THE END OF YEAR 5 so that you are indifferent between two options? (ANSWER:$873.58)

Now 1 2 3 4 5

Option A 200 200 200

Option B ?

Q14) Given that interest rate is 5%, how much of the money should I give to you AT THE END OF YEAR 5 so that you are indifferent between two options? (ANSWER:$695.13)

Now 1 2 3 4 5

Option A 200 200 200 200 Forever

Option B ?

Q15) Given that interest rate is 5%, how much of the money should I give to you NOW so that you are indifferent between two options? (ANSWER:$4,000)

( Effective Annual Rate

( Continuous Compounding

( Semiannual (or Quarterly, or Monthly, or Dalily) Compounding and Payment

( Find the amount to which $500 will grow under each of the following conditions.

12 percent compounded annually for 5 years

12 percent compounded semiannually for 5 years

12 percent compounded monthly for 5 years

12 percent compounded daily for 5 years [Assume 1 year =360 days]

12 percent compounded continuously for 5 years

( The First Bank pays 7% interest, compound annually, on time deposit. The Second Bank pays 6% interest, compounded quarterly. Which bank gives you higher interest? How can you justify your choice of the bank?

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