Lecture Notes

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**1. **Lecture Notes Economics 134a

**2. **October 6, 2005 2 Reminder First midterm is Tuesday
Practice exams are on the class page.
They may cover more material, since they were given later in the quarter
We will not have completed the first unit of the course. EXAM WILL COVER ONLY MATERIAL DISCUSSED IN CLASS.
RWJ up to and including Ch. 5

**3. **October 6, 2005 3 Overflow room Some of you should go to the overflow room (I don’t know what it is yet).
Check on the class page on Tuesday for the overflow room

**4. **October 6, 2005 4 Growing Payment Stream (Growing, that is, at a constant rate g)
Current value of the payment stream is C
Since we are valuing ex-dividend, the payment stream starts at C(1+g)
So we have the stream (C(1+g), C(1+g)2, C(1+g)3, …)

**5. **October 6, 2005 5 The Present Value is

**6. **October 6, 2005 6 To evaluate this, … Multiply the equation by (1+g)/(1+r) and subtract from the original equation.
You get

**7. **October 6, 2005 7 Simplifying results in

**8. **October 6, 2005 8 Dependence on g Must have g<r. Otherwise the stream would have infinite value.
Constant perpetuity is a special case (g=0)
Note the sensitivity of value to the estimate of g.
Suppose r=12%.
If g=6%, PV=17.7C
If g=8%, PV = 27C
This is bad news: in security valuation, g is usually hard to estimate precisely.

**9. **October 6, 2005 9 Growing Annuity Formula We did
(1) constant perpetuity
(2) constant annuity
(3) growing perpetuity
Could also do growing annuity,
But I’m not. It isn’t often needed.

**10. **October 6, 2005 10 Problem: The Power of Compound Interest Smith and Jones are 20.
Smith saves $1 per year year in his 20s, nothing in his 30s or 40s
Jones saves nothing in his 20, but saves $1 per year in his 30s and 40s.
Who will have more money at age 50?

**11. **October 6, 2005 11 Solve Using the Annuity formula Answer depends on what you assume the return to savings is. Try 10%
Use annuity formula for present value, multiply by (1+r)T to get future value.
Answer: Smith has more $ at age 50!
Even though he saved half as much, the fact that he saved earlier (so there’s more compounding) dominates.

**12. **October 6, 2005 12 Chapter 5 Valuing Stocks and Bonds
Stock (equity) – owner of the firm
Bond – creditor of the firm
The firm must pay interest and principal on bonds, or declare bankruptcy.

**13. **October 6, 2005 13 Types of Bonds Pure discount
No coupon
Market value of a pure discount bond
Coupon bonds
“Coupon” paid, generally twice annually, until the maturity date, at which time the principal must be repaid.

**14. **October 6, 2005 14 Coupon bonds The coupons are valued using the annuity formula
The principal repayment is valued as if it were a discount bond.
In both expressions, the interest rate used is the “yield”

**15. **October 6, 2005 15 Note We haven’t really explained the price of the bond
We’ve just shown the relation between the price and the yield (interest rate).
This is taking the face value (nominal value, par value) of the bond to be $1.
Typically it’s $1000.
Bonds typically quoted in units of 100 (not 1000 or 1). Interpret as % of face value

**16. **October 6, 2005 16 Effective Annual Rate Bond coupons are generally paid twice annually.
So interest is compounded twice annually
We will often assume once annually
Bond yields are quoted as “stated annual interest”.
So an 8% bond means coupon are 4% each 6-month period
To convert to effective annual interest, use the formula we derived:

**17. **October 6, 2005 17 Example An 8% bond trades at par
See below; this means that its market price = maturity value.
has an effective annual yield of (1.04)2-1, or 8.16%.

**18. **October 6, 2005 18 Price and Yield (yield = rate at which coupons and principal payment are discounted)
Suppose you have a bond yielding 6% (with annual payments)
And the yield is 6%
Then the current value = the nominal value (1)
Let’s verify this using the annuity formula
You can also use a spreadsheet.

**19. **October 6, 2005 19 From the annuity formula …
Here r is the annual yield/2
(Get the first term by substituting C=r in the annuity formula)
First term is from the annuity formula (since the coupon is r)
Second term is the discounted value of principal repayment

**20. **October 6, 2005 20 Conclusion If the yield = the coupon rate, the bond “trades at par”
That is, its market price = its face value.

**21. **October 6, 2005 21 Interest rate changes Coupon bonds are usually issued at par
Or as near as possible.
Meaning the coupon rate is set approximately equal to currently prevailing interest rates.
As time passes, the coupon stays the same, but interest rates rise or fall.
Suppose they rise. Then, from the bond valuation formula, value of bonds drops.
If not, whoever bought the bonds would earn less interest than could be earned on other investments
“discount bond”

**22. **October 6, 2005 22 Example Compute the value of a 5% 10-year bond if interest rates rise to 6%
Assume annual coupons
What’s the price?

**23. **October 6, 2005 23 Very important Note the negative relation between price and yield: yield rises, price drops!
“Discount bond”
Coupon rate is lower than discount rate
Price < face value
Here “yield” means “yield to maturity”
There are other types of yield.
Opposite case: “premium bond”

**24. **October 6, 2005 24 Going in reverse Given the price, you can do the same calculation in reverse to get the yield to maturity.
This involves solving a polynomial. Generally you need to do this numerically.
Guess the yield. If the implied price is too high (low), you guessed too low (high) a yield. Change your guess and try again. Keep going until you are very close to the price.
This is a natural for a spreadsheet.

**25. **October 6, 2005 25 Example A 10-year 9% bond (with annual coupons) has a price of 1.0671.
Note: bond prices are usually quoted in units of 100, so as to be interpreted as % of par value
So this would be 106.71.
If par is $1000, the actual price would be $1067.10.
What is its yield?
Since this is a premium bond, you know that the yield is lower than the coupon. It turns out to be 8%.

**26. **October 6, 2005 26 Valuing Stocks From PV formula, we have
This is equivalent to

**27. **October 6, 2005 27 return = div. yield + capital gain Which says that the return on stock (equal to the discount factor) equals the sum of the dividend yield
Dividends/price
Plus the rate of capital gain.

**28. **October 6, 2005 28 Constant dividend streams If dividends are (expected to be) constant, the perpetuity formula gives

**29. **October 6, 2005 29 Growing dividend streams If the dividend is expected to grow at rate g, the growing perpetuity formula gives

**30. **October 6, 2005 30 Where does g come from? Retained earnings
If
Where d is the dividend payout rate (D/E), and E is current earnings, then

**31. **October 6, 2005 31 Dividends or Earnings? This assumes that retained earnings generate a return at rate r
Then we also have

**32. **October 6, 2005 32 Proof

**33. **October 6, 2005 33 To prove this, start with the perpetuity formula (slide (28).
Use the expression for g in the preceding slide to Substitute out g in the denominator.
Since dE = D, you can cancel

**34. **October 6, 2005 34 Conclusion You can value stocks as a growing perpetuity starting from D(1+g)
Or a constant perpetuity starting from E(1+g)

**35. **October 6, 2005 35 Very important! Growing perpetuity formula applies to dividends, not earnings, because that’s what stockholders actually receive.
So price of stock = value of dividends, taking account of growth

**36. **October 6, 2005 36 Growth Opportunities If you think the firm has an investment opportunity that yields a return greater than r, use PV formula to value it!
NPVGO (net present value of growth opportunities)
Growth opportunities = investments that yield a return GREATER THAN R
(Investments that have return r have zero NPV)
Using perpetuity formula implicitly assume no growth opportunities
P = value from growing perpetuity formula + NPVGO

**37. **October 6, 2005 37 Zero Dividend firms What is the value of a zero-dividend firm?
Most zero dividend firms will eventually pay a dividend in the future.
Microsoft
What about a firm that promises (credibly) never to pay a dividend?
PV formula says its value is zero
Bubble: P > PV.

**38. **October 6, 2005 38 Bubbles Another way to think about NPV.
Start from slide 4:
Write this down again, replacing 0 by 1, 1 by 2

**39. **October 6, 2005 39 Substitute The second equation in the first:
Keep doing this. You get
P=PV + the limit of Pt/(1+r)t

**40. **October 6, 2005 40 Bubbles The limit of Pt/(1+r)t, if nonzero, is the bubble.
Price of an asset > NPV of dividends if there’s a bubble.
So in using NPV, we’re assuming there’s no bubble.