2. Valuing bonds and stocks Yields and growth

3. Exam (sub) question  r = 6%, compounded monthly.  Save $100 at the end of each month for 10 years.  Final value, in dollars of time 120?

4. Answer in two steps  Step 1. Find PDV of the annuity. .005 per month  120 months  PVAF = 90.073451  PVAF*100 = 9007.3451  Step 2. Translate to money of time 120.  [(1.005)^120]*9007.3451 = 16387.934

5. Present value of annuity factor

6. Example: Cost of College  Annual cost = 25000  Paid when?  Make a table of cash flows

7. Timing  Obviously simplified

8. Present value at time zero  25+25*PVAF(.06,3)  =91.825298

9. Spreadsheet confirmation

10. Saving for college  Start saving 16 years before matriculation.  How much each year?  Make a table.

11. The college savings problem

12. Solution outlined  Target = 91.825 dollars of time 16.  Discount to dollars of time 0.  Divide by (1.06) 16  Result 36.146687…, the new target  PV of savings =C+C*PVAF(.06,16)  Equate and solve for C.

13. Numerical Solution  PV of target sum = 36.146687  PV of savings = C+C*10.105895  Solve C*11.105695 = 36.14667  C = 3.2547298

14. Confirmation in an excel spread sheet.

15. Finish here 1/17/06

16. Apply the formula to a Bond This is a bond maturing T full years from now with coupon rate 2C/1000. C is the coupon payment.

17. Yield  Yield is a market rate now.  Coupon rate is written into the bond.  It is near the market rate when issued.  Yield and coupon rate are different.

18. Given the yield, r  Yield r for a bond with semi-annual coupons means r/2 each 6 months.  Value of the bond that matures in T years is  P = C*PVAF(r/2,2T) + 1000/(1+r/2) 2T

19. Given the price of the bond, P  Yield is the r that satisfies the valuation equation  P=C*PVAF(r/2,2T) + 1000/(1+r/2) 2T

20. A typical bond

21. Value at yield of 5%  Pure discount bond (the 1000): Value =1000/(1.025) 3 =928.599…  Strip: ( the coupon payments) 60*(1/.025)(1-1/(1.025) 3 )  =171.3614…  Total market value of bond =1099.96

22. Facts of bonds  They are called,  at the option of the issuer when interest rates fall.  or retired in a sinking fund,  as required to assure ultimate repayment.

23. More Facts  Yield > coupon rate, bond sells at a discount (P<1000)  Yield 1000)

24. Growing perpetuities  Thought to be relevant for valuing stocks  Present value of growing perpetuity factor PVGPF  g = growth rate (decimal)  r = interest rate (decimal)  PVGPF(r,g) = 1/(r-g)

25. Growing perpetuity

26. Riddle  What if the growth rate is above the discount rate?  Formula gives a negative value.  Correct interpretation is infinity.

27. More riddle: market response  An investment with growth rate above the interest rate.  Others copy the investment until competition drives the growth rate down  or until …  the opportunity drives the interest rate up.

28. Review question  A bond has a coupon rate of 8%.  The maturity is 10 years from now.  It sells today at par, that is, for $1000.  What is the yield?  Prove it.

29. Answer one  yield = coupon rate.  You must know that.

30. Answer two: proof  1000/(1.04) 20 + 40*(1/.04)[1-1/(1.04) 20 ] = 456.3869462+543.6130537 = 1000

31. Answer two: deeper proof  1000/(1.04) 20 + 40*(1/.04)[1-1/(1.04) 20 ]  1000/(1.04) 20 + 1000-1000/(1.04) 20  End terms cancel. Answer = 1000.

32. Growing perpetuity

33. Example: share of stock  The market expects a dividend of $4 in one year.  It expects the dividend to grow by 5% per year  The discount rate for such firms is 16%.  What is the price of a share?

34. Solution PP=4*(1/(.16-.05)) ==36.3636...

35. Decomposition of value  Absent growth, as a cash cow, value = 4*(1/.16)  = 25.  Remaining value of 36.3636… - 25 is net present value of growth opportunities (NPVGO).  =11.3636...

36. Example: whole firm  The market expects $30M in one year  and growth of 2% thereafter.  Discount rate = 17%.  Value of the firm is $200M.  That is 30M*(1/(.17-.02))

37. continued  A new line of business for the firm is discovered.  The market expects $20M in a year,  with growth at 7% thereafter.  Value of the new growth opportunity is $200M (at r = 17%).

38. Whole value: 400M = 200M + 200M  Note that the value is gross, not net.  Share price?  Divide by the number of shares.

39. Why should we be skeptical about the PV growing perpetuity  The value is coming from far distant years.