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Valuing bonds and stocks

Valuing bonds and stocks. Yields and growth. 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?. Answer in two steps. Step 1. Find PDV of the annuity. .005 per month 120 months PVAF = 90.073451

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Valuing bonds and stocks

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  1. Valuing bonds and stocks Yields and growth

  2. 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?

  3. 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

  4. Present value of annuity factor

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

  6. Timing • Obviously simplified

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

  8. Spreadsheet confirmation

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

  10. The college savings problem

  11. 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.

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

  13. Confirmation in an excel spread sheet.

  14. Finish here 1/17/06

  15. 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.

  16. 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.

  17. 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

  18. 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

  19. A typical bond

  20. 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

  21. 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.

  22. More Facts • Yield > coupon rate, bond sells at a discount (P<1000) • Yield < coupon rate, it sells at a premium(P>1000)

  23. 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)

  24. Growing perpetuity

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

  26. 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.

  27. 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.

  28. Answer one • yield = coupon rate. • You must know that.

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

  30. 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.

  31. Growing perpetuity

  32. 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?

  33. Solution • P=4*(1/(.16-.05)) • =36.3636...

  34. 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...

  35. 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))

  36. 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%).

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

  38. Why should we be skepticalabout the PV growing perpetuity • The value is coming from far distant years.

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