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 =  PVAF*100 =  Step 2. Translate to money of time 120.  [(1.005)^120]* =

Present value of annuity factor

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

Timing  Obviously simplified

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

Spreadsheet confirmation

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

The college savings problem

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

Numerical Solution  PV of target sum =  PV of savings = C+C*  Solve C* =  C =

Confirmation in an excel spread sheet.

Finish here 1/17/06

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.

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.

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) /(1+r/2) 2T

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

A typical bond

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

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.

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

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)

Growing perpetuity

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

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.

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.

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

Answer two: proof  1000/(1.04) *(1/.04)[1-1/(1.04) 20 ] = = 1000

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

Growing perpetuity

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?

Solution PP=4*(1/( )) ==

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

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/( ))

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%).

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

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