Valuing bonds and stocks Yields and growth
Dec 21, 2015
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 PVAF*100 = 9007.3451
Step 2. Translate to money of time 120.
[(1.005)^120]*9007.3451 = 16387.934
Spreadsheet confirmation
Time Start Pay End0 91.825 -25 66.8251 70.8345 -25 45.83452 48.58457 -25 23.584573 24.99964 -25 -0.000364 -0.00038 -0.00038
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.
Numerical Solution
PV of target sum = 36.146687 PV of savings = C+C*10.105895 Solve C*11.105695 = 36.14667 C = 3.2547298
Confirmation in an excel spread sheet.
0 3.25473 3.254731 3.25473 6.7047442 3.25473 10.36176… …15 3.25473 83.557116 3.25473 91.8253
Confirmation in an excel spread sheet.
Time contribution balance0 3.25473 =B21 3.25473 =1.06*C2+B3=A3+1 3.25473 =1.06*C3+B4=A4+1 3.25473 =1.06*C4+B5=A5+1 3.25473 =1.06*C5+B6=A6+1 3.25473 =1.06*C6+B7=A7+1 3.25473 =1.06*C7+B8=A8+1 3.25473 =1.06*C8+B9=A9+1 3.25473 =1.06*C9+B10
Apply the formula to a Bond
Time 0 0.5 1 1.5 … TCash flow 0 C C C … CCash flow 1000
This is a bond maturing T full yearsfrom now with coupon rate 2C/1000
Yield
Yield is the 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 is P = C*PVAF(r/2,2T) + 1000/(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) + 1000/(1+r/2)^2T
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
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 < coupon rate, it sells at a premium(P>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)
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%. It sells today at par, that is, for $1000. What is the yield? Prove it.
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.