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 PVAF*100 = 9007.3451

Step 2. Translate to money of time 120.

[(1.005)^120]*9007.3451 = 16387.934

Present value of annuity factor

Trr

TrPVAF)1(

11

1),(

Time 0 1 2 … T T+1Cash flow 0 1 1 … 1 0

Example: Cost of College

Annual cost = 25000 Paid when? Make a table of cash flows

Timing

Obviously simplified

Time 0 1 2 3 4Cash flow -25 -25 -25 -25 0

Present value at time zero

25+25*PVAF(.06,3) =91.825298

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

Saving for college

Start saving 16 years before matriculation.

How much each year? Make a table.

The college savings problem

Time 0 1 2 … 16Savings C C C … CFinal value 91.8253

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

A typical bond

T = 0 .5 1 1.5

Coupon 0 60 60 60

Principal 0 0 0 1000

Total 0 60 60 1060

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)

Growing perpetuity

Time 0 1 2 3 …Cash flow 0 1 (1+g) (1+g) 2̂ …

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 one

yield = coupon rate. You must know that.

Answer two: proof

1000/(1.04)^20 + 40*(1/.04)[1-1/(1.04)^20] = 456.3869462+543.6130537 = 1000

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.