Your monthly payments

No arbitrage pricing.

Key concepts

Real investment Financial investment

Interest rate defined

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pr

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Basic principle

Firms maximize value Owners maximize utility Separately

Justification

Real investment with positive NPV shifts consumption opportunities outward.

Financial investment satisfies the owner’s time preferences.

A typical bond

T = 0 .5 1 1.5

Coupon 0 60 60 60

Principal 0 0 0 1000

Total 0 60 60 1060

Note: Always start with the time line.

Definitions

Coupon -- the amount paid periodically Coupon rate -- the coupon times annual

payments divided by 1000

Two parts of a bond

Principal paid at maturity. A repeated constant flow -- an annuity

Strips

U.S. Treasury bonds Stripped coupon is an annuity Stripped principal is a payment of 1000

at maturity and nothing until then. Stripped principal is also called a pure

discount bond, a zero-coupon bond, or a zero, for short.

No arbitrage condition:

Price of bond = price of zero-coupon bond + price of stripped coupon.

Otherwise, a money machine, one way or the other.

Riskless increase in wealth

Pie theory

The bond is the whole pie. The strip is one piece, the zero is the

other. Together, you get the whole pie. No arbitrage pricing requires that the

values of the pieces add up to the value of the whole pie.

Yogi Berra on finance

Cut my pizza in four slices, please. I’m not hungry enough for six.

Why use interest rates?

In addition to prices? Answer: Coherence

Example: discount bonds

A zero pays 1000 at maturity. Price (value) is the PV of that 1000

cash flow, using the market rate specific to the asset and maturity.

Example continued

Ten-year maturity: price is 426.30576 Five-year maturity: price is 652.92095 Similar or different? They have the SAME discount rate

(interest rate) r = .089 (i.e. 8.9%)

Calculations

652.92095 = 1000 / (1+.089)5

Note: ^ is spreadsheet notation for raising to a power

426.30576 = 1000 / (1+.089)10

More realistically

For the ten-year discount bond, the price is 422.41081 (not 426.30576).

The ten-year rate is (1000/422.41081)1/10 - 1 = .09.

The 1/10 power is the tenth root. It solves the equation

422.41081 = 1000/(1+r)10

Annuity

Interest rate per period, r. Size of cash flows, C. Maturity T. If T=infinity, it’s called a perpetuity.

Market value of a perpetuity

Time 0 1 2 …

Cash flow

0 C C …

PV 0 C/(1+r) C/(1+r)2 …

Value of a perpetuity is C*(1/r)

In spreadsheet notation, * is the sign for multiplication.

Present Value of Perpetuity Factor, PVPF(r) = 1/r It assumes that C = 1.

For any other C, multiply PVPF(r) by C.

Finished here 1/12/06

Value of an annuity

C (1/r)[1-1/(1+r)T] Present value of annuity factor PVAF(r,T) = (1/r)[1-1/(1+r)T] or AT

r

Explanation

Annuity = difference in perpetuities. One starts at time 1, the other starts at time T + 1. Value = difference in values (no

arbitrage).

Explanation

Time 0 1 2 .. T-1 T T+1 T+2 …

Perp at 0 0 1 1 … 1 1 1 1 …-Perp at T 0 0 0 … 0 0 1 1 …

Annuity 0 1 1 … 1 1 0 0 …

Values

Value of the perpetuity starting at 1 is = 1/r … in time zero dollars Value of the perpetuity starting at T + 1 is =

1/r … in time T dollars, or (1/r)[1/(1+r)T] in time zero dollars. Difference is PVAF(r,T)= (1/r)[1-1/(1+r)T]

Compounding

12% is not 12% … ? … when it is compounded.

E.A.R. Equivalent Annual rate

Start Formula End E.A.R.

annual 1000 (1+.12)1 1120 0.12

monthly 1000 (1+.12/12)12 1126.825 0.12683

daily 1000 (1+.12/365)365 1127.475 0.12747continuous 1000 exp(.12) 1127.497 0.127497

Example: which is better?

Wells Fargo: 8.3% compounded daily World Savings: 8.65% uncompounded

Solution

Compare the equivalent annual rates World Savings: EAR = .0865 Wells Fargo: (1+.083/365)365 -1

= .0865314

Exam (sub) question

The interest rate is 6%, compounded monthly.

You set aside $100 at the end of each month for 10 years.

How much money do you have at the end?

Answer

t= 0 1 2 … 120

CF 0 100 100 100

Interest per period is .5% or .005.

Present value is PVAF(120,.005)*100 = 9007.3451

Future value is 9007.3451*(1.005)120 = 16387.934