Annuities and No Arbitrage
Pricing
Key concepts
 Real
investment
 Financial investment
Interest rate defined
 Premium
for current delivery
p0
r
1
p1
1
p1

1  r p0
equation of the budget
constraint:
c1
p0c0  p1c1  p0 c0  p1 c1
p0
slope  
 (1  r )
p1
(c0 , c1 ) = status quo
c0
Time zero cash flow
Financing possibilities,
not physical investment
c1
Withdrawal
(c0 , c1 )
deposit
c0
Time zero cash flow
c1
An investment opportunity
that increases value.
(c0 , c1 )
Time zero cash flow
NPV
c0
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.
Why use interest rates
 Instead
of just prices
 Coherence
Example: pure discount bond
 Definition: A pure
discount bond pays
1000 at maturity and has no interest
payments before then.
 Price is the PV of that 1000 cash flow,
using the market rate specific to the
asset.
Example continued
 Ten-year
discount bond: price is
426.30576
 Five-year discount bond: price is
652.92095
 Are they similar or different?
 Similar because they have the SAME
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 - 1 = .09
The .1 power is the tenth root.
 The longer bond has a higher interest
rate. Why?
 Because more time means more risk.
A typical bond
T=
0
.5
1
1.5
Coupon 0
60
60
60
Principal 0
0
0
1000
Total
60
60
1060
0
Definitions
 Coupon
-- the amount paid periodically
 Coupon rate -- the coupon times annual
payments divided by 1000
 Same as for mortgage payments
Pure discount bonds on 1/9/02
Matures
Feb 04
Feb 05
Feb 06
Feb 07
Feb 08
Feb 09
Feb10
Ask
98:20
96:13
93:12
89:15
85:17
80:15
76:30
Ask yield
1.27
1.75
2.23
2.73
3.09
3.46
3.73
No arbitrage principle
 Market
prices must admit no profitable,
risk-free arbitrage.
 No money pumps.
 Otherwise, acquisitive investors would
exploit the arbitrage indefinitely.
Example
 Coupons
sell for 450
 Principal sells for 500
 The bond MUST sell for 950.
 Otherwise, an arbitrage opportunity exists.
 For instance, if the bond sells for 920…
 Buy the bond, sell the stripped components.
Profit 30 per bond, indefinitely.
 Similarly, if the bond sells for 980 …
Two parts of a bond
 Pure
discount bond
 A repeated constant flow -- an annuity
Stripped coupons and principal
 Treasury
notes (and some agency
bonds)
 Coupons (assembled) sold separately,
an annuity.
 Stripped principal is a pure discount
bond.
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
 Start
with a perpetuity.
Time
0
1
2
…
Cash
flow
PV
0
C
C
…
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.
Justification
…
4
3
2
1
0
emiT
…
4^)r +1(/1 + 3^)r +1(/1 + 2^)r +1(/1 + )r +1(/1 +
0
= FPVP
…
…
3^)r +1(/1 + 2^)r +1(/1 + )r +1(/1 +
3^)r +1(/1- 2^)r +1(/1- )r +1(/1-
0
0
= )r +1(*FPVP
= FPVP-
1+
1
r/1
= r*FPVP
= FPVP
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 ArT
Explanation
 Value
of annuity =
 difference in values of perpetuities.
 One starts at time 1,
 the other starts at time T + 1.
Explanation
Time
0
1
2
..
T-1
T
T+1
T+2
…
Perp at 0
-Perp at T
0
0
1
0
1
0
…
…
1
0
1
0
1
1
1
1
…
…
Annuity
0
1
1
…
1
1
0
0
…
Values
 P.V.
of Perp at 0 = 1/r
 P.V. of Perp at T = (1/r) 1/(1+r)^T
 Value of annuity = difference = (1/r)[11/(1+r)^T ]
Compounding
 12%

is not 12% … ?
… when it is compounded.
Compounding: E.A.R.
Equivalent Annual rate
Start
annual 1000
monthly 1000
daily 1000
continuous 1000
Formula
End
E.A.R.
(1+.12)^1
1120
0.12
(1+.12/12)^12 1126.825 0.12683
(1+.12/365)^365 1127.475 0.12747
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
When to cut a tree
 Application
of continuous compounding
 A tree growing in value.
 The land cannot be reused.
 Discounting continuously.
 What is the optimum time to cut the
tree?
 The time that maximizes NPV.
Numerical example
 Cost
of planting = 100
 Value of tree -100+25t
 Interest rate .05
 Maximize (-100+25t)exp(-.05t)
 Check second order conditions
 First order condition .05 = 25/(-100+25t)
 t = 24 value = 500
Example continued
 Present
value of the tree =
500*exp(-.05*24) = 150.5971.
 Greater than cost of 100.
 NPV = 50.5971
 Market value of a partly grown tree at
time t < 24 is 150.5971*exp(.05*t)
 For t > 24 it is -100+25*t
Example: Cost of College
 Annual
cost = 25000
 Paid when?
 Make a table of cash flows
Timing
 Obviously
Time
Cash flow
simplified
0
-25
1
-25
2
-25
3
-25
4
0
Present value at time zero
 25+25*PVAF(.06,3)
 =91.825298
Spreadsheet confirmation
Start
91.825
70.8345
48.58457
24.99964
-0.00038
Pay
-25
-25
-25
-25
Balance
66.825
45.8345
23.58457
-0.00036
-0.00038
Saving for college
 Start
saving 16 years before
matriculation.
 How much each year?
 Make a table.
The college savings problem
Time
Savings
Final value
0
C
1
C
2
C
…
…
16
C
91.8253
Solution outlined
 Find
PV of target sum, that is, take
91.825 and discount back to time 0.
 Divide by (1.06)^16
 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
 C = 3.2547298
Balance = C + 1.06 previous
balance
1
2
…
15
16
3.25473
3.25473
…
3.25473
3.25473
6.704744
10.36176
83.5571
91.8253
Alternative solution outlined
 Need
91.825 at time 16.
 FV of savings =(1.06)^16
*(C+C*PVAF(.06,16))
 Equate and solve for C.
Numerical Solution
 Future
target sum = 91.825
 FV of savings =
(1.06)^16*(C+C*10.105895)
 91.825 = C*((1.06)^16)*(1+10.105895)
 C = 3.2547298
Review 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 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