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Topics Covered
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Future Values
Present Values
Multiple Cash Flows
Perpetuities and Annuities
Inflation & Time Value
Future Values
Example - Simple Interest
Interest earned at a rate of 6% for five years on a
principal balance of £100.
Today
Interest Earned
Value
100
1
6
106
Future Years
2
3
4
5
6
6
6
6
112 118 124 130
Value at the end of Year 5 = £130
Future Values
Example - Compound Interest
Interest earned at a rate of 6% for five years on
the previous year’s balance.
Today
Interest Earned
Value
100
Future Years
1
2
3
4
5
6.00 6.36 6.74 7.15 7.57
106.00 112.36 119.10 126.25 133.82
Value at the end of Year 5 = £133.82
Future Values
FV  £100  (1  r )
t
Example - FV
What is the future value of £100 if interest is
compounded annually at a rate of 6% for five years?
FV  £100  (1  .06)  £133.82
5
Manhattan Island Sale
Peter Minuit bought Manhattan Island for £16 in 1626.
Was this a good deal?
To answer, determine £16 is worth in the year 2001,
compounded at 8% and 5%.
FV  £16  (1  .08)
375
 £54.705 trillion
FV  £16  (1  .05)
375
 £1.412 trillion
Present Values
• Present Value: Value today of a future cash
flow
• Discount Rate: Interest used to calculate
value of future cash flows.
Present Values
Present Value = PV
PV =
Future Value after t periods
(1+r) t
Present Values
Example
You just bought a new computer for $3,000. The payment
terms are 2 years same as cash. If you can earn 8% on
your money, how much money should you set aside today
in order to make the payment when due in two years?
PV 
3000
(1.08)2
 $2,572
Time Value of Money
(applications)
• The PV formula has many applications.
Given any variables in the equation, you
can solve for the remaining variable.
PV  FV 
1
( 1 r ) t
PV of Multiple Cash Flows
Example
Your auto dealer gives you the choice to pay $15,500 cash
now, or make three payments: $8,000 now and $4,000 at
the end of the following two years. If your cost of money is
8%, which do you prefer?
Immediat epayment 8,000.00
PV1 
4 , 000
(1.08)1
 3,703.70
PV2 
4 , 000
(1.08) 2
 3,429.36
Total PV
 $15,133.06
PV of Multiple Cash Flows
• PVs can be added together to evaluate
multiple cash flows.
PV 
C1
( 1 r )
 (1 r ) 2 ....
C2
1
Perpetuities & Annuities
Perpetuity
A stream of level cash payments
that never ends.
Annuity
Equally spaced level stream of cash
flows for a limited period of time.
Perpetuities & Annuities
Example - Perpetuity
In order to create an endowment, which pays
£100,000 per year, forever, how much money must
be set aside today in the rate of interest is 10%?
Perpetuities & Annuities
Example - Perpetuity
In order to create an endowment, which pays
£100,000 per year, forever, how much money must
be set aside today in the rate of interest is 10%?
PV 
100, 000
.10
 £1,000,000
Perpetuities & Annuities
PV of Annuity Formula
PV  C

1
r

1
t
r ( 1 r )

C = cash payment
r = interest rate
t = Number of years cash payment is received
Arbitrage
• You are given the following prices Pt today
for receiving risk free £1 payments t
periods from now.
T=
Pt=
1
0.95
2
0.9
3
0.95
• How would you make a lot of money?
Compounding
• Natwest is offering loans at 10% interest
compounded quarterly.
• Barclays is offering loans at 10.5% interest
compounded annually.
• Which would you take?
Compounding Formula
• General Formula is
FVt  PV 1 

r nt
n
rt
• Continuous is FVt  PV (e )
Why? Look at % increase (slope/value)
• What is continuous rate for 10% annual?
• How long would it take to double your money?
Mortgage payments
• You want to buy a home for £100,000.
• Natwest offers you a mortgage: 0 down,
10% a year for 25 years.
• How much must you pay per year?
Save and Retire.
• You plan to save £4,000 every year for 20 years
and then retire. Given a 10% rate of interest, what
will be the FV of your retirement account?
Growth and Perpetuities
• What is the present value of a perpetuity
whose payment grows at a rate of (1+g) per
year?
Inflation
Inflation - Rate at which prices as a whole are
increasing.
Nominal Interest Rate - Rate at which money
invested grows.
Real Interest Rate - Rate at which the
purchasing power of an investment increases.
Inflation
1+ nominal interest rate
1  real interest rate =
1+inflation rate
approximation formula
Real int. rate  nominal int. rate - inflation rate
Bonds
Terminology
• Bond - Security that obligates the issuer to
make specified payments to the bondholder.
• Coupon - The interest payments made to the
bondholder.
• Face Value (Par Value or Maturity Value) - Payment
at the maturity of the bond.
• Coupon Rate - Annual interest payment, as a
percentage of face value.
Bonds
WARNING
The coupon rate IS NOT the discount rate
used in the Present Value calculations.
The coupon rate merely tells us what cash flow the
bond will produce.
Since the coupon rate is listed as a %, this
misconception is quite common.
Bond Pricing
The price of a bond is the Present Value of
all cash flows generated by the bond (i.e.
coupons and face value) discounted at the
required rate of return.
Bond Sensitivity
• A zero coupon bond pays £10000 in 10
years time.
• What is the PV of the bond if interest is
10% annual?
• What is the PV of the bond if interest falls
to 9% annual?
Bond Pricing
The price of a bond is the Present Value of
all cash flows generated by the bond (i.e.
coupons and face value) discounted at the
required rate of return.
cpn
cpn
(cpn  par )
PV 

....
1
2
t
(1  r ) (1  r )
(1  r )
Bond Pricing
• What is the price of a 6 % annual coupon bond,
with a $1,000 face value, which matures in 3
years? Assume a required return of 5%.
• What is the price if the return is 6%
Bond Yields
• Current Yield - Annual coupon payments
divided by bond price.
• Yield To Maturity - Interest rate for which
the present value of the bond’s payments
equal the price.
Bond Yields
Calculating Yield to Maturity (YTM=r)
If you are given the price of a bond (PV)
and the coupon rate, the yield to maturity
can be found by solving for r.
cpn
cpn
(cpn  par )
PV 

....
1
2
t
(1  r ) (1  r )
(1  r )
Interest Rate Risk
1,080
1,060
Premium Bond
1,040
Bond Price
1,020
1,000
980
960
Discount Bond
940
920
900
880
0
5
10
15
Time (Matures at 30)
20
25
30
Interest Rate Risk
3,000
2,500
30 yr bond
$ Bond Price
2,000
1,500
1,000
3 yr bond
500
0
2
4
Interest Rate
6
8
10
Default Risk
•
•
•
•
Credit risk
Default premium
Investment grade
Junk bonds
Default Risk
Moody' s
Standard
& Poor's
Aaa
AAA
Aa
AA
A
A
Baa
BBB
Ba
B
BB
B
Caa
Ca
C
CCC
CC
C
Safety
The strongest rating; ability to repay interest and principal
is very strong.
Very strong likelihood that interest and principal will be
repaid
Strong ability to repay, but some vulnerability to changes in
circumstances
Adequate capacity to repay; more vulnerability to changes
in economic circumstances
Considerable uncertainty about ability to repay.
Likelihood of interest and principal payments over
sustained periods is questionable.
Bonds in the Caa/CCC and Ca/CC classes may already be
in default or in danger of imminent default
C-rated bonds offer little prospect for interest or principal
on the debt ever to be repaid.
Corporate Bonds
• Zero coupons
• Floating rate bonds
• Convertible bonds
The Yield Curve
Term Structure of Interest Rates - A listing of
bond maturity dates and the interest rates
that correspond with each date.
Yield Curve - Graph of the term structure.
Question: If you knew interest rates won’t
change from now until one year from now,
what would that mean about the yield curve
of US treasury bonds?