Topics Covered

• 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 Future Years

1 2 3 4 5

Interest Earned 6 6 6 6 6

Value 100 106 112 118 124 130

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 Future Years

1 2 3 4 5

Interest Earned 6.00 6.36 6.74 7.15 7.57

Value 100 106.00 112.36 119.10 126.25 133.82

Value at the end of Year 5 = £133.82

Future Values

trFV )1(100£

Example - FV

What is the future value of £100 if interest is compounded annually at a rate of 6% for five years?

82.133£)06.1(100£ 5 FV

Manhattan Island Sale

Peter Minuit bought Manhattan Island for £16 in 1626. Was this a good deal?

trillion

FV

trillion

FV

412.1£

)05.1(16£

705.54£

)08.1(16£

375

375

To answer, determine £16 is worth in the year 2001, compounded at 8% and 5%.

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?

572,2$2)08.1(3000 PV

• The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable.

PV FVr t

1

1( )

Time Value of Money(applications)

Arbitrage

• You are given the following prices Pt today for receiving risk free £1 payments t periods from now.

• How would you make a lot of money?

T= 1 2 3

Pt= 0.95 0.9 0.95

PV of Multiple Cash Flows

ExampleYour 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?

$15,133.06 PVTotal

36.429,3

70.703,3

8,000.00

2

1

)08.1(

000,42

)08.1(

000,41

payment Immediate

PV

PV

PV of Multiple Cash Flows

• PVs can be added together to evaluate multiple cash flows.

PV C

r

C

r

1

12

21 1( ) ( )....

Review Question

• A consultant is paid a fee at the end of each of the next 5 years. At the 21% annual interest rate the present value of the payment is $1m. The contract is then renogotiated. Each payment remains the same but the first is received after 6 months and the rest follow at 12 monthly intervals. What is the new PV?

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%?

000,000,1£10.000,100 PV

Perpetuities & Annuities

PV of Annuity Formula

C = cash payment

r = interest rate

t = Number of years cash payment is received

PV C r r r t

1 11( )

Question

• A recent article in the NY times mentioned that a college degree increases one’s earning by one million dollars (in the US).

• How much should one be willing to pay for college tuition?

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?

Mortgage Question

• Assume mortgage interest is about 6% per year and mortgages last 25 years.

• If average mortgages is about 4 times annual income, what percent of income goes to the mortgage payment?

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

• Continuous is

Why? Look at % increase (slope/value)• What is continuous rate for 10% annual? • How long would it take to double your money?

nt

nr

t PVFV 1

)( rtt ePVFV

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 real interest rate = 1+nominal interest rate1+inflation rate

approximation formula

Real int. rate nominal int. rate - inflation rate

BondsTerminology

• 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

WARNINGWARNINGThe 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.

PVcpn

r

cpn

r

cpn par

r t

( ) ( )

....( )

( )1 1 11 2

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.

PVcpn

r

cpn

r

cpn par

r t

( ) ( )

....( )

( )1 1 11 2

Interest Rate Risk

880

900

920

940

960

980

1,000

1,020

1,040

1,060

1,080

0 5 10 15 20 25 30

Time (Matures at 30)

Bo

nd

Pri

ce

Premium Bond

Discount Bond

Interest Rate Risk

-

500

1,000

1,500

2,000

2,500

3,000

Interest Rate

$ B

on

d P

rice

30 yr bond

3 yr bond

Default Risk

• Credit risk

• Default premium

• Investment grade

• Junk bonds

Default RiskStandard

Moody' s & Poor's Safety

Aaa AAA The strongest rating; ability to repay interest and principalis very strong.

Aa AA Very strong likelihood that interest and principal will berepaid

A A Strong ability to repay, but some vulnerability to changes incircumstances

Baa BBB Adequate capacity to repay; more vulnerability to changesin economic circumstances

Ba BB Considerable uncertainty about ability to repay.B B Likelihood of interest and principal payments over

sustained periods is questionable.Caa CCC Bonds in the Caa/CCC and Ca/CC classes may already beCa CC in default or in danger of imminent defaultC C 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

• Green [1984] says they are useful to protect bondholders from the shareholders’ increasing the risk.

• Stein [1992] says they can signal the value of the firm.

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.

(see finance.yahoo.com)

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?

Orange County California

• Robert Citron lost $1.7 billion of Orange County’s $7.4 billion portfolio. He was not supposed to be taking risks.

• He was called a genius, but in 1994 the

• In “Big Bets Gone Bad” by Phillippe Jorion (Academic Press)