money and banking

Money and Banking

Mr. Vaughan

Time Value of

Money(Understanding Interest Rates)

Updated: 2/16/09

Learning ObjectivesTime Value of Money (TVM)

After this lecture, you will understand:

• How to put cash flows received/paid out at different times in common time denominator.

• How to use TVM framework to compute present/ value.

• How to price stocks.

• How to evaluate profitability of capital projects using net-present-value (NPV).

• How to price bonds.

• How to compute “yield to maturity” on bonds.

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Time Value of Money

• Firms/investors often receive/pay out cash flows at different times.

• Other things equal, people prefer cash flows received sooner to cash flows received later. This idea is called “time value of money (TVM).”

• Future value/present value calculations put cash flows in “common time denominator.”

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• A dollar in hand today is worth more than a dollar promised tomorrow because dollar today will earn interest.

• Amount an asset is worth at a specific time in the future is its future value.

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What is “future value?”

Example: Calculating Future Value

To calculate future value (FV) one period ahead:

• Multiply initial, or present value, of cash flow (PV) by appropriate interest rate (r).

• Add product (PV x r) to initial cash flow (PV).

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r PVPVFV x


• Factoring out initial cash flow (PV) yields:

• We can compute value of cash flow after more than one compounding period by recognizing interest will earn interest (i.e., compound).

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r1 PVFV x


• To account for compounding, multiply initial cash flow (PV) by one plus interest rate (1 + r) for each period.

• Therefore, future value of cash flow after “n” periods is given by:

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nr1 PV

...r1 r1 r1 PVFV

x

x x x


• Suppose $100 is invested at 5%, compounded yearly.

• After one year, investment will be worth:

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$105

0.051 $100FV x


• Over second year, $105 grows by another 5%, yielding:

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$110.25

0.051 $105FV x

• Note: Number could have obtained from general formula, with n=2:

$110.25

0.051 $100FV 2x


Note: Future value ($110.25) has three components:

• Original principal: $100

• Simple interest: 0.05 x $100 x 2 = $10

• Compound interest: $110.25 - $100 - $10 = 0.25

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Power of Compound Interest

Example

• In 1626, Peter Minuit of Dutch West Indies Company bought Manhattan island from Manhattan Indians for $24 worth of trinkets.

• Suppose Indians had invested funds and earned 10.1% annually (average nominal return on U.S. stocks from 1926 to 2006) since …

• What would $24 be worth today, assuming annual compounding?

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Power of Compound Interest

Future value = $24 (1.101)377

= $24 (1.431 x 1016)

= $3.434 x 1017

The $24 would now be worth $3.434 x 1017.

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Note: 17 Zeros!Enough to buy all real estate on earth, with change.

• Present value is inverse of future value.

• If we know when cash flow will be received, we can use present-value framework to calculate what cash flow is worth to us now.

• Present value allows us to determine “fundamental value” of assets.

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What is “present value?”

For single future payment (“n” periods in future), present value can be obtained by:

• Rearranging future-value equation.

• Plugging anticipated future cash flow in for FV and solving for present value.

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n

n

r1

FVPV

r1 PV FV x

Calculating “Present Value”

Example: Calculating Present Value

• What is present value of $100 to be received in one year if rate of return available on investments of comparable risk is 10%?

• What is intuition behind number?

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Solution:

Intuition:

$90.91 in hand today, invested at 10% (compounded

annually), would grow to $100 in one year.

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91.90$10.1

100$

PV


What about cash flows received over multiple periods?• Calculate present value of each cash flow (CF)

and sum discounted values:

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n

n

3

3

2

21

r1

CF...

r1

CF

r1

CF

r1

CFPV


What is present value of following stream of cash flows?• $100 received one year from now• $100 received two years from now

Assume: • return available on investments with comparable risk

is 10%. • annual compounding.

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Solution:

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55.173$

64.82$91.90$

10.1

100$

10.1

100$2

PV

PV

PV

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Like any financial asset, stocks are priced at present value of future cash flows [expected dividends (D) and expected resale price of stock (R)].

In well-functioning markets, resale price in year “n” depends on dividends to be received from “n+1” to infinity.

When dividends are expected to grow at a constant rate to infinity, the stock-pricing equation reduces to:

Applications of Present ValuePricing Stocks

nn

nn

r

R

r

D

r

D

r

DPV

11...

11 221Stock

Price=

gr

DPV

1Stock

Price=

where: “g” is the constant growth rate of dividends

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An increase in return on investments of comparable risk (r) causes present value of cash flows from stock (and, therefore, stock price) to fall.

Such an increase could be caused by rise in market interest rates

Bad economic news causes investors to revise expected dividends (D) downward, thereby lowering present value and stock price.

Economic News and Stock Prices

nn

nn

r

R

r

D

r

D

r

DPV

11...

11 221Stock

Price=

Note: Stock prices are more volatile than bond prices because numerator and denominator fluctuate with market conditions (only denominator fluctuates for bonds)

Applications of Present Value Evaluating Capital Projects

• A financial manager’s goal is: Maximizing shareholder wealth.

• To do this, he should undertake all capital projects that add to total value of firm.

• “Adding to value of the firm” means undertaking all capital projects with positive net present value.

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Present value of cash inflows from capital project (marginal benefit)

-Present value of cash outflows on capital project (marginal cost)

Net Present Value (NPV)

Example: Net Present Value (NPV)

Suppose you are considering buying a home computer system, so you can telecommute. You want to know whether this capital project makes sense from financial standpoint.

You should:

1. Calculate NPV.

2. Purchase computer system if NPV > 0

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Relevant information: • Cost of computer system: $3,000

• Life of computer: 4 years

• Salvage value of the computer: $0

• Yearly cash flows from computer: $900

• Appropriate discount rate: 5%

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Framework:NPV = Present value of inflows - Present value of outflows

where:CFn = Cash inflow in period “n”Co = Cash outflow at outset, year “0”r = Appropriate discount rate (opportunity cost of funds)

04

4

3

3

2

21

1111C

r

CF

r

CF

r

CF

r

CFNPV


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Framework:NPV = Present value of cash inflows - Present value of cash

outflows

Decision:You should buy computer because it is positive NPV project!

35.191$

000,3$43.740$45.777$33.816$14.857$

000,3$05.01

900$

05.01

900$

05.01

900$

05.01

900$432

NPV

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Economic News and Investment

• Higher interest rates or bad economic news reduce planned investment in structures and equipment.

Increase in current market interest rates reduces present value of cash inflows from project (marginal benefit) and hence, NPV.

Bad economic news reduces expected cash inflows from project (marginal benefit) and hence, NPV.

• A firm considering investment projects knows cash outlays and current discount rate but must estimate cash inflows.

Revisions in CFs due to economic news can be large (Keynes: “Animal Spirits”).

Business spending on structures and equipment can be quite volatile.

02

21

1...

11C

r

CF

r

CF

r

CFNPV n

n

Application of Present ValuePricing Bonds

In well-functioning bond market, price of bond equals present value of its cash flows:

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nn

n

n

r

P

r

C

r

C

r

C

r

CPV

11...

111Price 3

3

2

21

where:Cn = Coupon payment each period = coupon rate x face value (cash flows)Pn = Principal returned in “nth” year = face value (cash flow)r = Yield available on bonds with comparable risk

Applications of Present ValuePricing Bonds

• You hold a bond with 7% annual coupon and three years remaining until maturity.

• Face value of bond is $1,000, and yield available on comparable, newly issued bonds, is 10%.

• In efficient bond market, what should price of bond be?

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Applications of Present ValuePricing Bonds

Solution: In efficient bond market, price of bond equals present value of all cash flows from bond.

Intuition: $925.39 invested at 10% today would yield cash flows offered by this bond over next three years.

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$925.39

751.31$52.59$57.85$63.64$10.1

000,1$

10.1

70$

10.1

70$

10.1

70$Price 332

PV

Extension of Present Value Yield to Maturity

• Suppose you are considering buying a bond.

• You want to determine what the return would be if you bought bond at current market price and held to maturity.

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This is yield to maturity!

Example: Calculating “Yield to Maturity”

Use present-value framework and solve for “r”.

where:BP0 = Actual market price of bond in year “0”(given)

Cn = Expected coupon payment in year “n” (given)

Pn = Expected principal payment in the “nth” year (given)

r = Yield to maturity (unknown)

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n

n

n

n

2

21

0r1

P

r1

C...

r1

C

r1

CBP


Suppose you are considering purchase of bond with following characteristics: •Annual coupon: 7%•Years remaining until maturity: 3•Face value: $1,000•Current market price: $925.39

What is yield to maturity on bond?

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Solution:

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10% Maturity toYield

r)(1

$1,000

r)(1

$70

r)(1

$70

r)(1

$70$925.39

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• Firms selling bonds in primary markets set coupon payments (C) so yield-to-maturity (r) equals going rate on bonds of comparable risk.

• After initial sale, (C) and principal (P) remain fixed, but going rate on new bonds of comparable risk [r = opportunity cost of funds] fluctuates with market conditions.

• A rise in going rate on new bonds of comparable risk (due to rise in market interest rates or default/liquidity risk) cases present value to fall.

• Investors price at present value, so rise in opportunity cost of funds causes bond price to fall. Bond prices and interest rates move inversely.

Economic News and Bond Prices

nn

n

n

r

P

r

C

r

C

r

CPV

11...

11 2

21BondPrice

=

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Bond FactsBond Prices and Interest Rates

Fact: When coupon rate on bond “A” is less than yield available on new bonds of comparable risk, bond “A” will sell at discount.

Intuition: • Unattractive coupon rate reduces demand for bond “A.” • Falling demand depresses price of bond “A.”• Coupon rate on bond “A” was fixed at issue, so falling price implies

rising yield on bond “A.” • Price of bond “A” falls until yield on bond “A” equals market yield on

bonds of comparable risk.

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Questions over

Money and BankingMr. Vaughan

Time Value of

Money(Understanding Interest Rates)

money and banking

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