Understanding the Concept of Present Value

Interest Rates, Compounding, and Present Value

• In economics, an interest rate is known as the yield to maturity.

• Compounding is the process that gives us the value of a sum invested over time at a positive rate of interest.

• Present value is the process that tells us how much an expected future payment is worth today.

Compounding

• Assume you have $1 which you place in an account paying 10% annually.

• How much will you have in one year, two years, etc?– An amount of $1 at 10% interest– Year 1 2 3 n

• $1.10 $1.21 $1.33 $1(1 + i)n

• Formula: FV = PV(1 + i)

Compounding over Time

• Extending the formula over 2 years– FV = PV(1 + i) (1 + i) or FV = PV(1 + i)2

• 3 years– FV = PV(1 + i) (1 + i) (1 + i) = PV(1 + i)3

• n years– FV = PV(1 + i)n

Present Value

• Present value tells us how much an expected future payment is worth today.

• Alternatively, it tells us how much we should be willing to pay today to receive some amount in the future.– For example, if the present value of $1.10 at an

interest rate of 10% is $1, we should be willing to spend $1 today to get $1.10 next year.

Present Value Formula

• The formula for present value can be found by rearranging the compounding formula.– FV = PV(1 + i) solve for PV

• FV/(1 + i) = PV

Present Value over Time• Extending the formula over 2 years

– FV = PV(1 + i)2

– PV = FV/(1 + i)2

• 3 years– FV = PV(1 + i)3

– PV = FV/(1 + i)3

• n years– FV = PV(1 + i)n

– PV = FV/(1 + i)n

Things to Notice

• An increase in the interest rate causes present value to fall.– Higher rates of interest mean smaller amounts can

grow to equal some fixed amount during a specified period of time.

• A decrease in the interest rate causes present value to rise.– Lower rates of interest mean larger amounts are

needed to reach some fixed amount during a specified period of time.

Example:How much must I invest today to get $10,000 in five years if interest rates are 10%?

PV = FV/(1 + i)n

PV = $10,000/(1 + .10)5 = $10,000/1.6105 = $6,209.2

How much must I invest today to get $10,000 in five years if interestrates are 5%?

PV = FV/(1 + i)n

PV = $10,000/(1 + .05)5 = $10,000/1.2763 $7,835.15

More Things to Notice

• Present value is always less than future value.– (1 + i)n is positive so FV/(1 + i)n < FV

• In addition, PV4 < PV3 < PV2 < PV1

– (1 + i)1 < (1 + i)2

• The longer an amount has to grow to some fixed future amount, the smaller the initial amount needs to be.

Time Value of Money

• The longer the time to maturity, the less we need to set aside today. This is the principal lesson of present value. It is often referred to as the “time value of money.”

Example:

If I want to receive $10,000 in 5 years, how much do I have to investnow if interest rates are 10%?

$10,000 = PV(1 + .10)5

$10,000/1.5105 = $6209.25

If I want to receive $10,000 in 20 years, how much do I have to investnow if interest rates are 10%?

$10,000 = PV(1 + .10)20

$10,000/6.7275= $1486.44

Yield to Maturity

• Yield to maturity is the interest rate that equates the present value of payments received from a debt instrument with its value today.

• Yield to maturity can be calculated using the present value formula.– PV = FV/(1 + i)– i = FV - PV/PV

Simple Example:

• PV = FV/(1 + i)• PV(1 + i) = FV• PV + PVi = FV• PVi = FV - PV• i = FV - PV/PV

– $1.00 = $1.10/(1 + i)– $1.00 + $1.00i = $1.10– i = $1.10 - $1.00/$1.00 = 0.10 = 10%

Relationship between Yield to Maturity and Price

Yields to maturity on a 10% coupon rate bond with a face value of $1000 maturing in 10 years

Price of Bond Yield to Maturity 1200 7.13 1100 8.48 1000 10.00 900 11.75 800 13.81

Relationship between Yield to Maturity and Price

• Three interesting facts:– Price and yield are negatively related.– When the bond is at par, yield equals coupon

rate.– Yield is greater (less than) than the coupon rate

when the bond price is below (above) par value.

Current Yield

• In more complicated cases, yield to maturity can be difficult to calculate. Tables are available that can be used. And, of course, calculators do a fine job.

• There are also simple formulas that can approximate yield to maturity such as current yield.

Current Yield

• Current yield is an approximation for yield to maturity that is used to calculate the interest rate on a bond quickly.

• Formula:– Current yield = Coupon/Bond Price

Inverse Relationship

• We can use the current yield formula to see clearly the inverse relationship between interest rates and bond prices.– Current yield = Coupon/Bond Price

• The coupon is a fixed payment, it does not change. Therefore, if yields rise, bond prices must fall, and if yields fall, bond prices must rise.

Intuition

• Assume you buy a $1,000 bond today with a fixed coupon of $100. You are receiving a 10% return. Let a year pass, and you find you want to sell you bond. You call your broker and say, “Sell!” Your broker sighs and tells you that bonds just like yours now yield 12%. What price can you expect to receive?

Example

• Use the current yield formula:– 0.12 = $100/PB

– 0.12PB = $100

– PB = $100/.12 = $833.33

• You must reduce your price until $100 represents a 12% rate of return.

The Behavior of Interest Rates

The Bond Market Model

Understanding Interest Rates

• Economists use three different models to explain how interest rates are determined.

– The bond market model– The money demand/money supply model– The loanable funds model

The Bond Market Model

• The bond market model is useful because of the issues that can be considered within its framework.– The impact of changes in----

• Wealth

• Expected interest rates or expected return

• Expected inflation

• Riskiness of bonds relative to other assets

• Liquidity of bonds relative to other assets

The Bond Market Model

• The bond market can be modeled using the concepts of demand and supply.– The demand for bonds is determined by

individuals and institutions who wish to hold their wealth in bonds.

– The supply of bonds is provided by institutions that issue bonds to raise funds.

The Demand for Bonds

• The demand for bonds comes from savers, people who have funds in excess of their spending needs.

• They are willing to hold bonds for two reasons:– Interest earned– Potential capital gains

Bond Demand

• Rate of return– According to the asset theory of demand,

people compare one asset relative to another and choose the one that best suits their needs.

• As the opportunity cost of an asset increases, people find it increasingly unattractive.

Opportunity Cost

• The opportunity cost of an asset is defined as the difference between the rate of return received by the asset and the rate of return on an alternative asset.– When bond yields are high, people prefer bonds

because the opportunity cost of holding other assets is high.

– When bond yields are low, people prefer other assets because the opportunity cost of holding bonds is high.

Bond Demand

• Investors who demand bonds based on opportunity cost considerations prefer to buy when interest rates are high and sell when interest rates are low.

Bond Demand

• Speculation– When choosing an asset, investors also consider

risk.• Interest rate risk occurs when the market value of

a bond falls because interest rates rise.

– As we have seen, the existence of interest rate risk means investors face the possibility of capital losses when interest rates rise and capital gains when interest rates fall.

Speculation

• Investors who speculate in the bond market prefer to buy when interest rates are high and sell when interest rates are low.– When interest rates are high, people expect them

to fall. As they fall, bond prices rise, yielding a capital gain.

– When interest rates are low, people expect them to rise. As they rise, bond prices fall, diminishing capital gains or yielding a capital loss.

Bond Demand

• Both the opportunity cost motive and the speculative motive result in investors demanding bonds when interest rates are high and selling bonds when interest rates are low.

The Demand Curve for Bonds

• Let r = RET = (F - P)/P– If F = $1,000 and P = $950, r = 5.26%– If F = $1,000 and P = $900, r = 11.1%

• High bond prices are associated with low interest rates.

• Low bond prices are associated with high interest rates.

0

0Bond Price

Interest Rate

Demand

PBhigh

PBlow

ilow

ihigh

The Demand Curve for Bonds

When bond prices are high,interest rates are low, and bond demand is low.

When bond prices are low,interest rates are high, andbond demand is high.

QDlow QDhigh

Bond Supply• The supply of bonds comes from institutions,

governments (domestic and foreign), and businesses.

• The quantity of bonds supplied depends in part on the interest rate bond suppliers must pay to attract funds.– As interest rates increase, the quantity supplied falls.

– As interest rates decrease, the quantity supplied rises.

0

0Bond Price

Interest Rate

PBhigh

PBlow

ilow

ihigh

The Supply Curve for Bonds

QSlow QShigh

Supply

As bond prices rise, bond yieldsfall, and quantity supplied rises.

As bond prices fall, bond yieldsrise, and quantity supplied falls.

Equilibrium

• Equilibrium is a state of rest. Either there are no forces causing change or there are equal opposing forces.

• In the bond market, equilibrium occurs when the quantity of bonds demanded just equals the quantity of bonds supplied.

Equilibrium & Disequilibrium

0

0

BondPrice

Interest Rate

E

100 300 500

Excess supply occurs whenbond prices are high and interestrates are low.

Excess demand occurs whenbond prices are low and interestrates are high.

PBhigh

PBlow

ilow

ihigh

PBeq ieq

D

S

A B

F G

Disequilibrium

• Excess Supply– More people want to sell bonds than want to

buy them.• Bond prices fall and interest rates rise.

• Excess Demand– More people want to buy bonds than want to

sell them.• Bond prices rise and interest rates fall.

Mechanics of an Increase in Demand0

0

BondPrice

Interest Rate

a

Q1 Q2

P2

P1

D1

D2

b

S

i2

i1Increases in bond demand causebond prices to rise and bondyields to fall.

Mechanics of a Decrease in Demand0

0

BondPrice

Interest Rate

a

Q2 Q1

P2

P1

D1

D2

b

S

i2

i1

Decreases in bond demand causebond prices to fall and bondyields to rise.

Shifts in the Demand for Bonds

• According to the asset theory of demand, changes in bond demand are caused by changes in---– Wealth– Expected return on bonds relative to expected

returns on other assets– Expected riskiness of bonds relative to other

assets– Liquidity of bonds relative to other assets.

Bond Demand and Wealth

• Wealth is defined as a stock of assets that produce income. Wealth is not income.

• In a business cycle expansion, wealth grows, causing the demand for bonds to rise and the demand curve to shift to the right.

• In a business cycle contraction, wealth shrinks, causing the demand for bonds to fall and the demand curve to shift to the left.

Bond Demand and Expected Returns: Bonds

• Higher expected interest rates in the future decrease the demand for long-term bonds and shift the demand curve to the left.

• Lower expected interest rates in the future increase the demand for long-term bonds and shift the demand curve to the right.

Returns on Different Maturity 10% Coupon Rate Bonds

Term Initial i Initial P New i New P30 10% 1000 20% 50320 10% 1000 20% 51610 10% 1000 20% 597 5 10% 1000 20% 741 1 10% 1000 20% 1000

Bond Demand and Expected Returns: Other Assets

• Higher expected returns on other assets relative to bonds cause bonds to become less attractive and the bond demand curve shifts left.

• Lower expected returns on other assets relative to bonds cause bonds to become more attractive and the bond demand curve shifts right.

Bond Demand and Expected Returns: Inflation

• An increase in the expected rate of inflation will cause the demand for bonds to decline and the demand curve to shift to the left.

• A decrease in the expected rate of inflation will cause the demand for bonds to increase and the demand curve to shift to the right.

Bond Demand and Risk

• An increase in the riskiness of bonds causes the demand for bonds to fall and the demand curve to shift to the left.

• An increase in the riskiness of other assets causes the demand for bonds to rise and the demand curve to shift to the right.

Bond Demand and Liquidity

• Increased liquidity of bonds results in an increased demand for bonds and the demand curve shifts right.

• Increased liquidity of other assets results in a decreased demand for bonds and the demand curve shifts left.

Bond Demand and Liquidity

• Increased liquidity of bonds results in an increased demand for bonds and the demand curve shifts right.

• Increased liquidity of other assets results in a decreased demand for bonds and the demand curve shifts left.

Mechanics of an Increase in Supply

0

0

BondPrice

Interest Rate

a

Q1 Q2

P2

P1

S1

S2

b

D

i2

i1

An increase in the supply of bondscauses bond prices to fall and bondyields to rise.

Mechanics of an Decrease in Supply

0

0

BondPrice

Interest Rate

a

Q2 Q1

P1

P2

S1

S2

b

D

i1

i2

A decrease in the supply of bondscauses bond prices to rise and bondyields to fall.

Shifts in Supply

• Shifts in the supply curve for bonds are caused by changes in….– The expected profitability of investment

opportunities– Expected inflation– Government activities

Shifts in the Supply of Bonds

• Expected Profitability of Investment Opportunities– When an economy is growing rapidly, there are

many profitable investment opportunities. The supply of bonds increases and the supply curve shifts to the right.

– When an economy is contracting, there are fewer profitable opportunities. The supply of bonds decreases and the supply curve shifts left.

Shifts in the Supply of Bonds

• Expected Inflation– An increase in expected inflation causes the

supply of bonds to increase and the supply curve to shift right.

• Inflation causes the real cost of borrowing to fall.

Shifts in the Supply of Bonds

• Government Activities– Higher government deficits increase the supply

of bonds, causing the supply curve to shift right.

– Reductions in government deficits decrease the supply of bonds, causing the supply curve to shift left.

Real and Nominal Interest Rates

• Nominal interest rate is the rate of interest that makes no allowance for inflation.

• The real interest rate is the rate of interest that is adjusted for expected changes in the price level.– It more accurately reflects the true cost of

borrowing and lending.

Fisher Equation

• The Fisher equation states that the nominal interest rate equals the real interest rate plus the expected rate of inflation– in = ir +

• Rearranging terms we find:– ir = in -

Logic behind the Inflation Premium

• Lenders want to be compensated for the loss in buying power due to inflation.

• Buyers understand that they will be repaying debt with dollars that buy less.

• The interest rate must reflect these facts.

The Fisher Effect

• When expected inflation increases:– Bond supply increases and the supply curve

shifts right.– Bond demand decreases and the demand curve

shifts left.• As a result, bond prices fall and interest rates rise.

– When expected inflation rises, interest rates rise. This is the Fisher Effect.

The Fisher Effect

0

0

BondPrice

Interest Rate

a

Q

P

S

D

Your turn

i