Understanding the Concept of Present Value

1
Understanding the Concept of Present Value
2
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.

3
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)

4
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

5
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.

6
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

7
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

8
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.

9
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
interest rates are 5? PV FV/(1 i)n
PV 10,000/(1 .05)5 10,000/1.2763
7,835.15
10
More Things to Notice

• Present value is always less than future value.
• (1 i)n is positive so FV/(1 i)n lt FV
• In addition, PV4 lt PV3 lt PV2 lt PV1
• (1 i)1 lt (1 i)2
• The longer an amount has to grow to some fixed
  future amount, the smaller the initial amount
  needs to be.

11
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.

12
Example
If I want to receive 10,000 in 5 years, how much
do I have to invest now 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 invest now if
interest rates are 10? 10,000 PV(1 .10)20
10,000/6.7275 1486.44
13
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

14
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

15
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
16
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.

17
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.

18
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

19
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.

20
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?

21
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.

22
The Behavior of Interest Rates

• The Bond Market Model

23
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

24
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

25
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.

26
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

27
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.

28
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.

29
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.

30
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.

31
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.

32
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.

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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.

34
The Demand Curve for Bonds
0
Bond Price
Interest Rate
When bond prices are high, interest rates are
low, and bond demand is low. When bond prices
are low, interest rates are high, and bond demand
is high.
ilow
PBhigh
PBlow
ihigh
Demand
0
QDlow
QDhigh
35
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.

36
The Supply Curve for Bonds
0
Bond Price
Interest Rate
Supply
As bond prices rise, bond yields fall, and
quantity supplied rises. As bond prices fall,
bond yields rise, and quantity supplied falls.
ilow
PBhigh
PBlow
ihigh
0
QSlow
QShigh
37
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.

38
Equilibrium Disequilibrium
0
Bond Price
Interest Rate
S
Excess supply occurs when bond prices are high
and interest rates are low. Excess demand occurs
when bond prices are low and interest rates are
high.
A
B
ilow
PBhigh
E
PBeq
ieq
ihigh
PBlow
F
G
D
0
100 300 500
39
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.

40
Mechanics of an Increase in Demand
0
Bond Price
Interest Rate
S
b
Increases in bond demand cause bond prices to
rise and bond yields to fall.
P2
i1
a
P1
i2
D2
D1
0
Q1 Q2
41
Mechanics of a Decrease in Demand
0
Bond Price
Interest Rate
S
Decreases in bond demand cause bond prices to
fall and bond yields to rise.
a
P1
i2
b
P2
i1
D1
D2
0
Q2 Q1
42
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.

43
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.

44
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.

45
Returns on Different Maturity 10 Coupon Rate
Bonds
Term Initial i Initial P New i New
P 30 10 1000 20 503 20 10
1000 20 516 10 10 1000 20
597 5 10 1000 20 741 1 10
1000 20 1000
46
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.

47
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.

48
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.

49
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.

50
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.

51
Mechanics of an Increase in Supply
0
Bond Price
Interest Rate
S1
S2
An increase in the supply of bonds causes bond
prices to fall and bond yields to rise.
a
i2
P1
b
P2
i1
D
0
Q1 Q2
52
Mechanics of an Decrease in Supply
0
Bond Price
Interest Rate
S1
S2
A decrease in the supply of bonds causes bond
prices to rise and bond yields to fall.
b
i1
P2
a
P1
i2
D
0
Q2 Q1
53
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

54
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.

55
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.

56
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.

57
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.

58
Fisher Equation

• The Fisher equation states that the nominal
  interest rate equals the real interest rate plus
  the expected rate of inflation
• in ir p
• Rearranging terms we find
• ir in - p

59
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.

60
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.

61
The Fisher Effect
0
Bond Price
Interest Rate
S
Your turn
a
P
i
D
0
Q