Chapter 3 What Do Interest Rates Mean and What is Their Role in Valuation?

Chapter 3

What Do Interest Rates Mean and What is Their Role

in Valuation?

Chapter Preview

• We develop a better understanding of interest rates. We examine the terminology and calculation of various rates, and we show the importance of these rates in our lives and the general economy. Topics include:

– Measuring Interest Rates

– The Distinction Between Real and Nominal Interest Rates

– The Distinction Between Interest Rates and Returns

Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

3-2

The Big Picture

• Since interest rates are among the most closely watched variables in the economy, it is imperative that what exactly is meant by the phrase interest rates is understood. In this chapter, we will see that a concept known as yield to maturity (YTM) is the most accurate measure of interest rates.


3-3

The Big Picture

• Any description of interest rates will entail understanding certain vernacular and definitions, most of which will not only pertain directly to interest rates but will also be vital to understanding many other foundational concepts presented later in the text.


3-4

Present Value Introduction

• Different debt instruments have very different streams of cash payments to the holder (known as cash flows), with very different timing.

• All else being equal, debt instruments are evaluated against one another based on the amount of each cash flow and the timing of each cash flow.

• This evaluation, where the analysis of the amount and timing of a debt instrument’s cash flows lead to its yield to maturity or interest rate, is called present value analysis.


3-5

Present Value

• The concept of present value (or present discounted value) is based on the commonsense notion that a dollar of cash flow paid to you one year from now is less valuable to you than a dollar paid to you today. This notion is true because you could invest the dollar in a savings account that earns interest and have more than a dollar in one year.

• The term present value (PV) can be extended to mean the PV of a single cash flow or the sum of a sequence or group of cash flows.


3-6

Present Value Applications

• There are four basic types of credit instruments which incorporate present value concepts:

1. Simple Loan2. Fixed Payment Loan3. Coupon Bond4. Discount Bond


3-7

Present Value Concept: Simple Loan Terms

• Loan Principal: the amount of funds the lender provides to the borrower.

• Maturity Date: the date the loan must be repaid; the Loan Term is from initiation to maturity date.

• Interest Payment: the cash amount that the borrower must pay the lender for the use of the loan principal.

• Simple Interest Rate: the interest payment divided by the loan principal; the percentage of principal that must be paid as interest to the lender. Convention is to express on an annual basis, irrespective of the loan term.


3-8

Present Value Concept: Simple Loan


3-9

Simple loan of $100

Year: 0 1 2 3 n

$100 $110 $121 $133 100(1+i)n

PV of future $1 =$1

1+ i n

Present Value Concept: Simple Loan (cont.)

• The previous example reinforces the concept that $100 today is preferable to $100 a year from now since today’s $100 could be lent out (or deposited) at 10% interest to be worth $110 one year from now, or $121 in two years or $133 in three years.


3-10

Yield to Maturity: Loans

• Yield to maturity = interest rate that equates today's value with present value of all future payments

1. Simple Loan Interest Rate (i = 10%)


3-11

$100 $110 1 i

i $110 $100

$100

$10

$100.10 10%

Present Value of Cash Flows: Example


3-12

Present Value Concept: Fixed-Payment Loan Terms

• Simple Loans require payment of one amount which equals the loan principal plus the interest.

• Fixed-Payment Loans are loans where the loan principal and interest are repaid in several payments, often monthly, in equal dollar amounts over the loan term.


3-13

Present Value Concept: Fixed-Payment Loan Terms

• Installment Loans, such as auto loans and home mortgages are frequently of the fixed-payment type.


3-14

Yield to Maturity: Loans

1. Fixed Payment Loan (i = 12%)


3-15

$1000 $126

1 i

$126

1 i 2 $126

1 i 3 ... $126

1 i 25

LV FP

1 i

FP

1 i 2 FP

1 i 3 ... FP

1 i n

Yield to Maturity: Bonds

1. Coupon Bond (Coupon rate = 10% = C/F)


3-16

P $100

1 i

$100

1 i 2 $100

1 i 3 ... $100

1 i 10 $1000

1 i 10

P C

1 i

C

1 i 2 C

1 i 3 ... C

1 i n F

1 i n

Consol: Fixed coupon payments of $C forever

P C

ii

C

P

Yield to Maturity: Bonds

1. One-Year Discount Bond (P = $900, F = $1000)


3-17

$900 $1000

1 i

i $1000 $900

$900.111 11.1%

i F P

P

Relationship Between Price and Yield to Maturity

• Three interesting facts in Table 3-11. When bond is at par, yield equals coupon rate2. Price and yield are negatively related3. Yield greater than coupon rate when bond price

is below par valueCopyright © 2006 Pearson Addison-

Wesley. All rights reserved.3-18

Current Yield

• Two Characteristics1. Is better approximation to yield to maturity,

nearer price is to par and longer is maturity of bond

2. Change in current yield always signals change in same direction as yield to maturity


3-19

ic C

P

Yield on a Discount Basis


3-20

• One-Year Bill (P = $900, F = $1000)

idb (F - P)

F

360

(number of days to maturity)

idb $1000 - $900

$1000

360

365.099 9.9%

• Two Characteristics1. Understates yield to maturity; longer the maturity,

greater is understatement

2. Change in discount yield always signals change in same direction as yield to maturity

Bond Page of the Newspaper

Distinction Between Real and Nominal Interest Rates

• Real interest rate1. Interest rate that is adjusted for expected

changes in the price level


3-22

ir i e

1. Real interest rate more accurately reflects true cost of borrowing

2. When real rate is low, greater incentives to borrow and less to lend

Distinction Between Real and Nominal Interest Rates (cont.)

• If i = 5% and πe = 0% then


3-23

ir 5% 0% 5%

ir 10% 20% 10%

• If i = 10% and πe = 20% then

U.S. Real and Nominal Interest Rates


3-24

Figure 3-1 Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2004

Sample of current rates and indexeshttp://www.martincapital.com/charts.htm

http://www.martincapital.com/charts.htm

Distinction Between Interest Rates and Returns

• Rate of Return


3-25

RET C Pt1 Pt

Pt

ic g

where ic C

Pt

current yield

g pt1 Pt

Pt

capital gain

Key Facts about the Relationship Between Rates and Returns


3-26

Sample of current coupon rates and yields on government bondshttp://www.bloomberg.com/markets/iyc.html

http://www.bloomberg.com/markets/iyc.html

Maturity and the Volatility of Bond Returns

• Key findings from Table 3-2

1. Only bond whose return = yield is one with maturity = holding period

2. For bonds with maturity > holding period, i P implying capital loss

3. Longer is maturity, greater is price change associated with interest rate change


3-27

Maturity and the Volatility of Bond Returns (cont.)

• Key findings from Table 3-2 (continued)

1. Longer is maturity, more return changes with change in interest rate

2. Bond with high initial interest rate can still have negative return if i


3-28

Maturity and the Volatility of Bond Returns (cont.)

• Conclusion from Table 3-2 analysis

1. Prices and returns more volatile for long-term bonds because have higher interest-rate risk

2. No interest-rate risk for any bond whose maturity equals holding period


3-29

Reinvestment Risk

1. Occurs if hold series of short bonds over long holding period

2. i at which reinvest uncertain3. Gain from i , lose when i


3-30

Calculating Durationi =10%, 10-Year 10% Coupon Bond


3-31

Calculating Durationi = 20%, 10-Year 10% Coupon Bond


3-32

Formula for Duration

• Key facts about duration1. All else equal, when the maturity of a bond

lengthens, the duration rises as well2. All else equal, when interest rates rise, the

duration of a coupon bond fall


3-33

DUR tCPt

1 i tt 1

n

CPt

1 i tt 1

n

Formula for Duration

1. The higher is the coupon rate on the bond, the shorter is the duration of the bond

2. Duration is additive: the duration of a portfolio of securities is the weighted-average of the durations of the individual securities, with the weights equaling the proportion of the portfolio invested in each


3-34

Duration and Interest-Rate Risk

• i 10% to 11%:– Table 3-4, 10% coupon bond


3-35

%P DURi

1 i

%P 6.76 0.01

1 0.10

%P 0.0615 6.15%

Duration and Interest-Rate Risk (cont.)

• i 10% to 11%:– 20% coupon bond, DUR = 5.72 years


3-36

%P 5.72 0.01

1 0.10

%P 0.0520 5.20%

Duration and Interest-Rate Risk (cont.)

• The greater is the duration of a security, the greater is the percentage change in the market value of the security for a given change in interest rates

• Therefore, the greater is the duration of a security, the greater is its interest-rate risk


3-37

Chapter Summary

• Measuring Interest Rates: We examined several techniques for measuring the interest rate required on debt instruments.

• The Distinction Between Real and Nominal Interest Rates: We examined the meaning of interest in the context of price inflation.


3-38

Chapter Summary (cont.)

• The Distinction Between Interest Rates and Returns: We examined what each means and how they should be viewed for asset valuation.


3-39


4-40

Chapter Preview

• Although interest rates in the U.S. have been relatively stable in recent history, this has not always been the case. We examine the forces the move interest rates and the theories behind those movements. Topics include:– Determining Asset Demand

– Supply and Demand in the Bond Market

– Changes in Equilibrium Interest Rates


4-41

Determinants of Asset Demand• An asset is a piece of property that is a store of value. Facing

the question of whether to buy and hold an asset or whether to buy one asset rather than another, an individual must consider the following factors:

1. Wealth, the total resources owned by the individual, including all assets

2. Expected return (the return expected over the next period) on one asset relative to alternative assets

3. Risk (the degree of uncertainty associated with the return) on one asset relative to alternative assets

4. Liquidity (the ease and speed with which an asset can be turned into cash) relative to alternative assets


4-42

EXAMPLE 1: Expected ReturnWhat is the expected return on the Mobil Oil bond if the return is 12% two-thirds of the time and 8% one-third of the time?

Solution

The expected return is 10.68%.

Re = p1R1 + p2R2

where

p1 = probability of occurrence of return 1 = 2/3 =.67

R1 = return in state 1 = 12% = 0.12

p2 = probability of occurrence return 2 = 1/3 = .33

R2 = return in state 2 = 8% = 0.08

Thus

Re = (.67)(0.12) + (.33)(0.08) = 0.1068 = 10.68%


4-43

EXAMPLE 2: Standard Deviation (a)

• What is the standard deviation of the returns on the Fly-by-Night Airlines stock and Feet-on-the Ground Bus Company, with the same return outcomes and probabilities described above? Of these two stocks, which is riskier?


4-44

EXAMPLE 2: Standard Deviation (b)

• Solution– Fly-by-Night Airlines has a standard deviation of returns of 5%.


4-45

EXAMPLE 2: Standard Deviation (c)

• Feet-on-the-Ground Bus Company has a standard deviation of returns of 0%.


4-46

EXAMPLE 2: Standard Deviation (d)

• Fly-by-Night Airlines has a standard deviation of returns of 5%; Feet-on-the-Ground Bus Company has a standard deviation of returns of 0%

• Clearly, Fly-by-Night Airlines is a riskier stock because its standard deviation of returns of 5% is higher than the zero standard deviation of returns for Feet-on-the-Ground Bus Company, which has a certain return

• A risk-averse person prefers stock in the Feet-on-the-Ground (the sure thing) to Fly-by-Night stock (the riskier asset), even though the stocks have the same expected return, 10%. By contrast, a person who prefers risk is a risk preferrer or risk lover. Most people are risk-averse, especially in their financial decisions


4-47

Determinants of Asset Demand (2)

• The quantity demanded of an asset differs by factor.

1. Wealth: Holding everything else constant, an increase in wealth raises the quantity demanded of an asset

2. Expected return: An increase in an asset’s expected return relative to that of an alternative asset, holding everything else unchanged, raises the quantity demanded of the asset

3. Risk: Holding everything else constant, if an asset’s risk rises relative to that of alternative assets, its quantity demanded will fall

4. Liquidity: The more liquid an asset is relative to alternative assets, holding everything else unchanged, the more desirable it is, and the greater will be the quantity demanded


4-48

Determinants of Asset Demand (3)


4-49

Point B

i Re F P

P

P $950

i $1000 $950

$950.053 5.3%

Bd 100

P $900

i $1000 $900

$900.111 11.1%

Bd 200

Derivation of Demand Curve

Point A


4-50

Derivation of Demand Curve

• Point C: P = $850 i = 17.6% Bd = 300

• Point D: P = $800 i = 25.0% Bd = 400

• Point E: P = $750 i = 33.0% Bd = 500

• Demand Curve is Bd in Figure 1 which connects points A, B, C, D, E.– Has usual downward slope


4-51

Figure 4.1 Supply and Demand for Bonds

Supply and Demand Analysis of the Bond Market


4-52

Derivation of Supply Curve

• Point F: P = $750 i = 33.0% Bs = 100

• Point G: P = $800 i = 25.0% Bs = 200

• Point C: P = $850 i = 17.6% Bs = 300

• Point H: P = $900 i = 11.1% Bs = 400

• Point I: P = $950 i = 5.3% Bs = 500

• Supply Curve is Bs that connects points F, G, C, H, I, and has upward slope


4-53

Market Equilibrium

1. Occurs when Bd = Bs, at P* = 850, i* = 17.6%

2. When P = $950, i = 5.3%, Bs > Bd (excess supply): P to P*, i to i*

3. When P = $750, i = 33.0, Bd > Bs (excess demand): P to P*, i to i*


4-54

Market Demand ConditionsMarket equilibrium occurs when the amount that people are willing to buy (demand) equals the amount that people are willing to sell (supply) at a given price

Excess supply occurs when the amount that people are willing to sell (supply) is greater than the amount people are willing to buy (demand) at a given price

Excess demand occurs when the amount that people are willing to buy (demand) is greater than the amount that people are willing to sell (supply) at a given price


4-55

Loanable Funds Terminology

1. Demand for bonds = supply of loanable funds

2. Supply of bonds = demand for loanable funds

Figure 4.2 A Comparison of Terminology: Loanable Funds and Supply and Demand for Bonds


4-56

Shifts in the Demand Curve

Figure 4.3 Shifts in the Demand Curve for Bonds


4-57

How Factors Shift the Demand Curve

1. Wealth– Economy , wealth , Bd , Bd shifts out to right

2. Expected Return– i in future, Re for long-term bonds , Bd shifts

out to right– πe , relative Re , Bd shifts out to right


4-58

How Factors Shift the Demand Curve

1. Risk– Risk of bonds , Bd , Bd shifts out to right– Risk of other assets , Bd , Bd shifts out to right

2. Liquidity– Liquidity of bonds , Bd , Bd shifts out to right– Liquidity of other assets , Bd ,Bd shifts out to

right

Factors That Shift Demand Curve


4-60

Summary of Shifts in the Demand for Bonds

1. Wealth: in a business cycle expansion with growing wealth, the demand for bonds rises, conversely, in a recession, when income and wealth are falling, the demand for bonds falls

2. Expected returns: higher expected interest rates in the future decrease the demand for long-term bonds, conversely, lower expected interest rates in the future increase the demand for long-term bonds


4-61

Summary of Shifts in the Demand for Bonds (2)

1. Risk: an increase in the riskiness of bonds causes the demand for bonds to fall, conversely, an increase in the riskiness of alternative assets (like stocks) causes the demand for bonds to rise

2. Liquidity: increased liquidity of the bond market results in an increased demand for bonds, conversely, increased liquidity of alternative asset markets (like the stock market) lowers the demand for bonds


4-62

Shifts in the Supply Curve

1. Profitability of Investment Opportunities

– Business cycle expansion, investment opportunities , Bs , Bs shifts out to right

2. Expected Inflation– πe , Bs , Bs shifts out

to right

3. Government Activities– Deficits , Bs , Bs shifts out

to right

Figure 4.4 Shift in the Supply Curve for Bonds


4-63

Factors That Shift Supply Curve


4-64

Summary of Shifts in the Supply of Bonds

1. Expected Profitability of Investment Opportunities: in a business cycle expansion, the supply of bonds increases, conversely, in a recession, when there are far fewer expected profitable investment opportunities, the supply of bonds falls

2. Expected Inflation: an increase in expected inflation causes the supply of bonds to increase

3. Government Activities: higher government deficits increase the supply of bonds, conversely, government surpluses decrease the supply of bonds


4-65

Changes in πe: The Fisher Effect

• If πe 1. Relative Re ,

Bd shifts in to left

2. Bs , Bs shifts out to right

3. P , i

Figure 4.5 Response to a Change in Expected Inflation


4-66

Figure 4.6 Expected Inflation and Interest Rates (Three-Month Treasury Bills), 1953–2004

Evidence on the Fisher Effect in the United States


4-67

Summary of the Fisher Effect1. If expected inflation rises from 5% to 10%, the expected

return on bonds relative to real assets falls and, as a result, the demand for bonds falls

2. The rise in expected inflation also means that the real cost of borrowing has declined, causing the quantity of bonds supplied to increase

3. When the demand for bonds falls and the quantity of bonds supplied increases, the equilibrium bond price falls

4. Since the bond price is negatively related to the interest rate, this means that the interest rate will rise


4-68

Business Cycle Expansion

1. Wealth , Bd , Bd shifts out to right

2. Investment , Bs , Bs shifts right

3. If Bs shifts more than Bd then P , i

Figure 4.7 Response to a Business Cycle Expansion


4-69

Evidence on Business Cycles and Interest Rates

Figure 4.8 Business Cycle and Interest Rates (Three-Month Treasury Bills), 1951–2004


4-70

Profiting from Interest-Rate Forecasts

• Methods for forecasting1. Supply and demand for bonds: use Flow of

Funds Accounts and judgment

2. Econometric Models: large in scale, use interlocking equations that assume past financial relationships will hold in the future


4-71

Profiting from Interest-Rate Forecasts (cont.)

• Make decisions about assets to hold1. Forecast i , buy long bonds2. Forecast i , buy short bonds

• Make decisions about how to borrow1. Forecast i , borrow short2. Forecast i , borrow long

Web AppendixApplying the Asset Market Approach to a Commodity

Market: The Case of Gold


4-73

Re Pt1

e Pt

Pt

ge

Supply and Demand in Gold Market

• Deriving Demand Curve

– Pet+1 is held constant

– Pt , ge , Re Gd – Demand curve is downward sloping

• Deriving Supply Curve– Pt , more production, Gs – Supply curve is upward sloping


4-74

Supply and Demand in Gold Market

• Market Equilibrium

1. Gd = Gs

2. If Pt > P* = P1, Gs > Gd, Pt to P*

3. If Pt < P* = P1, Gs < Gd, Pt to P*


4-75

Changes in Equilibrium

• Factors That Shift Demand Curve for Gold1. Wealth2. Expected return on gold relative to alternative assets3. Riskiness of gold relative to alternative assets4. Liquidity of gold relative to alternative assets

• Factors That Shift Supply Curve for Gold1. Technology of mining2. Government sales of gold


4-76

Response of Gold Market to a Change in πe

• If πe 1. πe , Pe

t+1 ; at given Pt, ge Gd Gd shifts right

2. Go to point 2; Pt

3. Price of gold positively related to πe

4. Gold price is barometer of π- pressure

Figure 4.Web A1 A Change in the Equilibrium Price of Gold


4-77

Chapter Summary

• Determining Asset Demand: We examined the forces that affect the demand and supply of assets.

• Supply and Demand in the Bond Market: We examine those forces in the context of bonds, and examined the impact on interest rates.


4-78

Chapter Summary (cont.)

• Changes in Equilibrium Interest Rates: We further examined the dynamics of changes in supply and demand in the bond market, and the corresponding effect on bond prices and interest rates.

Documents

Chapter 3 What Do Interest Rates Mean and What is Their Role in Valuation?