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Chapter TwoDeterminants of

Interest Rates

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Interest Rate Fundamentals

• Nominal interest rates - the interest rate actually observed in financial markets– directly affect the value (price) of most

securities traded in the market

– affect the relationship between spot and forward FX rates

• Nominal interest rates - the interest rate actually observed in financial markets– directly affect the value (price) of most

securities traded in the market

– affect the relationship between spot and forward FX rates

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Time Value of Money and Interest Rates

• Assumes the basic notion that a dollar received today is worth more than a dollar received at some future date

• Compound interest– interest earned on an investment is reinvested

• Simple interest– interest earned on an investment is not

reinvested

• Assumes the basic notion that a dollar received today is worth more than a dollar received at some future date

• Compound interest– interest earned on an investment is reinvested

• Simple interest– interest earned on an investment is not

reinvested

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Calculation of Simple Interest

Value = Principal + Interest (year 1) + Interest (year 2)

Example: $1,000 to invest for a period of two years at 12 percent

Value = $1,000 + $1,000(.12) + $1,000(.12)

= $1,000 + $1,000(.12)(2)

= $1,240

Value = Principal + Interest (year 1) + Interest (year 2)

Example: $1,000 to invest for a period of two years at 12 percent

Value = $1,000 + $1,000(.12) + $1,000(.12)

= $1,000 + $1,000(.12)(2)

= $1,240

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Value of Compound Interest

Value = Principal + Interest + Compounded interest

Value = $1,000 + $1,000(.12) + $1,000(.12) + $1,000(.12)

= $1,000[1 + 2(.12) + (.12)2]

= $1,000(1.12)2

= $1,254.40

Value = Principal + Interest + Compounded interest

Value = $1,000 + $1,000(.12) + $1,000(.12) + $1,000(.12)

= $1,000[1 + 2(.12) + (.12)2]

= $1,000(1.12)2

= $1,254.40

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Present Value of a Cashflow

• PV function converts cash flows received over a future investment horizon into an equivalent (present) value by discounting future cash flows back to present using current market interest rate

• PVs decrease as interest rates increase

• PV function converts cash flows received over a future investment horizon into an equivalent (present) value by discounting future cash flows back to present using current market interest rate

• PVs decrease as interest rates increase

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Calculating the Present Value (PV) of a Cashflow

PV = FVn(1/(1 + i/m))nm = FVn(PVIFi/m,nm)

where:PV = present valueFV = future value (lump sum) received in n years i = simple annual interest rate earned n = number of years in investment horizon m = number of compounding periods in a year

i/m = periodic rate earned on investments nm = total number of compounding periods PVIF = present value interest factor of a lump sum

PV = FVn(1/(1 + i/m))nm = FVn(PVIFi/m,nm)

where:PV = present valueFV = future value (lump sum) received in n years i = simple annual interest rate earned n = number of years in investment horizon m = number of compounding periods in a year

i/m = periodic rate earned on investments nm = total number of compounding periods PVIF = present value interest factor of a lump sum

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Calculating Present Value of a Lump Sum

• You are offered a security investment that pays $10,000 at the end of 6 years in exchange for a fixed payment today.

• PV = FV(PVIFi/m,nm)

• at 8% interest - = $10,000(0.630170) = $6,301.70

• at 12% interest - = $10,000(0.506631) = $5,066.31

• at 16% interest - = $10,000(0.410442) = $4,104.42

• You are offered a security investment that pays $10,000 at the end of 6 years in exchange for a fixed payment today.

• PV = FV(PVIFi/m,nm)

• at 8% interest - = $10,000(0.630170) = $6,301.70

• at 12% interest - = $10,000(0.506631) = $5,066.31

• at 16% interest - = $10,000(0.410442) = $4,104.42

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Future Values

• Translate cash flows received during an investment period to a terminal (future) value at the end of an investment horizon

• FV increases with both the time horizon and the interest rate

• Translate cash flows received during an investment period to a terminal (future) value at the end of an investment horizon

• FV increases with both the time horizon and the interest rate

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Calculation of Future Value of a Lump Sum

• You invest $10,000 today in exchange for a fixed payment at the end of six years– at 8% interest = $10,000(1.586874) = $15,868.74

– at 12% interest = $10,000(1.973823) = $19,738.23

– at 16% interest = $10,000(2.436396) = $24,363.96

– at 16% interest compounded semiannually

• = $10,000(2.518170) = $25,181.70

• You invest $10,000 today in exchange for a fixed payment at the end of six years– at 8% interest = $10,000(1.586874) = $15,868.74

– at 12% interest = $10,000(1.973823) = $19,738.23

– at 16% interest = $10,000(2.436396) = $24,363.96

– at 16% interest compounded semiannually

• = $10,000(2.518170) = $25,181.70

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Relation between Interest Rates and Present and Future Values

Present Value(PV)

Interest Rate

FutureValue(FV)

Interest Rate

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Effective or Equivalent Annual Return (EAR)

Rate earned over a 12 – month period taking the compounding of interest into account.

EAR = (1 + r) c – 1

Where c = number of compounding periods per year

Rate earned over a 12 – month period taking the compounding of interest into account.

EAR = (1 + r) c – 1

Where c = number of compounding periods per year

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Loanable Funds Theory

• A theory of interest rate determination that views equilibrium interest rates in financial markets as a result of the supply and demand for loanable funds

• A theory of interest rate determination that views equilibrium interest rates in financial markets as a result of the supply and demand for loanable funds

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Supply of Loanable Funds

InterestRate

Quantity of Loanable FundsSupplied and Demanded

Demand Supply

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Funds Supplied and Demanded by Various Groups (in billions of dollars)

Funds Supplied Funds Demanded Net

Households $34,860.7 $15,197.4 $19,663.3Business - nonfinancial 12,679.2 30,779.2 -12,100.0Business - financial 31,547.9 45061.3 -13,513.4Government units 12,574.5 6,695.2 5,879.3Foreign participants 8,426.7 2,355.9 6,070.8

Funds Supplied Funds Demanded Net

Households $34,860.7 $15,197.4 $19,663.3Business - nonfinancial 12,679.2 30,779.2 -12,100.0Business - financial 31,547.9 45061.3 -13,513.4Government units 12,574.5 6,695.2 5,879.3Foreign participants 8,426.7 2,355.9 6,070.8

2-16McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Determination of Equilibrium Interest Rates

InterestRate

Quantity of Loanable FundsSupplied and Demanded

D S

I H

i

I L

E

Q

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Effect on Interest rates from a Shift in the Demand Curve for or Supply curve of

Loanable FundsIncreased supply of loanable funds

Quantity ofFunds Supplied

InterestRate DD SS

SS*

EE*

Q*

i*

Q**

i**

Increased demand for loanable funds

Quantity of Funds Demanded

DDDD* SS

EE*

i*

i**

Q* Q**

2-18McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Factors Affecting Nominal Interest Rates

• Inflation– continual increase in price of goods/services

• Real Interest Rate– nominal interest rate in the absence of inflation

• Default Risk– risk that issuer will fail to make promised payment

• Inflation– continual increase in price of goods/services

• Real Interest Rate– nominal interest rate in the absence of inflation

• Default Risk– risk that issuer will fail to make promised payment

(continued)

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• Liquidity Risk– risk that a security can not be sold at a

predictable price with low transaction cost on short notice

• Special Provisions – Taxability (Exempt = Lower rate paid)– Convertibility (Lower rate paid)– Callability (Higher rate paid)

• Time to Maturity

• Liquidity Risk– risk that a security can not be sold at a

predictable price with low transaction cost on short notice

• Special Provisions – Taxability (Exempt = Lower rate paid)– Convertibility (Lower rate paid)– Callability (Higher rate paid)

• Time to Maturity

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Inflation and Interest Rates: The Fischer Effect

The interest rate should compensate an investor for both expected inflation and the opportunity cost of foregone consumption (the real rate component)

i = Expected (IP) + RIR

Example: 5.08% - 2.70% = 2.38%

The interest rate should compensate an investor for both expected inflation and the opportunity cost of foregone consumption (the real rate component)

i = Expected (IP) + RIR

Example: 5.08% - 2.70% = 2.38%

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Default Risk and Interest Rates

The risk that a security’s issuer will default on that security by being late on or missing an interest or principal payment

DRPj = ijt - iTt

Example: DRPAaa = 7.55% - 6.35% = 1.20% DRPBbb = 8.15% - 6.35% = 1.80%

Both bonds are 30 year bonds.

The risk that a security’s issuer will default on that security by being late on or missing an interest or principal payment

DRPj = ijt - iTt

Example: DRPAaa = 7.55% - 6.35% = 1.20% DRPBbb = 8.15% - 6.35% = 1.80%

Both bonds are 30 year bonds.

2-22McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Tax Effects: The Tax Exemption of Interest on Municipal Bonds

Interest payments on municipal securities are exempt from federal taxes and possibly state and local taxes. Therefore, yields on “munis” are generally lower than on equivalent taxable bonds such as corporate bonds. iEXEMPT = iTAX *(1 - ts - tF)

iEXEMPT / (1 - ts - tF) = iTAX

Where: iTAX = Taxable equivalent rate iEXEMPT = Interest rate on a municipal bond ts = State plus local tax rate tF = Federal tax rate

Interest payments on municipal securities are exempt from federal taxes and possibly state and local taxes. Therefore, yields on “munis” are generally lower than on equivalent taxable bonds such as corporate bonds. iEXEMPT = iTAX *(1 - ts - tF)

iEXEMPT / (1 - ts - tF) = iTAX

Where: iTAX = Taxable equivalent rate iEXEMPT = Interest rate on a municipal bond ts = State plus local tax rate tF = Federal tax rate

2-23McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Term to Maturity and Interest Rates: Yield Curve

Yield toMaturity

Time to Maturity

(a)

(b)

(c)

(a) Upward sloping(b) Inverted or downward sloping(c) Flat

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Term Structure of Interest Rates

• Unbiased Expectations Theory– at a given point in time, the yield curve reflects the

market’s current expectations of future short-term rates

• Liquidity Premium Theory– an extension of the unbiased expectations theory,

namely, investors will only hold long-term maturities if they are offered a premium to compensate for future uncertainty in a security’s value

• Market Segmentation Theory– investors have specific maturity preferences and will

demand a higher maturity premium to move outside of that preferred maturity

• Unbiased Expectations Theory– at a given point in time, the yield curve reflects the

market’s current expectations of future short-term rates

• Liquidity Premium Theory– an extension of the unbiased expectations theory,

namely, investors will only hold long-term maturities if they are offered a premium to compensate for future uncertainty in a security’s value

• Market Segmentation Theory– investors have specific maturity preferences and will

demand a higher maturity premium to move outside of that preferred maturity

2-25McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Chapter ThreeInterest Rates and

Security Valuation

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Various Interest Rate Measures

• Coupon rate

• Required Rate of Return

• Expected rate of return

• Realized Rate of Return

• Coupon rate

• Required Rate of Return

• Expected rate of return

• Realized Rate of Return

2-27McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Bond Valuation

• The valuation of a bond instrument employs time value of money concepts– Reflects present value of all cash flows promised or

projected, discounted at the required rate of return (rrr)

– Expected rate of return (Err) is the interest rate that equates the current market price to the present value of all promised cash flows received over the life of the bond

– Realized rate of return (rr) on a bond is the actual return earned on a bond investment that has already taken place

• The valuation of a bond instrument employs time value of money concepts– Reflects present value of all cash flows promised or

projected, discounted at the required rate of return (rrr)

– Expected rate of return (Err) is the interest rate that equates the current market price to the present value of all promised cash flows received over the life of the bond

– Realized rate of return (rr) on a bond is the actual return earned on a bond investment that has already taken place

2-28McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Bond Valuation Formula

Vb = INT/2 + INT/2 + . . . + INT/2 __ (1 + id/2)1 (1 + id/2)2 (1 + id/2)2N

+ M_ _ _ (1 + id/2)2N

Where: Vb = Present value of the bond M = Par or face value of the bond INT = Annual interest (or coupon) payment per year on the bond; equals the par value of the bond times the (percentage) coupon rate N = Number years until the bond matures id = Interest rate used to discount cash flows on the bond

Vb = INT/2 + INT/2 + . . . + INT/2 __ (1 + id/2)1 (1 + id/2)2 (1 + id/2)2N

+ M_ _ _ (1 + id/2)2N

Where: Vb = Present value of the bond M = Par or face value of the bond INT = Annual interest (or coupon) payment per year on the bond; equals the par value of the bond times the (percentage) coupon rate N = Number years until the bond matures id = Interest rate used to discount cash flows on the bond

2-29McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Bond Valuation Example

Vb = 1,000(.1) (PVIFA8%/2, 12(2)) + 1,000(PVIF8%/2, 12(2)) 2

Where: Vb = $1,152.47 (solution) M = $1,000 INT = $100 per year (10% of $1,000) N = 12 years id = 8% (rrr) PVIF = Present value interest factor of a lump sum payment

PVIFA = present value interest factor of an annuity stream

Vb = 1,000(.1) (PVIFA8%/2, 12(2)) + 1,000(PVIF8%/2, 12(2)) 2

Where: Vb = $1,152.47 (solution) M = $1,000 INT = $100 per year (10% of $1,000) N = 12 years id = 8% (rrr) PVIF = Present value interest factor of a lump sum payment

PVIFA = present value interest factor of an annuity stream

2-30McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

A Better Example of Finding the Price of a Bond

1 CFt

Time(yrs) CFt (1 + 4%)2t (1 + 4%)2t

1 CFt

Time(yrs) CFt (1 + 4%)2t (1 + 4%)2t

.51

1.52

2.53

3.54

50505050505050

1,050

0.96150.92460.88900.85480.82190.79030.75990.7307

48.0846.2344.4542.7441.1039.5238.00

767.22

Coupon/PrincipalPayments

$1067.34

10% Coupon Bond, 8% Discount Rate, 4 Years to MaturityWhat’s the current price?

PV factor

PV of theCashflow

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Description of a Premium, Discount, and Par Bond

• Premium bond—when the coupon rate, INT, is greater then the required rate of return, rrr, the fair present value of the bond (Vb) is greater than its face value (M)

• Discount bond—when INT<rrr, then Vb <M

• Par bond—when INT=rrr, then Vb =M

• Premium bond—when the coupon rate, INT, is greater then the required rate of return, rrr, the fair present value of the bond (Vb) is greater than its face value (M)

• Discount bond—when INT<rrr, then Vb <M

• Par bond—when INT=rrr, then Vb =M

2-32McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Yield to Maturity

The return or yield the bond holder will earn on the bond if he or she buys it at its current market price, receives all coupon and principal payments as promised, and holds the bond until maturity

Vb = INT (PVIFAytm/m, Nm) + M(PVIFytm/m,Nm) m

The return or yield the bond holder will earn on the bond if he or she buys it at its current market price, receives all coupon and principal payments as promised, and holds the bond until maturity

Vb = INT (PVIFAytm/m, Nm) + M(PVIFytm/m,Nm) m

2-33McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Summary of Factors that Affect Security Prices and Price Volatility when Interest

Rates Change

• Interest Rate

• Time Remaining to Maturity

• Coupon Rate

• Interest Rate

• Time Remaining to Maturity

• Coupon Rate

2-34McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Impact of Interest Rate Changes on Security Values

Interest Rate

Bond Value

Interest Rate

Bond Value

12%

10%

8%

874.50 1,000 1,152.47

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Balance sheet of an FI before and after an Interest Rate Increase

(a) Balance Sheet before the Interest Rate Increase (a) Balance Sheet before the Interest Rate Increase Assets

Bond(8% requiredrate of return)

$1,152.47

Liabilities and Equity

Bond(10% requiredrate of return)

$1,000

Equity $152.47

(b) Balance Sheet after 2% increase in the Interest Rate Increase

Assets

$1,000Bond(10% requiredrate of return)

Liabilities and Equity

Bond(12% requiredrate of return)

Equity

$874.50

$125.50

2-36McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Impact of Maturity on Security Values

12 Years to Maturity 12 Years to Maturity 16 Years to Maturity

RequiredRate ofReturn

FairPrice*

PriceChange

PercentagePrice

Change

8% $1,152.47-$152.47 -13.23%

10% $1,000.00-$125.50 -12.55%

12% $874.50

FairPrice*

PriceChange

PercentagePrice

Change

$1,178.74-$178.74 -15.16%

$1,000.00-$140.84 -14.08%

$859.16

*The bond pays 10% coupon interest compounded semiannually and has a face value of $1,000

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Impact of a Bond’s Maturity on its Interest Rate Sensitivity

Absolute Value of Percent Change in aBond’s Price for aGiven Change inInterest Rates

Absolute Value of Percent Change in aBond’s Price for aGiven Change inInterest Rates

Time to Maturity

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Impact of a Bond’s Coupon Rate on Its Interest Rate Sensitivity

Interest Rate

Interest Rate

Bond Value

Low-Coupon Bond

High-Coupon Bond

2-39McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Duration: A Measure of Interest Rate Sensitivity

The weighted-average time to maturity on an investment

N N

CFt t PVt t t = 1 (1 + R)t t = 1

D = N = N

CFt PVt

t = 1 (1 + R)t t = 1

The weighted-average time to maturity on an investment

N N

CFt t PVt t t = 1 (1 + R)t t = 1

D = N = N

CFt PVt

t = 1 (1 + R)t t = 1

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Features of the Duration Measure

• Duration and Coupon Interest– the higher the coupon payment, the lower is a

bond’s duration

• Duration and Yield to Maturity– duration increases as yield to maturity increases

• Duration and Maturity– Duration increases with the maturity of a bond but

at a decreasing rate

• Duration and Coupon Interest– the higher the coupon payment, the lower is a

bond’s duration

• Duration and Yield to Maturity– duration increases as yield to maturity increases

• Duration and Maturity– Duration increases with the maturity of a bond but

at a decreasing rate

2-41McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Discrepancy Between Maturity and Duration on a Coupon Bond

0

1

2

3

4

5

6

7

1 2 3 4 5 6

Maturity Duration

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Example of Duration Calculation

1 CFt CFt X t Percent of Initialt CFt (1 + 4%)2t (1 + 4%)2t (1 + 4%)2t Investment Recovered

1 CFt CFt X t Percent of Initialt CFt (1 + 4%)2t (1 + 4%)2t (1 + 4%)2t Investment Recovered

.51

1.52

2.53

3.54

50505050505050

1,050

0.96150.92460.88900.85480.82190.79030.75990.7307

48.0846.2344.4542.7441.1039.5238.00

767.22

24.0446.2366.6785.48102.75118.56133.00

3,068.88

24.04/1,067.34 = 0.0246.23/1,067.34 = 0.0466.67/1,067.34 = 0.0685.48/1,067.34 = 0.08

102.75/1,067.34 = 0.10118.56/1,067.34 = 0.11133.00/1,067.34 = 0.13

3,068.88/1,067.34 = 2.88

Duration = 3,645.611,067.34

= 3.42 years

PV factor PV of Cashflow Time Weighted PV of Cashflow

$1067.34 3.42

10% Coupon Bond, 8% Discount Rate, 4 Years to Maturity

2-43McGraw-Hill/Irwin ©2007, The McGraw-Hill Companies, All Rights Reserved

Economic Meaning of Duration

• Measure of the average life of a bond

• Measure of a bond’s interest rate sensitivity (elasticity)

• Measure of the average life of a bond

• Measure of a bond’s interest rate sensitivity (elasticity)