## Chapter Two

### Determinants of

### Interest Rates

**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

**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

**Calculation of Simple Interest**

**Value = Principal + Interest**

### Example:

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

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

### = $1,240

**Value = Principal + Interest**

### Example:

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

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

**Value of Compound Interest**

**Value = Principal + Interest + Compounded interest**

### Value = $1,000 + $1,000(12)(2) + $1,000(12)(2)

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

### = $1,000[1 + 2(12) + (12)

### 2

_{] }

### = $1,000(1.12)

### 2

**Present Values**

### • 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

### – lump sum payment

### • a single cash payment received at the end of some

### investment horizon

### – annuity

### • a series of equal cash payments received at fixed

### intervals over the investment horizon

### • 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

### – lump sum payment

### • a single cash payment received at the end of some

### investment horizon

### – annuity

### • a series of equal cash payments received at fixed

### intervals over the investment horizon

**Calculating Present Value (PV) of a Lump **

**Sum**

**PV = FV**

_{n}

_{n}**(1/(1 + i/m))**

**(1/(1 + i/m))**

**nm**

**nm**

**= FV**

_{n}

_{n}**(PVIF**

_{i/m,nm}

_{i/m,nm}**)**

### where:

### PV = present value

*FV = future value (lump sum) received in n years*

*i = simple annual interest*

*n = number of years in investment horizon*

*m = number of compounding periods in a year*

### PVIF = present value interest factor of a lump sum

**PV = FV**

_{n}

_{n}**(1/(1 + i/m))**

**(1/(1 + i/m))**

**nm**

**nm**

**= FV**

_{n}

_{n}**(PVIF**

_{i/m,nm}

_{i/m,nm}**)**

### where:

### PV = present value

*FV = future value (lump sum) received in n years*

*i = simple annual interest*

*n = number of years in investment horizon*

*m = number of compounding periods in a year*

**Calculation of Present Value (PV) of an **

**Annuity**

### nm

**PV = PMT **

### ∑

### ∑

### ∑

### ∑

**(1/(1 + i/m))**

**(1/(1 + i/m))**

**t **

**t**

**= PMT(PVIFA**

_{i/m,nm}

_{i/m,nm}**)**

### t = 1

### where:

### PV = present value

### PMT = periodic annuity payment received

### during investment

*i = simple annual interest*

*n = number of years in investment horizon*

*m = number of compounding periods in a year*

### PVIFA = present value interest factor of an annuity

### nm

**PV = PMT **

### ∑

### ∑

### ∑

### ∑

**(1/(1 + i/m))**

**(1/(1 + i/m))**

**t **

**t**

**= PMT(PVIFA**

_{i/m,nm}

_{i/m,nm}**)**

### t = 1

### where:

### PV = present value

### PMT = periodic annuity payment received

### during investment

*i = simple annual interest*

*n = number of years in investment horizon*

*m = number of compounding periods in a year*

**Calculation of Present Value of an Annuity**

**You are offered a security investment that pays $10,000 on **

**the last day of every quarter for the next 6 years in **

**exchange for a fixed payment today.**

**PV = PMT(PVIFA**

_{i/m,nm}

_{i/m,nm}**)**

**at 8% interest - = $10,000(18.913926) = $189,139.26**

**at 12% interest - = $10,000(16.935542) = $169,355.42**

**at 16% interest - = $10,000(15.246963) = $152,469.63**

**You are offered a security investment that pays $10,000 on **

**the last day of every quarter for the next 6 years in **

**exchange for a fixed payment today.**

**PV = PMT(PVIFA**

_{i/m,nm}

_{i/m,nm}**)**

**at 8% interest - = $10,000(18.913926) = $189,139.26**

**at 12% interest - = $10,000(16.935542) = $169,355.42**

**at 16% interest - = $10,000(15.246963) = $152,469.63**

**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

**Future Values Equations**

**FV of lump sum equation**

### FV

_{n }

### = PV(1 + i/m)

### nm

_{= PV(FVIF }

### i/m, nm

### )

**FV of annuity payment equation**

*(nm-1)*

### FV

_{n}

### = PMT

### ∑

### ∑

### ∑

### ∑

### (1 + i/m)

### t

### = PMT(FVIFA

_{i/m, mn}

### )

*(t = 1)*

**FV of lump sum equation**

### FV

_{n }

### = PV(1 + i/m)

### nm

_{= PV(FVIF }

### i/m, nm

### )

**FV of annuity payment equation**

*(nm-1)*

### FV

_{n}

### = PMT

### ∑

### ∑

### ∑

### ∑

### (1 + i/m)

### t

### = PMT(FVIFA

_{i/m, mn}

### )

**Relation between Interest Rates and **

**Present and Future Values**

**Present **

**Value**

**(PV)**

**Interest Rate**

**Future**

**Value**

**(FV)**

**Interest Rate**

**Equivalent Annual Return (EAR)**

### Rate returned over a 12-month period

### taking the compounding of interest into

### account

### EAR = (1 + i/m)

### m

### - 1

### At 8% interest - EAR = (1 + .08/4)

### 4

### - 1 = 8.24%

### At 12% interest - EAR = (1 + .12/4)

### 4

_{- 1 = 12.55%}

### Rate returned over a 12-month period

### taking the compounding of interest into

### account

### EAR = (1 + i/m)

### m

### - 1

### At 8% interest - EAR = (1 + .08/4)

### 4

### - 1 = 8.24%

**Discount Yields**

### Money market instruments (e.g., Treasury

### bills and commercial paper) that are bought

### and sold on a discount basis

**i**

_{dy}

_{dy}

**= [(P**

_{t}

_{t}

**- P**

_{o}

_{o}

**)/P**

_{f}

_{f}

**](360/h)**

### Where:

*P*

_{f }

_{f }

### = Face value

*P*

_{o }

_{o }

### = Discount price of security

### Money market instruments (e.g., Treasury

### bills and commercial paper) that are bought

### and sold on a discount basis

**i**

_{dy}

_{dy}

**= [(P**

_{t}

_{t}

**- P**

_{o}

_{o}

**)/P**

_{f}

_{f}

**](360/h)**

### Where:

*P*

_{f }

_{f }

### = Face value

**Single Payment Yields**

### Money market securities (e.g., jumbo CDs,

### fed funds) that pay interest only once during

### their lives: at maturity

**i**

_{bey}

_{bey}

**= i**

_{spy}

_{spy}

**(365/360)**

### Money market securities (e.g., jumbo CDs,

### fed funds) that pay interest only once during

### their lives: at maturity

**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

**Supply of Loanable Funds**

**Interest**

**Rate**

**Quantity of Loanable Funds**

**Supplied and Demanded**

**Funds Supplied and Demanded by Various **

**Groups (in billions of dollars)**

### Funds Supplied

### Funds Demanded

### Households $31,866.4

### $ 6,624.4

### Business -- nonfinancial 7,400.0 30,356.2

### Business -- financial 27,701.9 29,431.1

### Government units 6,174.8

### 10,197.9

### Foreign participants 6,164.8

### 2,698.3

### Funds Supplied

### Funds Demanded

### Households $31,866.4

### $ 6,624.4

### Business -- nonfinancial 7,400.0 30,356.2

### Business -- financial 27,701.9 29,431.1

### Government units 6,174.8

### 10,197.9

### Foreign participants 6,164.8

### 2,698.3

**Determination of Equilibrium Interest **

**Rates**

**Interest**

**Rate**

**Quantity of Loanable Funds**

**Supplied and Demanded**

**D**

_{S}

_{S}

**I **

**I**

**H**

**H**

**i**

**i**

**I **

**I**

**L**

**L**

**E**

**E**

**Q**

**Q**

**Effect on Interest rates from a Shift in the **

**Demand Curve for or Supply curve of **

**Loanable Funds**

**Increased supply of loanable funds**

### Quantity of

### Funds Supplied

### Interest

### Rate

**DD**

**DD**

**SS**

**SS**

**SS***

**SS***

**E**

**E**

**E***

**E***

**Q***

**Q***

**i***

**i***

**Q****

**Q****

**i****

**i****

**Increased demand for loanable funds**

### Quantity of

### Funds Demanded

**DD**

**DD**

**DD***

**DD***

**SS**

**SS**

**E**

**E**

**E***

**E***

**i***

**i***

**i****

**i****

**Q* Q****

**Q* Q****

**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

**• Liquidity Risk**

### – risk that a security can not be sold at a

### predictable price with low transaction cost on

### short notice

**• Special Provisions**

### – taxability

### – convertibility

### – callability

**• 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

### – convertibility

### – callability

**• Time to Maturity**

**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**

**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**

**i = Expected (IP) + RIR**

**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

**DRP**

_{j}

_{j}**= i**

**= i**

_{jt}

_{jt}**- i**

**- i**

_{Tt}

_{Tt}### Example: DRP

_{Aaa}

_{Aaa}

### = 7.55% - 6.35% = 1.20%

### DRP

_{Bbb}

_{Bbb}

### = 8.15% - 6.35% = 1.80%

### The risk that a security’s issuer will default

### on that security by being late on or missing

### an interest or principal payment

**DRP**

_{j}

_{j}**= i**

**= i**

_{jt}

_{jt}**- i**

**- i**

_{Tt}

_{Tt}### Example: DRP

_{Aaa}

_{Aaa}

### = 7.55% - 6.35% = 1.20%

### DRP

_{Bbb}

_{Bbb}

### = 8.15% - 6.35% = 1.80%

**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.

**i**

**i**

_{m}

_{m}**= i**

**= i**

_{c}

_{c}**(1 - t**

**(1 - t**

_{s}

_{s}**- t**

**- t**

_{F}

_{F}**)**

### Where:

*i*

_{c}

_{c}

### = Interest rate on a corporate bond

*i*

_{m}

_{m}

### = Interest rate on a municipal bond

*t*

_{s}

_{s}

*= State plus local tax rate*

*t*

### = 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.

**i**

**i**

_{m}

_{m}**= i**

**= i**

_{c}

_{c}**(1 - t**

**(1 - t**

_{s}

_{s}**- t**

**- t**

_{F}

_{F}**)**

### Where:

*i*

_{c}

_{c}

### = Interest rate on a corporate bond

*i*

_{m}

_{m}

### = Interest rate on a municipal bond

*t*

_{s}

_{s}

*= State plus local tax rate*

*t*

_{F}

_{F}

### = Federal tax rate

**Term to Maturity and Interest Rates: **

**Yield Curve**

**Yield to**

**Maturity**

**Time to Maturity**

### (a)

### (b)

### (c)

### (d)

### (a) Upward sloping

### (b) Inverted or downward

### sloping

### (c) Humped

### (d) Flat

**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**

### – 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

**• 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**

### – 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

**Forecasting Interest Rates**

### Forward rate is an expected or “implied” rate

### on a security that is to be originated at some

### point in the future using the unbiased

### expectations theory

**_**

**_**

**R**

_{2}

_{2}

**= [(1 + R**

_{1}

_{1}

**)(1 + (f**

_{2}

_{2}

**))]**

**1/2**

**- 1**

### where

### f

_{2 }

### = expected one-year rate for year 2, or the implied

### forward one-year rate for next year

### Forward rate is an expected or “implied” rate

### on a security that is to be originated at some

### point in the future using the unbiased

### expectations theory

**_**

**_**

**R**

_{2}

_{2}

**= [(1 + R**

_{1}

_{1}

**)(1 + (f**

_{2}

_{2}