Week-3 Into to Interest Rates

Money and Banking Econ 311Tuesdays 7 - 9:45

Instructor: Thomas L. Thomas

Measuring Interest Rates

o Present Value:o A dollar paid to you one year from now is less

valuable than a dollar paid to you todayo Why?o A dollar deposited today can earn interest and

become $1 x (1+i) one year from today.

Discounting the Future

2

3

Let = .10

In one year $100 X (1+ 0.10) = $110

In two years $110 X (1 + 0.10) = $121

or 100 X (1 + 0.10)

In three years $121 X (1 + 0.10) = $133

or 100 X (1 + 0.10)

In years

$100 X (1 + ) n

i

n

i

Simple Present Value

n

PV = today's (present) value

CF = future cash flow (payment)

= the interest rate

CFPV =

(1 + )

i

i

Time Line

$100 $100

Year 0 1

PV 100

2

$100 $100

n

100/(1+i) 100/(1+i)2 100/(1+i)n

• Cannot directly compare payments scheduled in different points in the time line

Four Types of Credit Market Instruments

oSimple LoanoFixed Payment LoanoCoupon BondoDiscount Bond

Yield to Maturity

o The interest rate that equates the present value of cash flow payments received from a debt instrument with its value today

Simple Loan

1

PV = amount borrowed = $100

CF = cash flow in one year = $110

= number of years = 1

$110$100 =

(1 + )

(1 + ) $100 = $110

$110(1 + ) =

$100 = 0.10 = 10%

For simple loans, the simple interest rate equ

n

i

i

i

i

als the

yield to maturity

Fixed Payment Loan

2 3

The same cash flow payment every period throughout

the life of the loan

LV = loan value

FP = fixed yearly payment

= number of years until maturity

FP FP FP FPLV = . . . +

1 + (1 + ) (1 + ) (1 + )n

n

i i i i

Coupon Bond

2 3

Using the same strategy used for the fixed-payment loan:

P = price of coupon bond

C = yearly coupon payment

F = face value of the bond

= years to maturity date

C C C C FP = . . . +

1+ (1+ ) (1+ ) (1+ ) (1n

n

i i i i

+ )ni

Table 1 Yields to Maturity on a 10%-Coupon-Rate Bond Maturing in Ten Years (Face Value = $1,000)

o When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate

o The price of a coupon bond and the yield to maturity are negatively related

o The yield to maturity is greater than the coupon rate when the bond price is below its face value

Microsoft Excel Example

Consol or Perpetuity

o A bond with no maturity date that does not repay principal but pays fixed coupon payments forever

consol theofmaturity toyield

paymentinterest yearly

consol theof price

/

c

c

c

i

C

P

iCP

cc PCi /: thisasequation above rewritecan For coupon bonds, this equation gives the current yield, an easy to calculate approximation to the yield to maturity

Discount Bond

For any one year discount bond

i = F - P

PF = Face value of the discount bond

P = current price of the discount bond

The yield to maturity equals the increase

in price over the year divided by the initial price.

As with a coupon bond, the yield to maturity is

negatively related to the current bond price.

The Distinction Between Interest Rates and Returns

The payments to the owner plus the change in value

expressed as a fraction of the purchase price

RET = C

Pt

+ P

t1 - P

t

Pt

RET = return from holding the bond from time t to time t + 1

Pt = price of bond at time t

Pt1

= price of the bond at time t + 1

C = coupon payment

C

Pt

= current yield = ic

P

t1 - P

t

Pt

= rate of capital gain = g

• Rate of Return:

The Distinction Between Interest Rates and Returns (cont’d)

o The return equals the yield to maturity only if the holding period equals the time to maturity

o A rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if time to maturity is longer than the holding period

o The more distant a bond’s maturity, the greater the size of the percentage price change associated with an interest-rate change (also referred to as duration)

Table 2 One-Year Returns on Different-Maturity 10%-Coupon-Rate Bonds When

Interest Rates Rise from 10% to 20%

Interest-Rate Risk

o Prices and returns for long-term bonds are more volatile than those for shorter-term bonds

o There is no interest-rate risk for any bond whose time to maturity matches the holding period

Interest Risk - Duration

o Duration - is the weighted average time over which he cash flows form an investment are expected, where the weights are the relative present values of the cash flows.

o Focusing on maturity ignore the fact that some cash benefits are received before maturity (can be reinvested) and the benefits may be substantial.

Interest Risk - Duration

o Higher yields lead to lower durations. As the yield increases the present value of the distant cash flows gets exponentially smaller thus the weight given to distant time periods in the numerator get smaller lowering the duration.

Interest Risk - Duration

o The duration of any instrument is positively related to maturity, except for maturities in excess of 50 years. The duration of a bond increases as yield (coupon) increases.

Interest Risk - Duration

o Why is this important? The answer: for a given change in market yields, the percentage change in an asset’s price (PV) are proportional to the asset’s duration.

o Hence longer duration instruments are subject to greater price changes (exhibit greater price elasticity).

o This is expressed by the following formula:

-Duration × [Di ÷(1+i)]

Examples:

The Distinction Between Real and Nominal Interest Rates

o Nominal interest rate makes no allowance for inflation

o Real interest rate is adjusted for changes in price level so it more accurately reflects the cost of borrowing

o Ex ante real interest rate is adjusted for expected changes in the price level

o Ex post real interest rate is adjusted for actual changes in the price level

Fisher Equation

= nominal interest rate

= real interest rate

= expected inflation rate

When the real interest rate is low,

there are greater incentives to borrow and fewer incentives to lend.

The real inter

er

r

e

i i

i

i

est rate is a better indicator of the incentives to

borrow and lend.

Figure 1 Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2011

Sources: Nominal rates from www.federalreserve.gov/releases/H15 and inflation from ftp://ftp.bis.gov/special.requests/cpi/cpia.txt. The real rate is constructed using the procedure outlined in Frederic S. Mishkin, “The Real Interest Rate: An Empirical Investigation,” Carnegie-Rochester Conference Series on Public Policy 15 (1981): 151–200. This procedure involves estimating expected inflation as a function of past interest rates, inflation, and time trends and then subtracting the expected inflation measure from the nominal interest rate.

Determinants of Asset Demand

Wealth: the total resources owned by the individual, including all assets

Expected Return: the return expected over the next period on one asset relative to alternative assets

Risk: the degree of uncertainty associated with the return on one asset relative to alternative assets

Liquidity: the ease and speed with which an asset can be turned into cash relative to alternative assets

Theory of Portfolio Choice

Holding all other factors constant:1. The quantity demanded of an asset is positively related

to wealth

2. The quantity demanded of an asset is positively related to its expected return relative to alternative assets

3. The quantity demanded of an asset is negatively related to the risk of its returns relative to alternative assets

4. The quantity demanded of an asset is positively related to its liquidity relative to alternative assets

Supply and Demand in the Bond Market

At lower prices (higher interest rates), ceteris paribus, the quantity demanded of bonds is higher: an inverse relationship

At lower prices (higher interest rates), ceteris paribus, the quantity supplied of bonds is lower: a positive relationship

Supply and Demand for Bonds

Changes in Equilibrium Interest Rates

Shifts in the demand for bonds: Wealth: in an expansion with growing wealth, the

demand curve for bonds shifts to the right Expected Returns: higher expected interest rates in

the future lower the expected return for long-term bonds, shifting the demand curve to the left

Expected Inflation: an increase in the expected rate of inflations lowers the expected return for bonds, causing the demand curve to shift to the left

Risk: an increase in the riskiness of bonds causes the demand curve to shift to the left

Liquidity: increased liquidity of bonds results in the demand curve shifting right

Factors That Shift the Demand Curve for Bonds

Shifts in the Supply of Bonds

Expected profitability of investment opportunities: in an expansion, the supply curve shifts to the right

Expected inflation: an increase in expected inflation shifts the supply curve for bonds to the right

Government budget: increased budget deficits shift the supply curve to the right

Factors That Shift the Supply of Bonds