Money & Banking Notes Chapter 4

Measuring Interest Rates

Present Value (PV): A dollar paid to you one

year from now is less valuable than a dollar paid

to you today. Why?

- A dollar deposited today can earn interest and

become ($1)(1+ i) one year from today, where i is

the annual interest rate. For example,

Simple Present Value (example)

PV = today’s Present Value = ?

CF = future (yearly) cash flow payment = $250

i = the (usually annual) interest rate = 10 %

n = number of years (or periods) = 2 years

PV = $206.61

($100)(1 + i)n

from PV table:

i = 10%, n = 2 periods

Understanding Interest Rates

Year 0 Year 1 Year 2 Year 3 Year n

$100

($100)(1+0.10) = $110

($110)(1+0.10) = $121

($121)(1+0.10) = $133

double check

for yourself

Four Types of Credit Market Instruments - Simple Loan: the lender provides the borrower

with an amount of funds (the principal) which must

be repaid at the maturity date (the ending period of

the loan) together with the interest owed (e.g., a

commercial loan to a business).

- Fixed Payment Loan: the lender provides the

borrower with an amount of funds (the principal)

which must be repaid by making equal payments

(say, monthly) consisting of the principal

and interest owed for a set number of years (e.g., a

mortgage or auto loan).

- Coupon Bond: it pays the bondholder (the lender)

a fixed interest payment (called coupon payment)

every year (or six months) until the maturity date at

which point the face value (or par value) is repaid

(e.g., a 5-year $1,000 coupon bond with a coupon

rate of 7 % would pay its holder $70 every year for

5 years); e.g., Treasury bonds and corporate bonds.

- Discount Bond: also called a zero-coupon bond, it

is bought at a price below its face value (i.e., at a

discount) and the face value is repaid at maturity,

thus, it makes no interest payments (e.g., U.S.

Treasury-bills).

Yield to Maturity (ytm) = is the

interest rate at which the present value of cash

flow payments (received from a debt instrument)

is equal to its value today (PV).

where did this

come from?

When a bond sells “at par” = it means at face

value

Simple Loan

Fixed Payment 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

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

also PV

Coupon Bond

Discount Bond

(Bond prices and interest rates are negatively related.)

The Distinction Between Interest Rates and Returns

Rate of return = payments to the owner of the

security plus the change in its value, expressed as a

fraction of its purchase price (current yield +

capital gains yield).

also PV

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

also FV

The Distinction Between Interest Rates and

Returns (cont.)

The return on a bond equals the yield to

maturity only if the holding period equals the ytm

(put differently, the return on a bond will not

necessarily equal the ytm on that bond).

A rise in interest rates is associated with a fall in

bond prices, resulting in a capital loss if the ytm is

longer than the holding period.

current selling price

future selling price

The more distant a bond’s maturity, the greater

the size of the percentage change in price

associated with an interest-rate change.

The more distant a bond’s maturity, the lower

the rate of return that occurs as a result of an

increase in the interest rate.

Even if a bond has a substantial initial interest

rate, its return can be negative if interest rates

rise.

Interest-Rate Risk

Prices and returns for long-term bonds are

more volatile than those for shorter-term bonds

There is no interest-rate risk for any bond

whose time to maturity matches the holding

period (i.e., when you hold the bond until its

maturity, there is no interest-rate risk).

The Distinction Between Real and Nominal

Interest Rates

- Nominal interest rate makes no allowance

(adjustment) for inflation

- Real interest rate is adjusted for changes in

price level (inflation rate) so it more accurately

reflects the cost of borrowing

- Ex ante real interest rate is adjusted for

expected changes in the price level

- Ex post real interest rate is adjusted for actual

changes in the price level

Fisher Equation

Suppose:

e = 10 % = expected inflation rate

i = 8 % = nominal interest rate

What is the real interest rate? Answer = 2 %

When the real interest rate is low, there are greater

incentives to borrow and fewer incentives to lend. Why?

figure this out

Exercises

1. What is the present value of $500 to be

paid each time in two years (Year 1 and Year

2) if the interest rate is 5 percent? (Hint: you

can’t add up the two 500s.)

Fig. 1. Timeline of the present value problem

Setting up the equation,

So that,

PV = ?

today

$500 $500

cash flow 1 cash flow 2

Year 1 Year 2 Year 0

i = 5 %

21 )05.01(

500

)05.01(

500

PV

for denominator, use FV

interest factor from table

1025.1

500

05.1

500PV

FV interest factor,

period = 2, i = 5 %

FV interest factor,

period = 1, i = 5 %

Alternatively, use PV interest factor as shown

below

2. What is the present value of $500 to be

paid in 2 years (Year 2) if the interest rate is

5 percent?

)90703.0)(500()05.01(

5002

PV

Don’t believe this. Prove it to yourself.

PV interest factor,

period = 2, i = 5 %

PV interest factor,

period = 1, i = 5 %

PV = ?

today

$500

cash flow year 2

Year 1 Year 2 0

i = 5 %

Answer: $453.51

Answer: 476.19 + 453.51 = $929.7

PV = 500(0.95238) + 500(0.90703)

3. If a security pays $55 in year one and $133

in year three, its present value is $150 if the

interest rate is how much?

Setting up the equation,

What is the interest rate (i) that would make

the cash flows of $55 (in yr. 1) and $133 (in yr.

3) equal the present value of $150?

PV = $150

today

$133 $55

cash flow year 1 cash flow year 3

Year 1 Year 2 0

i = ? %

Year 3

31 )1(

133

)1(

55150

ii

Using a trial and error solution, first try i =

12%:

Should a higher or lower interest rate be

assumed next?

Now try i = 10%:

31 )12.01(

133

)12.01(

55150

40493.1

133

12.1

55150

FV interest factor,

period = 3, i = 12 %

667.94107.49150

?

?

?

774.143150

31 )10.01(

133

)10.01(

55150

331.1

133

10.1

55150

FV interest factor,

period = 3, i = 10 %

925.9950150

?

?

?

925.149150

Answer: i = 10 %

close enough

4. If a $5,000 coupon bond has a coupon rate

of 13 percent, then the coupon payment every

year is how much?

Coupon payment = (coupon rate)(bond’s face value)

= (0.13)($5,000)

= $650

5. An $8,000 coupon bond with a $400

coupon payment every year has a coupon rate

of how much?

coupon payment

Coupon rate = ---------------------------

face value

$400

= -------------

$8,000

6. For a 3-year simple loan of $10,000 at 10

percent, the amount to be repaid is how

much?

That is, after 3 years, how much is the

future value of $10,000 today at 10 %?

Answer: coupon rate = 0.05 or 5 %

face or par value

FV = $10,000(1 + 0.10)3

= $10,000(1.331)

This is like depositing money in a bank and

withdrawing the (accumulated) money after

3 years.

PV = $10,000

today

FV = ?

amt. to be repaid year 3

Year 1 Year 2 0

i = 10 %

Year 3

FV interest factor,

period = 3, i = 10 %

Answer: repayment = $13,310

7. If a security pays $110 next year and $121

the year after that, what is its yield to maturity

if it sells today for $200?

Solve by trial and error like in Problem #3.

Prove this answer to yourself.

PV = $200

today

$110

Year 1 Year 2 0

i = ? %

$121

21 )1(

121

)1(

110200

ii

PV

Answer: i = 10 %

8. What is the price of a perpetuity that has a

coupon (payment) of $100 per year and a

yield to maturity of 5 %?

The ytm formula for a perpetuity is:

where C = yearly (coupon) payment

P = price of perpetuity (consol)

i = yield to maturity (ytm)

which can be rewritten as:

Prove this answer to yourself.

9. What is the yield to maturity (ytm) of a

$1,000 discount bond that matures in one year

and currently sells for $900?

(Solve this problem and bring your solution to class.)

P

Ci

i

CP

Answer: P = $2,000