Financial Markets and Institutions (2-downloads)

loans, the simple interest rate equals the yield to maturity.

Hence the same

term i is used to denote both the yield to maturity and the simple interest rate.

Fixed-Payment Loan

Recall that this type of loan has the same cash flow payment

every year throughout the life of the loan. On a fixed-rate mortgage, for example, the

borrower makes the same payment to the bank every month until the maturity date,

when the loan will be completely paid off. To calculate the yield to maturity for a

fixed-payment loan, we follow the same strategy we used for the simple loan—we

equate today’s value of the loan with its present value. Because the fixed-payment loan

involves more than one cash flow payment, the present value of the fixed-payment loan

is calculated as the sum of the present values of all cash flows (using Equation 1).

Suppose the loan is $1,000, and the yearly cash flow payment is $85.81 for the

next 25 years. The present value is calculated as follows: At the end of one year, there

is a $85.81 cash flow payment with a PV of 

; at the end of two years,

there is another $85.81 cash flow payment with a PV of 

; and so on

until at the end of the 25th year, the last cash flow payment of $85.81 with a PV of

is made. Making today’s value of the loan ($1,000) equal to the sum

of the present values of all the yearly cash flows gives us

More generally, for any fixed-payment loan,

(2)

LV

⫽

FP

1

⫹ i

⫹

FP

11 ⫹ i2

2

⫹

FP

11 ⫹ i2

3

⫹ p ⫹

FP

11 ⫹ i2

n

$1,000

⫽

$85.81

1

⫹ i

⫹

$85.81

11 ⫹ i2

2

⫹

$85.81

11 ⫹ i2

3

⫹ p ⫹

$85.81

11 ⫹ i2

25

$85.81/

11 ⫹ i2

25

$85.81/

11 ⫹ i2

2

$85.81/

11 ⫹ i2

where

PV =

amount borrowed

= $100

CF =

cash flow in one year

= $110

n

=

number of years

= 1

Thus,

i

⫽ 1.10 ⫺ 1 ⫽ 0.10 ⫽ 10%

11 ⫹ i2 ⫽

$110

$100

11 ⫹ i2$100 ⫽ $110

$100

⫽

$110

11 ⫹ i2

Today

0

$100

$110

ˆ

 10%

Year

1

---

42

Part 2 Fundamentals of Financial Markets

where

LV = loan value

FP = fixed yearly cash flow payment

n = number of years until maturity

For a fixed-payment loan amount, the fixed yearly payment and the number

of years until maturity are known quantities, and only the yield to maturity is not.

So we can solve this equation for the yield to maturity i. Because this calculation

is not easy, many pocket calculators have programs that allow you to find i given

the loan’s numbers for LV, FP, and n. For example, in the case of the 25-year loan

with yearly payments of $85.81, the yield to maturity that solves Equation 2 is

7%. Real estate brokers always have a pocket calculator that can solve such equa-

tions so that they can immediately tell the prospective house buyer exactly what

the yearly (or monthly) payments will be if the house purchase is financed by tak-

ing out a mortgage.

Coupon Bond

To calculate the yield to maturity for a coupon bond, follow the same

strategy used for the fixed-payment loan: Equate today’s value of the bond with its

present value. Because coupon bonds also have more than one cash flow payment,

the present value of the bond is calculated as the sum of the present values of all

You decide to purchase a new home and need a $100,000 mortgage. You take out a loan

from the bank that has an interest rate of 7%. What is the yearly payment to the bank to

pay off the loan in 20 years?

Solution

The yearly payment to the bank is $9,439.29.

where