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Lecture 7: Measuring interest rate

Mishkin chapter 4 – part A

Page 67-78

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Future value Deposit $1 in bank, annual interest rate

i=0.1, how much you would get after 1 year? 2 years? … n years?

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1 (1 0.1) $1.1

1 (1 0.1) (1 0.1) 1 (1 0.1) $1.21

$1 has a future value of $[1 (1 ) ] in year nni

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Present value – discounting $1

1.33

1 2 3 n

1.211.1 1*(1 + .1)n

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(1 .1)n

1

2 3

1.1 1.21 1.331 ...

1 0.1 (1 0.1) (1 0.1)

$1 received after n years has a present value of:

1

(1 )ni

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Present value - meaning

A dollar paid to you one year from now is less valuable than a dollar paid to you today.

Present value of $1 is the minimum number of dollars that you would have to give up today in return for receiving $1 in year n.

Why? impatient forgone interest

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Present value – discounting a future cash flow

n

PV = today's (present) value

CF = future cash flow (payment)

= the interest rate

CFPV =

(1 + )

i

i

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Present value – discounting multiple future cash flows

2 3

100 150 200

(1 .1) (1 .1) (1 .1)PV

$100 $150 $200

Start date Maturity date

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Example

Your bank offers you a CD with 3% interest rate for five year investment. You wish to invest $1500 for five years.

How much your investment will be worth then? Known current value and need to calculate

future value.

5$1500 (1 0.03) $1738.91

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Example

You have been offered $40,000 to sell your printing business, payable in two years. Suppose the market interest rate is 8%. How much is the offer worth to you today?

(In other words, what is the offer’s present value?)

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$40,000$34,293.6

(1 0.08)

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Compare debt instruments

1. Simple loan2. Fixed payment loan3. Coupon bond4. Discount bond

Difference: repayment schemes. Calculate yield to maturity for each.

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Yield to maturity (YTM)

Yield to maturity (YTM) is the interest rate (i) that equates the present value of cash flow payments received from a debt instrument with its value today.

the most accurate measure of interest rates.

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Simple loan Payment scheme:

pay the loan value (LV) together with an interest payment (I) on the maturity date.

Time line:

Loan value (LV)

Loan value (LV)

+ Interest payment (I)

borrower receives

Lender receives

maturity date

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Simple loan - YTM

LV = loan value

I = Interest payment

n = years to mature(1 )n

LV ILV

i

Example:

borrow a simple loan of $100, interest rate is 0.1, or, need to pay interest of $10, mature in one year. What’s the yield to maturity?

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Example – Cont’d

1

1

$100

$10

1

(1 )

100 10100

(1 )

100 (1 ) 110

.1

LV

I

n

LV ILV

i

i

i

i

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Fixed payment loan Payment scheme:

makes periodic fixed payments to the lender until a specified maturity date.

These periodic fixed payments include both principal (loan value) and interest, so at maturity there is no lump-sum repayment of principal.

example: home mortgage

Loan value (LV)

Fixed payments (FP) FP FP

borrower receives

Lender receives

maturity date

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Fixed payment loan - YTM

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

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Example Consider a particular fixed-payment loan contract

with a loan value $5000, annual fixed payments $660.72, and a maturity of 20 years. What is the yield to maturity for this loan contract?

Answer: Yield to maturity for this fixed-payment loan contract is 0.12 or 12 percent.

2 20

2 20

FP FP FPPV = + +...+

(1+i) (1+i) (1+i)

600.72 600.72 600.725000= + +...+

(1+i) (1+i) (1+i)

0.12i

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Coupon bond Payment scheme:

(1)pay a fixed amount of funds (the coupon

payment) periodically; (2) pay the face value

(or par value) of the bond on maturity date. purchase price may not equal face value

Purchase price (P)

Coupon payments (C) C

Coupon payments (C)

+ Face value (F)

borrower receives

Lender receives

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Coupon bond - YTM

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

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Example Consider a coupon bond whose purchase price

is $94, face value is $100, whose coupon payment is $10 and maturity is 10 years. What’s yield to maturity i?

Cash flow is: ( $10, $10, $10, $10, $10, $10, $10, $10, $10, [$10 + $100] ).

2 9 10

10 10 10 (10+100)94 = + + ... + +

(1+i) (1+i) (1+i) (1+i)

0.11i

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1.When bond is at par (sold at face value), yield to maturity equals coupon rate.

2.Price and yield to maturity are negatively related.

Bond price and YTM

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Coupon rate and YTM

2 3...

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

C C C C FP

i i i i i

Other things unchanged:

1. P increase i decrease.

2. C increase i increase. Coupon rate or coupon payment is positively related to yield to maturity.

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Discount bond Payment scheme:

on maturity date, pay the face value (F) Sold at a discount: price < face value Time line:

Purchase price (P)

Face Value (F)

borrower receives

Lender receives

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Discount bond - YTM

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.

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Example

A discount bond selling for $15,000 with a face value of $20,000 in one year has a yield to maturity of _____?

2000015000 =

(1+i)

0.333i

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Recap

Present value Yield to maturity

definition

Calculate yield to maturity for: Simple Loan Fixed Payment Loan Coupon Bond Discount Bond