© 2013 Pearson Education, Inc. All rights reserved.4-1 Money & Banking Video 03—Interest Rates Understanding interest rates ( Chapter 4) Hal W. Snarr 8/20/2015

© 2013 Pearson Education, Inc. All rights reserved. 4-1

Money & Banking

Video 03—Interest Rates

Understanding interest rates (Chapter 4)

Hal W. Snarr8/20/2015

Chapter 4

Understanding Interest Rates


Measuring Interest Rates

Discount rate – the rate used to weigh a stream of future payments

• Most choices require decision-makers to trade-off costs and benefits at different points in time (e.g., savings, work effort, education, projects…)

• As the payment gets further and further into the future, people tend to place less and less weight on it

• People who are more worried about Right Now, discount the future more than those who are concerned more with the future

• In economics, the discount rate is determined by o market rate of interest io one’s time preference

http://www.youtube.com/watch?v=rMV-fenGP1g


If discount rate i represents a time preference, the present value of a stream of future payments is

Present Value

31 21 2 3

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

nn

P i

F

i

F

i i

F F

P = present value of a discounted stream of paymentsFi = dollar value of a future payment received at the END of period in = years to maturity date



P = present value of a discounted stream of paymentsFi = dollar value of a future payment received at the START of period in = years to maturity date

0 131 2

2 1 . . .

(1 ) (1 ) (1 ) (1 )nnP

i

F F

i

F

i i

F

Present Value



0 131 2

2 1 . . .

(1 ) (1 ) (1 ) (1 )nnP

i

F F

i

F

i i

F

Example: Suppose a person with a monthly discount rate of 0.4% wins a lottery. When the winner is announced, the winner can choose one check in the amount of $1,000,000, or checks in the amount of $10,000 each month for the next 10 years. Either way, the person leaves the building with a huge check in hand. Which option will the winner choose?

Present Value



0 1 2 119 . . .

(1 ) (1 ) (1 )

1000000

.004 .004 .0

0 0 0

(04 041 ).0P


Present Value



100 0000P


Present Value




0 1 2 119

10000 10000 100 . . .

(1 ) (1 ) (1 ).004 .

00 10000

004 . (1004 .004)P

M&B_lottery.xlsx

Present Value




955,365.92 P

M&B_lottery.xlsx

Present Value




955,365.92 P

The winner chooses the $1,000,000 check

M&B_lottery.xlsx

Present Value


18 6522Age

40,000

16,000

-5,000

Dollars/year

Goes to College

Only High School

0

A person who gets a HS diploma earns $16,000 a year from age 18 until he/she retires. If the person goes to college, • he/she pays $20,000 in tuition• foregoes earning $64,000 • but earns $1,032,000 more by graduating from college from age 22 to age 64.

(24,000/year)(43 years) = $1,032,000

$20,000 $64,000 + = $84,000

Why do people invest in HUMAN capital?


1 2 3 4

5,000 5,000 5,000 5,000$12,944

(1 0.2) (1 0.2) (1 0.2) (1 0.2)BS cost

womenPV

1 2 47

16,000 16,000 16,000... $79,985

(1 0.2) (1 0.2) (1 0.2)HS earnings

womenPV

The previous model ignores the discount rate• Swann (2003) estimates the annual discount rate of women at 20%

• Keane and Wolpin (1997) estimate the discount rate of young men at 28%

1 2 47

16,000 16,000 16,000... $57,142

(1 0.28) (1 0.28) (1 0.28)HS earnings

menPV

1 2 3 4

5,000 5,000 5,000 5,000$11,205

(1 0.28) (1 0.28) (1 0.28) (1 0.28)BS cost

menPV

Why do people invest in HUMAN capital?

5 6 47

40,000 40,000 40,000... $96,413

(1 0.2) (1 0.2) (1 0.2)BS earnings

womenPV

5 6 47

40,000 40,000 40,000... $53,217

(1 0.28) (1 0.28) (1 0.28)BS earnings

menPV

NPV = -15130

NPV = 3484

M&B_pv_college.xlsx


In Money & Banking, the discount rate is

• market rate of interest i on a loan:

• called the Yield to Maturity

Interest Rates

P = price of bond when auctioned/issuedF = face value of bond c = annual coupon rate (in percent) C = dollar value of annual coupon payment = cFn = years to maturity date

coupon bond

2 3 . . .

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

FP

C C

i

C

i i i

C

i


2 3 . . .

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

FP

C C

i

C

i i i

C

i

c = 10%, n = 10, F = $1,000

10.0713.0713 .0713.0713 .071310

Interest Rates

coupon bond

100

100

100

100

1000





2 3 . . .

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

FP

C C

i

C

i i i

C

i

c = 10%, n = 10, F = $1,000

10.0848.0848 .0848.0848 .084810

Interest Rates

coupon bond

100

100

100

100

1000





2 3 . . .

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

FP

C C

i

C

i i i

C

i

c = 10%, n = 10, F = $1,000

10.1000.1000 .1000.1000 .100010

Interest Rates

coupon bond

100

100

100

100

1000





2 3 . . .

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

FP

C C

i

C

i i i

C

i

c = 10%, n = 10, F = $1,000

10.1175.1175 .1175.1175 .117510

Interest Rates

coupon bond

100

100

100

100

1000





2 3 . . .

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

FP

C C

i

C

i i i

C

i

c = 10%, n = 10, F = $1,000

10.1381.1381 .1381.1381 .138110

M&B_bond_prices.xlsx

Interest Rates

coupon bond

100

100

100

100

1000





2 3 . . .

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

FP

C C

i

C

i i i

C

i

c = 10%, n = 10, F = $1,000

1010

Interest Rates

100

100

100

100

1000




coupon bond

1200


2 3 . . .

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

FP

C C

i

C

i i i

C

i

c = 10%, n = 10, F = $1,000

1010

Interest Rates

100

100

100

100

1000




coupon bond

1100


2 3 . . .

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

FP

C C

i

C

i i i

C

i

c = 10%, n = 10, F = $1,000

1010

Interest Rates

100

100

100

100

1000




coupon bond

1000


2 3 . . .

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

FP

C C

i

C

i i i

C

i

c = 10%, n = 10, F = $1,000

1010

Interest Rates

100

100

100

100

1000




coupon bond

900


2 3 . . .

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

FP

C C

i

C

i i i

C

i

c = 10%, n = 10, F = $1,000

1010

Interest Rates

M&B_yields.xlsx

100

100

100

100

1000




coupon bond

800


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates

fixed payment loan

0




P = payment to borrower after loan is signedF = 0 C = annual cash payment n = years to maturity date


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates

fixed payment loan

M&B_loan.xlsx

0




Example: You borrow $10,000 to buy a pickup. If you have to pay 60 monthly payments of $200, what is the interest rate? What is the annual rate of interest?


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates

fixed payment loan

M&B_loan.xlsx

0





12 (1 ) 1i 0.006183413


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates

fixed payment loan

M&B_loan.xlsx

0





7.68%i


Interest Rates

P = price of bond when auctioned/issuedF = 0 C = dollar value of annual coupon paymentn = infinity

2 3 . . .

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

FP

C C

i

C

i i i

C

i

perpetuity (or consol) bond





2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates







Interest Rates


1 2 31/(1 ) 1/(1 ) 1/(1 ) i i i P

C






Interest Rates


1 2 3x x x P

C






Interest Rates

0 1 2 31 x x x x P

C







Interest Rates

11

1

x

P

C







Interest Rates

1

x

x

P

C







Interest Rates

1/(1 )

1/(11 )i

i

P

C







Interest Rates

1/(1 )

/(1 )

i

i i

P

C







Interest Rates

1

i

P

C c







Interest Rates





ci P

C



Interest Rates





Example: A consol pays out $20 annually, and interest rates are 5%. Compute the price of the consol.

ci P

C


Interest Rates






.0520

P


Interest Rates






P 20

.05


Interest Rates






P 400


Interest Rates





Example: A perpetuity that pays $150 annually is currently has a price of $2500. Compute the current yield of the perpetuity.

ciC

P


Interest Rates






ci150

2500


Interest Rates






ci .06


Interest Rates






ci 6%





Interest Rates

2 3 . . .

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

FP

C C

i

C

i i i

C

i


Example: A perpetuity that pays $150 annually is currently has a price of $2500. Compute the current yield of the perpetuity using the above equation. To do this, assume the perpetuity ends in 5000 years and assume the future value is any value, say $1400. (Is 5000 years close to infinity? To the average human, yes.)





Interest Rates

2 3 3500 3500

14002500 . . .

1 (1 ) (1 ) (

150 150

1 ) (1 )

150 150

i i i i i







Interest Rates



ci 6%

M&B_perpetuity.xlsx


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates

P = price of bond when auctioned/issuedF = face value of bond C = dollar value of annual coupon paymentn = years to maturity date




zero-coupon bond


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates

P = price of bond when auctioned/issuedF = face value of bond C = 0n = years to maturity date

0 0 0 0




zero-coupon bond


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates

P = price of bond when auctioned/issuedF = face value of bond n = years to maturity date




(1 )n

F

i

zero-coupon bond


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates

P = price of bond when auctioned/issuedF = face value of bond n = years to maturity date




(1 )n

F

i

1/

1n

F

P

i

zero-coupon bond


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




(1 )n

F

i

1/

1n

F

P

i

zero-coupon bond

Example: Compute the interest rate on a $1000 bond that matures in nine years and costs $850 today.


Interest Rates




1/

1n

F

P

i

zero-coupon bond



Interest Rates




1/91000

1850

i

zero-coupon bond



Interest Rates




.0182216842i

zero-coupon bond



Interest Rates




1.822%i

zero-coupon bond



2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




(1 )n

F

i

1/

1n

F

P

i

zero-coupon bond

Example: If you require a minimum of 5% on safe investments, what price are you willing to pay for a $1000 bond that matures in nine years?


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




(1 )n

F

i

zero-coupon bond



2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




9

1000

(1 .05)

zero-coupon bond



2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




$644.61

zero-coupon bond



Interest Rates

P = payment to borrower after the loan is signedC = cash payoff of loan’s principal and interest n = years to maturity date

simple loan




2 3 . . .

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

FP

C C

i

C

i i i

C

i(1 )nC

i

1/

1n

P

iC


Interest Rates

simple loan




2 3 . . .

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

FP

C C

i

C

i i i

C

i(1 )nC

i

1/

1n

P

iC

Example: If the annual interest rate is 10%, do you prefer $500 five years from now or $800 ten years from now?

The present value of getting $500 five years is The present value of getting $800 ten years is


Interest Rates

simple loan




2 3 . . .

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

FP

C C

i

C

i i i

C

i5

5 0

(1

0

.10)


The present value of getting $500 five years is The present value of getting $800 ten years is


Interest Rates

simple loan




2 3 . . .

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

FP

C C

i

C

i i i

C

i310.46


The present value of getting $500 five years is $310.46 The present value of getting $800 ten years is $308.43


Interest Rates

simple loan




2 3 . . .

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

FP

C C

i

C

i i i

C

i10.11

80

( 0)

0




Interest Rates

simple loan




2 3 . . .

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

FP

C C

i

C

i i i

C

i



308.43


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates

P = price of bond when auctioned/issuedF = face value of bond n = 1 period




(1 )n

F

i

discount bond


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates





1(1 )

F

i

discount bond


Interest Rates





discount bond

2 3 . . .

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

FP

C C

i

C

i i i

C

i1

F

i


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates





discount bond

1

F

i 1

F

P i


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




discount bond

1

F

i 1

F

P i

Example: Compute the interest rate on a $1000 bond that matures in one year and costs $850 today.


Interest Rates




discount bond

1F

P i



Interest Rates




discount bond

1000 1

850 i



Interest Rates




discount bond

.1764705882i



Interest Rates




discount bond


17.647%i


Interest Rates




discount bond

Example: What is the interest rate on a 10-year, $1000 bond that you purchased for $850 nine years after it was issued by the Treasury Department?

2 3 . . .

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

FP

C C

i

C

i i i

C

i1

F

i 1

F

P i


Interest Rates




discount bond

1000 1

850 i



Interest Rates




discount bond


17.647%i


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




discount bond

1

F

i 1

F

P i

Example: What price are you willing to pay for a $1000bond that matures in one year, if you require 5% interest?


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




discount bond

1

F

i

Example: What price are you willing to pay for a $1000 bond that matures in one year, if you require 5% interest?


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




discount bond

1000

1 .05



2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




discount bond

$952.38



2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates

P = price of bond when auctioned/issuedC = cash payoff of loan’s principal and interestn = 1 period




1C

i 1

P i

C

simple loan


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




1C

i 1

P i

C

Example: If you borrow $200 and you agree to pay the lender $205 at the end of the month, what is the monthly interest rate? What is the annual interest rate?

simple loan


Interest Rates




1P

iC

simple loan



Interest Rates




205 1

200 i

simple loan



Interest Rates




5% 2.i

simple loan



Interest Rates




simple loan

12.025 (1 ) 1 i


5% 2.i


Interest Rates




simple loan

.3448888242i


5% 2.i


Interest Rates




simple loan


5% 2.i

34.489%i


2 3 . . .

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

FP

C C

i

C

i i i

C

i

Interest Rates




1C

i 1

P i

C

Example: If your mom lends you $200 and you agree to pay her $250 when you get paid at the end of the month, what is the monthly interest rate? What is the annual interest rate?

simple loan


Interest Rates




1P

iC


simple loan


Interest Rates




250 1

200 i

simple loan



Interest Rates




5% 2i

simple loan



Interest Rates




simple loan

12.2 (1 )5 1 i

5% 2i



Interest Rates




simple loan

13.55191523i

5% 2i



Interest Rates




simple loan

1355.192%i

5% 2i



1. The price of a bond falls

2 3 . . .

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

FP

C C

i

C

i i i

C

i

as interest rates rise.

Bond Properties



2. The rate of return (R) is the return on a bond held for a given period of time

• R = yield to maturity only if it is held for all n periods

2 3 . . .

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

FP

C C

i

C

i i i

C

i


no Interest-Rate Risk

Bond Properties





• R ≠ yield to maturity if the bond is held less than n periodso If a coupon bond is purchased and held for its final year,

1 1

FP

R R

C


i i

Bond Properties






1

FP

R

C


Bond Properties







1F

RP

C

Bond Properties






1F

RP

C


Bond Properties







F PR

P

C

Bond Properties







F PR

P P

C

Bond Properties





• R ≠ yield to maturity if the bond is held less than n periodso If a zero-coupon bond is purchased and held for its final year,


Bond Properties

0 F PR

P P





• R ≠ yield to maturity if the bond is held less than n periodso If a zero-coupon bond is purchased and held for its final year,


F PR

P

Bond Properties





• R ≠ yield to maturity if the bond is held less than n periodso If a coupon bond is purchased and held for its final year, and its

purchase price equals its face value,

F PR

P P

C


F

F

F

Bond Properties





• R ≠ yield to maturity if the bond is held less than n periodso If a coupon bond is purchased and held for its final year, and its

purchase price equals its face value,

FR

F

c F


F c

Bond Properties





• R ≠ yield to maturity if the bond is held less than n periodso If a coupon bond is held for one year and sold before it matures,


F PR

P P

C t

tt

Pt+1ci g

Bond Properties





• R ≠ yield to maturity if the bond is held less than n periodso If a zero-coupon bond is held for one year and sold before it matures,


F PR

P P

C t

tt

Pt+1g

Bond Properties





• R ≠ yield to maturity if the bond is held less than n periodso If a coupon bond is held for one year, and sold before it matures for the

same price it was purchased,


F PR

P P

Cci

Bond Properties


3. 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

4. The more distant a bond’s maturity, the greater the size of the percentage price change associated with an interest-rate change

5. The more distant a bond’s maturity, the lower the rate of return the occurs as a result of an increase in the interest rate

6. Even if a bond has a substantial initial interest rate, its return can be negative if interest rates rise

Bond Properties






2 3 4 5 5

1 (1 ) (1 ) (1 ) (1 ) (1 )

FP

C C C

i i i i i i

C C

.10

100

100

100

100

1000

100

.10 .10 .10 .10 .10= 1000

c = 10%, n = 5, F = $1,000

Bond Properties


= 1000





2 3 4 5 5

1 (1 ) (1 ) (1 ) (1 ) (1 )

FP

C C C

i i i i i i

C C

c = 10%, n = 5, F = $1,000

.20 .20 .20 .20 .20 .20= 741

100

100

100

100

1000

100

M&B_bond_properties.xlsx

Bond Properties






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

Interest-Rate Risk

Bond Properties

Documents

© 2013 Pearson Education, Inc. All rights reserved.4-1 Money & Banking Video 03—Interest Rates Understanding interest rates ( Chapter 4) Hal W. Snarr 8/20/2015