Understanding Interest Rates »... Wasn’t it Ben Franklin who said that???? A fool and his Money are soon Partying!!!! 1 Copyright © 2014 Diane Scott Docking

Understanding Interest Rates

» . . . Wasn’t it Ben Franklin who said that????

A fool and his

Money are soon

Partying!!!!

1Copyright © 2014 Diane Scott Docking

Copyright © 2014 Diane Scott Docking2

Learning Objectives

• Understand the different names for interest rates• Understand and compute the different ways

interest rates are quoted• Use quoted rates to calculate loan payments

and balances, future values, annuities


What are Interest Rates?

• To understand interest rates, it’s important to think of interest rates as a price—the price of using money.

• When you borrow money to buy a car, you are using the bank’s money now to get the car and paying the money back over time.

• The interest rate on your loan is the price you pay to be able to convert your future loan payments into a car today.

Copyright © 2014

Diane Scott Docki

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Various Types of Interest Rates

1. Coupon Rate, Nominal Rate, or Stated Rate

2. Simple Interest Rate

3. Yield to Maturity (YTM)

4. Current Yield (CY)

5. Internal Rate of Return (IRR)

6. Discount Rate (DR)

7. Effective Annual Return or Yield (EAR) or (EAY)

8. Annual Equivalent Rate (AER)

9. Annual Percentage Yield (APY)

10. Average Annual Percentage Yield (APY)

11. Annual Percentage Rate (APR)

Coupon Rate (aka: Stated rate, Nominal rate)

• A 2-year, $1,000 face value bond has a coupon of 5%. Interest is paid semi-annually.

• How much interest will you receive every 6 months?

• A: Face x (coupon/# payments per year) = Interest payment.

$1,000 x (.05/2) = $1,000 x .025 = $25.


Coupon Rate (aka: Stated rate, Nominal rate)

• A 2-year, $1,000 face value bond pays interest semi-annually. The current price of the bond is $1,019.04. The current market yield is 4%.

• What is the coupon rate on this bond?• A:


FV=1,000PV= -1,019.04n=2 x 2 = 4i/y = 4%/2 = 2%Pmt = $25 semiannual x 2 = $50/yearCoupon rate = 50/1,000 = 5%


Simple Interest Rate

• Simple Interest Rate: the interest payment divided by the loan principal;

the percentage of principal that must be paid as interest to the lender.

Convention is to express the interest rate on an annual basis, irrespective of the loan term.

• Example: Mary has a $10,000 3-year loan with a simple interest rate of 6%. Mary will pay $10,000 x 6% = $600 in interest every year for 3 years. At

the end of the third year she will repay the $10,000 principal balance.

Yield to Maturity (YTM)

• YTM most commonly refers to savings accounts Bonds

• In some in instances YTM is same as the internal rate of return (IRR) and the discount rate (DR). Depends on the context in which the term is used.

• Example: You invest $189.04 in a savings account today. In 2 years you withdraw $250. What is your YTM assuming annual compounding?


FV = 250PV = 189.04N = 2Pmt = 0Cpt i/y =14.99875 15%

ni

FVPV

1

%1599875.14104.189

250

1

25004.189 2

i

i


Current Yield (CY)

•

• Current yield (CY) is just an approximation for YTM – easier to calculate.

• It is usually used with bonds.• However, we should be aware of its properties:

1. If a bond’s price is near par and has a long maturity, then CY is a good approximation of YTM.

2. A change in the current yield always signals change in same direction as yield to maturity.

𝐶𝑌=𝑖𝑐=𝐶𝑜𝑢𝑝𝑜𝑛𝑃𝑟𝑖𝑐𝑒

Example: Current Yield

• A 2-year corporate bond, par value $1,000, is selling for $980. Its annual coupon rate is 6%. What is the bond’s current yield? What is the bond’s YTM assuming annual interest

payments? Semi-annual payments?


Solution to Example: Current Yield

• What is the bond’s current yield?

• What is the bond’s YTM assuming annual interest payments?

• Semi-annual payments?


%122.606122.980

60

P

Cic

FV=1000PV=980N=2Pmt=60therefor i/y = 7.1078%

FV=1000PV=980N=2 x 2 = 4Pmt=60/2 = 30therefore i/y = 3.545 x 2 = 7.090%


Internal Rate of Return (IRR)

• In Bonds: The IRR is the rate that forces the PV of expected future cash flows of interest and principal to equal the initial cost of the bond.

• In Corporate: The IRR is the rate that forces the PV of a project’s expected cash flows to equal its initial cost.

• IRR is the same as YTM in many instances.


Discount Rate (DR)

• The Discount Rate can mean many things: In Bonds: the discount rate can be the YTM In Corporate: the discount rate can be the IRR In Banking: the discount rate is the rate that the Federal

reserve loans money to commercial banks. In the Treasury Bill market: the discount rate is the rate

used to calculate the price of the T-Bill.


Effective Annual Rate, Return or Yield (EAR) or (EAY)

• The EAR or EAY is the return earned or paid over a 12-month period taking any within-year compounding of interest into account.

• EAR or EAY = (1 + r)c – 1where c = the number of compounding periods per year, and

r = the periodic rate. That is the nominal rate divided by the number of compounding periods in the year.

• Recall from Time Value of Money Lecture:

where m = the number of compounding periods per year

11

m

Nom

m

iEAR


Example 1: EAR or EAY

• Suppose your savings account pays interest at 6% (stated rate), compounded monthly. What is the Effective Annual Yield that you are

earning on your account?

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1005.1

112

06.1

11

12

12

m

Nom

m

iEAY


Example 2: EAR or EAY

• Suppose your bank account pays interest monthly with an effective annual rate of 6%. What is the “stated” or nominal interest rate your

bank is offering?


Solution to Example 2: EAR or EAY

• What is the “stated” or nominal interest rate your bank is offering?

%8416.5058416.0

12004868.12

1004868.1

121)06.1(

12106.1

112

106.0

121

12

12

Nom

Nom

Nom

Nom

Nom

Nom

i

i

i

i

i

i


Annual Equivalent Rate (AER)

• The AER is a way of quoting the actual interest earned each compounding period:

where:

r = stated annual interest rate

m = number of compounding periods per year

• AER is same as EAR

11

m

m

rAER


Annual Percentage Yield (APY) orAverage Annual Percentage Yield (APY)

• APY can mean two things.• Annual Percentage Yield APY is a way of quoting the actual

interest earned each compounding period:

where:

r = stated annual interest rate

m = number of compounding periods per year• In this case Annual Percentage Yield (APY) is same as Effective

Annual Yield (EAY)

11

m

m

rAPY

Copyright © 2014 Diane Scott Docking 21

11nm

m

rAPY 11

nm

m

rAPY

11nm

m

rAPY

• APY can also mean Average Annual Percentage Yield• If investment or loan is for more than 1 year then you calculate the

Average Annual Percentage Yield:

• where: r = annual interest rate m = number of times interest is compounded per year n = number of years

• Continuous compounding:

• APY can also mean Average Annual Percentage Yield• If investment or loan is for more than 1 year then you calculate the

Average Annual Percentage Yield:

• where: r = annual interest rate m = number of times interest is compounded per year n = number of years

• Continuous compounding:

nm

r

APY11

mn

n

eAPY

1nr

Annual Percentage Yield (APY) orAverage Annual Percentage Yield (APY)


Example 1: APY

• Mary has invested $10,000 in a savings account that is paying interest at a 6% (stated rate), compounded quarterly. What is the Annual Percentage Yield (APY) that Mary is earning on her

investment? If rates remain constant, how much will Mary have in her account at the

end of 3 years? What is the Average Annual Percentage Yield (APY) that Mary earned

on her initial investment over the 3 year period?


Solution to Example 1: APY

• What is the Annual Percentage Yield that Mary is earning on her investment?

%1364.61061364.1

1015.1

14

06.1

11

4

4

m

Nom

m

iAPY



• If rates remain constant, how much will Mary have in her account at the end of 3 years?

18.956,11$)195618.1(000,10

015.1000,10

4

06.1000,10

1

12

34

nm

Nom

m

iPVFV

PV = -$10,000N = 3 x 4 = 12I/Y = 6%/4 = 1.5Pmt = 0Cpt FV = $11,956.18



• What is the Average Annual Percentage Yield (APY) that Mary earned on her initial investment over the 3 year period?

%5206.6065206.03

195618.0

3

1195618.13

1015.1

3

1406.

1

11

12

43

mn

nmr

APY


Annual Percentage Rate (APR)

• The Annual Percentage Rate (APR) is a way of quoting the actual interest cost of funds over the term, INCLUDING any closing costs and fees.


Example: Computing a Loan APR with and without upfront closing costs

1) Mary agrees to a 15-year, $200,000 mortgage loan, with a rate of 5%. Her upfront closing costs are estimated to be $0. What are Mary’s monthly loan payments? What is the APR on this loan?

2) Assume Mary agrees to a 15-year, $200,000 mortgage loan, with a rate of 5% with estimated upfront closing costs to be $2,000. What are Mary’s monthly loan payments? What is the APR on this loan?


1) Mary agrees to a 15-year, $200,000 mortgage loan, with a rate of 5%. Her upfront closing costs are estimated to be $0. What are Mary’s monthly loan payments?

What is the APR on this loan?

5%

FV= 0PV=200,000N=15 x 12 = 180I/Y = 5%/12 = 0.4166%therefore Pmt = $1,581.59



2) Mary agrees to a 15-year, $200,000 mortgage loan, with a rate of 5%. Her upfront closing costs are estimated to be $2,000. What are Mary’s monthly loan payments?

What is the APR on this loan?

FV= 0PV=200,000N=15 x 12 = 180I/Y = 5%/12 = 0.4166%therefore Pmt = $1,581.59


FV= 0PV=200,000 - $2,000 closing costs = $198,000N=15 x 12 = 180Pmt = $1,581.59 CPT I/Y = APR = 0.429411 x 12 = 5.153%

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Understanding Interest Rates »... Wasn’t it Ben Franklin who said that???? A fool and his Money are soon Partying!!!! 1 Copyright © 2014 Diane Scott Docking