### Understanding Interest Rates

```Understanding Interest Rates
A fool and his
Money are
soon
Partying!!!!
»
. . . Wasn’t it Ben Franklin who said that????
1
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
2
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.
3
Copyri
2014
Diane
Scott
Docki
ng4
Various Types of Interest Rates
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Coupon Rate, Nominal Rate, or Stated Rate
Simple Interest Rate
Yield to Maturity (YTM)
Current Yield (CY)
Internal Rate of Return (IRR)
Discount Rate (DR)
Effective Annual Return or Yield (EAR) or (EAY)
Annual Equivalent Rate (AER)
Annual Percentage Yield (APY)
Average Annual Percentage Yield (APY)
Annual Percentage Rate (APR)
5
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.
6
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=1,000
PV= -1,019.04
n=2 x 2 = 4
i/y = 4%/2 = 2%
Pmt = \$25 semiannual x 2 = \$50/year
Coupon rate = 50/1,000 = 5%
7
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.
8
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 = 250
PV = 189.04
N=2
Pmt = 0
Cpt i/y =14.99875  15%
PV 
FV
1  i n
189 . 04 
i
250
1  i 2
250
 1  14 . 99875  15 %
189 . 04
9
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.
10
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?
11
Solution to Example: Current Yield
• What is the bond’s current yield?
ic 
C
P

60
 . 06122  6 . 122 %
980
• What is the bond’s YTM assuming annual interest payments?
FV=1000
PV=980
N=2
Pmt=60
therefor i/y = 7.1078%
• Semi-annual payments?
FV=1000
PV=980
N=2 x 2 = 4
Pmt=60/2 = 30
therefore i/y = 3.545 x 2 = 7.090%
12
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.
13
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.
14
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 – 1
where 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:
EAR
i


  1  Nom 
m 

m
1
 where m = the number of compounding periods per year
15
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
i Nom 

EAY   1 

m 

. 06 

 1 

12 

 1 . 005
12
m
1
12
1
1
 1 . 061678  1  6 . 1678 %
16
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?
17
Solution to Example 2: EAR or EAY
• What is the “stated” or nominal interest rate your bank is offering?
i


0 . 06   1  Nom 
12 

12
i Nom 

1 . 06   1 

12 

12
1
(1 . 06 )
12
 1
1
i Nom
12
1 . 004868  1 
i Nom
12
. 004868  12  i Nom
i Nom  0 . 058416  5 . 8416 %
18
Annual Equivalent Rate (AER)
• The AER is a way of quoting the actual interest
earned each compounding period:
m
r 

AER   1 
 1
m

 where:
r = stated annual interest rate
m = number of compounding periods per year
• AER is same as EAR
19
Annual Percentage Yield (APY) or
Average 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:
m
r 

APY   1    1
m

 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)
20
Annual Percentage Yield (APY) or
Average Annual Percentage Yield (APY)
• 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:
r 

r 

1



 1    1
m
m


APY 
n
n m
nm
APY
1
r 

APY   1  
m

nm
1
• where: r = annual interest rate
m = number of times interest is compounded per year
n = number of years
r 

APY   1  
m

nm
• Continuous compounding:
1
APY 
e
 r n 
1
n
21
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?
22
Solution to Example 1: APY
• What is the Annual Percentage Yield that Mary is earning on
her investment?
i Nom 

APY   1 

m 

m
1
4
. 06 

 1 
 1
4 

 1 . 015
4
1
 1 . 061364  1  6 . 1364 %
23
Solution to Example 1: APY
• If rates remain constant, how much will Mary have in her
account at the end of 3 years?
i Nom 

FV  PV  1 

m 

m n
. 06 

 10 , 000  1 

4 

 10 , 000 1 . 015
43
PV = -\$10,000
N = 3 x 4 = 12
I/Y = 6%/4 = 1.5
Pmt = 0
Cpt FV = \$11,956.18

12
 10 , 000 (1 . 195618 )  \$ 11 , 956 . 18
24
Solution to Example 1: APY
• What is the Average Annual Percentage Yield (APY) that Mary
earned on her initial investment over the 3 year period?
r 

1



m


APY 
n
n m
. 06 

1 

4 


3

1 . 015 12
1
34
1
1
3

1 . 195618  1
3
 0 . 065206

0 . 195618
3
 6 . 5206 %
25
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.
26
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?
27
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?
FV= 0
PV=200,000
N=15 x 12 = 180
I/Y = 5%/12 = 0.4166%
therefore Pmt = \$1,581.59
 What is the APR on this loan?
5%
28
Example: Computing a Loan APR with
and without upfront closing costs
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?
FV= 0
PV=200,000
N=15 x 12 = 180
I/Y = 5%/12 = 0.4166%
therefore Pmt = \$1,581.59
 What is the APR on this loan?
FV= 0
PV=200,000 - \$2,000 closing costs = \$198,000
N=15 x 12 = 180
Pmt = \$1,581.59
CPT I/Y = APR = 0.429411 x 12 = 5.153%