Chapter 5
Interest Rates
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5.1 How Interest Rates Are Quoted:
Annual and Periodic Interest Rates
Annual percentage rate (APR):
Yearly rate earned by investing in assets or charged
for borrowing (the most commonly quoted rate)
Lenders often charge interest on a non-annual basis.
Periodic interest rate (PIR): r =
[Tab 5.1]
where m = the number of compounding periods
per year (C/Y).
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5-2
5.1 How Interest Rates Are Quoted:
Annual and Periodic Interest Rates
The more compounding periods per year,
the more interest earned or paid.
m increases = more compounding -> interest increases (by the interest on interest)
(ex) $500 CD for 1 year with 5% APR
Annual (m=1): PIR = 0.05, Ending balance = $525
*(개인 필기) no difference in APR and PIR
Quarterly (m=4): PIR = 0.0125,
Ending balance = 500(1+PIR)^4
= 525.48
[Tab 5.2]
Conclusion : if the m goes larger, the ending balance becomes larger 5-3
!!
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5.1 How Interest Rates Are Quoted:
Annual and Periodic Interest Rates
Effective(=Actual) annual rate (EAR):
m > 1 => EAR and APR are different from each other.
The interest rate actually paid or earned per year
with compounding
Also known as the annual percentage yield (APY=EAR).
Given APR and m,
The meaning of -1 : ‘net’
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5-4
5.1 How Interest Rates Are Quoted:
Annual and Periodic Interest Rates
[Tab 5.2] $500 CD with APR = 0.05. (=PIR)
Monthly ending balance = 500 * (1.004167)
505.58
Quarterly ending balance = 525.47
Annual EAR = 0.05000 = APR
Quarterly EAR = (1+APR / 4)^4 -1=0.05094 (PIR = 0.0125)
Monthly EAR = (1+APR/12)^12 - 1= 0.05116
(farther higher than quarterly EAR)
Daily EAR = (1+APR/365)^365 -1 = 0.05127
Ending balance = PV × ( 1 + EAR ) = PV * (1+APR/m)^m
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Since EAR = (1+APR/m)^m =1
5-5
5.1 How Interest Rates Are Quoted:
Annual and Periodic Interest Rates
In U.S. -> deposit account, the bank would account EAR
For borrowers, they apply and prefer APR to lower their burden by loan.
Ex1: Calculating an EAR (or APY)
It is different in case of ‘non-annual’ base
The reason why it is different is dependent on its commercial purpose.
----
A bank lends $100,000 for up to 3 years
at an APR of 8.5% (compounded monthly).
Why?
youasborrow
EAR >IfAPR,
m>1
$100,000 for 1 year,
how much interest will you have paid
and what is the bank's APY (=EAR)?
the EAR(APY) for deposits. (savers)
※ Banks quotes
the APR for loans. (borrowers)
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5-6
5.2 Effect of Compounding Periods
on the TVM Equations
TVM equation requires the periodic rate (r = PIR)
and the number of compounding periods (n),
where n = years × m.
* The interest rate (r) should be consistent
with the frequency of compounding
and the number of payments.
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5-7
5.2 Effect of Compounding Periods
on the TVM Equations
Ex2: Effect of Payment Frequency on Total Payment
Jim needs to borrow $50,000 for a business expansion
project. His bank agrees to lend him the money over a 5year term at an APR of 9% and accepts annual, quarterly, or
monthly payments with no change in the APR. Calculate the
periodic payment under each alternative and compare the
total amount paid each year under each option.
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5-8
Ex2: Effect of Payment Frequency on Total Payment
Loan = $50,000; Period = 5 years; APR = 9%
PMT = 50,000 * 0.09 / 1- [1/(1.09)^5]
Annual: PV = 50000; n=5; I=9;
CPT PMT = 12,854.62
FV=0; P/Y=1; C/Y=1;
Periodic repayment of amortized loan
Quarterly: PV = 50000; n=20; I=9; FV=0; P/Y=4; C/Y=4;
CPT PMT = 3,132.10 PMT = 50,000 * [ 0.0225 / 1- {1/(1.0225)^20} ]
Total annual payment = 3132.1 * 4 = $12,528.41
Monthly: PV = 50000; n=60; I=9; FV=0; P/Y=12; C/Y=12;
PMT = 50,000 * 0.0075 / 1- [1/(1.0075)^60]
CPT PMT = 1,037.92
Total annual payment=1.0372.92 * 12=12,455.04
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5-9
추가 슬라이드
Annual: Annual total = 12,854.62
Quarterly: Annual total = $12,528.41
Monthly: Annual total = $12,455.04
☼ As m increases -> Annual total cash flow goes down.
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5-10
5.2 Effect of Compounding Periods
on the TVM Equations
Ex3: Comparing Annual and Monthly Deposits
Joshua (currently 25 years old) wants to invest money into a
retirement fund so as to have $2,000,000 saved up when he
retires at age 65.
If he can earn 12% per year in an equity fund, calculate the
amount of money he would have to invest in equal annual
amounts and alternatively, in equal monthly amounts.
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Ex3: Comparing Annual and Monthly Deposits
With annual deposits:
With monthly deposits:
FV = $2,000,000
FV = $2,000,000
N = 40 (years)
N = 12 x 40 = 480 (months)
I/Y = APR = 12%
I/Y = APR = 12%
PV = 0
PV = 0
C/Y = 1
C/Y = 12
P/Y = 1
P/Y = 12
PMT = $2,607.25
PMT = $169.99
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-> $2,039.88 yearly 5-12
5.3 Consumer Loans
and Amortization Schedules
Interest is charged only
on the outstanding balance of a consumer loan.
(remaining)
Increases in 1) frequency and 2) size of payments
-> result in 1) reduced interest charges and
2) a quicker payoff. Over time, outstanding balance 감소
1) interest expense 감소
2) principal repayment 증가
Amortization schedules help in planning and analysis
of consumer loans.
[Tab 5.3] Monthly Amortization Schedule for $25,000 loan
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(6 years at APR = 8%)
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5.3 Consumer Loans
and Amortization Schedules
Ex4: Paying Off a Loan Early!
Kay has just taken out a $200,000, 30-year, 5% mortgage.
She has heard from friends that if she increases the size of her
monthly payment by one-twelfth of the monthly payment, she
will be able to pay off the loan much earlier and save a
bundle on interest costs.
Use the necessary calculations to help convince her
that this is, in fact, true.
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5-14
Ex4: Paying Off a Loan Early!
Monthly payment:
PV = $200,000; I/Y = 5; N (= 30 x 12) = 360; FV = 0;
C/Y = 12; P/Y = 12; PMT = PV * r / 1 – [ 1 / (1+r)^n) ]
PIR = r = 0.00416
= 200,000 * 0.004167 / 1- [1 / (1.004167)^360
The monthly payment increased by 1/12
è PMT =
1,073.64 * (13/12) = $1,163.11
PV=$200,000; I/Y=5; C/Y=12; P/Y=12;
N= - [ln (1- {PV * r / PMT } ] / ln (1+r)
N = 303.13 months or 25.26 years.
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5-15
Ex4: Paying Off a Loan Early!
[Interest saved]
Within the minimum monthly payments:
Total paid = 360 * 1,073.64 = $386,510.40
Amount borrowed = $200,000.00
Interest paid. = $186,510.40
With the higher monthly payments:
Total paid = 303.13 * 1,163.11 = $353,573.53
Amount borrowed = $200,000.00
Interest paid = $153,573.53
Interest saved = $32,936.87
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It is true!
5-16
5.4 Nominal and Real Interest Rates
The nominal rate is the rate of interest earned
on a investment
such as bank CD or treasury security.
(Compensation paid for giving up of current consumption)
- Nominal rate (r ): the growth rate of actual dollars
- Real rate (r* ): the growth rate of the purchasing power
of the dollars
(adjusted for the erosion of purchasing
power caused by inflation)
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5-17
5.4 Nominal and Real Interest Rates
The Fisher Effect shows the relationship between
the real rate (r* ), the inflation rate (h ),
and the nominal interest rate (r ).
(1 + r) = (1 + r*) x (1 + h) → r = r* + h + (r* x h)
r* X h = Compensation for inflation erosion of the interest
Approximately, r = r* + h
with r* & h not too high
(그래서 생략 가능)
The nominal interest rate includes the real interest rate
and the inflation premium.
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5-18
5.4 Nominal and Real Interest Rates
Ex5: Calculating nominal & real interest rates
Jill has $100 and is tempted to buy 10 T-shirts,
with each one costing $10.
However, she realizes that if she saves the money in a bank
account, she should be able to buy 11 T-shirts.
If the cost of the T-shirt increases by the rate of inflation, i.e.,
by 4%, how much would her nominal and real rates of return
have to be?
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5-19
5.4 Nominal and Real Interest Rates
Ex5: Calculating nominal & real interest rates
r* = (FV/PV)1/n – 1 = (11 shirts/10 shirts)1/1 – 1 = 10%
Price of a shirt next year = $10 x (1.04) = $10.4
Cost of 11 shirts next year = 10.4 * 11 = $114.4 (=FV)
PV = $100; n = 1; I/Y = r = (FV/PV) – 1
= (114.4/100) – 1 = 14.4%
r = r* + h + (r* x h) = 0.1 + 0.04 (=Inflation premium)
+ [0.1 * 0.04 ] (=the compensation)
= 14.4%
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5-20
5.5 Risk-Free Rate and Premiums
⑴ Default premium (dp): Compensation for the risk
associated with varying types of
collateral & frequency of default
* Risk-free rate on the U.S. Treasury bill
(base rate)
T-bill rate = r(f) = r* + h + 0(=dp).
<- dp가 0이기 때문에 T-bill의 r(f)가 두개로만 구성됨
⑵ Maturity premium (mp): Compensation for additional
waiting time
[Tab 5.4]
Different maturity preferences => Different maturities
Variety of interest rates ⇐
Risk
Length
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of investment or loan
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Secured
Collateral(resale) value (implies d
of premium..?)
=> dp
ecured
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5-22
Waiting time => mp
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5-23
5.5 Risk-Free Rate and Premiums
The rate of return on all other riskier investments would
have to include a default
and maturity premium
r = ( r* + h ) + dp + mp + …
(many other things could be contained)
“Term structure of interest rates”
(Ex) 30-year corporate bond yield > 30-year T-bond yield
(due to the higher default risk on corporate bonds)
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5-24
5.6 A Brief History of Interest Rates
The four components of interest rates vary over time.
(r = r* + h + dp + mp)
[Fig 5.2] Inflation in the U.S. (1950-1999)
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5-25
5.6 A Brief History of Interest Rates
The four components of interest rates vary over time.
( r = r* + h + dp + mp )
[Fig 5.3] 3-month T-bill rate in the U.S. (1950-1999)
mp = dp = 0,
risk –free
=> r (f)
Fig 5.2 : shows actual inflation rate
While Fig 5.3 shows nominal rate with expected inflation
=> Discrepancy
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5-26
5.6 A Brief History of Interest Rates
[Fig 5.3] 3-month T-bill rate in the U.S. (1950-1999)
Data shows the average values over 1950-1999 as:
r = 5.23%, h = 4.05% ⇒ r* = 1.18%
h increases -> h(e) increases
-> Uncertainty increase -> Premiums also increase
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5-27
5.6 A Brief History of Interest Rates
[Tab 5.5] Bond yields (1953-1999)
dp(default premium) = 7.13 – 6.64 = 0.49 (=49 basis point)
(for AAA-bond over T-bond)
mp = 6.64 – 5.23 = 1.41%
(for T-bond over T-bill, due to 20-year maturity difference)
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5-28
5.6 A Brief History of Interest Rates
[Tab 5.5] Bond yields (1953-1999)
Corp-bond yield = 7.13 %
= 1.18 + 4.05 + 0.49 + 1.41
(r= r* + h + dp + mp)
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5-29
5.6 A Brief History of Interest Rates
The four components of interest rates vary over time.
(r = r* + h + dp + mp)
[Fig 5.2] Inflation in the U.S. (1950-1999)
[Fig 5.3] 3-month T-bill rate in the U.S. (1950-1999)
Data shows the average values over 1950-1999 as:
r = 5.23%, h = 4.05% ⇒ r* =
[Tab 5.5] Bond yields (1953-1999)
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