Interest Rate
FINA 3702C
Investment Analysis and Portfolio Management
Arkodipta Sarkar
1
Special Multiple Cash Flows: Perpetuities and Annuities
• Perpetuity is a constant stream of cash flows that lasts forever
• Growing perpetuity is a stream of cash flows that grows at a constant rate forever
• Annuity is a stream of constant cash flows that lasts for a fixed number of periods
(Annuity due)
• Growing annuity is a stream of cash flows that grows at a constant rate for a fixed
number of periods
2
Formula Refresher
• Present Value: 𝑃𝑉0 = σ𝑇𝑡=1
• Perpetuity: 𝑃𝑉0 =
𝐶
𝑟
• Growing Perpetuity: 𝑃𝑉0 =
• Annuity: 𝑃𝑉0 = 𝐶
𝐶𝑡
1+𝑟 𝑡
1
𝑟
𝐶
𝑟−𝑔
1
−
𝑟 1+𝑟 𝑡
• Growing Annuity: 𝑃𝑉0 =
𝐶
𝑟−𝑔
1−
1+𝑔 𝑡
1+𝑟
3
Topics
• Define annual percentage rate (APR) and effective annual rate (EAR)
• Learn how to convert APR and EAR
• Find period interest rates for non-annual cash flows from APR/EAR
• Compute loan payments and balances based on quoted rates
4
Topics
• Define annual percentage rate (APR) and effective annual rate (EAR)
• Learn how to convert APR and EAR
• Find period interest rates for non-annual cash flows from APR/EAR
• Compute loan payments and balances based on quoted rates
5
Effective Annual Rate (EAR)
• EAR (effective annual rate) is the rate that is actually earned/paid at the end of
one year, including compounding.
• For example, suppose that you deposited $100 in your bank two years ago. The
deposit pays out interest once a year. After two years, you have $104.04 in your
bank account. What's the EAR?
• Answer:
6
Effective Annual Rate (EAR)
• EAR (effective annual rate) is the rate that is actually earned/paid at the end of
one year, including compounding
• For example, suppose that you deposited $100 in your bank two years ago. The
deposit pays out interest once a year. After two years, you have $104.04 in your
bank account. What's the EAR?
• Answer: The EAR is the annual rate of return that we obtain on our investment,
and it solves
$100 × 1 + 𝐸𝐴𝑅 2 = 104.04
𝐸𝐴𝑅 = 2%
7
Effective Annual Rate (EAR)
• Suppose that you deposited $100 in your bank. The deposit pays out a 2% interest
every three months. What is the EAR? That is, earning a 2% interest rate every
three months is equal to earning how much interest for a year?
• Answer:
8
Effective Annual Rate (EAR)
• Suppose that you deposited $100 in your bank. The deposit pays out a 2% interest
every three months. What is the EAR? That is, earning a 2% interest rate every
three months is equal to earning how much interest for a year?
• Answer: If compounding is quarterly, it means that
$100 × 1 + 2% 4 = $100 × 1 + 𝐸𝐴𝑅
same as compounding every year for 4 years
• Solving for EAR we get,
𝐸𝐴𝑅 = 1.024 = 1 ≈ 8.24%
9
Effective Annual Rate (EAR)
• Suppose that you deposited $100 in your bank. The banker tells you that you are
earning interest every three months, and that the EAR is 10%? How much interest
are you earning every three months (call it 𝑟𝑞 )?
• Answer:
10
Effective Annual Rate (EAR)
• Suppose that you deposited $100 in your bank. The banker tells you that you are
earning interest every three months, and that the EAR is 10%? How much interest
are you earning every three months (call it 𝑟𝑞 )?
• Answer: If compounding is quarterly, it means that
$100 × 1 + 𝑟𝑞
4
= $100 × (1 + 𝐸𝐴𝑅)
Solving,
𝑟𝑞 = 1 + 𝐸𝐴𝑅
1
4
−1=
1
1.14
− 1 = 2.41%
11
Effective Annual Rate (EAR): General Formula
• EAR is the annual interest rate that solves
1 + 𝐸𝐴𝑅 = 1 + 𝑟𝑝𝑒𝑟𝑖𝑜𝑑
𝑛
• Where 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 is the interest rate that is earned every period, and n is the number
of periods (measured as fractions/ multiples of a year) that the interest rate is
earned
• If n>1 it means that 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 is earned more than once every year
– E.g., n=12 is monthly compounding
– E.g., n=52 is daily compounding
weekly
• If n < 1, it means that 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 is earned at higher frequencies than a year
– E.g., n=1/2 means that 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 is earned once every two years
12
Annual Percentage Rate (APR): Definition
• APR (annualized percentage rate) is a rate that ignores compounding
– It is given by APR = 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 × 𝑛
• E.g., If you earn interest of 3% every two months, then 𝐴𝑃𝑅 = 3% × 6 = 18%.
• Why do we need this?
– Used in contracts and quoted by banks
• Other definitions (we are not going to use them, but you might see them around)
– APR = Nominal rate
– APR = Quoted rate
– APR = Stated Annual Interest Rate
13
How do we know if we have an APR?
• As noted on the prior slide, if it’s a financial product there’s a good
chance it is an APR.
• Look for key tells:
– “This CD has an 8% APR with monthly compounding.”
– “This credit card charges 12% interest with quarterly compounding.”
– “This semi-annual bond has a yield of 8%.”
• In almost all cases, if its an APR it will tell you. The only real exception is
bonds, but it’s easy enough to remember that bonds are quoted in
APRs.
Page 14
EAR and APR: Example 1
• Suppose that you can either earn 3% per quarter in Bank A, or 1% per month in
Bank B. What is the APR for each bank?
• Answer
• What is the EAR for each bank? Which bank should you put your money in?
• Answer
15
EAR and APR: Example 1
• Suppose that you can either earn 3% per quarter in Bank A, or 1% per month in
Bank B. What is the APR for each bank?
• Answer: 1 + 𝐸𝐴𝑅 = 1 + 𝑟𝑝𝑒𝑟𝑖𝑜𝑑
𝑛
= 1 + 3%
4
= 1.1255; 𝐸𝐴𝑅 = 12.56%
• What is the EAR for each bank? Which bank should you put your money in?
• Answer: 1 + 𝐸𝐴𝑅 = 1 + 𝑟𝑝𝑒𝑟𝑖𝑜𝑑
𝑛
= 1 + 1%
12
= 1.1268; 𝐸𝐴𝑅 = 12.68%
16
Topics
• Define annual percentage rate (APR) and effective annual rate (EAR)
• Learn how to convert APR and EAR
• Find period interest rates for non-annual cash flows from APR/EAR
• Compute loan payments and balances based on quoted rates
17
Translating APR into EAR
• A stated rate of 12% with monthly compounding means you pay 1% a
month for twelve months!
0
1%
1
2
3
12
• From the stated rate (APR), the only thing useful is the periodic rate:
rPeriodic
rAPR .12
=
=
= .01
m
12
# of compounding periods/year
Page 18
Does 12% really mean 12%?
• If we deposit $1 today in a bank account that has a stated rate of 12%
with monthly compounding, what will we have at the end of the year?
• FV = 1(1.01)12 = $1.1268 => 12.68% (effective rate)
• So what’s the rule?
Page 19
EAR and APR: Conversions
• From APR to EAR
• where m is the number of compounding periods per year
– For example, m = 4 for quarterly compounding, m = 12 for monthly
compounding
• From EAR to APR
Page 20
10% APR
Compounding
m
Formula
Annual
1
Semiannual
2
Monthly
12
 0.10 
1 +
 −1
1


2
 0.10 
1 +
 −1
2 

12
 0.10 
1 +
 −1
12


Daily
365
1
 0.10 
1 +

365 

365
−1
EAR
10.00%
10.25%
10.47%
10.52%
• These are different EARs for the same APR of 10%
– The only thing that changes is the compounding frequency m
• Note for m > 1, EAR > APR. This is the effect of compounding
Page 21
Topics
• Define annual percentage rate (APR) and effective annual rate (EAR)
• Learn how to convert APR and EAR
• Find period interest rates for non-annual cash flows from APR/EAR
• Compute loan payments and balances based on quoted rates
22
Finding Period Interest Rates
• Watch out for the frequency of cash flows
• How do we adjust to find the period interest rate?
– By definition, APR is the number of compounding periods m times the interest rate
earned in each period
– Interest rate per compounding period 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 is then:
– 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 =
𝐴𝑃𝑅
𝑚
– Note that if m = 1, then 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 = 𝐴𝑃𝑅
• Wrapping up:
23
Topics
• Define annual percentage rate (APR) and effective annual rate (EAR)
• Learn how to convert APR and EAR
• Find period interest rates for non-annual cash flows from APR/EAR
• Compute loan payments and balances based on quoted rates
24
Problem Setup
• You just graduated from NUS and you are considering buying a house. The
house you are looking at costs $1mln. To finance this purchase, DBS offers you a
20-year mortgage with an APR of 12% and bi-monthly payments (i.e. you will
pay a fixed amount every two months).
• What is the value of the bi-monthly payments?
• In 10 years from now, you might want to have a kid and move out of the house.
How much mortgage outstanding will you have then?
25
Solution Setup
• First, we need to find the bi-monthly interest rate 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 (for simplicity, I will
just call this 𝑟 ) associated with an APR of 12%
• Then, we will use this r and the annuity formula to compute the value of the biis an annuity as you are paying the same amount (a constant) for a fixed period of
monthly payments today ittime
• Finally, we will use r and the monthly payments in the annuity formula to
calculate the value of the residual mortgage in ten years
26
Step 1: Finding r
• This is easy, since
𝐴𝑃𝑅
𝑟=
𝑚
0.12
=
= 2%
6
• The only wrinkle is that these are bi-monthly payments, so there are six of them
each year
27
Step 2: Finding the Payment Value
• Recall the annuity formula
• Here, the number T of bi-monthly payments is 𝑇 = 20 × 6 = 120, and we just
found that r = 2%. Moreover, we are borrowing $1mln, so that the present value
of the mortgage is equal to $1mln. We need to find C that solves
• Which yields C ≈ $22,048. Therefore, the mortgage will require 120 payments of
$22,048 every two months.
28
Step 3: Finding the Mortgage Value in Ten Years
• Recall the annuity formula
• In ten years, the mortgage will only have 60 payments of $22,048 left. Therefore
in ten years, T = 60, r = 2%, C = $22,048. We need to find 𝑃𝑉10 (the present value
in ten years).
• Therefore, in ten years we will have $766,408 left to pay in mortgage.
29
Step 3: Finding the Mortgage Value in Ten Years
• Why do we still owe $766,411 after 10 years? The answer is interest
• The first bi-monthly payment of $22,048 is comprised of $1,000,000 × 0.02 =
20,000 interest payment. The residual $2,048 goes to repay the principal. The
payment outstanding is $1,000,000 − $2,048 = $997,952
• The second payment is comprised of $997,952 × 0.02 = 19,959.04 interest
payment. The residual $2089 goes to repay the principal, and so on
• At the beginning, we are borrowing more and most of our payments are
comprised of interest payments
• As time goes by and we start paying off the principal, we will pay less and less in
interest until the last payment is only comprised of principal repayments
30
Amortization Table
31
Outstanding Balance of a Mortgage
32
Proportion of Payments in a Mortgage
33
Summary
• Annual percentage rate (APR) is conventionally used by banks
• Effective annual rate (EAR) can be used to compare investments with
different compounding horizons
• The main difference between APR and EAR is that APR is based on simple
interest, while EAR takes compound interest into account:
• Conversion between APR and EAR
34