COMM298-CLASS04.pdf

8/9/2019 COMM298-CLASS04.pdf

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COMM 298 – Introduction to Finance

Cornelia Kullmann

Class 4: Compounding Frequencies: APRs vs. EARs

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iClicker Example: Multiple Cash Flows

Tim’s aunt and uncle give him $500 for his birthday each year (they are verypredictable). It’s Tim’s 19th birthday and he badly wants to buy a new car. Agood used car, in his opinion, costs about $1500, but all he has are the $500 he just got.

In comes Tim’s older brother, who generously offers to lend him money inexchange for Tim’s next three birthday gifts from their aunt and uncle. Theinterest he is willing to lend Tim the money at is 20% per year.

–

Will Tim be able to buy a used car if he accepts his brother’s offer?

Timeline!

a) No, even with his brother’s money Tim would only have $1,053.24 today.

b) No, even with his brother’s money Tim would only have $1,294.37 today.c) Yes, with his brother’s money Tim would have $1553.24 today.

d) Yes, with his brother’s money Tim would have $1,941.90 today.

e) 20%? Isn’t that usury?

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iClicker Example: Multiple Cash Flows (cont’d)

Tim’s aunt and uncle give Tim $500 for his birthday each year (they are verypredictable). It’s Tim’s 19th birthday and he wants to buy a new car. A goodused car, in his opinion, costs about $1,500, but all he has are the $500 he just got.

In comes Tim’s older brother, who generously offers to lend him money in

exchange for Tim’s next three birthday gifts from their aunt and uncle. Theinterest he is willing to lend Tim the money at is 20% per year.

– Will Tim be able to buy a used car now if he accepts his brother’s offer?

-1 0 1 2 3 4 Years

---|-------|-------|-------|-------|-------|----

$500 $500 $500 $500 CFs

The correct answer is c).

Together with the $500 you got today, you have $1,553.24.

24.053,12.1

500

2.1

500

2.1

500)(

32 =++= payments future PV

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Objectives

Today we start by using our compounding/discounting formula to solvefor r and n.

The next topic will deal with nominal (stated, quoted) and effective

interest rates. Annual Percentage Rates: APRs (nominal, stated, quoted rates)

–

These are always denoted by ‘i m’ or ‘ APR’ in this course, where ‘m’indicates the number of compounding periods per year.

Effective rates:

–

These are denoted by r m in this course, where ‘m’ indicates the number ofcompounding periods per year.

Effective Annual Rates: EARs or r Yr – It’s important to understand that you always have to use effective interest rate for

discounting or compounding.

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Solving for other things

Recall our FV Formula

One equation with four variable

=> If we know three out of the four variables, we can solve for the 4th

FV n = PV

0 * (1+ r)n

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Example: Solving for the Interest Rate r

A company has the option of buying a machine for $355,000 today.The government promises to buy all the output of the machine infour years for $400,000.

–

The machine is worth 0 at this point.

–

What is the rate of return of this investment?4)1(*355400 r +=

%0286.31030286.1

1355

4001

355

400

)1(355

400

)1(355

400

44

1

4

1

4

=−==>

−=−⎟⎠

⎞⎜⎝

⎛==>

+=⎟⎠

⎞⎜⎝

⎛=>

+==>

r

r

r

r

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Solving for the Interest Rate r

The general equation for solving for the interest rate r is:

Which can also be written as:

1

1

−⎟⎠

⎞

⎜⎝

⎛=

n

PV

FV r

1−= n

PV

FV r

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Example: Solving for the Number of Periods n

You just met the girl of your dreams.

You currently have $10,000 in your Engagement Ring Fund, whichearns 12% per year.

The engagement ring you want costs $20,000.

How long do you have to wait before you can ‘pop the question?’ n)12.01(*000,10000,20 +=

1163.6

)12.1ln(

)2ln(

)2ln()12.1ln(*

)2ln()12.1ln(

2)12.1(

)12.1(000,10

000,20

)12.01(*000,10000,20

==>

==>

==>

==>

==>

==>

+==>

n

n

n

n

n

n

n

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Aside: The Rule of 72 again

According to the this rule, it would have taken youapproximately

years to double your money at an annual rate ofinterest of 12%, which is actually pretty close to what

we found on the previous slide.

72

r *100=

72

12=

6

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Solving for the Number of Periods n

FV n = PV *(1+ r)

n

! (1+ r)n =FV

n

PV

! ln(1+ r)n = FV nPV

! n " ln(1+ r) = ln FV

n

PV

#

$%

&

'(

! n =ln

FV n

PV

#

$%

&

'(

ln(1+ r)

10

Of course one can also use ‘log’ instead of ln


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Compounding Frequencies: APRs vs. EARs

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12

Compounding Frequencies

So far we (implicitly) assumed that interest was paid/earned once perperiod. The interest rates were effective per period rates.

Interest rates are usually stated on an annual basis. – What happens if interest is compounded more frequently than once per period? (E.g. if

banks pay interest more than once per year).

Example: APRs

A)

Bank A: 12% compounded annually (or i yr = 12%)

B)

Bank B: 12% compounded semi-annually (or i semi = 12%)

C) Bank C: 12% compounded quarterly (or i q = 12%)

D) Bank D: 12% compounded monthly (or i mo = 12%)

Rates stated like this are called quoted, stated, or nominal rates. – They are denoted by ‘i k’, where k indicates the number of compounding periods per year

What are the differences between these rates?


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13

Compounding Frequencies: Comparing Returns (1)

How do you compare these interest rates?

–

We first find out by how much $1 grows over the course of one year foreach of these rates.

– Then we can solve for the annual interest rate that would give investorsthe same return.

– This rate is called the Effective Annual Rate (EAR) or r Yr

Bank A: iYr = 12%

– Since Bank A pays 12% per year, there is no compounding doneduring the year and the stated/quoted rate (APR) is equal to the

EAR.

If interest is compounded once per year, the APR is the EAR.

– Or i Yr = r Yr


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14

Compounding Frequency: Comparing Returns (2)

Bank B: i semi = 12% (12% compounded semi-annually) What does that mean?

–

By convention, this means that you earn every sixmonths, where r semi is the effective 6-months rate.

–

Timeline: 1 Years

0 1 2 Periods

– --------|------------------|-------------------|----------

$1 $1*(1+r semi) $1*(1+r semi)2

–

After six months, you have:

–

After one year, you have:

$1*(1+ isemi2 ) = $1*(1+ r

semi) = $1*(1+ 0.06)= $1.06

$1*(1+ rsemi

)(1+ rsemi

)= $1*(1+ rsemi

)2= $1!(1+

isemi

2 ) = $1*(1.06)

2= $1.1236

isemi

2 = r

semi = 6%


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15

Compounding Frequeny: Comparing Returns (3)

Bank B: isemi = 12% continued –

What effective annual rate of return (EAR) would give you the samereturn as Bank B’s offer of 6% every 6 month?

So putting things together, we get the following general rule:

– This shows how to compute the EAR r Yr that corresponds to a stated rate of return(or APR) of 12% compounded semi-annually (which is the same as a semi-annualeffective rate of return of r semi = 6%).

$1*(1+ rYr) = $1.1236

! rYr = 1.1236"1

! rYr = 0.1236 =12.36%

$1*(1+ isemi2 )2 = $1*(1+ r

semi)2 = $1*(1+0.06)2 = $1.1236 = $1*(1+0.1236)= $1*(1+ r

Yr)

! 1+ isemi2( )

2

= (1+ rsemi

)2 = (1+ rYr)!

EAR = rYr = (1+ r

semi)2 "1= 1+ isemi

2( )2

"1


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16

iClicker Example: Comparing Returns

What is the effective annual interest rate (EAR) thatyields the same return as Bank C’s offer of 12%compounded quarterly?

a) 12%

b) 12.36%

c) 12.48%

d)

12.55%

e)

None of the above

Correct answer is d). Solution next two slides.


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17


Bank C: 12% compounded quarterly –

Timeline: 1 Years

0 1 2 3 4 Periods (Quarters)

–

---|-------------|-------------|-------------|-------------|--------

$1 $1(1+r q) $1(1+r Q)2 $1(1+r Q)

3 $1(1+r Q)4

– After three months, you have:

– After six months, you have:

–

After nine months, you have:

– After one year, you have:

$1*(1+iQ

4) = $1*(1+ r

Q) = $1*(1+ 0.03) = $1.03

0609.1$)03.1(*1$)1(*1$)1)(1(*1$ 22 ==+=++ QQQ r r r

092727.1$)03.1(*1$)1(1$ 33 ==+ Qr

$1! (1+iQ

4)4 = $1(1+ rQ )

4= $1*(1.03)4 = $1.1255

iQ =

12%!iQ

4 = rQ = 3%


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18


Bank C: continued

– The effective annual interest rate that yields the same return as anominal rate of 12% compounded quarterly (which is the same asan effective quarterly interest rate of 3%) is equal to:

– And in general (note that the $1 ‘cancels’ out, so this works forany amount):

$1*(1+iQ

4 )4

= $1*(1+ rQ )4

= $1*(1.03)4

= $1.1255 = $1*(1+ rYr )!1.1255 =1+ rYr

!rYr =

12.55%

1+iQ

4( )4

= (1+ rQ)4=1+ r

Yr

! rYr =

(1+ rQ )4"1= 1+

iQ

4( )4

"1

iQ =

12%!iQ

4 = rQ = 3%


19/22

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Bank D: 12% compounded monthly (i mo = 12%)

– 12% compounded monthly means the effective monthly rate r mo isequal to 1%.

– By similar reasoning as above, one can show that $1 invested atan effective monthly rate of r mo = 1% grows to $1.1268 after oneyear, i.e.:

– So the effective annual interest rate that yields the same return asa nominal rate of 12% compounded monthly (which is equal to an

effective monthly rate r m of 1%) is given by:

1268.1$)01.01(1$ 12 =+

1+0.12

12

!

"#

$

%&

12

= (1+ 0.01)12

=1.1268= (1+ rYr)

' rYr =12.68%


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20

Turning Things Around

We have seen that: –

For a given nominal rate compounded semi-annually, we can calculate thecorresponding effective annual rate (EAR) as follows:

Now for a given r Yr , we can use this equation to determine thecorresponding effective semi-annual rate and the nominal ratecompounded semi-annually:

This implies that

1+ isemi

2( )2

= (1+ rsemi

)2=1+ r

Yr ! EAR = r

Yr = (1+ r

semi)2"1= 1+

isemi

2( )2

"1

1+ rYr = 1+

isemi

2( )2

= (1+ rsemi

)2!

isemi

2 = r

semi = (1+ r

Yr)

12"1= 1+ r

Yr "1

When would one want to do that? When one has the EAR, but cash flows that occur at a

higher frequency than once per year.

isemi

2= (1+ r

Yr)

12 !1= 1+ r

Yr !1= r

semi

" isemi

= 2# (1+ rYr)

12 !1$%

&'


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21

How Interest Rates are Annualized

Interest rates are normally quoted on an annual basis.

So how should one annualize interest rates that arecompounded more than once per year?

– One could calculate the corresponding effective annual rate(EAR) as we did above or

– One could just multiply by the number of compoundingperiods per year.

For a variety of reasons, including tradition, government legislation,and attempts to mislead investors, these rates are typicallyannualized by multiplying them by the number of compounding

periods per year. These artificially created annual rates are called nominal rates, or

quoted rates, or stated rates.

This is how Annual Percentage Rates (APRs) are calculated (by law)

–

Said differently, APRs are nominal rates!

– I usually call them nominal interest compounded m times

per year, or i m. (note: book uses q instead of i)


22/22

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APRs

Banking regulation requires lenders to disclose the AnnualPercentage Rate (APR) on virtually all consumer loans.

–

The rate must be displayed on a loan document ‘in a prominent andunambiguous way.’

– The law also stipulates that the APR is to be computed by

multiplying the effective per period interest rate r k by the numberof compounding periods in a year k.

This means APRs are nominal rates. And yes, I know I am repeating myself.

–

Since they are nominal rates, they do not reflect the actual cost ofborrowing when interest is compounded more than once per year.

Ironically, this means that legislation forces banks and other lenders

to report a rate that does not reflect the actual cost of borrowingmoney!

Documents

COMM298-CLASS04.pdf