CFA Level 1 Time Value of Money LO c: Impact of frequency of compounding on EAR

Hello and welcome back. 

In the previous blog we talked about the components of an interest rate. If you haven’t read the previous blog, please see it first.

In this blog, we will talk about the impact of frequency of compounding on effective annual rate (EAR). Compound interest, as you should know by now, is the interest on interest, i.e. the money that we earn on our interest from the previous period.

Let’s take a simple example. 

Example 1)

Suppose you have a 1000$ and you want to invest it for 2 years. You have 2 options:

1.) You can invest it on Simple Interest, i.e. you earn fixed interest every period. Let’s say the rate of interest is 10%. So, the Simple interest for one year will be 10% of 1000$, i.e. 100$. So, over 2 years you will earn a total of 200$.

2.) You can invest it on Compound Interest, i.e. you earn progressively increasing interest as you earn interest on interest. Let’s say the interest is compounded annually.

So, after 1 year, you earn 100$ on your investment.

Now, at the end of 1 year, your new Principal is 1100$. So, at the end of 2^nd year you earn 100$ on your initial investment and 10$ (10% of 100$) on the interest you earned after first year. So, after 2 years, you earn a total of 210$.

Let’s see this again.

We know that FV = PV(1 + I/Y)^N  where

• PV is used for Present value or Current Value or Value at time 0

• FV is used for Future value or Value at time t

• N signifies Number of compounding periods

• I/Y is used for Interest rate per compounding period

If this is new to you, please read our blog Introduction to Time Value of Money (TVM) calculations for CFA Level 1

So, at the end of 1^st year, FV=1000(1.1)¹ = 1000*1.1 = 1100

At the end of 2^nd year, FV=1000(1.1)² = 1000*1.21 = 1210

You need to get comfortable doing these calculations and you should be able to do these on BA II plus calculator.

PV= -1000 (negative as this is a cash outflow, you are investing money)

I/Y= 10 (rate of interest per compounding period, you do not need to change it to decimal)

N = 2 (number of compounding periods)

PMT = 0 (Regular payments, not present in this case)

CPT à FV  to get the answer $1210.

Now, let’s take another example:-

Example 2)     

Let’s say you have $1000 and you can invest it on either simple interest or compound interest (compounded annually) for 20 years. The rate of interest is 10%. What will be the value of your investment after 20 years?

Again, for SI: Interest per year = 1000 * 0.1 = $100

In 20 years, we will earn $100 * 20 = $2000

Total Amount after 20 years = $1000 (Principal) + $2000 = $3000

For Compound Interest PV= -1000, N=20, I/Y=10, PMT=0 CPTàFV to get $6727.5

So, as you can see, by investing on Compound Interest, you receive more than twice of what you would have received at Simple Interest. This is the Power of Compounding.

Example 3)

Let’s say you have 1000$ to invest for 2 years. The frequency of compounding is semi-annually. What is the value of the investment after 2 years?

In this case, PV = -1000,

I/Y = 5 (Remember, we are compounding semi-annually, so the interest rate we will use will be half of stated annual interest rate)

N = 4 (Again, number of compounding periods will be 4 in 2 years as we are compounding semi-annually)

PMT = 0

CPT à FV to get the answer $1215.51

So, as you can see, we are earning more money than when we were compounding annually.

What happens if we start compounding quarterly?

PV = -1000

I/Y = 2.5 (As now we have 4 quarters per year)

N = 8 (We have 8 quarters in 2 years)

PMT = 0

CPT à FV to get the answer $1218.40

Let’s go a bit further. What if we start compounding monthly?

PV = -1000

I/Y = 0.833 (As now we have 12 months per year)

N = 24 (We have 24 months in 2 years)

PMT = 0

CPT à FV to get the answer $1220.38

Few points to note:-

1.) Both Compound Interest and Simple Interest values are exactly the same after the end of 1^st compounding period. This is because you have not started earning interest on interest.

2.) Higher the frequency of compounding, greater is the amount of interest earned.

3.) Compound interest effects are pronounced when the interest rates are high and the duration of investment is long.

Now we can start talking about Effective Annual rate. 

Traditionally, all interest rates are quoted as annual interest rates and the frequency of compounding is mentioned.

For example, you will see interest rate mentioned as 10% compounded quarterly, rather than 2.5% per quarter.

So, in reality an investor will receive a different interest rate because of the effects of compounding. This actual interest rate received by an investor is called Effective Annual Rate (EAR).

Let’s look at an example.

Example 4)

Let’s say you want to invest 100$ at 10% interest rate compounded annually.

So, you will receive 1.1*100 = 110$ at the end of 1^st year.

What if the frequency of compounding was semi-annually?

PV = -100, N =2, I/Y=5, PMT=0, CPTàFV to get $110.25

So, the effective annual rate in this case is 10.25%

What if the frequency of compounding was quarterly?

PV = -100, N =4, I/Y=2.5, PMT=0, CPTàFV to get $110.38

So, the effective annual rate in this case is 10.38%

In general, you can calculate Effective Annual Rate using this formula:-

So, for our previous example:-

EAR _{Semi Annual Compounding} = (1 + .05) ² – 1 = 0.1025 = 10.25%

EAR _{Quarterly Compounding} = (1 + .025) ⁴ – 1 = 0.1038 = 10.38%

As you must have realized by now, Stated Annual Interest Rate and EAR will be equal if and only if interest is being compounded annually. In all other cases, EAR will be higher than Stated Annual Interest rate and the difference between these two will increase with the increase in frequency of compounding.

Finally, let’s talk about the extreme case of continuous compounding. In case of continuous compounding,

So, if r = 10%

EAR _{Continuous Compounding =} e ^0.1 – 1 = 1.105171- 1= .105171 = 10.51%

Before we end this topic, let’s see a couple of examples:-

Example 5)

You are an investor who has been provided with two similar investment options with the same risk level. The 1^st investment option is offering you 10.4 % interest rate compounded semi-annually, while the 2^nd investment is offering you 10% interest rate compounded monthly. Which is a better investment option, considering all other factors are same for these two investments?

To solve these problems, it is imperative that we convert all the interest rates to EARs so that we can do a proper comparison.

In our current example,

For Semi-Annual compounding, EAR = (1 + 0.104/2)² – 1

                                                              = (1.052)² – 1

                                                              = 1.106704 – 1 = 0.106704 = 10.67%

For monthly compounding, EAR  = (1 + 0.10/12)¹² – 1

   = (1.008333)¹² – 1

                                                       = 1.104709 – 1 = 0.104709 = 10.47%

As we can see, the 1^st investment option with Semi-Annual compounding is better in this case.

Example 6)

You are told that the EAR for an investment is 12.55% with quarterly compounding. Find the periodic rate for this investment.

We know that EAR =   (1 + Periodic Rate) ^m -1

0.1255 =  (1 + Periodic Rate) ⁴ – 1

1.1255 =  (1 + Periodic Rate) ⁴  

   (1.1255)^1/4  =  (1 + Periodic Rate)

           1.0299 =  (1 + Periodic Rate)

Periodic Rate = .0299 = .03 = 3%

By now you should be comfortable handling Effective Annual Rates, Stated Annual Rate and Periodic Rates. 

You can watch a video explaining the points discussed in this blog here:-

This brings us to the end of learning outcome c for this reading. In the next blog, we will talk about learning outcome d: Solving TVM problems for different frequencies of compounding.

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