# Suppose earning an APR of 7% for 5 years, with interest compounded quarterly. What is the EAR? What is the Total Return?

Suppose you earn an APR of 7% for 5 years, with interest compounded quarterly. What is your EAR? What is your Total Return?

The part I am confused is "with interest compounded quarterly"

The formulas I use is Annual Percentage rate(APR) = (1+ Effective Annual Rate (EAR))^T - 1)/ Time(T)

The way I calculate is 0.07(5) + 1 = (1 + EAR)^5 >>> EAR = .0619 ~ 6.19%. The total return is 1.0619...Is this right way to calculate...

If not, what did I do wrong? If so, how do I amend my calculation if interest is compounded annually? Monthly?

Thank you

To get the quarterly effective interest rate, the quoted interest rate should be divided by four thus `1.75%`.

To get the annual effective interest rate, the effective quarterly interest rate should be indexed, added to `1`, and taken the fourth power thus `7.19%`:

``````( 1 + 0.0175 ) ^ 4
``````

To get the total return, the effective annual interest rate should be indexed, added to `1`, and taken to the fifth power thus `41.5%`:

``````( 1 + 7.19 ) ^ 5
``````

Relationship between the effective annual and compounded interest rate

The spread between the APR & EAR is proportional to compounding frequency and to the APR. The EAR will always be greater in magnitude than the APR and will always have the same direction because of the nature of the formula:

``````EAR = ( 1 + APR / compounding period ) ^ ( compounding period )
``````

Because the EAR is a geometric representation of the interest rate while the APR is the arithmetic representation. The arithmetic mean is usually the lowest, the geometric the highest, and the harmonic in between.

Change in terminology

In the US, the APR is fast becoming interpreted as the EAR because EAR disclosure is now mandatory for most products, but this has not historically been the case and probably still isn't the case everywhere else.

Effective from compound rates

Compounded annual rates are usually the effective rates multiplied by the number of compounding periods.

Historically, arithmetic averages were preferred over their more accurate geometric counterparts because of ease of calculation. When interest rates are small and calculators expensive, this is sufficiently accurate. This explains the strange math behind compound rates, which the US has more or less abandoned.

Now that anyone can buy a pocket calculator for next to nothing, it pays to be more precise with interest rate calculation especially at near-0 rates since a 0.1% error while paying 1% is a possible 10% increase in debt costs.

• what do you mean mathematically "effective quarterly interest rate should be indexed"? – afsdf dfsaf Feb 15 '14 at 5:48
• To check my understanding, if the time horizon is 10 instead of 5, the EAR is the same right. In general, as interest rate increase, the difference EAR and APR increases right? Also, as interest is compounded more frequent, the difference EAR and APR increases right? Last, EAR >= APR right? – afsdf dfsaf Feb 15 '14 at 6:23
• I understood that terms APR and EAR and EAY all refer to the same annual effective yield. – user11906 Feb 15 '14 at 7:06
• Isn't the quarterly rate `1.07^(1/4)` or ~1.7%? If it were 1.75%, that would make the APR `1.0175^4` or ~7.2%, no? – Daniel Lubarov Feb 15 '14 at 17:46
• As you stated nicely,"Because the EAR is a geometric representation of the interest rate while the APR is the arithmetic representation. [The arithmetic mean is usually the lowest, the geometric the highest, and the harmonic in between.] " So EAR >= APR should be right? – afsdf dfsaf Feb 15 '14 at 18:23

If the APR - annual percentage rate is already stated then that is the same as EAR - effective annual rate thus there is nothing else left to annualize.

You would simply use the 7% in your calculations as follows

``````=(1+7%/4)^(5*4)
=(1+0.0175)^20
=(1.0175)^20
=1.4148

41.48% is the return
``````
• Given the definition of APR as identical to EAR (effective annual rate), the calculation is incorrect, in that it uses the APR as the nominal annual rate... – DJohnM Feb 15 '14 at 23:34
• I think I screwed up :( I guess got to take those pills before posting a reply – user11906 Feb 16 '14 at 1:04