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Calculating the Annual Percentage Yield (APY) And Continuous Compounding
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Introduction
In calculating the Annual Percentage Yield (APY) over the course of one year, it is usual to take a nominal interest rate and derive the effective interest rate for a given number of compounding periods. When the number of compounding periods per year is one then the effective interest rate is identical to the nominal interest rate. When the number of compounding periods is infinite then you have continuous compounding, and the effective interest rate is maximised for the same nominal interest rate. For a discussion on the distinction between APY and exponential growth see my article What Is Exponential Growth? for more.
In the tables below I have represented a broad range of positive nominal interest rates (Table1 and Table 2) and negative nominal interest rates (Table 3 and Table 4). In each table, I have calculated the number of compounding periods per year, month, week, day, hour, minute and second. That is, the number of years, months, weeks, days, hours, minutes or seconds in one year.
The results are not the effective interest rates, nor the Annual Percentage Yield (APY), but rather what I call the exponential factor. The formula used is:
F = (1 + (p/C)) ^{C}
F = Exponential Factor
C = number of compounding periods
p = percentage (e.g. 1% = 1/100)
Example:
Nominal Interest Rate = 1%
Number of Compounding Periods = 12
F= (1+(0.01/12)) ^{12 }= (1 + 0.0008333333) ^{12 } = 1.0008333333 ^{12} = 1.0100459609
Annual Percentage Yield
To calculate the APY, simply subtract 1 from exponential factors listed in the tables below. Or, use the formula:
APY = (1 + (p/C)) ^{C } 1
APY = Annual Percentage Yield
C = number of compounding periods
p = percentage (e.g. 1% = 1/100)
Example:
Nominal Interest Rate = 1%
Number of Compounding Periods = 12
APY= (1+(0.01/12)) ^{12 } 1 = (1 + 0.0008333333) ^{12 }  1 = 1.0008333333 ^{12}  1 = 1.0100459609  1 =
0.0100459609
Effective Interest Rate
This is another name for Annual Percentage Yield.
What is the exponential factor?
Essentially, the exponential factor is the factor by which a population or investment would grow for a given effective interest rate.
The reason why I refer to this factor as the exponential factor is because it is directly linked the number e. In fact, as will be demonstrated, for any given Nominal Interest Rate the limit to the exponential factor is represented by e (represented by the MS Excel function Exp) to the power of that Nominal Interest Rate.
Positive Rates
Nominal Interest Rate 

Compounding
periods for 1 year: 
Equates to:  1%  2%  3%  4%  5%  6%  7% 
1  Annually  1.0100000000  1.0200000000  1.0300000000  1.0400000000  1.0500000000  1.0600000000  1.0700000000 
12  Monthly  1.0100459609  1.0201843557  1.0304159569  1.0407415429  1.0511618979  1.0616778119  1.0722900809 
52  Weekly  1.0100491960  1.0201974172  1.0304456200  1.0407947700  1.0512458419  1.0617998195  1.0724576961 
365  Daily  1.0100500287  1.0202007810  1.0304532636  1.0408084931  1.0512674965  1.0618313107  1.0725009832 
8,760  Every Hour 
1.0100501613  1.0202013167  1.0304544810  1.0408106791  1.0512709464  1.0618363284  1.0725078813 
525,600  Every Minute 
1.0100501670  1.0202013397  1.0304545331  1.0408107726  1.0512710939  1.0618365430  1.0725081763 
31,536,000  Every second 
1.0100501665  1.0202013388  1.0304545321  1.0408107716  1.0512710935  1.0618365428  1.0725081842 
exp(0.01)  exp(0.02)  exp(0.03)  exp(0.04)  exp(0.05)  exp(0.06)  exp(0.07)  
Infinite  Continuous Compounding  1.0100501671  1.0202013400  1.0304545340  1.0408107742  1.0512710964  1.0618365465  1.0725081813 
Table 1: Calculating the Exponential Factor for the given compounding periods for Nominal Interest Rates 1% to 7%. For any combination of Nominal Interest Rate and compounding periods, the limit to the exponential factor is represented by e (represented by the MS Excel function Exp) to the power of that Nominal Interest Rate.
Nominal Interest Rate 

Compounding
periods for 1 year: 
Equates to:  10%  50%  100%  200%  250%  500%  1000% 
1  Annually  1.1000000000  1.5000000000  2.0000000000  3.0000000000  3.5000000000  6.0000000000  11.0000000000 
12  Monthly  1.1047130674  1.6320941327  2.6130352902  6.3585995587  9.6881549067  65.3449611308  1441.7740923459 
52  Weekly  1.1050647928  1.6447879216  2.6925969544  7.1170766277  11.4932869399  118.3909828469  9379.8087823034 
365  Daily  1.1051557816  1.6481572517  2.7145674820  7.3488253366  12.0791068159  143.4609683096  19253.8327075863 
8,760  Every Hour 
1.1051702873  1.6486977455  2.7181266916  7.3873695491  12.1781496382  148.2016137944  21901.1971246215 
525,600  Every Minute 
1.1051709075  1.6487208786  2.7182792426  7.3890279820  12.1824215285  148.4096295635  22024.3705571415 
31,536,000  Every second 
1.1051709198  1.6487212679  2.7182817813  7.3890556356  12.1824927001  148.4131003298  22026.4309204041 
exp(0.1)  exp(0.5)  exp(1)  exp(2)  exp(2.5)  exp(5)  exp(10)  
Infinite  Continuous Compounding  1.1051709181  1.6487212707  2.7182818285  7.3890560989  12.1824939607  148.4131591026  22026.4657948067 
Table 2: Calculating the Exponential Factor for the given compounding periods for Nominal Interest Rates 10%, 50%, 100%, 200%, 250%, 500% and 1000%. For any combination of Nominal Interest Rate and compounding periods, the limit to the exponential factor is represented by e (represented by the MS Excel function Exp) to the power of that Nominal Interest Rate.
For a nominal growth rate of 100%, e (represented here by the MS Excel function exp(1)) is defined as the limit to the exponential factor (and hence the effective interest rate) for an infinite number of compounding periods. Thus, as the number of compounding periods increases, so the nominal 100% interest rate approaches the limit e.
Negative Rates
Nominal Interest Rate 

Compounding
periods for 1 year: 
Equates to:  1%  2%  3%  4%  5%  6%  7% 
1  Annually  0.9900000000  0.9800000000  0.9700000000  0.9600000000  0.9500000000  0.9400000000  0.9300000000 
12  Monthly  0.9900457063  0.9801823186  0.9704090818  0.9607252460  0.9511300672  0.9416228069  0.9322027322 
52  Weekly  0.9900488817  0.9801949024  0.9704371323  0.9607746503  0.9512065440  0.9417319095  0.9323498514 
365  Daily  0.9900496981  0.9801981362  0.9704443370  0.9607873332  0.9512261666  0.9417598888  0.9323875606 
8,760  Every Hour 
0.9900498281  0.9801986509  0.9704454837  0.9607893514  0.9512292888  0.9417643401  0.9323935591 
525,600  Every Minute 
0.9900498336  0.9801986729  0.9704455327  0.9607894377  0.9512294223  0.9417645304  0.9323938156 
31,536,000  Every second 
0.9900498342  0.9801986742  0.9704455350  0.9607894376  0.9512294232  0.9417645331  0.9323938198 
exp(0.01)  exp(0.02)  exp(0.03)  exp(0.04)  exp(0.05)  exp(0.06)  exp(0.07)  
Infinite  Continuous Compounding  0.9900498337  0.9801986733  0.9704455335  0.9607894392  0.9512294245  0.9417645336  0.9323938199 
Table 3: Calculating the Exponential Factor for the given compounding periods for negative Nominal Interest Rates 1% to 7%. For any combination of Nominal Interest Rate and compounding periods, the limit to the exponential factor is represented by e (represented by the MS Excel function Exp) to the power of that Nominal Interest Rate.
Nominal Interest Rate 

Compounding
periods for 1 year: 
Equates to:  10%  50%  100%  200%  250%  500%  1000% 
1  Annually  0.9000000000  0.5000000000  0.0000000000  1.0000000000  1.5000000000  4.0000000000  9.0000000000 
12  Monthly  0.9044583741  0.6000661541  0.3519956280  0.1121566548  0.0606049798  0.0015523925  0.0000000005 
52  Weekly  0.9047503069  0.6050650129  0.3643135196  0.1300967228  0.0771429914  0.0052109241  0.0000150241 
365  Daily  0.9048250208  0.6063227895  0.3673749207  0.1345930428  0.0813820180  0.0065090447  0.0000394869 
8,760  Every Hour 
0.9048369016  0.6065220046  0.3678584425  0.1353043836  0.0820557157  0.0067283356  0.0000451413 
525,600  Every Minute 
0.9048374094  0.6065305155  0.3678790912  0.1353347683  0.0820845106  0.0067377868  0.0000453956 
31,536,000  Every second 
0.9048374188  0.6065306578  0.3678794349  0.1353352749  0.0820849905  0.0067379443  0.0000453999 
exp(0.1)  exp(0.5)  exp(1)  exp(2)  exp(2.5)  exp(5)  exp(10)  
Infinite  Continuous Compounding  0.9048374180  0.6065306597  0.3678794412  0.1353352832  0.0820849986  0.0067379470  0.0000453999 
Table 4: Calculating the Exponential Factor for the given compounding periods for negative Nominal Interest Rates 10%, 50%, 100%, 200%, 250%, 500% and 1000%. For any combination of Nominal Interest Rate and compounding periods, the limit to the exponential factor is represented by e (represented by the MS Excel function Exp) to the power of that Nominal Interest Rate.
Only 1 Compounding Period
When the number of compounding periods for the year is 1, clearly the exponential factor is easily derived simply by adding the nominal interest rate to 1.
However, this should not be confused with simple interest which gives precisely the same result when restricted to just one year's growth. As explored in my article Linear Growth versus Exponential Growth (and Couttsian Growth), once we extend the growth period beyond just one year the difference between compound interest and simple interest becomes starkly obvious.
Conclusion
Clearly, the more frequent the compounding periods, the closer we get to the limit (imposed by e) to the effective interest rate. Continuous compounding is yet another example of the direct and binding link between compound growth and exponential growth. The two are intrinsically linked, as explored in my article Compound Growth versus Exponential Growth (and Couttsian Growth).
The fact is that anything that grows via compound interest grows exponentially. That includes money earned in the form of investments, or owed in the form of loans. It also includes all living creatures, all of which are alive today as the result of a replication event, and the majority of which are individually or jointly capable of a replication event.