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Calculating the Annual Pecentage Yield (APY) And Continuous Compounding
Negative Growth - Constant Rate
Positive Growth - Constant Rate
Positive and Negative Growth Compared

External Links:

Math Centre - Continuous Compounding

Brothers Technology - Academic Press: Daily InScight - Math Buffs Find An Easier "e"

University Of Toronto - Answers and Explanations - The Number e as a Limit

Annual Percentage Rate - Wikipedia

Annual Percentage Yield - Wikipedia

Ivar's Peterson's Mathtrek - Hunting e

Calculating the Annual Percentage Yield (APY) And Continuous Compounding

Note: This page is best printed in landscape and in colour.

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 peri
ods
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 peri
ods
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.

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Copyright 2005 David A. Coutts
Last modified: 05 January, 2011