To properly answer this question definitively, we would need the results of a five-year long study. That would be very expensive and of limited value, and I was doubtful that one would exist. Proving the non-existence of studies is practically impossible, but a technique that we have accepted here before at Skeptics.SE is finding an appropriate expert who has performed a literature search and who explained that such studies are missing or lacking.

Fortunately, the 2009 book Contraceptive Technology by James Trussell and Anita L. Nelson, M.D., 19th Edition, covers this topic.

In Chapter 3 (page 28) they explain:

We confine attention to the first-year probabilities of pregnancy solely because probabilities for longer durations are generally not available.

So, as of 2009, there is no definitive answer to your question...

... however, as a bonus, the authors do go on (in the same passage) to warn that the simplistic mathematical models being proposed in the question are inappropriate. [Emphasis theirs, paragraph breaks mine.]

There are three main points to remember about the effectiveness of contraceptive methods over time.

The first and last points support the simple multiplication method, with caveats.

First, the risk of pregnancy during either perfect or typical use of a method should remain constant over time for an individual woman with a specific partner, providing that her underlying fecundity and frequency of intercourse do not change (although it is possible that the risk for a woman could decline during typical use of certain methods because she learns to use her method correctly and consistently).

The second point, however, shows it to be overly-simplistic.

Second, in contrast, the risk of pregnancy during typical use of a method will decline over time for a group of users, primarily because those users prone to fail do so early, leaving a pool of more diligent contraceptive users, those who are relatively infertile, or those who have lower coital frequency. This decline will be far less pronounced among users of those methods with little scope for imperfect use. The risk of pregnancy during perfect use for a group of users should decline as well, but this decline will not be as pronounced as that during typical use, because only the relatively more fecund and those with higher coital frequency are selected out early. For these reasons, the probability of becoming pregnant for a group of users during the first year of use of a contraceptive method will be higher than the probability of becoming pregnant during the second year of use.

 

Third, probabilities of pregnancy cumulate over time. Suppose that 15%, 12% and 8% of women using a method experience contraceptive failure during years 1, 2, and 3, respectively. The probability of not becoming pregnant within 3 years is calculated by multiplying the probabilities of not becoming pregnant for each of the 3 years: 0.85 times 0.88 times 0.92, which equals 0.69. Thus, the percentage of becoming pregnant within 3 years is 31% (=100% - 69%)

 

The lesson here is that differences among probabilities of pregnancy for various methods will increase over time. For example, suppose that each year the typical proportion of women becoming pregnant while taking the pill is 8% and while using the diaphragm is 16%. Within 5 years, 34% of pill users and 58% of diaphragm users will become pregnant.

To properly answer this question definitively, we would need the results of a five-year long study. That would be very expensive and of limited value, and I was doubtful that one would exist. Proving the non-existence of studies is practically impossible, but a technique that we have accepted here before at Skeptics.SE is finding an appropriate expert who has performed a literature search and who explained that such studies are missing or lacking.

Fortunately, the 2009 book Contraceptive Technology by James Trussell and Anita L. Nelson, M.D., 19th Edition, covers this topic.

In Chapter 3 (page 28) they explain:

We confine attention to the first-year probabilities of pregnancy solely because probabilities for longer durations are generally not available.

So, as of 2009, there is no definitive answer to your question...

... however, as a bonus, the authors do go on (in the same passage) to warn that the simplistic mathematical models being proposed in the question are inappropriate. [Emphasis theirs, paragraph breaks mine.]

There are three main points to remember about the effectiveness of contraceptive methods over time.

The first and last points support the simple multiplication method, with caveats.

First, the risk of pregnancy during either perfect or typical use of a method should remain constant over time for an individual woman with a specific partner, providing that her underlying fecundity and frequency of intercourse do not change (although it is possible that the risk for a woman could decline during typical use of certain methods because she learns to use her method correctly and consistently).

The second point, however, shows it to be overly-simplistic.

Second, in contrast, the risk of pregnancy during typical use of a method will decline over time for a group of users, primarily because those users prone to fail do so early, leaving a pool of more diligent contraceptive users, those who are relatively infertile, or those who have lower coital frequency. This decline will be far less pronounced among users of those methods with little scope for imperfect use. The risk of pregnancy during perfect use for a group of users should decline as well, but this decline will not be as pronounced as that during typical use, because only the relatively more fecund and those with higher coital frequency are selected out early. For these reasons, the probability of becoming pregnant for a group of users during the first year of use of a contraceptive method will be higher than the probability of becoming pregnant during the second year of use.

Third, probabilities of pregnancy cumulate over time. Suppose that 15%, 12% and 8% of women using a method experience contraceptive failure during years 1, 2, and 3, respectively. The probability of not becoming pregnant within 3 years is calculated by multiplying the probabilities of not becoming pregnant for each of the 3 years: 0.85 times 0.88 times 0.92, which equals 0.69. Thus, the percentage of becoming pregnant within 3 years is 31% (=100% - 69%)

The lesson here is that differences among probabilities of pregnancy for various methods will increase over time. For example, suppose that each year the typical proportion of women becoming pregnant while taking the pill is 8% and while using the diaphragm is 16%. Within 5 years, 34% of pill users and 58% of diaphragm users will become pregnant.

To properly answer this question definitively, we would need the results of a five-year long study. That would be very expensive and of limited value, and I was doubtful that one would exist. Proving the non-existence of studies is practically impossible, but a technique that we have accepted here before at Skeptics.SE is finding an appropriate expert who has performed a literature search and who explained that such studies are missing or lacking.

Fortunately, the 2009 book Contraceptice Technology by James Trussell and Anita L. Nelson, M.D., 19th Edition, covers this topic.

In Chapter 3 (page 28) they explain:

We confine attention to the first-year probabilities of pregnancy solely because probabilities for longer durations are generally not available.

So, as of 2009, there is no definitive answer to your question...

... however, as a bonus, the authors do go on (in the same passage) to warn that the simplistic mathematical models being proposed in the question are inappropriate. [Emphasis theirs, paragraph breaks mine.]

There are three main points to remember about the effectiveness of contraceptive methods over time.

The first and last points support the simple multiplication method, with caveats.

First, the risk of pregnancy during either perfect or typical use of a method should remain constant over time for an individual woman with a specific partner, providing that her underlying fecundity and frequency of intercourse do not change (although it is possible that the risk for a woman could decline during typical use of certain methods because she learns to use her method correctly and consistently).

The second point, however, shows it to be overly-simplistic.

Second, in contrast, the risk of pregnancy during typical use of a method will decline over time for a group of users, primarily because those users prone to fail do so early, leaving a pool of more diligent contraceptive users, those who are relatively infertile, or those who have lower coital frequency. This decline will be far less pronounced among users of those methods with little scope for imperfect use. The risk of pregnancy during perfect use for a group of users should decline as well, but this decline will not be as pronounced as that during typical use, because only the relatively more fecund and those with higher coital frequency are selected out early. For these reasons, the probability of becoming pregnant for a group of users during the first year of use of a contraceptive method will be higher than the probability of becoming pregnant during the second year of use.

Third, probabilities of pregnancy cumulate over time. Suppose that 15%, 12% and 8% of women using a method experience contraceptive failure during years 1, 2, and 3, respectively. The probability of not becoming pregnant within 3 years is calculated by multiplying the probabilities of not becoming pregnant for each of the 3 years: 0.85 times 0.88 times 0.92, which equals 0.69. Thus, the percentage of becoming pregnant within 3 years is 31% (=100% - 69%)

The lesson here is that differences among probabilities of pregnancy for various methods will increase over time. For example, suppose that each year the typical proportion of women becoming pregnant while taking the pill is 8% and while using the diaphragm is 16%. Within 5 years, 34% of pill users and 58% of diaphragm users will become pregnant.

To properly answer this question definitively, we would need the results of a five-year long study. That would be very expensive and of limited value, and I was doubtful that one would exist. Proving the non-existence of studies is practically impossible, but a technique that we have accepted here before at Skeptics.SE is finding an appropriate expert who has performed a literature search and who explained that such studies are missing or lacking.

Fortunately, the 2009 book Contraceptive Technology by James Trussell and Anita L. Nelson, M.D., 19th Edition, covers this topic.

In Chapter 3 (page 28) they explain:

We confine attention to the first-year probabilities of pregnancy solely because probabilities for longer durations are generally not available.

So, as of 2009, there is no definitive answer to your question...

... however, as a bonus, the authors do go on (in the same passage) to warn that the simplistic mathematical models being proposed in the question are inappropriate. [Emphasis theirs, paragraph breaks mine.]

There are three main points to remember about the effectiveness of contraceptive methods over time.

The first and last points support the simple multiplication method, with caveats.

First, the risk of pregnancy during either perfect or typical use of a method should remain constant over time for an individual woman with a specific partner, providing that her underlying fecundity and frequency of intercourse do not change (although it is possible that the risk for a woman could decline during typical use of certain methods because she learns to use her method correctly and consistently).

The second point, however, shows it to be overly-simplistic.

Second, in contrast, the risk of pregnancy during typical use of a method will decline over time for a group of users, primarily because those users prone to fail do so early, leaving a pool of more diligent contraceptive users, those who are relatively infertile, or those who have lower coital frequency. This decline will be far less pronounced among users of those methods with little scope for imperfect use. The risk of pregnancy during perfect use for a group of users should decline as well, but this decline will not be as pronounced as that during typical use, because only the relatively more fecund and those with higher coital frequency are selected out early. For these reasons, the probability of becoming pregnant for a group of users during the first year of use of a contraceptive method will be higher than the probability of becoming pregnant during the second year of use.

Third, probabilities of pregnancy cumulate over time. Suppose that 15%, 12% and 8% of women using a method experience contraceptive failure during years 1, 2, and 3, respectively. The probability of not becoming pregnant within 3 years is calculated by multiplying the probabilities of not becoming pregnant for each of the 3 years: 0.85 times 0.88 times 0.92, which equals 0.69. Thus, the percentage of becoming pregnant within 3 years is 31% (=100% - 69%)

The lesson here is that differences among probabilities of pregnancy for various methods will increase over time. For example, suppose that each year the typical proportion of women becoming pregnant while taking the pill is 8% and while using the diaphragm is 16%. Within 5 years, 34% of pill users and 58% of diaphragm users will become pregnant.