Problem 56135. Ordinary Annuity : Rate (Future Value)

Created by Mehmet OZC

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<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 229.233px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 114.617px; transform-origin: 407px 114.617px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 150px 8px; transform-origin: 150px 8px; unicode-bidi: normal; "><span style="">Theorem : Future Value of an Ordinary Annuity </span></span></div><div style="block-size: 36.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.25px; text-align: left; transform-origin: 384px 18.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-15px"><img src="data:image/png;base64,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" style="width: 117.5px; height: 36.5px;" width="117.5" height="36.5"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; "><span style="">where;</span></span></div><ul style="block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 40.8667px; transform-origin: 391px 40.8667px; margin-top: 10px; margin-bottom: 20px; "><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; "><span style="block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.5px 8px; transform-origin: 50.5px 8px; unicode-bidi: normal; "><span style="">F : Future value</span></span></li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; "><span style="block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 65px 8px; transform-origin: 65px 8px; unicode-bidi: normal; "><span style="">A : Periodic payment</span></span></li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; "><span style="block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 58px 8px; transform-origin: 58px 8px; unicode-bidi: normal; "><span style="">r : Rate per period</span></span></li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; "><span style="block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 105.5px 8px; transform-origin: 105.5px 8px; unicode-bidi: normal; "><span style="">n : Number of periods (payments)</span></span></li></ul><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 242px 8px; transform-origin: 242px 8px; unicode-bidi: normal; "><span style="">Calculate r, given, F, A and n. Assume interest rate is a positive real number. </span></span></div></div></div>

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William on 1 Oct 2022

The "correct" values of r used in these tests all appear to have errors greater than the tolerance of 1e-4, as you can verify by plugging them in to the formula for Future Value.

Mehmet OZC on 1 Oct 2022

Thanks for the catch. I have updated the test suite.

William on 1 Oct 2022

Thanks Mehmet, but there are still big errors. The correct answers are:
Test 1: r = 0.08837372469
Test 2: r = 0.06774487509
Test 3: r = 0.01422142597

Mehmet OZC on 1 Oct 2022

I fixed the errors. I also updated the assert statements.

Rafael S.T. Vieira on 3 Nov 2022

Good problem.

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Problem 56135. Ordinary Annuity : Rate (Future Value)

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Problem 56135. Ordinary Annuity : Rate (Future Value)

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