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8527 - Exam FM Raise Your Odds with Adapt INTEREST...

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www.coachingactuaries.com Copyright © 2016 Coaching Actuaries. All Rights Reserved. 1 Raise Your Odds® with Adapt Exam FM Effective Rate of Interest ࠵ౖ࠵? " = ࠵౎࠵? ࠵ౡ࠵? − ࠵౎࠵? ࠵ౡ࠵? − 1 ࠵౎࠵? ࠵ౡ࠵? − 1 Effective Rate of Discount ࠵౑࠵? " = ࠵౎࠵? ࠵ౡ࠵? − ࠵౎࠵? ࠵ౡ࠵? − 1 ࠵౎࠵? ࠵ౡ࠵? Simple Interest ࠵౎࠵? ࠵ౡ࠵? = 1 + ࠵ౖ࠵?࠵ౡ࠵? Force of Interest ࠵໿࠵? " = ࠵౎࠵? + ࠵ౡ࠵? ࠵౎࠵? ࠵ౡ࠵? ࠵౎࠵? ࠵ౡ࠵? = exp ࠵໿࠵? / ࠵౑࠵?࠵౑࠵? " 2 Compound Interest ࠵౎࠵? ࠵ౡ࠵? = 1 + ࠵ౖ࠵? ࠵ಕ࠵? Relationships ࠵ౣ࠵? = 1 1 + ࠵ౖ࠵? = 1 − ࠵౑࠵? ࠵౑࠵? = ࠵ౖ࠵? 1 + ࠵ౖ࠵? = ࠵ౖ࠵?࠵ౣ࠵? 1 + ࠵ౖ࠵? = 1 + ࠵ౖ࠵? 5 ࠵ౚ࠵? 5 = 1 − ࠵౑࠵? 78 = 1 − ࠵౑࠵? 5 ࠵ౚ࠵? 75 = ࠵౒࠵? : Annuity-Immediate ࠵ృ࠵?࠵ృ࠵? = ࠵౎࠵? =| = ࠵ౣ࠵? + ࠵ౣ࠵? ? + ⋯ + ࠵ౣ࠵? = = 1 − ࠵ౣ࠵? = ࠵ౖ࠵? ࠵ఴ࠵?࠵ృ࠵? = ࠵ౠ࠵? =| = 1 + 1 + ࠵ౖ࠵? + ⋯ + 1 + ࠵ౖ࠵? =78 = 1 + ࠵ౖ࠵? = − 1 ࠵ౖ࠵? Annuity-Due ࠵ృ࠵?࠵ృ࠵? = ࠵౎࠵? =| = 1 + ࠵ౣ࠵? + ࠵ౣ࠵? ? + ⋯ + ࠵ౣ࠵? =78 = 1 − ࠵ౣ࠵? = ࠵౑࠵? ࠵ఴ࠵?࠵ృ࠵? = ࠵ౠ࠵? =| = 1 + ࠵ౖ࠵? + 1 + ࠵ౖ࠵? ? + ⋯ + 1 + ࠵ౖ࠵? = = 1 + ࠵ౖ࠵? = − 1 ࠵౑࠵? Immediate vs. Due ࠵౎࠵? =| = ࠵౎࠵? =| 1 + ࠵ౖ࠵? = 1 + ࠵౎࠵? =78| ࠵ౠ࠵? =| = ࠵ౠ࠵? =| 1 + ࠵ౖ࠵? = ࠵ౠ࠵? =C8| − 1 Deferred Annuity m-year deferred n-year annuity-immediate: ࠵ృ࠵?࠵ృ࠵? = ࠵౎࠵? 5| =| = ࠵ౣ࠵? 5 ⋅ ࠵౎࠵? =| = ࠵౎࠵? 5C=| − ࠵౎࠵? 5| Perpetuity Perpetuity-immediate ࠵ృ࠵?࠵ృ࠵? = ࠵౎࠵? E| = ࠵ౣ࠵? + ࠵ౣ࠵? ? + ⋯ = 1 ࠵ౖ࠵? Perpetuity-due ࠵ృ࠵?࠵ృ࠵? = ࠵౎࠵? | = 1 + ࠵ౣ࠵? + ࠵ౣ࠵? ? + ⋯ = 1 ࠵౑࠵? ࠵౎࠵? E| = 1 + ࠵౎࠵? E| j-effective method is used when payments are more or less frequent than the interest period. “j-effective” Method Convert the given interest rate to the equivalent effective interest rate for the period between each payment. Example: To find the present value of ࠵౛࠵? monthly payments given annual effective rate of i , define ࠵౗࠵? as the monthly effective rate where ࠵౗࠵? = 1 + ࠵ౖ࠵? 8 8? − 1 . Then apply ࠵ృ࠵?࠵ృ࠵? = ࠵౎࠵? =| using j . Payments in Arithmetic Progression PV of n-year annuity-immediate with payments of ࠵ృ࠵?, ࠵ృ࠵? + ࠵ౄ࠵?, ࠵ృ࠵? + 2࠵ౄ࠵?, … , ࠵ృ࠵? + ࠵౛࠵? − 1 ࠵ౄ࠵? ࠵ృ࠵?࠵౎࠵? =| + ࠵ౄ࠵? ࠵౎࠵? =| − ࠵౛࠵?࠵ౣ࠵? = ࠵ౖ࠵? Calculator-friendly version: ࠵ృ࠵? + ࠵ౄ࠵? ࠵ౖ࠵? ࠵౎࠵? =| + ࠵ౄ࠵?࠵౛࠵? ࠵ౖ࠵? ࠵ౣ࠵? = N = ࠵౛࠵?, I Y = ࠵ౖ࠵? in % , PMT = ࠵ృ࠵? + ࠵ౄ࠵? ࠵ౖ࠵? , FV = − ࠵ౄ࠵?࠵౛࠵? ࠵ౖ࠵? PV of n-year annuity-immediate with payments of 1, 2, 3, … , ࠵౛࠵? Textbook formula: ࠵఼࠵?࠵౎࠵? =| = V W| 7=X W Y P&Q version: ࠵ృ࠵? = 1, ࠵ౄ࠵? = 1, ࠵ు࠵? = ࠵౛࠵?
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