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Week 06b - QMA 2006S2 (General &amp; Deferred Annuities, Depreciation Methods) ver2

# Week 06b - QMA 2006S2 (General & Deferred Annuities, Depreciation Methods) ver2

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Jessica Chung & Derek Hui Week 06 QMA PASS | 2006 S2 | Tues, 3-4pm | QUAD G042 Questions? Email us: [email protected] 1/4 G ENERAL A NNUITIES Annuities with different compounding and payment periods. Step One ° Find the effective rate of interest for the corresponding payment period 1 1 k p A r r k ° ± = + - ² ³ ´ µ Step Two ° Identify the correct annuity formula to use (i.e. PV/FV and ordinary/due/deferred) ° N OTE : The total periods in the investment (n) is calculated by time in years (t) multiplied by the payment periods per year (p) – not the interest compounding periods (k)!! E XAMPLE Find the PV of an investment receiving monthly payments of \$150 for 10 years, with interest compounded semi-annually at a 5% nominal rate. \$150 2 0.05 R k r = = = 10 12 120 t p n t p = = = × = ( ) 2 12 0.05 1 1 2 0.00412 E r = + - = (10 12) 1 (1 ) . . \$150 \$14,175.69 E E r PV r - × - + = × = D EFERRED A NNUITIES Annuities where the first payment has been postponed for a length of time (k) after the first interest or compounding period. Although payments are not initially being made/received, interest is still calculated on the principal sum during this time E XAMPLE Find the value of the initial amount invested into a venture that will yield annual payments of \$120 for 6 years if the first payment is made 2 years from now and the interest is 8.0% per annum compounded annually Method One – Ordinary Annuity ° To transform it into an ordinary annuity, we take the P.V. at time = 1 and then discount this P.V. to time = 0 5 1 1 0 1 1 (1 0.08) \$120 0.08 \$479.1252 (1 0.08) \$443.63 PV PV PV - - + = × = = + = Method Two – Annuity Due ° To transform into an annuity due, we take the P.V. at time = 2 and then discount this P.V. to time = 0 5 2 2 0 2 1 (1 0.08) (1 0.08) \$120 0.08 \$517.4552 (1 0.08) \$443.63 PV PV PV - - + = + × × = = + = If you are more confident, the steps above can be combined into a single formula based on the ordinary annuity method but *** be careful of ‘k’ *** 1 (1 ) 1 (1 ) n k i PV R i i - · - + = × ¸ ¹ º » + (1 ) 1 n i FV R i · + - = × ¸ ¹ º » Note that the future value of a deferred annuity is just the future value formula. It does not change . (This is because if you defer the start, you are ‘earning interest’ on zero dollars in your bank account. As this is just zero, no adjustment to the ordinary future value annuity needs to be made.) T RICKY A NNUITIES When you are given a question where the interest rate changes during the period of the annuity, you must identify whether the annuity in question is a SAVING annuity (i.e. payments made to achieve a future value) or a LOAN REPAYMENT annuity (i.e. payments made to reduce original principal) Consider the following timeline: ° SAVING: If you were making payments into the annuity, S 10 would continue earning interest at the new interest rate ( i=7.0% ). Assuming compounding annually, the future value at t=20 would thus be [S 10 (1.07) 10 ] + [annuity payments from t=10 to t=20 at i =7.0%] ° LOAN REPAYMENT: If you were making payments to pay off a loan, S 10 would be unaffected by the new rate as it represents an amount which you have already paid off in the annuity. Only the annuity payments beyond t=10 would be affected.

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Week 06b - QMA 2006S2 (General & Deferred Annuities, Depreciation Methods) ver2

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