1. **Problem:** Monthly payments of 2000 for 5 years with interest rate of 12% compounded annually. - This is a **Simple Annuity** with monthly payments but interest compounded annually. - Formula for Future Value (FV) of an ordinary annuity: $$FV = P \times \frac{(1 + i)^n - 1}{i}$$ - Here, $P=2000$, $n=5$ years $\times 12 = 60$ payments, but interest rate per period $i$ must be adjusted. - Since interest is compounded annually, effective annual rate $i_a=0.12$. - Monthly interest rate $i_m$ is not directly given; since compounding is annual, we treat interest annually for calculation. - We calculate the future value at the end of 5 years by summing payments grown annually. - Each payment grows for a different number of years depending on when it was made. - Total FV = $2000 \times \sum_{k=0}^{59} (1+0.12)^{\frac{5 - \frac{k}{12}}{1}}$ - Calculating this sum gives the accumulated value. 2. **Problem:** Yearly payment of 15000 for 10 years with interest rate of 8% compounded annually. - This is a **Simple Annuity**, ordinary annuity. - Using formula for FV: $$FV = 15000 \times \frac{(1 + 0.08)^{10} - 1}{0.08}$$ - Calculate: $$FV = 15000 \times \frac{(1.08)^{10} - 1}{0.08} = 15000 \times \frac{2.1589 - 1}{0.08} = 15000 \times 14.486 = 217290$$ 3. **Problem:** Lump sum 1,500,000, interest 5%, payout 4%, annuitization 10 years. - This is a **General Annuity** (annuitization phase). - Present Value (PV) of annuity payout: $$PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}$$ - Here, $PMT$ is annual payout, $r=0.05$, $n=10$. - We find $PMT$ such that $PV = 1,500,000$: $$1,500,000 = PMT \times \frac{1 - (1.05)^{-10}}{0.05}$$ - Calculate denominator: $$\frac{1 - (1.05)^{-10}}{0.05} = \frac{1 - 0.6139}{0.05} = 7.7217$$ - So, $$PMT = \frac{1,500,000}{7.7217} = 194,256.5$$ - Annual payout is approximately 194,257. 4. **Problem:** Varying premiums over 5 years, interest 4%, payout 3.5%, accumulation 15 years, annuitization 10 years. - This is a **General Annuity** with varying payments. - Accumulated value at end of 15 years is sum of each premium grown for remaining years: - Calculate accumulation for each premium: Year 1: 20,000 \times (1.04)^{14} = 20,000 \times 1.747 = 34,940 Year 2: 25,000 \times (1.04)^{13} = 25,000 \times 1.680 = 42,000 Year 3: 30,000 \times (1.04)^{12} = 30,000 \times 1.615 = 48,450 Year 4: 45,000 \times (1.04)^{11} = 45,000 \times 1.552 = 69,840 Year 5: 50,000 \times (1.04)^{10} = 50,000 \times 1.480 = 74,000 - Total accumulated value: $$34,940 + 42,000 + 48,450 + 69,840 + 74,000 = 269,230$$ - Annual income payment during 10-year annuitization at 3.5%: $$PMT = \frac{269,230 \times 0.035}{1 - (1.035)^{-10}}$$ - Calculate denominator: $$1 - (1.035)^{-10} = 1 - 0.7089 = 0.2911$$ - So, $$PMT = \frac{9,423}{0.2911} = 32,370$$ - Annual income payment is approximately 32,370. 5. **Simple Annuity:** Present value and amount of ordinary annuity P5,000 annually for 10 years, 8% compounded semi-annually. - Interest per period $i = \frac{0.08}{2} = 0.04$, number of periods $n=10 \times 2=20$. - Present Value (PV): $$PV = 5000 \times \frac{1 - (1 + 0.04)^{-20}}{0.04}$$ - Calculate: $$PV = 5000 \times \frac{1 - 0.4564}{0.04} = 5000 \times 13.59 = 67,950$$ - Future Value (FV): $$FV = 5000 \times \frac{(1 + 0.04)^{20} - 1}{0.04} = 5000 \times 29.778 = 148,890$$ 6. **Simple Annuity:** Ruben's loan with 8 quarterly payments of 24,491.28 at 12% compounded quarterly. - Interest per period $i=0.12/4=0.03$, $n=8$. - Present Value (loan amount): $$PV = 24491.28 \times \frac{1 - (1 + 0.03)^{-8}}{0.03}$$ - Calculate: $$PV = 24491.28 \times 6.805 = 166,600$$ 7. **Simple Annuity:** Save 50,000 in 5.5 years monthly at 0.25% monthly interest. - $i=0.0025$, $n=5.5 \times 12=66$. - Future Value formula: $$FV = P \times \frac{(1 + i)^n - 1}{i}$$ - Solve for $P$: $$P = \frac{FV \times i}{(1 + i)^n - 1} = \frac{50000 \times 0.0025}{(1.0025)^{66} - 1}$$ - Calculate denominator: $$(1.0025)^{66} - 1 = 1.177 - 1 = 0.177$$ - So, $$P = \frac{125}{0.177} = 706.21$$ - Monthly deposit is approximately 706.21. 8. **Simple Annuity:** Car buyer pays 160,000 cash and 12,000 monthly for 5 years at 10% compounded monthly. - $i=0.10/12=0.008333$, $n=5 \times 12=60$. - Present Value of payments: $$PV = 12000 \times \frac{1 - (1 + 0.008333)^{-60}}{0.008333}$$ - Calculate: $$PV = 12000 \times 49.72 = 596,640$$ - Total cash price = 160,000 + 596,640 = 756,640. 9. **Simple Annuity:** TV for 13,490 cash or 2,500 monthly for 6 months at 9% compounded monthly. - $i=0.09/12=0.0075$, $n=6$. - Present Value of installments: $$PV = 2500 \times \frac{1 - (1 + 0.0075)^{-6}}{0.0075} = 2500 \times 5.85 = 14,625$$ - Since 14,625 > 13,490, cash is preferable. 10. **General Annuity:** P5,000 semi-annually for 10 years, 8% compounded annually. - Interest per semi-annual period $i=0.08/2=0.04$, $n=20$. - Present Value: $$PV = 5000 \times \frac{1 - (1 + 0.04)^{-20}}{0.04} = 67,950$$ - Future Value: $$FV = 5000 \times \frac{(1 + 0.04)^{20} - 1}{0.04} = 148,890$$ 11. **General Annuity:** Ruben's debt with 8 quarterly payments of 24,491.28 at 12% compounded semi-annually. - Semi-annual rate $i=0.12/2=0.06$, quarterly payments, so adjust periods. - Number of semi-annual periods $n=4$ (8 quarters = 4 semi-annual periods). - Present Value: $$PV = 24491.28 \times \frac{1 - (1 + 0.06)^{-4}}{0.06} = 24491.28 \times 3.465 = 84,860$$ 12. **General Annuity:** Car buyer pays 160,000 cash and 12,000 monthly for 5 years at 10% compounded annually. - Annual rate 10%, monthly payments. - Monthly rate $i = (1 + 0.10)^{1/12} - 1 = 0.00797$ approx. - $n=60$. - Present Value: $$PV = 12000 \times \frac{1 - (1 + 0.00797)^{-60}}{0.00797} = 12000 \times 49.12 = 589,440$$ - Total cash price = 160,000 + 589,440 = 749,440. 13. **General Annuity:** TV 13,490 cash or 2,500 monthly for 6 months at 9% compounded monthly. - Same as #9, PV = 14,625 > 13,490, cash preferred. 14. **Deferred Annuity:** Monthly payments 10,000 for 8 years starting 6 months from now. - $i=0.12/12=0.01$, $n=96$ payments. - Present Value at start of payments: $$PV = 10000 \times \frac{1 - (1 + 0.01)^{-96}}{0.01} = 10000 \times 62.3 = 623,000$$ - Discount PV back 6 months: $$PV_0 = \frac{623,000}{(1 + 0.01)^6} = 623,000 / 1.0615 = 586,800$$ 15. **Deferred Annuity:** Semi-annual payments 15,000 for 10 years starting 5 years from now. - $i=0.12/2=0.06$, $n=20$. - PV at start of payments: $$PV = 15000 \times \frac{1 - (1 + 0.06)^{-20}}{0.06} = 15000 \times 11.47 = 172,050$$ - Discount back 5 years (10 semi-annual periods): $$PV_0 = \frac{172,050}{(1.06)^{10}} = 172,050 / 1.791 = 96,060$$ 16. **Deferred Annuity:** Payments 5,000 every 4 months for 10 years starting 5 years from now. - Payments per year = 3, total $n=30$. - Interest per 4 months $i = (1 + 0.12)^{1/3} - 1 = 0.0387$ approx. - PV at start of payments: $$PV = 5000 \times \frac{1 - (1 + 0.0387)^{-30}}{0.0387} = 5000 \times 19.56 = 97,800$$ - Discount back 5 years (15 periods): $$PV_0 = \frac{97,800}{(1.0387)^{15}} = 97,800 / 1.797 = 54,400$$ 17. **Deferred Annuity:** Annual payments 600 for 29 years starting 10 years from now. - $i=0.12$, $n=29$. - PV at start of payments: $$PV = 600 \times \frac{1 - (1 + 0.12)^{-29}}{0.12} = 600 \times 7.84 = 4,704$$ - Discount back 10 years: $$PV_0 = \frac{4,704}{(1.12)^{10}} = 4,704 / 3.106 = 1,515$$ 18. **Deferred Annuity:** Payments 3,000 every 3 years for 12 years starting at end of 8 years. - Number of payments $n=4$ (12/3). - Interest per 3 years $i = (1 + 0.12)^3 - 1 = 0.4049$. - PV at start of payments: $$PV = 3000 \times \frac{1 - (1 + 0.4049)^{-4}}{0.4049} = 3000 \times 2.53 = 7,590$$ - Discount back 8 years (2.67 periods): $$PV_0 = \frac{7,590}{(1.4049)^{2.67}} = 7,590 / 3.02 = 2,513$$ 19. **Deferred Annuity:** Loan repaid quarterly for 5 years starting end of 2 years, interest 6% quarterly, payment 10,000. - $i=0.06$, $n=20$. - PV at start of payments: $$PV = 10000 \times \frac{1 - (1 + 0.06)^{-20}}{0.06} = 10000 \times 11.65 = 116,500$$ - Discount back 2 years (8 quarters): $$PV_0 = \frac{116,500}{(1.06)^8} = 116,500 / 1.593 = 73,100$$ 20. **Deferred Annuity:** Car purchased with monthly payments 17,000 for 4 years starting end of 4 months, interest 12% monthly. - $i=0.12/12=0.01$, $n=48$. - PV at start of payments: $$PV = 17000 \times \frac{1 - (1 + 0.01)^{-48}}{0.01} = 17000 \times 38.6 = 656,200$$ - Discount back 4 months: $$PV_0 = \frac{656,200}{(1.01)^4} = 656,200 / 1.041 = 630,000$$ **Summary:** Problems 1,2,5-9 are Simple Annuities; 3,4,10-13 are General Annuities; 14-20 are Deferred Annuities. 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