1. **Problem:** Monthly payments of 2000 for 5 years with 12% interest compounded annually. - This is an **annuity due** because payments are monthly but interest is compounded annually. - Formula for future value of annuity due: $$FV = P \times \frac{(1 + r)^n - 1}{r} \times (1 + r)$$ - Here, $P=2000$, $r=0.12$ (annual), $n=5$ years. - Since payments are monthly but interest compounded annually, convert payments to annual total: $2000 \times 12 = 24000$ per year. - Calculate: $$FV = 24000 \times \frac{(1 + 0.12)^5 - 1}{0.12} \times (1 + 0.12)$$ $$= 24000 \times \frac{1.7623 - 1}{0.12} \times 1.12 = 24000 \times 6.3525 \times 1.12 = 170,832$$ 2. **Problem:** Yearly payment of 15000 for 10 years with 8% interest compounded annually. - Ordinary annuity with $P=15000$, $r=0.08$, $n=10$. - Future value formula: $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ $$= 15000 \times \frac{(1.08)^{10} - 1}{0.08} = 15000 \times 14.4866 = 217,299$$ 3. **Problem:** Lump sum 1,500,000, interest 5%, payout 4%, amortization 10 years. - Present value $PV=1,500,000$, interest $i=0.05$, payout rate $p=0.04$, amortization $n=10$. - Annual payout $A$ formula: $$A = PV \times \frac{i - p}{1 - (1 + p)^{-n}}$$ $$= 1,500,000 \times \frac{0.05 - 0.04}{1 - (1.04)^{-10}} = 1,500,000 \times \frac{0.01}{1 - 0.6756} = 1,500,000 \times 0.0319 = 47,850$$ 4. **Problem:** Varying premiums over 5 years, interest 4%, payout 3.5%, amortization 10 years. - Calculate accumulated value at end of 15 years (5 years premiums + 10 years accumulation). - Use compound interest for each premium: $$FV = \sum_{k=0}^4 P_k (1 + r)^{15 - k}$$ Where $P_k$ are premiums for years 1 to 5, $r=0.04$. Calculate each term: Year 1: $500,000 \times (1.04)^{14} = 500,000 \times 1.747 = 873,500$ Year 2: $400,000 \times (1.04)^{13} = 400,000 \times 1.680 = 672,000$ Year 3: $300,000 \times (1.04)^{12} = 300,000 \times 1.615 = 484,500$ Year 4: $200,000 \times (1.04)^{11} = 200,000 \times 1.552 = 310,400$ Year 5: $100,000 \times (1.04)^{10} = 100,000 \times 1.480 = 148,000$ Total accumulated value = $2,488,400$ - Annual income payment during 10-year amortization: $$A = FV \times \frac{r - p}{1 - (1 + p)^{-n}} = 2,488,400 \times \frac{0.04 - 0.035}{1 - (1.035)^{-10}} = 2,488,400 \times 0.005 / 0.3021 = 41,200$$ 5. **Simple Annuity 1:** Present and future value of ordinary annuity P15,000 annually for 10 years at 8% compounded semi-annually. - Effective annual rate: $$r = (1 + 0.08/2)^2 - 1 = 0.0816$$ - Present value: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r} = 15000 \times \frac{1 - (1.0816)^{-10}}{0.0816} = 15000 \times 6.710 = 100,650$$ - Future value: $$FV = PV \times (1 + r)^n = 100,650 \times (1.0816)^{10} = 100,650 \times 2.219 = 223,300$$ 6. **Simple Annuity 2:** Ruben's loan with 8 quarterly payments of 24,491.28 at 12% compounded quarterly. - $r=0.12/4=0.03$, $n=8$, $P=24491.28$ - Present value: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r} = 24491.28 \times \frac{1 - (1.03)^{-8}}{0.03} = 24491.28 \times 6.805 = 166,600$$ 7. **Simple Annuity 3:** Save 50,000 in 5.5 years monthly at 0.25% monthly interest. - $r=0.0025$, $n=5.5 \times 12=66$, solve for $P$: $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ $$P = \frac{FV \times r}{(1 + r)^n - 1} = \frac{50000 \times 0.0025}{(1.0025)^{66} - 1} = \frac{125}{0.177} = 706.21$$ 8. **Simple Annuity 4:** Car buyer pays 160,000 cash + 12,000 monthly for 5 years at 10% compounded monthly. - $r=0.10/12=0.008333$, $n=60$, find present value of installments: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r} = 12000 \times \frac{1 - (1.008333)^{-60}}{0.008333} = 12000 \times 44.955 = 539,460$$ - Total cash price = 160,000 + 539,460 = 699,460 9. **Simple Annuity 5:** TV for 13,498 cash or 2,500 monthly for 6 months at 9% compounded monthly. - $r=0.09/12=0.0075$, $n=6$ - Present value of installments: $$PV = 2500 \times \frac{1 - (1.0075)^{-6}}{0.0075} = 2500 \times 5.82 = 14,550$$ - Prefer cash price 13,498 over installments 14,550. 10. **General Annuity 1:** Present and future value of ordinary annuity 5,000 semi-annually for 10 years at 8% compounded annually. - Semi-annual rate $r=0.08/2=0.04$, $n=20$ payments. - Present value: $$PV = 5000 \times \frac{1 - (1.04)^{-20}}{0.04} = 5000 \times 13.590 = 67,950$$ - Future value: $$FV = PV \times (1.04)^{20} = 67,950 \times 2.191 = 148,800$$ 11. **General Annuity 2:** Ruben's debt with 8 quarterly payments of 24,491.28 at 12% compounded semi-annually. - Semi-annual rate $r=0.12/2=0.06$, quarterly payments, convert rate to quarterly: $$r_q = (1 + 0.06)^{1/2} - 1 = 0.02956$$ - Present value: $$PV = 24491.28 \times \frac{1 - (1 + 0.02956)^{-8}}{0.02956} = 24491.28 \times 6.805 = 166,600$$ 12. **General Annuity 3:** Car buyer pays 160,000 cash + 12,000 monthly for 5 years at 10% compounded annually. - Annual rate $r=0.10$, monthly payments $n=60$. - Monthly rate: $$r_m = (1 + 0.10)^{1/12} - 1 = 0.00797$$ - Present value of installments: $$PV = 12000 \times \frac{1 - (1.00797)^{-60}}{0.00797} = 12000 \times 49.11 = 589,320$$ - Total cash price = 160,000 + 589,320 = 749,320 13. **General Annuity 4:** TV for 13,498 cash or 2,500 monthly for 6 months at 9% compounded monthly. - Same as Simple Annuity 5, prefer cash price 13,498. 14. **Deferred Annuity 1:** Monthly payments 10,000 for 8 years starting 6 months from now. - $r=0.12/12=0.01$, $n=96$ payments. - Present value of annuity at start: $$PV = 10000 \times \frac{1 - (1.01)^{-96}}{0.01} = 10000 \times 56.15 = 561,500$$ - Discount back 6 months: $$PV_0 = \frac{561,500}{(1.01)^6} = 561,500 / 1.0615 = 529,000$$ 15. **Deferred Annuity 2:** Semi-annual payments 15,000 for 10 years starting 5 years from now. - $r=0.12/2=0.06$, $n=20$ payments. - Present value at start: $$PV = 15000 \times \frac{1 - (1.06)^{-20}}{0.06} = 15000 \times 11.4699 = 172,050$$ - Discount back 5 years: $$PV_0 = \frac{172,050}{(1.06)^{10}} = 172,050 / 1.7908 = 96,100$$ 16. **Deferred Annuity 3:** Payments 5,000 every 4 months for 10 years starting 5 years from now. - Payments per year: 3, total $n=30$ payments. - Interest rate per 4 months: $r = (1 + 0.12)^{1/3} - 1 = 0.0387$ - Present value at start: $$PV = 5000 \times \frac{1 - (1.0387)^{-30}}{0.0387} = 5000 \times 19.56 = 97,800$$ - Discount back 5 years (15 periods): $$PV_0 = \frac{97,800}{(1.0387)^{15}} = 97,800 / 1.788 = 54,700$$ 17. **Deferred Annuity 4:** Annual payments 600 for 20 years starting 10 years from now. - $r=0.12$, $n=20$. - Present value at start: $$PV = 600 \times \frac{1 - (1.12)^{-20}}{0.12} = 600 \times 7.469 = 4,481$$ - Discount back 10 years: $$PV_0 = \frac{4,481}{(1.12)^{10}} = 4,481 / 3.106 = 1,443$$ 18. **Deferred Annuity 5:** Payments 3,000 every 3 years for 12 years starting at end of 8 years. - Number of payments $n=4$, interest rate per 3 years: $$r = (1 + 0.12)^3 - 1 = 0.4049$$ - Present value at start: $$PV = 3000 \times \frac{1 - (1.4049)^{-4}}{0.4049} = 3000 \times 2.487 = 7,461$$ - Discount back 8 years (2.67 periods): $$PV_0 = \frac{7,461}{(1.4049)^{2.67}} = 7,461 / 3.02 = 2,470$$ 19. **Deferred Annuity 6:** Loan repaid quarterly for 5 years starting end of 2 years, interest 6% quarterly, loan 10,000. - $r=0.06/4=0.015$, $n=20$ payments. - Present value at start of payments: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r}$$ $$10,000 = P \times 16.351 \Rightarrow P = \frac{10,000}{16.351} = 611.5$$ - Discount back 2 years (8 quarters): $$PV_0 = \frac{10,000}{(1.015)^8} = 10,000 / 1.126 = 8,880$$ 20. **Deferred Annuity 7:** Car purchased with monthly payments 17,000 for 4 years starting end of 4 months, interest 12% monthly. - $r=0.12/12=0.01$, $n=48$ payments. - Present value at start: $$PV = 17000 \times \frac{1 - (1.01)^{-48}}{0.01} = 17000 \times 36.785 = 625,345$$ - Discount back 4 months: $$PV_0 = \frac{625,345}{(1.01)^4} = 625,345 / 1.0406 = 600,900$$ **Final answers summarized:** 1. 170,832 2. 217,299 3. 47,850 4. Accumulated 2,488,400; Annual income 41,200 5. PV=100,650; FV=223,300 6. Borrowed 166,600 7. Deposit 706.21 monthly 8. Cash price 699,460 9. Prefer cash 13,498 10. PV=67,950; FV=148,800 11. Borrowed 166,600 12. Cash price 749,320 13. Prefer cash 13,498 14. PV now 529,000 15. PV now 96,100 16. PV now 54,700 17. PV now 1,443 18. PV now 2,470 19. Quarterly payment 611.5 20. Cash value 600,900