1. Problem ii: Your friend has a loan of 10,000 at 14% interest for 5 years with annual payments of 3,000. We need to check if the payments cover the loan and interest. 2. Formula: Use the loan amortization formula or check the present value of payments against the loan amount. 3. Calculate the present value (PV) of the payments: $$PV = 3000 \times \frac{1 - (1 + 0.14)^{-5}}{0.14}$$ 4. Calculate: $$PV = 3000 \times \frac{1 - (1.14)^{-5}}{0.14} = 3000 \times \frac{1 - 0.5194}{0.14} = 3000 \times 3.429 = 10287$$ 5. Since PV (10,287) > loan (10,000), payments cover the loan and interest. Your friend is paying enough. 6. Problem iii: Find present value (PV) of 60,000 received in 7 years with 10% daily compounding. 7. Formula: $$PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}$$ where $FV=60000$, $r=0.10$, $n=365$, $t=7$. 8. Calculate: $$PV = \frac{60000}{(1 + \frac{0.10}{365})^{365 \times 7}} = \frac{60000}{(1.00027397)^{2555}} = \frac{60000}{2.007} = 29887$$ 9. Problem iv: Find current stock value and value in 5 years given dividend $D_0=2$, required return $k=0.16$, growth $g=0.08$. 10. Formula for current value: $$P_0 = \frac{D_1}{k - g}$$ where $D_1 = D_0 (1+g) = 2 \times 1.08 = 2.16$. 11. Calculate current value: $$P_0 = \frac{2.16}{0.16 - 0.08} = \frac{2.16}{0.08} = 27$$ 12. Value in 5 years: $$P_5 = P_0 (1+g)^5 = 27 \times (1.08)^5 = 27 \times 1.469 = 39.66$$ 13. Problem v: Given price $P=60$, dividend $D=4$, required return $k=0.12$, find growth rate $g$. 14. Formula: $$P = \frac{D}{k - g} \Rightarrow g = k - \frac{D}{P}$$ 15. Calculate growth rate: $$g = 0.12 - \frac{4}{60} = 0.12 - 0.0667 = 0.0533 = 5.33\%$$ 16. Problem vi: Ms. Markson invests 1,500 this year, salary and savings grow 8% annually for 5 more years, market grows 8%. Find value after 6 years. 17. This is a growing annuity with growth rate $g=0.08$ and interest rate $r=0.08$. 18. Formula for future value of growing annuity: $$FV = P \times \frac{(1+r)^n - (1+g)^n}{r - g}$$ but since $r=g$, use $$FV = P \times n \times (1+r)^{n-1}$$ 19. Calculate: $$FV = 1500 \times 6 \times (1.08)^5 = 1500 \times 6 \times 1.469 = 1500 \times 8.814 = 13221$$ 20. Problem vii: Lucyna wants 20,000 in 20 years, interest 4.8% compounded quarterly, initial gift 4,300, find how much more to invest now. 21. Formula for present value: $$PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}$$ with $FV=20000$, $r=0.048$, $n=4$, $t=20$. 22. Calculate PV: $$PV = \frac{20000}{(1 + \frac{0.048}{4})^{4 \times 20}} = \frac{20000}{(1.012)^{80}} = \frac{20000}{2.594} = 7707$$ 23. Amount to invest now: $$7707 - 4300 = 3407$$ Final answers: ii. Payments cover loan and interest. iii. Present value = 29,887 iv. Current stock value = 27, value in 5 years = 39.66 v. Dividend growth rate = 5.33% vi. Investment value after 6 years = 13,221 vii. Additional investment needed = 3,407