1. **Problem Statement:** Advise a start-up entrepreneur on 1) choosing between two loan options using interest calculations, 2) recommending stock or bond investments with expected returns, and 3) preparing a savings or annuity plan for long-term security. 2. **Loan Comparison:** - Assume Loan A: Simple Interest, Principal $P=1,000,000$, Rate $r=8\%$ yearly, Time $t=3$ years. - Simple Interest formula: $$I = P \times r \times t$$ - Calculate interest for Loan A: $$I = 1,000,000 \times 0.08 \times 3 = 240,000$$ - Total repayment Loan A: $$1,000,000 + 240,000 = 1,240,000$$ - Assume Loan B: Compound Interest, Principal $P=1,000,000$, Rate $r=7\%$ yearly compounded annually, Time $t=3$ years. - Compound Interest formula: $$A = P (1 + r)^t$$ - Calculate amount for Loan B: $$A = 1,000,000 (1 + 0.07)^3 = 1,000,000 \times 1.225043 = 1,225,043$$ - _Choice:_ Loan B costs less at $1,225,043$ vs $1,240,000$. Recommend Loan B. 3. **Investment Recommendation:** - Suppose investing $500,000$ in stocks with expected annual return $10\%$. - Future value after 5 years: $$FV = 500,000 (1 + 0.10)^5 = 500,000 \times 1.61051 = 805,255$$ - For bonds with yield $6\%$ compounded annually: $$FV = 500,000 (1 + 0.06)^5 = 500,000 \times 1.3382 = 669,100$$ - _Recommendation:_ Stocks offer higher expected return but higher risk; bonds are safer with lower returns. Choose based on risk preference. 4. **Savings or Annuity Plan:** - Suppose a monthly annuity deposit $PMT = 10,000$, annual interest rate $r=5\%$ compounded monthly, duration $n=10$ years. - Number of payments: $$N = 10 \times 12 = 120$$ - Monthly interest rate: $$i = 0.05/12 \approx 0.004167$$ - Future value formula for ordinary annuity: $$FV = PMT \times \frac{(1 + i)^N - 1}{i}$$ - Calculate: $$FV = 10,000 \times \frac{(1 + 0.004167)^{120} - 1}{0.004167} = 10,000 \times 155.29 = 1,552,900$$ - This plan helps secure long-term business stability. **Final Summary:** - Choose Loan B (compound interest) for lower repayment. - Invest in stocks for higher returns or bonds for lower risk. - Establish a monthly annuity plan to build savings over 10 years. All decisions are justified with clear financial computations.