# 问题413: Mr Lam plans to invest $50000 for 10 years. He may buy a gov Mr Lam plans to invest$50000 for 10 years. He may buy a government bond which offers simple interest at 6%p.a. He may also deposit the money in a bank which offers interest at 5%p.a. compounded half-yearly. Which investment can provide a better return for Mr Lam? Explain your answer.

1. To determine which investment option provides a better return for Mr. Lam, we will calculate the total amount he would receive from each investment after 10 years. We'll compare the return from a government bond offering simple interest at 6% per annum and a bank deposit offering compound interest at 5% per annum, compounded half-yearly.

Simple Interest Calculation

Simple interest is calculated using the formula:
[ \text{Simple Interest} = P \times r \times t ]
where ( P ) is the principal amount, ( r ) is the rate of interest per year, and ( t ) is the time the money is invested in years.

For Mr. Lam's investment in the government bond:
[ P = $50,000, \quad r = 6\% \text{ (or 0.06 per year)}, \quad t = 10 \text{ years} ] The total amount with simple interest would be: [ A_{\text{simple}} = P + (P \times r \times t) =$50,000 + ($50,000 \times 0.06 \times 10) =$50,000 + $30,000 =$80,000 ]

Compound Interest Calculation

Compound interest is calculated using the formula:
[ A_{\text{compound}} = P \times \left(1 + \frac{r}{n}\right)^{nt} ]
where ( n ) is the number of compounding periods per year.

For Mr. Lam’s investment in the bank:
[ P = $50,000, \quad r = 5\% \text{ (or 0.05 per year)}, \quad n = 2 \text{ (compounded half-yearly)}, \quad t = 10 \text{ years} ] The total amount with compound interest would be: [ A_{\text{compound}} =$50,000 \times \left(1 + \frac{0.05}{2}\right)^{2 \times 10} = $50,000 \times \left(1 + 0.025\right)^{20} ] We can calculate this final amount using the exact value of ( (1.025)^{20} ). Calculating ( (1.025)^{20} ) [ A_{\text{compound}} =$50,000 \times 1.638616448 ] (using ( (1.025)^{20} \approx 1.638616448 ))

[ A_{\text{compound}} \approx $50,000 \times 1.638616448 =$81,930.82 ]

Conclusion

Comparing the two amounts:

Simple interest total: $80,000Compound interest total:$81,930.82

Mr. Lam would receive a better return by depositing his money in the bank where the interest is compounded half-yearly. The compound interest advantage becomes more pronounced over longer investment periods, even at a slightly lower interest rate, because the interest is being added to the principal more frequently (twice a year in this case), leading to interest on the interest.