3. (i) Address the following points related to risk-free assets using clear expla- nation and precise...

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3. (i) Address the following points related to risk-free assets using clear expla- nation and precise mathematical notation. (a) Explain the three methods of simple interest, periodic compounding and continuous compounding to compute the interest of a risk-free asset. State what are the growth and discount factors corresponding to each of the three methods. (b) Prove why, given the same interest rater, the continuous com- pounding method always produces a higher future value. (ii) After successfully using your laptop during the remote lectures of this academic year, you feel that you deserve an upgrade to a high-spec laptop that costs £1,600. However, during the next year you can only set aside a maximum of £200 each month to fund your new laptop. (a) Your first option is to pay it via 12 equal monthly payments. Assum- ing that the dealer sets a (periodically compounded) annual interest rate of r₁ = 3%, what is the amount of the monthly payments for your new laptop? (b) Your second option is to pay your new laptop in full in one year's time at the price of £1,685. What is the equivalent annual interest rate re of this purchase? (c) A building society is offering a savings account with a (periodically compounded) annual interest rate of r2 = 5% and a maximum monthly deposit of £250; such an investment can be cashed after one year from now. Given also this saving account option, what is the best strategy between this and (ii)(a, b) to buy your new laptop? 3. (i) Address the following points related to risk-free assets using clear expla- nation and precise mathematical notation. (a) Explain the three methods of simple interest, periodic compounding and continuous compounding to compute the interest of a risk-free asset. State what are the growth and discount factors corresponding to each of the three methods. (b) Prove why, given the same interest rater, the continuous com- pounding method always produces a higher future value. (ii) After successfully using your laptop during the remote lectures of this academic year, you feel that you deserve an upgrade to a high-spec laptop that costs £1,600. However, during the next year you can only set aside a maximum of £200 each month to fund your new laptop. (a) Your first option is to pay it via 12 equal monthly payments. Assum- ing that the dealer sets a (periodically compounded) annual interest rate of r₁ = 3%, what is the amount of the monthly payments for your new laptop? (b) Your second option is to pay your new laptop in full in one year's time at the price of £1,685. What is the equivalent annual interest rate re of this purchase? (c) A building society is offering a savings account with a (periodically compounded) annual interest rate of r2 = 5% and a maximum monthly deposit of £250; such an investment can be cashed after one year from now. Given also this saving account option, what is the best strategy between this and (ii)(a, b) to buy your new laptop?