Saving, investing, and financial products

Name the three functions of money and classify these situations according to the function they illustrate: paying for a coffee with a coin, comparing the price of two cell phones, and keeping bills in a piggy bank for a trip.

You deposit $\$20\,000$ in an account paying simple interest of $4\ \%$ per year.

1. a) What interest do you receive after one year? And after two years?
2. b) What total amount do you have at the end of the second year?

The figure compares how $\$1,000$ at $5\ \%$ per year evolves over ten years with simple interest and with compound interest. One of the lines (blue) and the other (orange) each correspond to one type of interest.

1. a) Which line (the blue one or the orange one) corresponds to each type of interest? Justify your answer.
2. b) Read from the graph, approximately, the accumulated difference at the end of the period and explain what causes it.

You deposit $\$20\,000$ at $5\ \%$ per year with compound interest for three years.

1. a) Calculate the balance at the end of each year, rounding to the nearest cent.
2. b) Explain why the interest in the third year is greater than in the first.

A financial advisor proposes two options to Lucía to generate returns on $\$10,000$ that she will not need for the next $15$ years:

• Option A: A guaranteed product that pays the returns annually into her checking account at $4\ \%$ per year, money that Lucía does not reinvest but leaves idle at $0\ \%$.

• Option B: An accumulation fund that automatically reinvests the annual returns at that same rate of $4\ \%$ compounded annually.

Lucía states: "After $15$ years, I will have earned exactly the same profit from both products, $\$6,000$, because the interest rate of $4\ \%$ and the term of $15$ years are identical."

1. a) Numerically demonstrate Lucía's mistake by calculating the final accumulated capital and the net profit obtained in each option after $15$ years. Explain the reason for the mathematical divergence between both results.
2. b) Assume that average inflation over that period is $2.5\ \%$ per year. Evaluate whether Lucía has preserved the real purchasing power of her initial $\$10,000$ with each option. Based on your results, justify the practical relevance of reinvesting returns for a long-term saver against monetary erosion.

A couple takes out a mortgage of $\$2\,400\,000$ for $25$ years to buy an apartment.

1. a) Explain what the principal, the interest rate, and the term of this loan are.
2. b) If the rate rises from $3\ \%$ to $4\ \%$, explain without performing calculations what happens to the monthly installment and to the total interest paid.

Cell phone insurance costs $\$160$ a month and home insurance costs $\$300$ a month.

1. a) Explain what an insurance premium is and what risk each of these two transfers.
2. b) Reason why the insurer can charge such different premiums and what factors it will take into account when setting them.

A financial platform offers three investment alternatives to an individual:

• Asset Alpha: A $10$-year German sovereign bond with an annual coupon of $2.3\ \%$.

• Asset Beta: Shares of an emerging biotechnology company that will multiply its revenue if a medical patent is approved, but which is currently running losses.

• Asset Gamma: A corporate bond issued by a hotel company under bankruptcy proceedings (insolvency) offering an annual interest rate of $11.5\ \%$.

An inexperienced saver argues: "The smartest thing to do is to invest everything in Asset Gamma, since being fixed income, it contractually ensures that I receive more than four times what Germany offers, avoiding the uncertainty of the biotech company."

1. a) Refute the saver's claim by distinguishing between market risk and credit (or default) risk. Justify why the market sets such disparate interest rates between Asset Alpha and Asset Gamma.
2. b) Analyze the return-risk-liquidity trilemma across the three assets and explain, using the mechanism of capital supply and demand, why it is unfeasible for a product to simultaneously offer high returns, zero risk, and immediate liquidity.
3. c) If this saver needed full access to their money in exactly $18$ months to buy a vehicle, design and justify a suitable asset allocation strategy among these three assets.

Mateo, aged $17$, has saved $\$60\,000$ from summer jobs and wants to use this money, along with any returns, to pay for a master's degree in six years.

1. a) Design a simple financial plan (how much he saves, in what type of product, and why), justifying each decision.
2. b) State what risks he assumes and how he could reduce them.