Saving, investment and financial products

Your account pays $1\ \%$ per year and inflation is $3\ \%$.

1. a) Calculate the approximate real return on your savings.
2. b) Explain why your balance rises in dollars and yet you lose purchasing power. What advice would you give to a saver in this scenario?

You deposit $\$1,000$ 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 $\$1,000$ 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.

Distinguish between saving and investment, and classify the following decisions: putting money into a savings account, buying a new oven for a bakery, buying shares in a company, and keeping bills in a piggy bank. Which of them creates productive capacity?

You deposit $\$2,000$ at $3\ \%$ 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) Calculate what balance you would have with simple interest and explain why the difference grows every year.

A financial advisor proposes two options to Emma 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 Emma 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.

Emma 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 Emma'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 Emma 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.

What is interest and why does someone who lends money charge it? Calculate the interest on lending $\$500$ for one year at $3\ \%$ simple interest and explain what that payment compensates the lender for.

Explain the difference between a checking account and a certificate of deposit (CD): how do they differ in availability of the money (liquidity) and in the interest they pay? What type of saving would you use each one for?

Mateo is $17$ years old, has saved $\$3,000$ from summer jobs, and estimates that in $6$ years he will need to gather $\$4,500$ to pay for a master's degree tuition. To reach that amount, he plans to contribute an additional $\$25$ per month during that period. He is presented with three alternatives:

1. Keep the $\$3,000$ and the monthly contributions in a checking account at a bank at $0\ \%$ interest.

2. Invest everything in an international equity index fund (average historical return of $7\ \%$ per year, but with drops of up to $-25\ \%$ in adverse years).

3. Design a mixed strategy: secure a base in a certificate of deposit (CD) or interest-bearing account at $2.8\ \%$ per year and channel the remainder (including monthly contributions) into a moderate mixed fund (expected return of $4.5\ \%$ with moderate volatility).

1. a) Determine whether Option 1 allows him to reach the nominal goal of $4,500\ \text{\$.}$ If the average annual inflation rate is $2.5\ \%$, calculate the loss of real purchasing power that this idle money would experience after $6$ years.
2. b) Analyze the specific risks of Option 2, taking into account that the start date of the master's degree is fixed and does not allow postponing the disinvestment if the stock market goes through a bear market in year $6$.
3. c) Design and numerically justify a reasonable allocation of Mateo's initial capital and recurring contributions, showing how it balances the need to beat inflation with the requirement of safety at the deadline.

A student plans to take an international specialization course in $3$ years, with a current tuition cost of $\$3,600$. She earns a stable income of $\$150$ per month by giving private tutoring lessons.

1. a) If the expected average annual inflation is $3\ \%$, calculate the nominal cost that the course will have in three years. What exact monthly amount should she save if she kept her money in cash without any return?
2. b) To achieve the goal, she decides to allocate $\$100$ per month to a savings account earning $2\ \%$ annually and the remaining $\$50$ to current consumption. However, right at the end of month $18$, an unexpected event forces her to withdraw $\$500$ from her saved funds.

• Assess the impact of this unforeseen shock on achieving her goal.

• Design a financial contingency plan for the remaining $18$ months (adjusting the monthly installment, reducing expenses, or seeking higher returns), evaluating the economic feasibility and the risks assumed with each alternative.

A laptop costs $\$500$ upfront, but the store offers to let you pay for it in $12$ monthly installments of $\$45$.

1. a) Calculate how much you pay in total with the financing and how much the financing costs you.
2. b) What questions should you ask before agreeing to pay in installments?

A couple takes out a mortgage of $\$120,000$ 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.

One bank offers a loan at a nominal rate of $5\ \%$ with no fees; another, at a nominal $4.5\ \%$ with a $1\ \%$ origination fee.

1. a) Explain what the APR is and why it is a better reference than the nominal rate for comparing both offers.
2. b) What other elements, besides the rate, should you compare before signing (term, monthly payment, bundled products)?

Define in your own words the three elements of an insurance policy: the premium, the risk covered and the claim. Explain how it is possible for an insurer to pay for an $\$800$ cell phone in exchange for premiums of $\$8$ a month: who really pays for that phone?

Three options for $\$5,000$: a guaranteed certificate of deposit (CD) at $1\ \%$, a mixed fund with an expected return of $4\ \%$ but possible losses some years, and stocks with an expectation of $7\ \%$ and high volatility.

1. a) Order the options from lowest to highest risk and explain the relationship between risk and expected return.
2. b) If the money is to pay for college in two years, which would you choose? Justify your answer using the time horizon.

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 car, design and justify a suitable asset allocation strategy among these three assets.

Noa works as an engineer at a large publicly traded tech company and has accumulated a financial net worth of $\$50,000$, invested entirely in her own company's shares. She justifies her stance by stating: “I know the company from the inside, I fully trust my colleagues, and that way I win twice over when the company thrives.”

1. a) Identify and classify the financial and asset risks Noa assumes by concentrating her savings and her source of earned income in the same company. Analyze the impact on her solvency if the company undergoes financial restructuring due to an industry crisis.
2. b) Refute Noa's statement by distinguishing between specific (diversifiable) risk and systematic (market) risk. Why should a well-informed investor never expect an extra return premium for taking on diversifiable risk?
3. c) Design a restructuring proposal for her $\$50,000$ portfolio by applying the principle of diversification across asset classes, geographic areas, and economic sectors. Justify the percentages allocated to each category for a moderate risk profile, and explain how it reduces the correlation between her paycheck and her wealth.

A digital entity promotes an investment opportunity on social media: "Guaranteed $2.5\ %$ monthly return through high-frequency arbitrage algorithms with no market risk. Limited spots for the first $50$ investors with a minimum contribution of $\$3,000$".

1. a) Demonstrate numerically how much an initial investment of $\$3,000$ would amount to after $5$ years if that monthly rate were compounded. Based on the result, refute the credibility of the offer using the concept of financing cost: if the company had access to that risk-free return, why wouldn't it take out a bank loan at $4\ %$ or $5\ %$ per year instead of raising retail money at $34\ %$ compounded annually?
2. b) Model the internal financial structure of a pyramid scheme or Ponzi scheme. Explain how return payouts to early depositors are funded, why the system inevitably collapses when the pace of new deposits declines or massive redemption requests occur, and what psychological role the "limited spots" clause plays.
3. c) Describe the formal verification protocol that an investor in the United States should follow to check the legitimacy of a financial intermediary before transferring capital, indicating which official regulatory bodies you must consult (such as the SEC, FINRA, and your state securities regulator) and what warnings about unauthorized entities (fraudulent investment firms or boiler rooms) you must verify.