Question

The market interest rate is 5 percent and is expected to stay at that level. Consumers can borrow and lend all they want at this rate. Explain your choice in each of the following situations:

1. Would you prefer a \(500 gift today or a \)540 gift next year?
2. Would you prefer a \(100 gift now or a \)500 loan without interest for four years?
3. Would you prefer a \(350 rebate on an \)8000 car or one year of financing for the full price of the car at 0-percent interest?
4. You have just won a million-dollar lottery and will receive \(50,000 a year for the next 20 years. How much is this worth to you today?
5. You win the “honest million” jackpot. You can have \)1 million today or \(60,000 per year for eternity (a right that can be passed on to your heirs). Which doyou prefer?
6. In the past, adult children had to pay taxes on gifts of over \)10,000 from their parents, but parents could make interest-free loans to their children. Why did some people call this policy unfair? To whom were the rules unfair?

Short answer

1. A $540 gift next year will be preferred.
2. A $100 gift will be preferred.
3. Financing the car will be preferred.
4. The present worth will be $654,266.04.
5. $60,000 per year for eternity will be preferred.
6. To make the gift amount tax-free requires a huge amount for the parents to pay their children; thus, parents with moderate income will not prefer the policy.

Step-by-step solution

Explanation for part (a)

The present value of $500 on the same day will remain $500.

The present value of $540 will be:

$$\begin{gathered} \text{FV = \$ 100} \\ \text{r =}\frac{\text{5}}{\text{100}} \\ \text{= 0.05} \\ \text{n = 1} \\ \text{PV = FV}{\left(\text{1 + r}\right)}^{\text{- n}} \\ =500{\left(1+0.05\right)}^{-1} \\ =50\times 0.9524 \\ =\$476.20 \end{gathered}$$

The present value of $500 is greater than the present value of $540. Thus, a gift of $500 will be preferred as the higher present value is chosen.

Explanation for part (b)

The future value of $100 and $500 for four years are calculated below:

$$\begin{gathered} \text{P = \$ 100,\$ 500} \\ \text{r = 0.05} \\ \text{n = 4} \\ \text{FV = P}{\left(\text{1 + r}\right)}^{\text{n}} \\ {\text{FV}}_{\text{P = 100}}\text{= 100}{\left(\text{1 + 0.05}\right)}^{\text{4}} \\ \text{=100×1.2155} \\ =\text{\$ 121.55} \\ {\text{FV}}_{\text{P =500}}\text{=500}{\left(\text{1 + 0.05}\right)}^{\text{4}} \\ \text{=500×1.2155} \\ =\text{\$607.75} \end{gathered}$$

If a loan of $500 is taken for four years, after payment, there will be a surplus of $107.75 (=$607.75 – 5). Thus, the future value of $100 is greater than the $500 as a loan. Hence, $100 will be preferred.

Explanation for part (c)

5% of $8000 will be $400 (=4000*0.05). The rebate amount is $350; thus, the interest is greater than the rebate amount. Hence, financing the car will be preferred.

Explanation for part (d)

The lottery first payment is made immediately.

The present value of a lottery that pays for the next 20 years is calculated below:

$$\begin{gathered} FV=\$50,000 \\ PV=50,000+50,000{\left(1+0.05\right)}^{-1}+50,000{\left(1+0.05\right)}^{-2}+........+50,000{\left(1+0.05\right)}^{-19} \\ =50,000+47619.05+45351.47+43191.88+41135.12+ \\ 39176.31+37310.77+35534.07+33841.97+32230.45+ \\ 30695.66+29233.96+27841.87+26516.07+25253.40+ \\ 24050.85+22905.58+21814.83+20776.03+19786.70 \\ =\$654,266.04 \end{gathered}$$

The present worth of the lottery will be $654,266.04.

Explanation for part (e)

The present value of the perpetuity will be $1,200,000 (=60,000/0.05). The $60,000 every year is greater than $1 million today; thus, a rational investor will choose $60,000 every year. The $60,000 every year benefits the investor’s heirs.

Explanation for part (f)

To be saved from the taxation, the amount is given as a loan, i.e., $M should be equal $\frac{\$M\left(1+r\right)}{r}$.

Suppose the gift amount is $20,000 and has to be saved from the taxation, then the parent should give a loan of $420,000, i.e., $\left(\frac{\$20,000\left(1+0.05\right)}{0.05}=\$420,000\right)$. If the child could earn interest from the loan of $21,000 (=420,000*0.05), a year later, the child would return the loan to amount to the parent. The present value of the interest amount will be $20000, i.e., $\left(21000{\left(1+0.05\right)}^{-1}=21000\times 0.9524=\$20,000\right)$. Hence, a gift of $20,000 without any tax can be obtained by the child.

A moderate-income earner could not have a huge amount to give; thus, they would find the policy unfair. They could give the gift amount, but that will be taxed and will not have such a huge capital, i.e., $420,000 to give to their child.