Let’s suppose you win the daily grand lottery, which gives you $1000 every day for the rest of your life, or a lump sum of $7 million. Which option should you choose? To make the informed decision, we need to evaluate the “VALUE” of each option. The second one is straightforward, as it is a lump sum of $7 million. The first one, however, requires us to calculate the present value of the annuity (the daily payments). Suppose you are 35 years old when you win the lottery, and let’s say you will live until 85 years old. This means you will receive $1000 every day for 50 years. Let’s say your bank offers interest rate of 3% per year, and for the sake of simplicity, we will assume that the interest is compounded annually, and it is fixed for the next 50 years. Let’s assume we deposit money at the end of each year, meaning: \(A=1000 \times 365 = 365000\)
\(PV = A \times \frac{(1+r)^n - 1}{r} \times \frac{1}{(1+r)^n}\)
\(PV = 365000 \times \frac{(1+0.03)^{50} - 1}{0.03} \times \frac{1}{(1+0.03)^{50}}\)
\(PV = 365000 \times \frac{(1.03)^{50} - 1}{0.03} \times \frac{1}{(1.03)^{50}}\)
\(PV = 365000 \times \frac{4.384 - 1}{0.03} \times \frac{1}{4.384}\)
\(PV = 365000 \times \frac{3.384}{0.03} \times \frac{1}{4.384}\)
\(PV = 365000 \times 112.8 \times 0.228\)
\(PV \approx 9387216\)
So, this shows that the present value of the annuity is approximately $9.4 million, which is greater than the lump sum of $7 million. Therefore, you should choose the daily payments of $1000 for life.
NOTE: This was a simplified calculation. In reality, you may need to think about different things like
- Do I live till 85 years old? If not, then is it better to take the lump sum?
- What if bank changes the interest rate?
I’ve not considered these factors here, but they are important to consider in real life. How? do not know! But I think it is important to think about them!!