When Alfred Nobel​ died, he left the majority of his estate to fund five prizes to be made annually starting one year after he died​ (the sixth​ one, in​ economics, was added​ later).

**a.** If he wanted the cash award of each of the five prizes to be 

​$36 comma 00036,000 

and his estate could earn 

88​% 

per​ year, how much would he need to fund his​ prizes?

**b.** If he wanted the value of each prize to grow by 

44​% 

per year​ (perhaps to keep up with​ inflation), how much would he need to​ leave?

**c.** His heirs were surprised by his will and fought it. If they had been able to keep the amount of money you calculated in​ (b), and had invested it at 

88​% 

per​ year, how much would they have in​ 2018, 122 years after he​ died?

Question content area bottom

Part 1

**a.**  To fund the five​ prizes, he would need 

​$enter your response here. 

 ​(Round *to the nearest*​ cent.)

Part 2

**b.**  He would need to leave 

​$enter your response here*.* 

​ (Round *to the nearest*​ cent.)

Part 3

**c.**  In​ 2018, 122 years after his​ death, his heirs would have 

​$enter your response here*.* 

​ (Round *to the nearest*​ dollar.)

Let's break down the problem step by step to answer each part.

### Part 1: Funding the Prizes with No Growth
**a.** If Alfred Nobel wanted each of the five prizes to be $36,000 and his estate could earn 8% per year, we need to calculate the total amount required to fund these prizes indefinitely, starting one year after his death.

This is a classic example of a perpetuity, where the cash flow occurs once a year and starts one year from now.

The formula for the present value of a perpetuity is:

\[
PV = \frac{C}{r}
\]

where:
- $ C $ is the cash flow per period (\$36,000 per prize),
- $ r $ is the interest rate (8% or 0.08).

Since there are five prizes, the total annual cash flow is:

\[
C_{\text{total}} = 5 \times 36,000 = \$180,000
\]

The required fund amount is:

\[
PV = \frac{180,000}{0.08} = \$2,250,000
\]

**Answer for (a):** To fund the five prizes, he would need **$2,250,000**.

### Part 2: Funding the Prizes with Growth
**b.** If Nobel wanted the value of each prize to grow by 4% per year, we are dealing with a growing perpetuity. The formula for the present value of a growing perpetuity is:

\[
PV = \frac{C}{r - g}
\]

where:
- $ C = 180,000 $ (total cash flow),
- $ r = 0.08 $ (interest rate),
- $ g = 0.04 $ (growth rate).

The required fund amount is:

\[
PV = \frac{180,000}{0.08 - 0.04} = \frac{180,000}{0.04} = \$4,500,000
\]

**Answer for (b):** He would need to leave **$4,500,000**.

### Part 3: Value of the Fund After 122 Years
**c.** If his heirs had invested the amount calculated in (b) at an 8% annual interest rate, we need to calculate the future value after 122 years.

The formula for future value is:

\[
FV = PV \times (1 + r)^n
\]

where:
- $ PV = 4,500,000 $,
- $ r = 0.08 $,
- $ n = 122 $.

Let's calculate the future value:

\[
FV = 4,500,000 \times (1.08)^{122}
\]

Now, let's compute it:

\[
FV = 4,500,000 \times 2340.43 \approx \$10,531,935,000
\]

**Answer for (c):** In 2018, 122 years after his death, his heirs would have **$10,531,935,000** (rounded to the nearest dollar).

Would you like any further details or have any questions?

Here are 5 related questions you might find useful:
1. What would be the future value if the heirs invested at a lower interest rate, say 6%?
2. How does the required amount change if the annual prize amount increases to $50,000?
3. What would be the effect on the required fund if the interest rate was 10% instead of 8%?
4. How much would the heirs have in 2018 if they invested only $2,250,000 at 8% interest?
5. What would be the required fund amount if Nobel wanted the prize to grow by 6% annually?

**Tip:** When planning for perpetual cash flows, the interest rate and growth rate differences play a crucial role in determining the necessary initial investment.

Calculate Alfred Nobel's prize fund requirements and future value using compound interest and perpetuity formulas. Learn how much he needed to leave to fund annual prizes and how much his heirs would have after 122 years.

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