Question

As a winner of a breakfast cereal competition, you can choose one of the following prizes:

• $100,000 now.
• $100,000 at the end of 5 years.
• $10,000 a year forever (starting from this year).
• $11,000 a year forever (starting from next year).
• $2,000 next year and increasing thereafter by 2% a year forever
• $6,000 starting this year, growing at 1% per annum
• $800,000 at the end of 20 years
• $5,000 a year for 5 years starting next year
• $1,000 a year, starting this year for 12 payments
• $10,000 in year 2 and $50,000 in year 5

If the interest rate is 5%, which is the most valuable prize? Show your working for each.

Answer 1

1. $100,000 now.

Present value = $100,000

2. $100,000 at the end of 5 years.

Present value = Future Value / (1 + interest rate%)^n

Here , future value =100000

Interest rate= 5%

N= 5

PV= 100000/(1+0.05)^5

PV= $78,352.62

3. $10,000 a year forever (starting from this year).

In case of infinite cash flow, Present value = Cash flow / interest rate

But for this the cash flow must be at the end of year,

So the present value of the cash flow from the next years is:

Here cash flow = 10000

Interest rate= 5%

PV= 10000/5%

PV= $200,000

Plus, the cash flow of this year = $10000

So the total cash flow = $200000+10000= $210,000

4. $11,000 a year forever (starting from next year).

In case of infinite cash flow, Present value = Cash flow / interest rate

Here cash flow = 11000

Interest rate= 5%

PV= 11000/5%

PV= $220,000

5. $2,000 next year and increasing thereafter by 2% a year forever

Here, cash flow = $2000

Growth g= 2%

R= 5%

So the Present value formula will be = cash flow /(r-g)

PV= 2000/(0.05-0.02)

PV= $66.666.67

6. $6,000 starting this year, growing at 1% per year

In this case,

PV= cash flow*(1+g) /(r-g)

PV= 6000*(1+0.01)/(0.05-0.01)

PV= $151,500

7. $800,000 at the end of 20 years

PV= Future Value / (1 + interest rate%)^n

PV= 800000/ (1+0.05)^20

PV=$ 301,511.59

8. $5,000 a year for 5 years starting next year

In case of continuous cash flow,

Present value= P* (1- (1+r)^-n)/r

P= 5000

N= 5

R= 5%

PV= 5000*(1-(1+0.05)^-5)/0.05

PV= $21,647.38

9. $1,000 a year, starting this year for 12 payments

Present value= P* (1- (1+r)^-n)/r

P= 1000

N= 11

R= 5%

PV= 1000*(1-(1+0.05)^-11)/0.05

PV=8306

Total PV= 1000+8306= 9,306

10.$10,000 in year 2 and $50,000 in year 5

Firstly for 10000

PV= 10000/(1+0.05)^2= 9070

And for 50000

PV= 50000/(1+0.05)^5

PV= 39176

So total PV= 48,246.

So from all the options available option 7 ,where $800000 20 years from now is the most profitable one .