Question

As a winner of a breakfast cereal competition, you can choose one of the following prizes: a) 11,000 a year (forever) and starting NEXT year b) 12,000 a year (forever) and starting THIS year c) 3,000 next year and increasing thereafter by 2% a year forever d) 7,000 starting this year, geowing at 1% per annum e) 800,000 at the end of 20 years f) 6,000 a year for 5 years starting next year g) 1,000 a year starting this year for 12 payments h) 10,000 in year 2 and 50,000 in year 5

*If the interest rate is 5%, which is the most valuable prize? please show all work

Answer 1

Answer>

In order to determine which one of these options is the most lucrative one, we will need to find the present value of each cashflows and compare them.

When, the number of payments is finite over several years , Present Value is calculated as = CF / ((1+r)^n)

When the number of payments is infinite (forever) - it is known as perpetuity - PV of perpetuity = CF/r,

where,

CF = cashflows

r = interest rate

n = no. of years from now in which the cashflow is received/paid

I have calculated the PV of each of the cashflows using the above formulaes:

a> 11,000 a year forever starting next year

PV = CF / r

PV = 11000 / 0.05 = 220000

b> 12000 a year (forever) starting this year:

PV = CF/r + CF for this year

PV = ( 12000 / 0.05 ) + 12000 = 240000 + 12000 = 252000

c> 3,000 next year and increasing thereafter by 2% a year forever

PV for increasing perpetuity is given by:

CF / (r-g),

Where, g = growth rate of cash flow

Therefore,

PV = 3000 / (0.05 - 0.02)

PV = 3000 / 0.03 = 100000

d> 7,000 starting this year, growing at 1% per annum

PV for increasing perpetuity is given by:

CF / (r-g),

Where, g = growth rate of cash flow

Therefore,

PV = 7000 / (0.05 - 0.01)

PV = 7000 / 0.04 = 175000

e> 800,000 at the end of 20 years

PV = 800000 / ((1+0.05)^20)

PV = 301511.32

f> 6,000 a year for 5 years starting next year

PV = 6000 *(1/(1.05) + 1/1.05^2 + 1/1.05^3 + 1/1.05^4 + 1/1.05^5)

PV = 6000 * (0.952 + 0.907 + 0.864 + 0.823 + 0.784)

PV = 6000 * 4.33 = 25980

g> 1,000 a year starting this year for 12 payments

PV = 1000 *(1/(1.05) + 1/1.05^2 + 1/1.05^3 + 1/1.05^4 + 1/1.05^5 + 1/1.05^6 + 1/1.05^7 + 1/1.05^8 + 1/1.05^9 + 1/1.05^10 + 1/1.05^11 + 1/1.05^12)

PV = 1000*(0.952 + 0.907 + 0.864 + 0.823 + 0.784 + 0.746 + 0.711 + 0.677 + 0.645 + 0.614 + 0.585 + 0.557)

PV = 1000 * 8.865 = 8865

h> 10,000 in year 2 and 50,000 in year 5

PV = 10000 / (1.05^2) + 50000/ (1.05^5)

PV = 9070.29 + 39176.31 = 48246.6

Based on the calculations shown above the most valuable option is to receive 800,000 at the end of 20 years

Hence correct answer is option e