Question

1. You have just won the prize in the State lottery. A recent innovation is to offer prize winners a choice of payoffs. You must choose one of the following prizes:

a. $1,000,000 paid immediately
b. $600,000 paid exactly one year from today, and another $600,000 paid exactly 3 years from today
c. $70,000 payment at the end of each year forever (first payment occurs exactly 1 year from today)
d. An immediate payment of $600,000, then beginning exactly 5 years from today, an annual payment of $50,000 forever

e. An annual payment of $200,000 for the next 7 years (first payment occurs exactly 1 year from today)
You believe that 8% p.a. compounded annually is an appropriate discount rate. Assuming you wish to maximize your current wealth, which is the best prize?

2. Kate's financial advisor tells her that she wil need $2 million to fund her retirement. She plans to work for another 30 years before retiring. She will make 30 contributions to a pension. How much will each contribution be, if the interest rate is 9% p.a?

3. Mary has just retired and has $1 miliom in her retirement account. Her bank offers an arrangement whereby the bank takes her $1 million now and pays her $110,000 at the end of each year for the next 20 years. Is it a fair deal, if the offered rate is 10%p.a?

Answer 1

1. For the value of current wealth, firstly we have to calculate Present Value (PV) in each prizes.

a. This is paid immediately so the Present Value (PV) is same $1,000,000.

b. Payment 1: $600,000 (after 1 year)

  Present Value Factor = 1 / (1+r)ⁿ [ r = discount rate or rate of return , n = number of periods ]

Present Value (PV) = Future Value (FV) * Present Value Factor (PVF)

  = $600,000 * [1/(1+0.08)¹]

= $600,000 * 0.9259

= $555,540

  Payment 2: $600,000 (after 3 years)

PV = FV * PVF

= $600,000 * [1/(1+.08)³]

= $600,000 * 0.7938

= $476,280

Thus, I would have a total amount $1,031,820 ( $555,540 + $476,280 )

c. $70,000 payment at the end of each year forever. This is perpetuity. Perpetuity is infinite stream of cash flows.

  Present Value of Perpetuity = CF / r [ CF = Cash flow, r = discounted rate or rate of return ]

  = $70,000 / 8 %

= $875,000

d. Payment 1: immediate payment of $600,000,

Payment 2: an annual payment of $50,000 forever, beginning after 5 years

Value of Perpetuity ( after 5 years ) = $50,000 / 8% ( same formula applicable use above)

= $625,000

  Present value of perpetuity = Future Value * PVF (we calculate this because the above value of perpetuity is after 5 years, but we want present timr value)

  = $625,000 * [ 1/(1+0.08)⁵]

= $625,000 * 0.6806

= $425,375  

Thus, I would have a total amount $1,025,375 ( $600,000 + $425,375 )

e. An annual payment of $200,000 for the next 7 years (first payment occurs exactly 1 year from today)

Case (i) if we assume $200,000 receives after 7 years, then Present value is:

PV = $200,000 * [ 1/(1+0.08)⁷]

= $200,000 * 0.5835

= $116,700

Maximum Current Wealth in Prize (b), so we should choose prize b.

Case (ii) if we assume $200,000 receives end of every year for 7 years.

PV = CF *  [1 - {1 / (1 + r)ⁿ }] / r (this is total value of all years)

= $200,000 * [1 - {1 / (1+0.08)⁷ }] / 0.08

= $200,000 * 5.2064    

= $1,041,280

Maximum Current Wealth in Prize (e), so we should choose prize e.