Question

(a) You have won a lottery worth $1,000,000. The amount will be paid to you in equal installments over 20 years. If the interest rate is 10% compounded annually, how much will you be paid at the end of each year?

(b) You have just joined the investment banking firm of Mckenzie & Co. They have offered you two different salary arrangements. You can have $75,000 per year for the next two years, or you can have $55,000 per year for the next two years, along with a $30,000 signing bonus today. If the interest rate is 12% compounded monthly, which is a better offer? NB: first convert the annual percentage rate of 12% to EAR and use the EAR as the discount rate.

Answer 1

a) Amount of lottery = $1,000,000

Interest rate = 10 %

Number of payout periods= 20 years

The value of annuity payout can be calculated by the following formula:

PVA of ordinary annuity =  $1,000,000.

The value of annuity payout will be :

PVA of ordinary annuity/​ [1 – (1 + r)^-n]

Hence, the value of annuity payout = 10% * $1,000,000 / [1 – (1 + 10%)^-20] = $1,17,459.62

Therefore, the amount that will be paid to you at the end of each year would be $1,17,459.62

The payout schedule may be illustrated as below.

b) The effective annual rate or EAR may be calculated as below.

i=(1+r/m)^m - 1

=(1+ 0.12/12)^12 -1

Therefore, EAR =12.6825%

Offer 1 : $75,000 per year for a duration of two years

$75,000 per year for a duration of two years

Interest rate percent compounded monthly = 12%

EAR= 12.6825%

Amount =$75,000

i = 12.6825%

The present value of an annuity is denoted as:

PV of ordinary annuity = (C) *[1- (1+ r) ^ (-n)/ r ],

p – Equated annual payments

r – Effective annual rate or discount rate

n – time period in years

Therefore,

P = $75000[(1 – (1/(1+0.1268)^(-24))/.0.1268]

= $41,325,41

Offer 2: $55,000 per year for a duration of two years + Signing Bonus of $30,000

Effective Value =PV of $55,000 permonth+ $30,000 signing bonus

PV of $55,000 per year

P = $ 55,000 [(1 – (1/(1+.0.1268)^ (-24))/.0.1268]

= $30,305.37

Effective Value

= $30,305.37 + $30,000

= 60,305.37

The above particulars may be illustrated as below:

Hence, the second offer should be accepted.