The $24.0 million lottery payment that you have just won actually pays $1.2 million per year for 20 years. The interest rate is 8%.

The $24.0 million lottery payment that you have just won actually pays $1.2 million per year for 20 years. The interest rate is 8%.

a. If the first payment comes in 1 year, what is the present value of the winnings?



b. What is the present value if the first payment comes immediately?







 
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers should be used for actual calculations.

a.

PVOA	=	C((1 / r) – {1 / [r(1 + r)t]})
	=	$1,200,000 × ((1 / .08) – {1 / [.08(1.08)20]})
	=	$11,781,776.89

b.

PVAD	=	[C((1 / r) – {1 / [r(1 + r)t]})] × (1 + r)
	=	$1,200,000 × ((1 / .08) – {1 / [.08(1.08)20]}) × 1.08
	=	$12,724,319.04

You can also calculate this using the following:

PVAD	=	PVOA × (1 + r)
	=	$11,781,776.89 × 1.08
	=	$12,724,319.04

In this case, you are receiving the payments. With the annuity due, you receive each payment one year sooner which makes the annuity due worth more today than the ordinary annuity.

Calculator computations:

a.
Calculator payments set to "END".

Enter

20

8

-1,200,000

N

I/Y

PV

PMT

FV

Solve for

11,781,776.89

b.
Calculator payments set to "BGN".

Enter

20

8

-1,200,000

N

I/Y

PV

PMT

FV

Solve for

12,724,319.04