Question

Alysha has just won a lottery. She will receive a payment of $8,000 at the end of each year for 9 years. As an alternative, she can choose an immediate payment of $55,000.

A. Which alternative should she pick if the interest rate is 4 percent: make a payment at the end of each year or an immediate payment? (choose)

B. What would the interest rate have to be for Alysha to be indifferent about the two alternatives? (Round answer to 4 decimal places, e.g. 25.2341%. Do not round your intermediate calculations.)

Answer 1

Part A:

Select the alternative with higher PV.

PV of 55000 received today is $ 55000

PV of $ 8000 for 9 Years:

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here cash flows are happened at the end of the period. PV of annuity is current value of cash flows to be received at regular intervals discounted at specified int rate or discount rate to current date.

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period
n - No. of periods

Particulars	Amount
Cash Flow	$            8,000.00
Int Rate	4.0000%
Periods	9

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
= $ 8000 * [ 1 - [(1+0.04)^-9]] /0.04
= $ 8000 * [ 1 - [(1.04)^-9]] /0.04
= $ 8000 * [ 1 - [0.7026]] /0.04
= $ 8000 * [0.2974]] /0.04
= $ 59482.65

Suggested $ 8000 for 9 Years at Int rate of 4%, as it has higher PV than receiving $ 55000 today.

Part B:

Particulars	Amount
PV Annuity	$      55,000.00
Time Period	9.00
Cash Flow	$         8,000.00

PV of Annuity = Cash flow * PVAF(r%, n)
PVAF(r%, n ) = PV of Annuity / Cash Flow
= $ 55000 / $ 8000
= 6.875

PVAF = SUm [ PVF(r%, n) ]
PVF(r%, n) = 1 / ( 1 + r)^n
r = Int rate per period
n = No. of periods

How to calculate PVAF using Excel:
=PV(Rate,NPER,-1)
Rate = Disc Rate
NPER = No.of periods

The Rate at which PVAF for 9 Periods will be equal to 6.875 will be the answer.
PVAF(5.75%9) = 6.8763
PVAF(5.8%9) = 6.8613

Required Rate = 5.75 % + [ [ 6.8763 - 6.875 ] / [ 6.8763 - 6.8613 ] ] * 0.05 %
= 5.75 % + [ [ 0.0013 ] / [ 0.015 ] ] * 0.05 %
= 5.75 % + [ 0.0867 ] * 0.05 %
= 5.75 % + 0.004335 %
= 5.754335 %

If the Int Rate is 5.754335%, both options are indifferent. Difference any may be due to Roundig off problem.

Proof:

Particulars	Amount
Cash Flow	$            8,000.00
Int Rate	5.7543%
Periods	9

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
= $ 8000 * [ 1 - [(1+0.0575)^-9]] /0.0575
= $ 8000 * [ 1 - [(1.0575)^-9]] /0.0575
= $ 8000 * [ 1 - [0.6044]] /0.0575
= $ 8000 * [0.3956]] /0.0575
= $ 55000.1