Question

Professor’s Annuity Corp. offers a lifetime annuity to retiring professors. For a payment of $75,000 at age 65, the firm will pay the retiring professor $475 a month until death.

a. If the professor’s remaining life expectancy is 20 years, what is the monthly interest rate on this annuity? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

b. What is the effective annual interest rate? (Use the monthly rate computed in part (a) rounded to 2 decimal places when expressed as a percent. Enter your answer as a percent rounded to 2 decimal places.)

c. If the monthly interest rate is 0.50%, what monthly annuity payment can the firm offer to the retiring professor? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Answer 1

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

Part A:

Particulars	Amount
PV Annuity	$      75,000.00
Time Period	240.00
Cash Flow	$            475.00

PV of Annuity = Cash flow * PVAF(r%, n)
PVAF(r%, n ) = PV of Annuity / Cash Flow
= $ 75000 / $ 475
= 157.8947

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 240 Periods will be equal to 157.8947 will be the answer.
PVAF(0.37%240) = 158.8776
PVAF(0.38%240) = 157.2592

Required Rate = 0.37 % + [ [ 158.8776 - 157.8947 ] / [ 158.8776 - 157.2592 ] ] * 0.01 %
= 0.37 % + [ [ 0.9829 ] / [ 1.6184 ] ] * 0.01 %
= 0.37 % + [ 0.6073 ] * 0.01 %
= 0.37 % + 0.006073 %
= 0.376073 %

Int rate per monthis 0.376073% I.e 0.38%

Part B:

Effective Annual Rate = ( 1 + r ) ^ n - 1
r = Int Rate per period
n = No.of periods per anum

Particulars	Amount
Ret period	0.3761%
No. of periods	12.0000

EAR = [ ( 1 + r ) ^ n ] - 1
= [ ( 1 + 0.003761 ) ^ 12 ] - 1
= [ ( 1.003761 ) ^ 12 ] - 1
= [ 1.0461 ] - 1
= 0.04607
I.e EAR is 4.607 %

I.e Effective annual Rate is 4.61%

Part C:

Particulars	Amount
PV Annuity	$          75,000.00
Int Rate	0.5000%
Periods	240

Cash Flow = PV of Annuity / [ 1 - [(1+r)^-n]] /r
= $ 75000 / [ 1 - [(1+0.005)^-6]] /0.005
= $ 75000 / [ 1 - [(1.005)^-6]] /0.005
= $ 75000 / [ 1 - 0.3021 ] /0.005
= $ 75000 / [0.6979 / 0.005 ]
= $ 75000 / 139.5808
= $ 537.32

Monthly withdrawl is $ 537.32