Question

You are offered an annuity that will pay you 10,000 at the end of each year for 20 years, with the first payment being in 10 years from today. If the interest rate is 12% annually, what is this annuity worth to you today?

Martha receives $1000 on the first of each year. Stewart receives $100 on the last day of each year. Both Martha and Stewart will receive payments for 11 years. At an 8% discount rate, what is the difference in the present value of these two sets of payments?

What is the present value of a perpetuity that will pay $10,000 in one year and grow at 3% annually, if the discount rate is 10%. (first payment at the end of this year).

Ben invested $5,000 twenty years ago in an account that has paid him 5% with semiannual compounding. How much does Ben have in his account today.?

You plan on investing $10,000 at the end of each year for the next 20 years. You found an investment that will pay you 8% annual percentage rate with daily compounding. How much will you end up in your investment account at the end of 20 years?

Answer 1

Ans a.

First we need to find the PV of $10,000 anuuity 10 years from

now.

Formula for present value of an anuuity = PV= A [ {(1+k)n-1}/k(1+k)n]

PV = Present value of Annuity =??

A = Annual annuity payment=$10,000

K=interest rate=12% pa

n=20 years

PV =10,000*[(1.12^20-1)/12%*1.12^20]

PV=$74,694.44

So PV of the Annuity after 10 years =$74,694.44

Annuity worth today is =74694.44/1.12^10=$24,049.61

Ans b.
Martha receives an Annuity due while Stewart receives an
ordinary Annuity.
Martha's Annuity PV Calculation:
Present Value of Annuity Due
PV=A+A*[1-(1+k)^-(n-1)]/k
A = Annual annuity payment=$1000
K=interest rate=8% pa
n=11 years
PV =1000+1000*[1-1.08^-10]/8%
PV =$7,710.08
So PV of Martha's Annuity =$7,710.08
PV of Stewart's Ordinary Annuity
Formula for present value of an anuuity = PV= A [ {(1+k)n-1}/k(1+k)n]
PV = Present value of Annuity =??
A = Annual annuity payment=$100
K=interest rate=8% pa
n=11 years
PV =100*[1.08^11-1]/(8%*1.08^11)
PV =$713.90
So PV of Stewart's Annuity =$713.90
So differemce between PV of Martha & Stewart's Annuity=$7,710.08-$713.90
=	$         6,996.18

Ans c.

PV of Growing perpetual Annuity =D/r-g

Where D=annuity in first years =$10,000

r=discount rate =10%

g=annuity growth rate =3%

PV =10,000/(10%-3%) =$142,857

So PV of growing Perpetuity =$142,857

Ans d.

Interest rate =5% pa with semi annual compounding

EAR =(1+5%/2)^2-1=5.06% pa

We need to consider Effctive Annual Ineterst rate =5.06% pa

Principal invested $5,000

Maturity period =20 years

Compounding factor =(1+5.06%)^20

So Maturity Value after 20 years=5000*(1.0506)^20=$13,419

Ans e.

We have to find the FV of $10,000 annuity after 20 years.

Interest 8% pa with daily compounding

EAR =(1+8%/365)^365-1=8.33% pa

So Effective Annual Ineterst rate =8.33% pa

Formula for future value of Annuity :

FV= A [ {(1+k)n-1}/k]

FV = Future annuity value

A = Annual nvestment=$10,000

K=interest rate=8.33% pa

N=periods=20 years

FV =10000*[1.0833^20-1]/8.33%

FV =$494,277

So the Future Value of Annuity after 20 yrs=$494,277