i.Find the future values of the following ordinary annuities FV of $400 each 6 m
ID: 2784411 • Letter: I
Question
i.Find the future values of the following ordinary annuities
FV of $400 each 6 months for 5 years at a nominal rate of 12%, compounded semiannually
FV of $200 each 3 months for 5 years at a nominal rate of 12%, compounded quarterly
The annuities described in parts a and b have the same total amount of money paid into them during the 5-year period, and both earn interest at the same nominal rate, yet the annuity in part b earns $101.75 more than the one in part a over the 5 years. Why does this occur?
ii.What is the present value of a perpetuity of $100 per year if the appropriate discount rate is 7%? If interest rates in general were to double and the appropriate discount rate rose to 14%, what would happen to the present value of the perpetuity?
iii.Ralph Renner just borrowed $30,000 to pay for a new sports car. He took out a 60-month loan and his car payments are $761.80 per month. What is the effective annual rate (EAR) on Ralph’s loan?
iv.Joe Ferro’s uncle is going to give him $250 a month for the next two years starting today. If Joe banks every payment in an account paying 6% compounded monthly, how much will he have at the end of three years?
Explanation / Answer
(i) Find the future values of the following ordinary annuities
(a) FV of $400 each 6 months for 5 years at a nominal rate of 12%, compounded semiannually
Ordinary Annuity formula for future value calculation
FV = PMT *{(1+i) ^n1} / i
Where
Future Value FV =?
PMT is periodic payment = $400
And n is number of payments = 5 years * 2 = 10 semi-annual payments
Semiannual Interest rate i = annual interest rate/2 = 12%/2 = 6% or 0.06 interest per payment period
Putting all the values in Ordinary Annuity formula for future value calculation
FV = $400 *{(1+ 6%) ^ 10 – 1} /6%
= $5,272.32
(b) FV of $200 each 3 months for 5 years at a nominal rate of 12%, compounded quarterly
Ordinary Annuity formula for future value calculation
FV = PMT *{(1+i) ^n1} / i
Where
Future Value FV =?
PMT is periodic payment = $200
And n is number of payments = 5 years * 4 = 20 quarterly payments
Quarterly Interest rate i = annual interest rate/4 = 12%/4 = 3% or 0.03 interest per payment period
Putting all the values in Ordinary general Annuity formula for future value calculation
FV = $200 *{(1+ 3%) ^ 20 – 1} /3%
= $5,374.07
The annuities described in parts a and b have the same total amount of money paid into them during the 5-year period, and both earn interest at the same nominal rate, yet the annuity in part b earns $101.75 more than the one in part a over the 5 years. Why does this occur?
Yes, it is true that the annuity in part b earns $101.75 more than the one in part a over the 5 years
$5,374.07-$5,272.32 = $101.75
It is because of more frequent compounding of interest rate which helps quarterly compounding earn more interest in comparison of semiannual compounding
ii. What is the present value of a perpetuity of $100 per year if the appropriate discount rate is 7%? If interest rates in general were to double and the appropriate discount rate rose to 14%, what would happen to the present value of the perpetuity?
Present value of a perpetuity = C/ r
Where,
Annual payment C = $100
Interest rate r = 7%
Therefore,
Present value of a perpetuity = $100 /7% = $100/0.07 = $1,428.57
If interest rates in general were to double and the appropriate discount rate rose to 14%
Present value of a perpetuity = $100 /14% = $100/0.14 = $714.29
iii. Ralph Renner just borrowed $30,000 to pay for a new sports car. He took out a 60-month loan and his car payments are $761.80 per month. What is the effective annual rate (EAR) on Ralph’s loan?
We can use PV of an Annuity formula to calculate the effective annual rate (EAR) of loan amount
PV = PMT * [1-(1+i) ^-n)]/i
Where,
Loan amount (PV) =$30,000
PMT = Monthly payment =$761.80
n = N = number of payments = 60 months
i = I/Y = interest rate per year =? Monthly interest rate = i/12 =?
Therefore,
$30,000 = $761.80* [1- (1+i) ^-60]/i
By trial and error method, we can calculate the value of i
Here i = 0.015 or 1.5% per month
Therefore, effective annual rate (EAR) of loan amount = 1.5%*12 = 18%
iv. Joe Ferro’s uncle is going to give him $250 a month for the next two years starting today. If Joe banks every payment in an account paying 6% compounded monthly, how much will he have at the end of three years?
For first two years, we can use FV of an Annuity formula to calculate the loan amount at the end of year two
FV = PMT *{(1+i) ^n1} / i
Where,
Loan amount after two years FV =?
PMT = Monthly payment =$250
n = N = number of payments = 2*12 = 24 months
i = I/Y = interest rate per year =6 therefore Monthly interest rate = 6%/12 =0.5%
Therefore,
FV = $250* *{(1+0.5%) ^241} / 0.5%
= $6,389.78
But we have to calculate the amount at the end of year 3, therefore
Amount at end of year 3 = $6,389.78 *(1+0.5%) ^12 = = $6750.14
Where 0.5% is monthly interest rate and 12 is the number of months
Therefore Amount at end of year 3 is $6750.14
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