Attempts: Keep the Highest: /4 4. Present value of fixed-payment security Aa Aa
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Attempts: Keep the Highest: /4 4. Present value of fixed-payment security Aa Aa Avangard is an art school that borrows money to expand its studio. The loan agreement requires paying a fixed payment (F) of $1,500 at the end of each year for the next five years, with an annual interest rate (i) of 15%. Avangard calculates the present value of the loan (PV) using the following formula: O Fx (1 -[1/(1 + )5/1) O Ix(1-1/( / F Avangard's present value of the loan is Suppose that the loan agreement requires Avangard to make semiannual flxed payments of $750 for the next five years, while the annual interest rate stays at 15%. Avangard calculates the present value of the loan at using the following formula: O t1 1/ (1 + Interest Rate per six months)10) / Interest Rate per six months O Interest Rate per six months x (1-[1/(1+ Fixed Payment)110)/ Fixed Payment O Fixed Payment x (1-1/(+Interest Rrate per six months))30) / Interest Rate per six months NA 3.16 2004-2016 Aplia All ights reserve 2 2013 Cengage Lriaig ecept as noted. Al nges reserved. Continue withour savingExplanation / Answer
Given that,
Fixed payment at the end of each year=F
=$ 1500
Period of payment of loan=N= 5 years
Annual rate of interest =i= 15%
Present value (PV) of the money to be paid at the end of first year=$ F/(1+i)
Present Value (PV) of the money to be paid at the end of second year=$ F/(1+i)^2
Similarly,PV of the money to be paid at the end of 'n'th year=$ F/(1+i)^n
Present value of the money to be paid in all these years combined is
PV=F/(1+i)^1+F/(1+i)^2+...+F/(1+i)^n
The above expression is the sum of n terms in a Geometric Progression (G.P) with the initial term a=F/(1+i) and common ratio r=1/(1+i).
The formula for sum of n terms in G.P
S(n)=a[(1-r^n)/(1-r)]
Putting the values of a and r in the formula for n=5,we get,
S(5)=(F/(1+i))[(1-(1/(1+i))^5)/(1-1/(1+i))]
=((F/(1+i))/[((1+i)-1)/(1+i)])(1-(1/(1+i))^5)
=(F/i)(1-(1/(1+i))^5) or F(1-(1/(1+i))^5)/i
Therefore the first option is the right answer.
The present value PV for the given values is
PV=1500(1-(1/(1+0.15))^5)/(0.15)=$ 5028.24
which is approximately equal to $ 5029
Now when the installments get semiannual over same 5 year period and the fixed payment becomes $ 750,the above equation changes to
S(n')=F'(1-(1/(1+i'))^n')/i'
Where F'=$ 750,i'=i/2=0.075 is the interest rate per six months and n'=2n=10 is the number of semiannual installments over the period of 5 years
S(10)=F'(1-(1/(1+0.075))^10)/0.075
Therefore the present value of loans is calculated as Fixed payment*
(1-(1/(1+interest rate per six months))^10)/interest rate per six months
Therefore the third option is the right answer.
The present value PV for the given values is
PV=750(1-(1/(1+0.075))^10)/(0.075)
=$ 5148.06 which is approximately equal to $ 5149
Thus we come to a conclusion that the more frequently the amount is compounded the greater will be the present value of loan to be paid.
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