Answer to Question #265415 in Financial Math for piumini Ireshika

Question

1. A company will have to spend $300000 on new plant in two years from now.

Currently investment rates are at a nominal 10%. (a) What single sum should now be invested, if compounding is six-monthly?

(b) What is the APR?

1. Find the effective rate (APR) for a nominal rate at

(a) 7% p.a. compounded annually.

(b) 8% p.a. compounded semi-annually.

(c) 10% p.a. compounded quarterly.

(d) 9% p.a. compounded monthly.

Answer 1

1a.

Present Value = $300,000

Time Period = 2 Years

Interest Rate = 10% 

$Future \space Value = Present\space  Value ×(1+Interest\space Rate)^{Time\space Period}\\ = \$300,000 ×(1+\frac{0.1}{2})^{2×2}\\ = \$300,000 ×(1+0.05)^{4}\\ = \$300,000 ×(1.05)^4\\ = \$300,000 ×1.21550625 \\ = \$364,651.875 \space or\space \$364,651.88$

b.

$Annual \space Percentage\space Rate\\ =\frac{2×Time Period×Interest rate}{Time Period +1}\\ =\frac{2×2×10\%}{2 +1}\\ =\frac{0.4}{3}\\ =0.1333333333 \space or \space 13.33\%$

1.

Answer A.

Under annual compounding, the number of compounding periods will be equal to 1.

Effective Annual Rate = Nominal Rate = 7%

Answer B.

Nominal Rate = 8%

Compounded Semi Annually

Number of compounding periods = 2

Effective Interest Rate =(1+(Nominal Rate$\div$ Number of compounding periods))

^{Number of compounding periods} − 1

$= (1+(\frac{8\%}{2}))^2 − 1\\ =1.0816− 1\\ = 8.16\%$

Answer C.

Nominal Rate = 10%

Compounded Quarterly

Number of compounding periods = 4

Effective Interest Rate =(1+(Nominal Rate$\div$ Number of compounding periods))

^{Number of compounding periods} − 1

$= (1+(\frac{10\%}{4}))^4 − 1\\ =1.1038− 1\\ = 10.38\%$

Answer D.

Nominal Rate = 9%

Compounded monthly

Number of compounding periods = 12

Effective Interest Rate =(1+(Nominal Rate$\div$ Number of compounding periods))

^{Number of compounding periods} − 1

$= (1+(\frac{9\%}{12}))^{12} − 1\\ =1.0938− 1\\ = 9.38\%$