Question

Investment A offers to pay you $10,000 per year for 8 years, whereas Investment B offers to pay you $12,500 per year for 5 years.

(a) Determine if Investment A or Investment B has the higher present value if the interest rate is 5%.

(b) Determine if Investment A or Investment B has the higher present value if the interest rate is 15%. Show all your calculations.

Answer 1

Assuming the payments from both investments, A and B, are made at the end of each year, they represent ordinary annuities. The present values required are therefore present values of the given ordinary annuities. The formula for the present value$(PV)$ of an ordinary annuity is given as follows:

$PV = PMT \left[ \dfrac {(1+i)^n - 1}{i(1+i)^n} \right]$ ,

where $PMT$ is the annual payment, $i$ is the annual interest rate and $n$ is the number of years.

For investments A and B, the following calculations hold.

$\bold {Question \space (a)}$

$\bold {Answers}$

Project A, whose present value is $\$64,632.12$ , has a higher present value than project B, whose present value is $\$54,118.46$

$\bold {Solutions}$

For project A:

PMT = $10,000

i = 5%

n = 8 years

$PV = 10,000 \left[ \dfrac {(1+0.05)^8 - 1}{0.05(1+0.05)^8} \right]$

$PV = 10,000 \left[ \dfrac {1.05^8- 1}{0.05(1.05)^8} \right]$

$=10,000 × 6.4632127596$

$= \$64,632.127596$

$=\bold { \$64,632.13}$

For project B:

PMT = $12,500

i = 5%

n = 5 years

$PV = 12,500 \left[ \dfrac {(1+0.05)^5 - 1}{0.05(1+0.05)^5} \right]$

$PV = 12,500 \left[ \dfrac {1.05^5- 1}{0.05(1.05)^5} \right]$

$= 12,500 × 4.3294766706$

$= 54,118.458383$

$= \bold {\$54,118.46}$

$\bold {Question (b)}$

$\bold {Answers}$

Project A, whose present value is $\$44,873.22,$ has a higher present value than project B, whose present value is $\$41,901.94$

$\bold {Solution}$

For project A:

PMT = $10,000

i = 15%

n = 8 years

$PV = 10,000 \left[ \dfrac {(1+0.15)^8 - 1}{0.15(1+0.15)^8} \right]$

$= 10,000 \left[ \dfrac {1.15^8 - 1}{1.15(1.15)^8} \right]$

$= 10,000× 4.4873215075$

$= \$44,873.215075$

$=\bold {44,873.22}$

For project B:

PMT = $12,500

i = 15%

n = 5 years

$PV = 12,500 \left[ \dfrac {(1+0.15)^5 - 1}{0.15(1+0.15)^5} \right]$

$= 12,500 \left[ \dfrac {1.15^5- 1}{0.15(1.15)^5} \right]$

$= 12,500 × 3.352155098$

$= \$41,901.938725$

$= \bold {\$41,901.94}$