Part I: This Part of the Assignments Tests Your Ability to Calculate Present Value

Part I: This part of the assignments tests your ability to calculate present value.

Suppose your bank account will be worth $15,000.00 in one year. The interest rate (discount rate) that the bank pays is 7%. What is the present value of your bank account today? What would the present value of the account be if the discount rate is only 4%?

Formula for present value:

PV = FV /(1 +i)n

PV (Present Value) = ?

FV (Future Value) = $15,000.00

i (Interest/Discount Rate) = 7%

n (The number of years) = 1

PV = $15,000.00 / (1 + .07)1

PV = $15,000.00/ (1.07)1

PV = $14,018.69

With the rate of 4%:

PV = $15,000.00 / (1 + .04)1

PV = $15,000.00 /(1.04)1

PV = $14,423.08

Suppose you have two bank accounts, one called Account A and another Account B. Account A will be worth $6,500.00 in one year. Account B will be worth $12,600.00 in two years. Both accounts earn 6% interest. What is the present value of each of these accounts?

Formula for Present Value:

PV = FV /(1 + i)n

Account A:

PV=?

FV = $6,500.00

i = 6%

n = 1

PV = $6,500.00 / (1+.06)1

PV = $6,500.00 / (1.06)1

PV = $6,132.08

Account B:

PV = ?

FV = $12, 600.00

i = 6 %

n = 2

PV = $12,600.00 / (1 + .06)2

PV = $12,600.00 / (1.06)2

PV = $11, 213.96

C. Suppose you just inherited a gold mine. This gold mine is believed to have three years worth of gold deposit. Here is how much income this gold mine is projected to bring you each year for the next three years:

Year 1: $49,000,000

Year 2: $61,000,000

Year 3: $85,000,000

Compute the present value of this stream of income at a discount rate of 7%. Remember, you are calculating the present value for a whole stream of income, i.e. the total value of receiving all three payments (how much you would pay right now to receive these three payments in the future). Your answer should be one number - the present value for this gold mine at a 7% discount rate but you have to show how you got to this number.

Formula for present value:

PV = FV /(1 +i)n

PV = ?

FV = $49,000,000 $61,000,000 $85,000,000

n = 1 2 3

i = 7 %

PV = $49,000,000/(1+.07)1 + $61,000,000/(1 + .07)2 +$85,000,000/(1 + .07)3

PV = $49,000,000/(1.07)1 + $61,000,000/(1.07)2 + $85,000,000/(1.07)3

PV = $45,794,392.52 + $53,279,762.42 + $69,385,319.54

PV = $168,459,474.48

Now compute the present value of the income stream from the gold mine at a discount rate of 5%, and at a discount rate of 3%. Compare the present values of the income stream under the three discount rates and write a short paragraph with conclusions from the computations.

At 5%:

PV = $49,000,000 / (1 + .05)1 + $61,000,000 / (1 +.05)2 + $85,000,000 / (1 +.05)3

PV = $49,000,000 / (1.05)1 + $61,000,000 / (1.05)2 + $85,000,000 / (1.05)3

PV = $46,666,666.67 + $55,328,798.19 + $73,426,195.88

PV = $175,421,660.74

At 3%

PV = $49,000,000 / (1 + .03)1 + $61,000,000 / (1 + .03)2 + $85,000,000 / (1 + .03)3

PV = $49,000,000 / (1.03)1 + $61,000,000 / (1.03)2 + $85,000,000 / (1.03)3

PV = $47,572,815.53 + $57,498,350.46 + $77,787,041.05

PV = $182,858,207.04

The present value is lower when the discount rate is higher. With a discount rate of 7 % it would cost $168,459,474.48 to receive the payments in the future. A discount rate at 5% results in paying $175,421,660.74 presently to receive the payments in the future. The 3% discount resulted in paying $182,858,207.04, which is higher than the other two rates. The present value is higher for longer periods.

Part II: Capital Budgeting Practice Problems

A. Consider the project with the following expected cash flows:

Year Cash flow

0 - $400,000

1 $100,000

2 $120,000

3 $850,000

If the discount rate is 0%, what is the project's net present value?

PV = -$400,000/ (1 + .0)0 + $100,000/(1 +.0)1 + $120,000/(1 +.0)2 + $850,000/( 1 + .0)3

PV = -$400,000/ (1.0)0 + $100,000/(1.0)1 + $120,000/(1.0)2 + $850,000/( 1.0)3

PV = -$400,000 + $100,000 + $120,000 + $850,000

PV = $670,000.00

If the discount rate is 2%, what is the project's net present value?

PV = -$400,000/(1 + .02)0 + $100,000/ (1 + .02)1 + $120,000/(1 + .02)2 + $850,000/(1 + .02)3

PV = -$400,000/(1.02)0 + $100,000/ (1.02)1 + $120,000/(1 + 1.02)2 + $850,000/(1 + 1.02)3

PV =-$400,000 + $98,039.22 + $115,340.25 + $800,973.98

PV = $614,353.45

If the discount rate is 6%, what is the project's net present value?

PV = -$400,000/(1 + .06)0 + $100,000/ (1 + .06)1 + $120,000/(1 + .06)2 + $850,000/(1 + .06)3

PV = -$400,000/(1.06)0 + $100,000/ (1.06)1 + $120,000/(1.06)2 + $850,000/(1.06)3

PV = -$400,000 + $94,339.62 + $106,799.57 + $713,676.39

PV = $514,815.59

If the discount rate is 11%, what is the project's net present value?

PV = -$400,000/(1 + 0.11)0 + $100,000/ (1 + 0.11)1 + $120,000/(1 + 0.11)2 + $850,000/(1 + 0.11)3

PV = -$400,000/(1.11)0 + $100,000/ (1.11)1 + $120,000/(1.11)2 + $850,000/(1.11)3

PV = -$400,000 + $90,090.09 + $97,394.69 + $621,512.67

PV = $408,997.46

With a cost of capital of 5%, what is this project's modified internal rate of return?

Now draw (for yourself) a chart where the discount rate is on the horizontal axis (the "x" axis) and the net present value on the vertical axis (the Y axis). Plot the net present value of the project as a function of the discount rate by dots for the four discount rates. Connect the four points using a free hand 'smooth' curve. The curve intersects the horizontal line at a particular discount rate. What is this discount rate at which the graph intersects the horizontal axis?

[You can't upload the graph unto Coursenet. Look at the graph you draw and write a short paragraph stating what the graph 'shows"]..

B. Consider a project with the expected cash flows:

Year Cash flow

0 -$815,000

1 $141,000

2 $320,000

3 $440,000

What is this project's internal rate of return?
If the discount rate is 1%, what is this project's net present value?
If the discount rate is 4%, what is this project's net present value?
If the discount rate is 10%, what is this project's net present value?
If the discount rate is 18%, what is this project's net present value?

[Observe the graph and write a short paragraph stating what the graph shows]

A project requiring a $4.2 million investment has a profitability index of 0.94. What is its net present value? (Remember: Profitability Index is defined as Present Value of the proceeds divided by the initial investment)

Net Present Value?

Initial Investment: 4,200,000

Profitability Index: 0.94

0.94 * 4,200,000 = 3,948,000 Present Value

-4,200,000 + 3,948,000 = -252,000 Net Present Value