### Sample Question

**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%?
- 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?
- Suppose you just inherited an 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.

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.

### SAMPLE ANSWER

- Present value =Future value (FV)/ (1+r) ^Y, where r is the annual interest rate and Y is the number of years invested. Therefore, to calculate the present value of the bank:

Considering discount rate as 7%,

PV = $15,000/ (1+0.07) ^1

= $140186.70.

Considering the interest rate as 4%,

PV= $15,000/ (1+0.04) ^1

= $14423.10.

- Considering account A, the present value is: PV=FV/ (1+r) ^Y, $6,500/ (1+0.06) ^1 which is equal to $6,132.10. Considering account B, PV=$12,600/ (1+0.06) ^2 which is equal to $11,214.

- Since what is needed is the present value for the whole stream of income, what is used as the future value is that obtained in the third year because of exponents rule, and taking discount rate as 7%, and the number of years as three. Therefore, PV=$85,000,000/ (1+0.07) ^3. Which is equal to $69,385,319.54.

The other part is computing the present value of the income stream from the gold mine at a discount rate of 5% and 3%.Taking the discount rate as 5%, and the future value as that obtained in the third year, and the number of years as three, PV = $85,000,000/ (1+0.05) ^3, which is equal to $73,426,195.88. Taking the discount rate as 3%, future value as $85,000,000 and the number of years as 3, PV=$85,000,000. / (1+0.03) ^3.The final answer is 77,787,041.05.

When a discount rate of 7% is used, the present value achieved is $69,385,319.54, while when the rate is 5%; the value is $73,426,195.88 and $77,787,041.05 for 3% discount rate. This explains the fact that present value is the existing value of a yet to come amount of cash, considering indicated rate of incomes. It also shows that yet to come income is discounted at the stated discount rates. Finally, it elaborates that the greater the discount rates, the lower the existing value of the future cash streams.

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