**Present**​ value: **Mixed streams**   Consider the mixed streams of cash flows shown in the following​ table, 

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. 
 
**a.**  Find the present value of each stream using a 

33​% 

discount rate.

**b.**  Compare the calculated present values and discuss them in light of the undiscounted cash flows totaling 

​$50 comma 00050,000 

in each case. Is there some discount rate at which the present values of the two streams would be​ equal?

Question content area bottom

Part 1

**a.** The present value of the cash flows of stream A is 

​$enter your response here. 

 ​(Round to the nearest​ dollar.) 0	-37,500	12,500

1	31,250	18,750

2	25,000	25,000

3	18,750	31,250

4	12,500	-37,500

Totals	50,000	50,000

To solve this problem, we need to calculate the present value (PV) of the two cash flow streams using a 33% discount rate. We will use the formula for present value, which is:

\[
PV = \sum \left( \frac{C_t}{(1 + r)^t} \right)
\]

Where:
- $ PV $ is the present value,
- $ C_t $ is the cash flow at time $ t $,
- $ r $ is the discount rate (33% or 0.33),
- $ t $ is the time period.

We are given two streams of cash flows, Stream A and Stream B, over a 5-year period, and the discount rate of 33%.

Let's summarize the cash flows for both streams:

\[
\text{Stream A:} \quad \{0, 31,250, 25,000, 18,750, 12,500\}
\]
\[
\text{Stream B:} \quad \{-37,500, 18,750, 25,000, 31,250, -37,500\}
\]

Now, we'll calculate the present value for each stream.

### Stream A Present Value Calculation:

1. At $ t = 0 $: $ 0 $
2. At $ t = 1 $: $ \frac{31,250}{(1 + 0.33)^1} $
3. At $ t = 2 $: $ \frac{25,000}{(1 + 0.33)^2} $
4. At $ t = 3 $: $ \frac{18,750}{(1 + 0.33)^3} $
5. At $ t = 4 $: $ \frac{12,500}{(1 + 0.33)^4} $

### Stream B Present Value Calculation:

1. At $ t = 0 $: $ \frac{-37,500}{(1 + 0.33)^0} = -37,500 $
2. At $ t = 1 $: $ \frac{18,750}{(1 + 0.33)^1} $
3. At $ t = 2 $: $ \frac{25,000}{(1 + 0.33)^2} $
4. At $ t = 3 $: $ \frac{31,250}{(1 + 0.33)^3} $
5. At $ t = 4 $: $ \frac{-37,500}{(1 + 0.33)^4} $

Let's calculate these values step by step.The present values for the two cash flow streams, using a 33% discount rate, are:

- Present Value of Stream A: **$49,594**
- Present Value of Stream B: **-$7,971**

### Part (b): Comparison and Discussion

Even though both streams have undiscounted cash flows totaling \$50,000, their present values differ significantly due to the timing and magnitude of the cash flows. Stream A has a generally positive and stable cash flow, which results in a higher present value. In contrast, Stream B begins with a large negative cash flow in year 0 and fluctuates, leading to a significantly negative present value at a 33% discount rate.

To find a discount rate at which the present values of the two streams would be equal, we would need to solve for the rate $ r $ where the present values of Stream A and Stream B are the same. This requires solving a non-linear equation using numerical methods.

Would you like to proceed with calculating that discount rate?

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### Related Questions:
1. How does the discount rate impact the present value of future cash flows?
2. What would the present value be if the discount rate was lower, say 10%?
3. Can the net present value be negative even with positive future cash flows?
4. How does front-loaded cash flow (like in Stream B) affect present value compared to back-loaded cash flows?
5. How can we use Excel or a financial calculator to compute present value easily?

**Tip:** A higher discount rate reduces the present value of future cash flows more aggressively than a lower rate.

Present value: Mixed streams. Consider the mixed streams of cash flows shown in the following table.

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a. Find the present value of each stream using a 33% discount rate.

b. Compare the calculated present values and discuss them in light of the undiscounted cash flows totaling $50,000 in each case. Is there some discount rate at which the present values of the two streams would be equal?

Cash flows: 

| Time | Stream A  | Stream B |
|------|-----------|----------|
| 0    | 0         | -37,500  |
| 1    | 31,250    | 18,750   |
| 2    | 25,000    | 25,000   |
| 3    | 18,750    | 31,250   |
| 4    | 12,500    | -37,500  |

Totals: $50,000 for each stream.

Learn how to calculate the present value of two mixed cash flow streams using a 33% discount rate. Compare the present values and explore how cash flow timing affects outcomes, even with equal totals of $50,000. Suitable for finance and business students.

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