In supply chain management, what is the NPV (net present value) of a cash stream that is equal to $75 per period for 5 periods with a rate of return of 15% per period?

Steps to Solve

1. Identify NPV Formula

The NPV formula is given by:

$$ NPV = \sum_{t=0}^{n} \frac{C_t}{(1 + r)^t} $$

where:

• $C_t$ is the cash flow at time $t$
• $r$ is the discount rate
• $n$ is the total number of periods

In this problem, $C_t = 75$, $r = 0.15$, and $n = 5$.

2. Calculate Present Value for Each Cash Flow

Now we will calculate the present value for each of the 5 cash flows.

• For $t=0$: $$ PV_0 = \frac{75}{(1 + 0.15)^0} = 75 $$

• For $t=1$: $$ PV_1 = \frac{75}{(1 + 0.15)^1} = \frac{75}{1.15} $$

• For $t=2$: $$ PV_2 = \frac{75}{(1 + 0.15)^2} = \frac{75}{1.3225} $$

• For $t=3$: $$ PV_3 = \frac{75}{(1 + 0.15)^3} = \frac{75}{1.520875} $$

• For $t=4$: $$ PV_4 = \frac{75}{(1 + 0.15)^4} = \frac{75}{1.749} $$

• For $t=5$: $$ PV_5 = \frac{75}{(1 + 0.15)^5} = \frac{75}{2.011357} $$

3. Sum Up All Present Values

Now, we sum all the present values calculated:

$$ NPV = PV_0 + PV_1 + PV_2 + PV_3 + PV_4 + PV_5 $$

4. Perform the Calculations

Substituting the values:

• $ PV_1 = 65.217 \approx 65.22 $
• $ PV_2 = 56.781 \approx 56.78 $
• $ PV_3 = 49.251 \approx 49.25 $
• $ PV_4 = 42.926 \approx 42.93 $
• $ PV_5 = 37.248 \approx 37.25 $

Now we sum these values:

$$ NPV \approx 75 + 65.22 + 56.78 + 49.25 + 42.93 + 37.25 $$

5. Calculate Final NPV

Finally, calculate the total:

$$ NPV \approx 75 + 65.22 + 56.78 + 49.25 + 42.93 + 37.25 = 326.53 $$