Make the external rate explicit
External rate of return accounts directly for a specified reinvestment/discount rate. Use the separate positive and negative net-flow convention stated here.
Period 0: −$10,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
| Bucket | Equivalent amount ($) |
|---|---|
| Outflows at 0 | 16,350.66 |
| Inflows at 3 | 13,478.40 |
Under this convention, discount net negative cash flows and compound net positive cash flows using the stated external rate.
Separate positive and negative NET cash flows at each date.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
What the model means
At each date first form NET cash flow. Discount negative net flows to time 0 at ε; compound positive net flows to year N at ε. Define these positive magnitudes as P_E and F_R, then solve P_E(1+ERR)ᴺ=F_R.
Build the calculation
When the external rate equals a compatible IRR root, the constructions agree. In general ERR changes with ε. Some conventions use different finance and reinvestment rates or a project-balance method; state the convention before comparing results.
Set up. Calculate. Explain.
Rebuild the starting scenario
External reinvestment rate: 8 %; Terminal cleanup: 8000 $. Calculate the ERR in percent.
Setup hint
Separate positive and negative NET cash flows at each date.
Open the visual solution
- Separate positive and negative NET cash flows at each date.
- Discount all negatives to 0 at ε; accumulate positives to year 3 at ε.
- Bridge these two equivalent totals over 3 periods. ERR can be negative and changes with the external rate.
Period 0: −$10,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
| Bucket | Equivalent amount ($) |
|---|---|
| Outflows at 0 | 16,350.66 |
| Inflows at 3 | 13,478.40 |
Under this convention, discount net negative cash flows and compound net positive cash flows using the stated external rate.
Separate positive and negative NET cash flows at each date.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.
Transfer the idea to a new scenario
For −$10,000 now, +$6,000 at years 1 and 2, and −$5,000 at year 3, use external rate 10% and calculate ERR.
Setup hint
Separate positive and negative NET cash flows at each date.
Open the visual solution
- Separate positive and negative NET cash flows at each date.
- Discount all negatives to 0 at ε; accumulate positives to year 3 at ε.
- Bridge these two equivalent totals over 3 periods. ERR can be negative and changes with the external rate.
Period 0: −$10,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
| Bucket | Equivalent amount ($) |
|---|---|
| Outflows at 0 | 13,756.57 |
| Inflows at 3 | 13,860.00 |
Under this convention, discount net negative cash flows and compound net positive cash flows using the stated external rate.
Separate positive and negative NET cash flows at each date.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.
Which rate compounds interim receipts to the final date in this ERR convention?
Can you defend your setup?
Can you place each amount on the timeline, match the interest period, calculate the equivalent quantity, and state the decision or limitation?
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