Why Cash-Flow Dates Matter
A project may generate cash flows at the end of a quarter, halfway through a year, or on other dates that do not coincide with annual periods. The NPV calculation still adds the present values of the dated cash flows, but the discount factor must reflect the time until each cash flow and the convention used to state the interest rate. Using an annual rate as though it applied over six months, for example, changes the valuation.
Measure each cash-flow date from time 0, the valuation date. A date of \(t=0.5\) means six months after that date when time is measured in years. A cash flow at time 0 is not discounted; a cash flow received later is discounted back to time 0.
Here, \(D(t)\) is the present-value factor for a cash flow at time \(t\), with \(D(0)=1\). The sum includes every project cash flow, positive or negative. This tutorial focuses on determining the correct factor when a date is not an integer number of years from time 0.
Match the Discount Factor to the Rate Convention
For a stated effective annual rate \(i\), the usual fractional-year convention assumes that the accumulation over \(t\) years is \((1+i)^t\). Under this convention, discounting a cash flow at \(t\) years uses the factor \((1+i)^{-t}\). This fractional-period extension is a valuation convention: an effective annual rate alone does not dictate every possible way to interpolate between annual dates. Use the convention specified in the problem.
A nominal annual rate convertible \(m\) times per year, written \(i^{(m)}\), specifies a rate of \(i^{(m)}/m\) for each conversion period. If the cash-flow date is aligned with those periods, and \(k=mt\) is an integer, discount with \(k\) periods:
For a continuous compounding convention with force of interest \(\delta\), the accumulation factor over \(t\) years is \(e^{\delta t}\), so the discount factor is \(e^{-\delta t}\). Do not substitute this factor for an effective annual rate unless the problem states a continuous convention.
The nominal-rate formula above applies directly when each cash-flow date falls on a compounding date. If a date falls between conversion dates, do not silently round the period count or assume a fractional-period rule. Apply the off-period convention stated in the problem; if no such convention is supplied, identify the ambiguity rather than inventing one.
| Stated convention | Discount factor for an aligned date \(t\) | Timing to check |
|---|---|---|
| Effective annual rate \(i\), with fractional-year convention | \((1+i)^{-t}\) | Measure \(t\) in years, including fractions. |
| Nominal rate \(i^{(m)}\), convertible \(m\) times per year | \(\left(1+i^{(m)}/m\right)^{-mt}\) | Confirm that \(mt\) is a whole number of periods. |
| Force of interest \(\delta\) | \(e^{-\delta t}\) | Measure \(t\) in years. |
Why the Discounting Formula Holds
The present value is the amount at time 0 that grows, under the stated accumulation convention, to the cash flow on its payment date. With an effective annual rate \(i\) and the fractional-year convention, one dollar at time 0 accumulates to \((1+i)^t\) dollars at time \(t\). If \(C_t\) is the cash flow due then, its present value \(P_t\) must satisfy \(P_t(1+i)^t=C_t\). Dividing both sides by the accumulation factor gives the discounting formula.
The reasoning is the same for the other conventions: present value multiplied by the appropriate accumulation factor must equal the dated cash flow. The key is to use the accumulation rule actually stated, not just the numerical rate without its timing convention.
A Consistent Procedure
Before summing present values, make a timeline and label each date in years and, when relevant, in compounding periods. Then select the rate-specific factor for each date.
Write the initial investment at time 0 and list every later cash flow with its date and sign.
Determine whether the rate is effective annual, nominal with a specified conversion frequency, or continuous. Note any stated convention for fractional or off-period dates.
Use the factor that matches both the date and convention, include the time-0 flow without discounting, and check that the final sum uses consistent signs and units.
Worked Examples
Worked Example: Effective Annual Rate and Half-Year Dates
A hypothetical equipment project requires an immediate outlay of $10,000. It is expected to return $6,000 after six months and another $6,000 after 18 months. The discount rate is an effective annual rate of 8%, and the stated convention uses \((1+i)^t\) for fractional years. Calculate NPV to the nearest cent.
Method. Express both dates in years: six months is \(t=0.5\), and 18 months is \(t=1.5\). Apply the effective annual discount factor to each receipt separately.
At six months, the discount factor is \(1.08^{-0.5}=0.962250449\). The present value of that receipt is \(6{,}000(0.962250449)=\$5{,}773.50\).
At 18 months, the discount factor is \(1.08^{-1.5}=0.890972638\). The present value of that receipt is \(6{,}000(0.890972638)=\$5{,}345.84\).
Include the initial outlay at time 0 without discounting:
Check. The two positive cash flows have total present value of $11,119.34, which exceeds the $10,000 outlay by $1,119.34. The common mistake this method avoids is applying 8% as a six-month rate instead of using the six-month exponent.
Worked Example: Quarterly Nominal Rate and Aligned Dates
A hypothetical software project has an initial cost of $18,000 and expects receipts of $7,000 after three months, $8,000 after nine months, and $8,000 after 18 months. The discount rate is 12% nominal, convertible quarterly. Find NPV to the nearest cent.
Method. The quarterly rate is \(0.12/4=0.03\), or 3% per quarter. The three cash flows occur at \(1\), \(3\), and \(6\) quarters, respectively, so each date is aligned with a conversion period.
Discount the first receipt for one quarter: \(7{,}000/(1.03)^1=\$6{,}796.12\). Discount the second for three quarters: \(8{,}000/(1.03)^3=\$7{,}321.13\). Discount the third for six quarters: \(8{,}000/(1.03)^6=\$6{,}699.87\).
The present value of all receipts is \(6{,}796.12+7{,}321.13+6{,}699.87=\$20{,}817.12\). Subtract the time-0 cost:
Check. The time conversions are \(3/3=1\), \(9/3=3\), and \(18/3=6\) quarters. The common mistake this method avoids is treating 12% as the rate per quarter; the specified quarterly rate is 3%.
Worked Example: Continuous Discounting at Fractional Dates
A hypothetical service project costs $30,000 immediately and is expected to produce cash flows of $12,000 after six months, $12,000 after one year, and $9,000 after two years. The force of interest is \(\delta=0.06\) per year. Calculate NPV to the nearest cent.
Method. Because the stated rate is a force of interest, use \(e^{-\delta t}\), not an effective annual discount factor. The dates are \(0.5\), \(1\), and \(2\) years.
At \(t=0.5\), the present value is \(12{,}000e^{-0.06(0.5)}=12{,}000e^{-0.03}=\$11{,}645.35\). At \(t=1\), it is \(12{,}000e^{-0.06}=\$11{,}301.17\). At \(t=2\), it is \(9{,}000e^{-0.12}=\$7{,}982.28\).
The receipts have total present value \(11{,}645.35+11{,}301.17+7{,}982.28=\$30{,}928.80\). Including the initial outlay gives:
Check. All three receipts occur after time 0, so each is discounted; the initial cost is not. The common mistake this method avoids is using a discrete-rate factor for a rate stated as a continuous force.
Common Mistake: Treating an Annual Rate as a Half-Year Rate
Suppose $6,000 is due in six months, and the stated rate is an effective annual rate of 8% with the fractional-year convention. A mistaken calculation discounts for a full year:
That calculation uses the wrong time exponent. The correct calculation uses \(t=0.5\):
The difference is not a rounding issue: discounting for a full year applies a longer period of interest than the cash flow actually experiences. A related trap is converting a nominal annual rate to a periodic rate but then counting the wrong number of periods. State the rate per period and the number of periods before calculating.
What to Check Before Reporting NPV
A nonannual-date calculation is reliable only when the timing and rate convention are both clear. Check the following before finalizing the sum:
- Each date is measured from the same valuation date, in years or in the matching compounding periods.
- Each cash flow is discounted using the rate convention stated in the problem.
- A nominal rate is divided by its conversion frequency to find the periodic rate.
- Cash flows between nominal conversion dates use an explicitly stated off-period convention; they are not assigned an arbitrary fractional period.
- The initial investment and later cash flows have consistent signs and monetary units.
These checks are useful whenever the project timeline does not consist solely of year-end payments. Matching each date to its factor prevents a timing convention from being mistaken for a change in the size of a cash flow. This treatment of dated cash flows is aligned with the SOA VEE Accounting and Finance syllabus.
Check Your Understanding
Use the stated rate convention in each problem. Round money to the nearest cent.
- A project costs $5,000 now and pays $5,300 after six months. The effective annual rate is 6%, using the fractional-year convention. Find NPV. (Answer: $147.82)
- A project costs $12,000 now and pays $7,000 after six months and $7,000 after 18 months. The effective annual rate is 10%, using the fractional-year convention. Find NPV. (Answer: $741.73)
- A project costs $15,000 now and pays $8,000 after six months and $9,000 after 15 months. The rate is 8% nominal, convertible quarterly. Find NPV. (Answer: $840.93)
- A project costs $25,000 now and pays $9,000 after six months, $9,000 after one year, and $9,000 after 18 months. The force of interest is 5% per year. Find NPV. (Answer: $688.55)
- A $5,000 payment is due after three months. The effective annual rate is 12%, and the fractional-year convention applies. What is its present value at time 0? (Answer: $4,860.33)