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Combining Cash Flows at the Same Maturity

Combine all signed cash flows due on each date, then apply that date’s discount factor to the net amount.

Intermediate 10 min read

What You'll Learn

  • Represent receipts and costs as positive and negative cash flows.
  • Aggregate cash flows that are due on the same date.
  • Apply the discount factor for that date’s maturity to the net amount.
  • Explain why aggregating before discounting gives the same result as discounting each flow separately.
  • Avoid netting cash flows from different dates before applying maturity-specific discount factors.

Introduction

A cash-flow schedule may list several amounts due on the same date: a receipt, a fee, a tax, or a maintenance cost. The valuation can be organized by first combining those amounts into one signed net cash flow for that date, then applying that date’s discount factor. This keeps the timing and the direction of every payment visible.

The key restriction is that only cash flows with the same maturity can be combined before discounting. Amounts due on different dates generally have different discount factors, so they must remain in separate maturity groups. This tutorial develops that rule and applies it to receipts, costs, and net payments.

Definition: Treat a receipt as a positive cash flow and a cost or payment as a negative cash flow. For a date \(t\), the net cash flow at \(t\) is the sum of all signed cash flows due on that date. If \(D(t)\) is the discount factor from date \(t\) to the valuation date, the present value of that date’s net cash flow \(C(t)\) is \(C(t)D(t)\).

Grouping Cash Flows by Date

Suppose a project has several receipts and costs. Write every amount with its sign, group amounts by their due date, and add the amounts within each group. Then discount each group using the factor for its own maturity. A negative net cash flow remains negative after discounting when the discount factor is positive.

For a term structure with annual effective spot rate \(i_t\) for a maturity of \(t\) years, the discount factor for that maturity is \(D(t)=(1+i_t)^{-t}\). The spot rate is maturity-specific: \(i_t\) need not equal the spot rate for another maturity. If discount factors are given directly, use the factor associated with each date instead of converting it to a rate.

$$ C(t)=\sum_{j:\,t_j=t} C_j \qquad\text{and}\qquad \operatorname{PV}=\sum_t C(t)D(t) $$

Here, \(C_j\) is a signed individual cash flow due at time \(t_j\). The sum over \(j\) forms one net amount for each distinct payment date; the sum over \(t\) then adds the present values of those dated net amounts.

Worked Example: Combining Receipts and a Cost at Two Years

A hypothetical project will receive $900 and $760, and pay a fee of $240, all exactly two years from now. The annual effective two-year spot rate is 5%. Find the present value of these cash flows. State the method and check the result by discounting each signed cash flow separately.

Method. Assign positive signs to receipts and a negative sign to the fee. Aggregate the amounts at the common two-year maturity, then use the two-year discount factor. This avoids applying a discount factor to unsigned totals or using the wrong maturity.

Step 1: Find the net cash flow due in two years.

$$ C(2)=900+760-240=1{,}420 $$

Step 2: Find the two-year discount factor. With a 5% annual effective spot rate for two years,

$$ D(2)=(1.05)^{-2}=\frac{1}{1.1025} $$

Step 3: Discount the net amount.

$$ \operatorname{PV}=\frac{1{,}420}{1.1025} =1{,}287.981859\ldots $$

Rounded to the nearest cent, the present value is $1,287.98.

Check. Discounting the signed amounts separately gives \(900/1.1025+760/1.1025-240/1.1025\). The numerator after using the common denominator is \(900+760-240=1{,}420\), so this is exactly the same calculation and gives $1,287.98. The check confirms both the signs and the aggregation.

Why Aggregation Works

The justification is the distributive property of multiplication. Every cash flow due at a particular date has the same discount factor, so that factor can be multiplied by the sum of the cash flows or applied to each one separately. These are algebraically identical operations.

$$ \sum_{j:\,t_j=t} C_jD(t) = D(t)\sum_{j:\,t_j=t}C_j = D(t)C(t) $$

Applying this identity at each distinct date gives the total present value as the sum of the discounted net cash flows. No approximation is involved. Aggregating first is a way to organize the arithmetic, not a different valuation rule.

The equality depends on the cash flows in a group sharing the same date and therefore the same discount factor. If two payments are due on different dates, even if the dates are close together, they cannot be combined into a single amount and discounted at one of the dates without changing the calculation.

Several Dates Require Several Net Amounts

When a schedule has multiple dates, make a separate net amount for each date. Then apply the discount factor for that maturity to that net amount. In particular, do not add all future receipts and costs in nominal dollars first: a dollar due sooner and a dollar due later are generally not worth the same amount at the valuation date.

Worked Example: Netting Three Maturities on a Project Schedule

A hypothetical project has these cash flows: at year 1, a $1,500 receipt and a $500 cost; at year 2, a $1,800 receipt, a $400 maintenance cost, and a $190 tax; at year 3, a $2,600 receipt, a $700 cost, and a $200 fee. The annual effective spot rates are 4% for one year, 5% for two years, and 6% for three years. Find the total present value.

Method. Net the signed cash flows within each year, and then discount each net amount using that year’s spot rate. The main mistake to avoid is applying one spot rate to every date.

Step 1: Net the year 1 cash flows and discount them.

$$ C(1)=1{,}500-500=1{,}000 $$
$$ \operatorname{PV}_1=\frac{1{,}000}{(1.04)^1} =\frac{1{,}000}{1.04} =961.538461\ldots $$

Step 2: Net the year 2 cash flows and discount them.

$$ C(2)=1{,}800-400-190=1{,}210 $$
$$ \operatorname{PV}_2=\frac{1{,}210}{(1.05)^2} =\frac{1{,}210}{1.1025} =1{,}097.505668\ldots $$

Step 3: Net the year 3 cash flows and discount them.

$$ C(3)=2{,}600-700-200=1{,}700 $$
$$ \operatorname{PV}_3=\frac{1{,}700}{(1.06)^3} =\frac{1{,}700}{1.191016} =1{,}427.35278\ldots $$

Step 4: Add the present values.

$$ \operatorname{PV} =961.538461\ldots+1{,}097.505668\ldots+1{,}427.35278\ldots =3{,}486.39691\ldots $$

Rounded only at the final step, the total present value is $3,486.40. Each maturity has its own net amount and discount factor; the year 1, year 2, and year 3 net cash flows were not combined with one another.

Negative Net Cash Flows

A maturity group can have a negative net amount when its costs exceed its receipts. That amount is a net payment at that date, and its present value is negative under a positive discount factor. It should be included with its negative sign when adding the present values of all dates.

Worked Example: A Net Payment Followed by a Net Receipt

A hypothetical venture has a $1,400 receipt and a $1,850 cost due in one year. In two years, it has a $5,200 receipt, a $3,400 cost, and a $600 fee. The annual effective spot rate is 10% at both one-year and two-year maturities. Find the net present value of this schedule. State the method and identify the error that would result from combining the dates first.

Method. Find the signed net amount at each date before discounting. Retain the negative sign on the one-year net payment. The common trap is to combine amounts due at different dates as if they had one maturity.

Step 1: Net the one-year cash flows.

$$ C(1)=1{,}400-1{,}850=-450 $$

Its present value is

$$ \operatorname{PV}_1=\frac{-450}{1.10}=-409.090909\ldots $$

Step 2: Net the two-year cash flows.

$$ C(2)=5{,}200-3{,}400-600=1{,}200 $$

Its present value is

$$ \operatorname{PV}_2=\frac{1{,}200}{(1.10)^2} =\frac{1{,}200}{1.21} =991.735537\ldots $$

Step 3: Add the signed present values.

$$ \operatorname{PV} =-409.090909\ldots+991.735537\ldots =582.644628\ldots $$

The net present value is $582.64, rounded to the nearest cent.

Incorrect approach and correction. Adding the two date-specific net amounts first gives \(-450+1{,}200=750\). Discounting that total at the two-year rate would give \(750/1.21=$619.83\), which is incorrect: the $450 net payment is due in one year, not two. The correct calculation discounts \(-450\) at one year and \(1{,}200\) at two years before adding, giving $582.64.

Common Exam Traps

The arithmetic of grouping is simple, but a few details determine whether the setup is valid. Before calculating, check the due date and sign of every item. A list that gives receipts and costs as unsigned amounts still requires each cost to be entered as negative when forming a net cash flow.

  • Netting across dates. Combine amounts only if they have the same payment date. Separate dates can have different discount factors.
  • Using the wrong maturity’s spot rate. Match each date to its own discount factor. A spot rate at one maturity does not automatically apply at another.
  • Losing the sign of a payment. A cost reduces the date’s net amount. If costs exceed receipts, keep the resulting negative net cash flow negative through discounting.
  • Rounding too early. Carry full precision through the discounting and addition, then round the final money amount to cents.

This grouping method is useful whenever a cash-flow schedule has multiple line items due together. It reduces repeated arithmetic and makes the timing of each net payment or receipt explicit. The central check is simple: every group must contain one date only, and every group must use the discount factor for that date.

Key takeaway: Add signed cash flows that share a maturity, discount the resulting net amount using that maturity’s discount factor, and then add the present values across dates. The grouping is valid because all amounts in a date group have the same discount factor.

Check Your Understanding

For each schedule, combine cash flows by date before discounting. Round final dollar amounts to the nearest cent.

  1. At year 2, a hypothetical arrangement pays a $900 receipt, a $250 rebate, and a $500 cost. The annual effective two-year spot rate is 4%. Find the present value. (Answer: $600.96)
  2. At year 1, a project receives $2,500 and pays $800. At year 2, it receives $1,200 and pays costs of $300 and $200. The annual effective spot rates are 3% for one year and 4% for two years. Find the total present value. (Answer: $2,297.67)
  3. At 18 months, a hypothetical contract receives $3,000 and pays costs of $500 and $1,700. The annual effective spot rate for 18 months is 6%, with fractional-year accumulation \((1.06)^{1.5}\). Find the present value. (Answer: $733.05)
  4. At year 1, a project receives $1,000 and pays $200. At year 2, it receives $2,100 and pays $900 and $300. At year 3, it receives $2,500 and pays $700 and $500. The annual effective spot rates are 5%, 6%, and 7% for maturities of one, two, and three years, respectively. Find the total present value. (Answer: $2,624.09)

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