Matching Cash-Flow Timing and Reinvestment Needs

Why Payment Dates Matter

A portfolio can have assets whose value is adequate relative to its liabilities and still face a cash shortage on a particular date. The issue is timing: an asset payment that arrives after a liability is due cannot pay that liability unless funds are borrowed or another source of cash is available. Conversely, a payment received early creates a balance that must be held or reinvested until it is needed.

The calculations in this tutorial track cash at each payment date. Assume first that all cash flows occur at the stated annual dates, that interest is earned or charged at the stated effective rate for each year, and that there are no fees, taxes, defaults, or other cash flows. A positive balance is available cash; a negative balance is a funding deficit. When borrowing is shown, assume the fund can borrow the required amount at the specified rate. These assumptions matter: if borrowing and reinvestment rates differ, or if transaction costs apply, each must be included separately.

Definition: Let $B_k$ be the fund balance immediately after all cash flows at time $k$. Let $C_k$ be the net cash flow at that time, defined as asset payments received minus liability payments made. If the effective rate from time $k-1$ to $k$ is $i_k$, then $$ B_k=B_{k-1}(1+i_k)+C_k. $$ A positive $B_k$ is a surplus; a negative $B_k$ is a shortfall. If a shortfall is funded by borrowing, its absolute value is the amount that must be borrowed at that date.

The timing convention is important: the existing balance earns interest over the interval, and then the cash flows at the end of that interval are added or subtracted. An asset payment and a liability due at the same date can therefore be netted in $C_k$, provided both are available at that time.

Carrying a Balance Between Payment Dates

When a cash payment arrives before it is needed, the balance does not stay unchanged. It accumulates at the applicable reinvestment rate. When a liability arrives before the assets intended to fund it, the balance becomes negative unless the fund has other cash or borrows. The same balance equation handles both situations.

B_k=B_{k-1}(1+i_k)+C_k

For a constant annual effective rate $i$ and dates one year apart, this becomes $B_k=B_{k-1}(1+i)+C_k$ at every date. If rates vary by year, use the corresponding $i_k$ for each interval. The balance at one date is the starting balance for the next interval; it should not be replaced by a fresh present-value comparison that ignores the cash already carried forward.

Derivation: Over the interval from $k-1$ to $k$, the balance $B_{k-1}$ earns $i_kB_{k-1}$ in interest. Immediately before the time-$k$ flows, it is therefore $B_{k-1}+i_kB_{k-1}=B_{k-1}(1+i_k)$. Adding the time-$k$ net cash flow $C_k$ gives the stated recurrence. Repeated substitution gives $$ B_n=B_0\prod_{j=1}^{n}(1+i_j)+\sum_{k=1}^{n}C_k\prod_{j=k+1}^{n}(1+i_j). $$ The empty product for $k=n$ is 1, since a cash flow received at time $n$ earns no interest before the balance is measured at that same time. This expression shows why an early payment is multiplied by the accumulation factors for all later intervals, while a payment at the final date is not accumulated.

Worked Examples

Worked Example: An Early Payment and a Changing Reinvestment Rate

A fund receives $12,000 at time 1 and must pay $13,230 at time 3. There are no other cash flows. The effective reinvestment rate is 4% from time 1 to time 2 and 5% from time 2 to time 3. Find the balance after the liability is paid. Then find the rate required in the second year to meet the liability exactly if the first-year rate remains 4%.

Method. Carry the early asset payment forward one year at a time, then subtract the liability at its due date. The rates are specific to each interval, so do not apply one rate for both years.

At time 1, the fund receives $12,000, so $B_1=$12{,}000$. At time 2, it earns 4%:

B_2=12{,}000(1.04)=$12{,}480.

The balance then earns 5% from time 2 to time 3. Immediately before the liability is paid, it is

$$ 12{,}480(1.05)=$13{,}104. $$

After paying $13,230, the balance is

B_3=13{,}104-13{,}230=-$126.

The fund has a $126 shortfall at time 3. To find the second-year rate $r$ that would exactly fund the liability, use the time-2 balance:

12{,}480(1+r)=13{,}230, \qquad r=\frac{13{,}230}{12{,}480}-1 =\frac{750}{12{,}480} \approx 0.060096.

Thus the required second-year effective rate is approximately 6.01%, rounded to two decimal places. Check: $12{,}480(1.06009615)\approx $13{,}230$, so the payment is met. A common mistake is to apply 5% in both years: that gives $12{,}000(1.05)^2=$13{,}230$, but it ignores the specified 4% first-year rate.

Worked Example: Two Asset Payments Before One Liability

A fund receives $5,000 at time 1 and $7,000 at time 2. It owes $13,000 at time 3. Assume a 6% effective annual rate throughout, and that any balance is reinvested at that rate. Find the balance immediately after the time-3 liability is paid.

Method. Update the fund balance at each date, carrying the first receipt forward before adding the second. This avoids treating cash available at time 2 as though it had been available for the whole three years.

At time 1, the first asset payment gives $B_1=$5{,}000$. At time 2, that balance has earned one year of interest, and the second payment is then received:

B_2=5{,}000(1.06)+7{,}000 =5{,}300+7{,}000 =$12{,}300.

From time 2 to time 3, the balance earns another year of interest:

$$ 12{,}300(1.06)=$13{,}038. $$

After the $13,000 liability is paid, the balance is

B_3=13{,}038-13{,}000=$38.

The result is a $38 surplus at time 3, not a shortfall. Check: the time-2 balance of $12,300 grows by $738, and $12{,}300+738-13{,}000=38$. The common mistake is to compare $12,300 directly with the liability and call the $700 difference a shortage. That comparison is at the wrong date; the liability is not due until time 3, so the balance must first accumulate for one year.

Worked Example: Borrowing to Bridge a Timing Gap

A fund receives $4,000 at time 1, owes $8,200 at time 2, and receives $4,200 at time 3. The fund earns 5% on positive balances and can borrow at 5%. Determine the amount it must borrow at time 2 and whether the time-3 asset payment repays that borrowing in full.

Method. First accumulate the time-1 asset payment to time 2. Borrow only the amount needed to bring the available cash up to the liability. Then accumulate the loan debt to its repayment date and compare it with the later asset payment.

The $4,000 received at time 1 grows for one year to

$$ 4{,}000(1.05)=$4{,}200 $$

at time 2. The liability is $8,200, so the cash deficit at that date is

$$ 8{,}200-4{,}200=$4{,}000. $$

Borrowing $4,000 makes exactly $8,200 available to pay the liability. The fund has no remaining cash immediately after that payment. The loan accrues 5% for one year, so the amount due at time 3 is

$$ 4{,}000(1.05)=$4{,}200. $$

The time-3 asset payment is also $4,200, so it repays the loan exactly and leaves a zero balance. Check: the liability is fully paid at time 2, and the later asset payment equals the loan repayment at time 3. The common mistake is to compare the later $4,200 asset payment with the $4,000 loan principal and conclude that $200 is left over. Interest is also owed, so the relevant comparison is with the $4,200 loan balance due at time 3.

Practical Checks and Limits

A cash-flow schedule should make the date of every balance explicit. In particular, distinguish a balance immediately before a payment from the balance immediately after it. A fund may have enough cash just before a liability is paid and have a zero balance just afterward; those are different statements about its position.

For each date, apply the same sequence: accumulate the previous balance over the interval, add asset receipts, subtract liabilities, and record the resulting balance. If the result is negative, that is the amount of financing needed at that date under the stated model. If the fund borrows, continue carrying the negative balance forward at the borrowing rate. Do not assume that a later asset payment cancels an earlier deficit without accounting for interest on the borrowing.

List the dates and cash flows.
Record receipts as positive amounts and payments as negative amounts. Keep every amount attached to its actual payment date.

Accumulate the existing balance.
Apply the rate for the interval that has just elapsed. If rates vary, use each interval's stated rate rather than a single rate for the whole horizon.

Add the flows at the date.
Calculate the post-flow balance. A negative result is a funding gap; a positive result is cash that must be held or reinvested.

Carry any financing forward.
Accumulate a loan at its borrowing rate and compare the debt due with later asset receipts. State any residual surplus or shortfall.

This date-by-date analysis answers a liquidity question: can cash be made available when a payment is due? Present-value and duration comparisons are useful for assessing values and sensitivity, but they do not by themselves establish that cash arrives on every liability date. A portfolio can have a favorable value comparison and still require interim borrowing if its asset payments arrive too late. Conversely, early asset receipts create reinvestment exposure: the eventual funding result depends on the rates earned before the liability date.

The simple recurrence assumes the stated rates are actually available for the relevant interval and that the fund can reinvest or borrow as modeled. If reinvestment and borrowing rates differ, use the reinvestment rate for positive balances and the borrowing rate for negative balances. If the balance changes sign within an interval, the calculation must reflect the dates of those changes rather than applying one rate indiscriminately. Likewise, irregular payment dates require rates and accumulation periods appropriate to those intervals.

Key takeaway: Match cash-flow timing by tracking the balance at each payment date. Early receipts must earn enough before the liability date; late receipts require a bridge whose accumulated repayment must be checked against the later cash available.

Check Your Understanding

For each problem, track the cash balance to the date stated. Assume there are no unlisted flows or costs.

A fund receives $6,000 at time 1 and owes $6,615 at time 3. It reinvests at an effective annual rate of 5% in both intervals. What is the balance after the liability is paid? (Answer: $0.)
A fund receives $8,000 at time 1 and owes $8,500 at time 2. The effective annual rate is 4%. What is the funding shortfall at time 2? (Answer: $180.)
A fund receives $3,000 at time 1 and $4,000 at time 2, then owes $7,500 at time 3. The effective annual rate is 3%. Find the balance after the time-3 payment. (Answer: a $197.30 shortfall.)
A fund has $2,500 at time 0, receives $3,000 at time 1, and owes $5,700 at time 2. The effective rate is 4% from time 0 to 1 and 6% from time 1 to 2. Find the balance after the liability is paid. (Answer: a $236 surplus.)
A fund receives $6,000 at time 1 and owes $10,000 at time 2. It can borrow at 5% and receives $4,400 at time 3. How much must it borrow at time 2, and what is the balance after the time-3 receipt repays the loan? (Answer: borrow $3,700 at time 2; the loan is $3,885 at time 3, leaving a $515 surplus.)

What You'll Learn