Building a Liability-Funding Ladder

Why Build a Liability-Funding Ladder?

A portfolio can have enough assets in total and still lack cash on a particular liability date. A maturity ladder makes that timing issue visible: at each date, it places asset cash available alongside the liability due, then records the resulting surplus or shortfall. This is useful when liabilities occur successively and asset maturities do not line up exactly with them.

A ladder is a cash-flow schedule, not by itself a proof that a portfolio is safe under every interest-rate outcome. Its conclusions depend on the assumed dates and amounts of the cash flows, and on what rate is available to reinvest a surplus or finance a deficit between dates. In the examples below, a single effective annual rate is used as a clear benchmark.

Definition: At liability date $t_k$, let $A_k$ be the amount of asset cash flow available at that date and $L_k$ the liability payment due. The bucket surplus is $C_k=A_k-L_k$. A positive $C_k$ is an excess of cash at that date; a negative $C_k$ is a funding shortfall. A liability-funding ladder lists these dated asset and liability amounts, bucket surpluses, and—when relevant—the cumulative surplus or shortfall carried from one date to the next.

Set Up the Buckets Before Calculating

Choose the valuation date and list liability dates in chronological order. For each date, include asset proceeds that are actually available then, not merely assets whose market value appears sufficient in aggregate. Subtract the liability payment from the available asset cash flow to obtain that date’s bucket surplus. If there are several payments on the same date, add the asset cash flows together and add the liability payments together before taking the difference.

A bucket surplus is local to one date. It does not automatically pay a liability due later: the cash must be held, reinvested, or otherwise made available. Likewise, a deficit at one date cannot be cured by an asset maturing later unless a borrowing or bridge-funding arrangement covers the intervening period.

Date	Asset cash available	Liability due	Bucket surplus
$t_1$	$A_1$	$L_1$	$C_1=A_1-L_1$
$t_2$	$A_2$	$L_2$	$C_2=A_2-L_2$
$\cdots$	$\cdots$	$\cdots$	$\cdots$
$t_n$	$A_n$	$L_n$	$C_n=A_n-L_n$

Worked Example: Reading the Surplus in Each Bucket

A hypothetical fund has liability payments of $5,000 at year 1, $6,000 at year 2, and $4,000 at year 3. Its assets are scheduled to pay $5,300, $5,700, and $4,500 at those respective dates. Find each bucket surplus and explain what the signs mean. No interest assumption is needed to calculate these date-specific differences.

Method. Subtract the liability due from the asset cash available at each date:

\begin{aligned} C_1&=5{,}300-5{,}000=300,\\ C_2&=5{,}700-6{,}000=-300,\\ C_3&=4{,}500-4{,}000=500. \end{aligned}

At year 1 there is $300 of excess cash. At year 2 there is a $300 shortfall on that date, and at year 3 there is $500 of excess cash. These are bucket results; without a reinvestment or financing assumption, they should not be added as though each dollar had the same value and availability at every date.

Check and mistake avoided. The total asset cash flows are $15,500 and total liabilities are $15,000, a nominal difference of $500. That total does not show the year-2 shortfall. Treating the $500 overall excess as proof that every liability is funded on its due date would ignore the timing gap.

Carrying a Surplus or Shortfall Between Dates

To evaluate the effect of timing gaps, suppose the dates are annual and any positive balance earns an effective annual rate $i$. For this benchmark, suppose a negative balance can also be financed at that same rate. Let $B_k$ be the cumulative balance immediately after the cash flows at date $k$, with $B_0=0$. A positive balance is accumulated surplus; a negative balance is a cumulative funding shortfall.

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

This recursion first carries the previous balance forward for one year, then adds the current bucket surplus. If dates are separated by different numbers of years, the carry factor for that interval is $(1+i)^{t_k-t_{k-1}}$, assuming $t_k$ is measured in years and the same effective annual rate applies throughout.

Assumption: The carry calculation is a benchmark, not a guarantee. It assumes that surplus cash can earn $i$ and any deficit can be financed at $i$, with no transaction costs, liquidity constraints, or change in the rate. If actual reinvestment and borrowing rates differ, or financing is unavailable, the balance must be evaluated under those actual conditions instead.

Why the Accumulated-Balance Formula Works

The recursion can be expanded to show how each dated bucket contributes to the balance at the final date. At date 1, $B_1=C_1$. At date 2, the first bucket has earned one year of interest before $C_2$ is added, so $B_2=C_1(1+i)+C_2$. Repeating the same reasoning gives the general expression:

B_n=C_1(1+i)^{n-1}+C_2(1+i)^{n-2}+\cdots+C_{n-1}(1+i)+C_n.

Each bucket is carried only for the time remaining until date $n$. This explains why adding dated surpluses without adjustment can misstate the final funding position: an earlier surplus has more time to earn interest, while an earlier deficit has more time to accumulate financing cost.

Worked Example: Carrying a Surplus Through a Later Deficit

Use the three annual buckets from the preceding example: $300 at year 1, $-\$300$ at year 2, and $500 at year 3. Assume the effective annual rate is $i=4\%$, and assume both reinvestment and financing occur at that rate. Find the cumulative balance after each date and at year 3.

Method. Apply the annual carry recursion, keeping the sign of each bucket:

\begin{aligned} B_1&=0(1.04)+300=300,\\ B_2&=300(1.04)-300=12,\\ B_3&=12(1.04)+500=512.48. \end{aligned}

The initial $300 earns $300(0.04)=\$12$ over the first year. At year 2, the $300 shortfall uses all but $12 of that accumulated amount. The remaining $12 then earns $12(0.04)=\$0.48$, and the year-3 bucket adds $500. The final cumulative balance is therefore $512.48.

Check. Carry each bucket directly to year 3:

$$ 300(1.04)^2-300(1.04)+500 =324.48-312+500 =512.48. $$

Common mistake avoided. Adding $300-300+500$ gives $500, but that ignores interest earned on the year-1 surplus. The carry calculation correctly accounts for the different dates.

What a Maturity Gap Means in Practice

A gap occurs when asset cash is not available on a liability date. If assets mature before a liability, the proceeds must be held or reinvested until the liability is due. The amount available later then depends on the reinvestment terms. If assets mature after a liability, the liability needs another source of cash on its due date; the later asset proceeds may repay that bridge funding, but they do not remove the need to fund the earlier gap.

A ladder therefore helps distinguish two questions. First, does each date’s scheduled asset cash flow cover the liability due on that date? Second, if not, what amount must be reinvested or financed, and for how long? The second question is sensitive to the assumed rate. A flat-rate carry calculation is useful for measuring the gap under a stated assumption, but it is not a claim that the same rate will actually be available.

Worked Example: Measuring a Gap Before an Asset Matures

A hypothetical fund owes $4,000 at year 1, $3,000 at year 2, and $4,000 at year 3. Its assets pay $4,200 at year 1, nothing at year 2, and $9,000 at year 3. Assume an effective annual borrowing and reinvestment rate of $5\%$. Find the cumulative balance after each date and identify the bridge-funding need.

Method. First calculate the buckets:

C_1=4{,}200-4{,}000=200,\qquad C_2=0-3{,}000=-3{,}000,\qquad C_3=9{,}000-4{,}000=5{,}000.

Now carry the balance at $5\%$ per year:

\begin{aligned} B_1&=200,\\ B_2&=200(1.05)-3{,}000=210-3{,}000=-2{,}790,\\ B_3&=-2{,}790(1.05)+5{,}000=-2{,}929.50+5{,}000=2{,}070.50. \end{aligned}

At year 2, only $210 is available from carrying the year-1 surplus, so the fund needs $2,790 of bridge financing to pay the $3,000 liability. Under the stated assumption, that deficit grows to $2,929.50 by year 3. The year-3 net inflow of $5,000 repays it and leaves $2,070.50.

Check. Carry the three dated buckets directly to year 3:

200(1.05)^2-3{,}000(1.05)+5{,}000 =220.50-3{,}150+5{,}000 =2{,}070.50.

Common mistake avoided. The assets pay $13,200 in total, more than the $11,000 total liabilities. That does not mean the year-2 payment is available without borrowing: no asset matures at year 2, and the carried year-1 surplus is only $210.

Build and Review a Ladder Systematically

A reliable ladder calculation separates the date-by-date cash-flow comparison from the assumption used to move balances across dates. The following process helps make the timing and signs explicit.

List the dates.
Order all liability payment dates chronologically and specify the time between successive dates.

Record cash available.
Include only asset proceeds available by each date, together with any cash reserve assigned to the ladder.

Calculate local buckets.
For each date, subtract the liability due from asset cash available. Preserve negative signs for deficits.

State the carry assumption.
Specify the effective rate and whether it applies to reinvested surplus, financed deficit, or both.

Track cumulative balances.
Carry the prior balance over the interval, then add the current bucket. Identify the date and size of any financing need.

Check the final balance.
Accumulate each bucket separately to the final date and compare that result with the recursion.

The final check is especially useful in a multi-date problem. A wrong sign on a deficit, or an incorrect number of periods of accumulation, can produce a plausible-looking but incorrect total. Writing the carried value of each bucket makes both errors easier to detect.

Key takeaway: A ladder reveals whether cash is available on each liability date, while the cumulative balance measures the effect of carrying earlier surpluses or deficits under stated rate assumptions. Aggregate asset value alone does not establish date-by-date funding.

Check Your Understanding

For each problem, calculate bucket surpluses first. Apply the stated carry assumption only where requested.

A fund has liabilities of $2,000 at year 1 and $3,000 at year 2, and asset proceeds of $2,200 and $2,700 at those dates. Find both bucket surpluses and the year-2 cumulative balance, assuming 4% effective annually for carrying the year-1 balance. (Answer: $C_1=\$200$, $C_2=-\$300$, and $B_2=\$200(1.04)-\$300=-\$92$.)
Bucket surpluses are $150 at year 1, $-\$100$ at year 2, and $250 at year 3. At 3% effective annually, find the cumulative balance after year 3. (Answer: $150(1.03)^2-100(1.03)+250=\$306.14$.)
A liability of $5,000 is due at year 1. No asset matures then; an asset pays $5,400 at year 2, with no liability due at year 2. Assuming a 5% annual borrowing rate and no other cash, what is the year-2 balance after the asset proceeds arrive? (Answer: $-\$5{,}000(1.05)+\$5{,}400=\$150$.)
Three annual bucket surpluses are $400, $-\$500$, and $200. Find the year-3 cumulative balance at an effective annual rate of 6%. (Answer: $400(1.06)^2-500(1.06)+200=\$119.44$.)

Date	Asset cash available	Liability due	Bucket surplus
\(t_1\)	\(A_1\)	\(L_1\)	\(C_1=A_1-L_1\)
\(t_2\)	\(A_2\)	\(L_2\)	\(C_2=A_2-L_2\)
\(\cdots\)	\(\cdots\)	\(\cdots\)	\(\cdots\)
\(t_n\)	\(A_n\)	\(L_n\)	\(C_n=A_n-L_n\)

What You'll Learn