Reinvestment Exposure in Matched Portfolios

Why Reinvestment Exposure Matters

A portfolio can have enough total value to meet a liability and still depend on what happens to cash received before the liability is due. If an asset pays a coupon or principal amount before the liability date, that cash must either be used to meet a liability at the same date or be held, invested, or otherwise funded until it is needed. The return on that interim amount can change the portfolio’s eventual surplus.

The distinction is between matching cash flows on each date and having asset proceeds that must be carried from one date to another. In the first case, corresponding asset and liability payments offset directly. In the second, the terminal result can depend on the rates earned on interim positive balances, or the rates charged to fund interim shortfalls.

Definition: Let $A_t$ be the asset cash flow and $L_t$ the liability cash flow at the end of period $t$. Define the net residual cash flow as $C_t=A_t-L_t$. A positive $C_t$ is an excess asset cash flow; a negative $C_t$ is a shortfall. Exact cash-flow matching over dates $1,\ldots,H$ means $A_t=L_t$, and therefore $C_t=0$, at every date.

The calculations below treat cash flows as certain amounts in the same currency, occurring at stated dates. They exclude default, transaction costs, taxes, and other frictions. A reinvestment rate is an effective rate for the period specified. Unless stated otherwise, a negative residual is assumed to be financed at the same rate used to accumulate positive residuals. If borrowing and investment rates differ, each must instead be modeled with its own rate.

Exact Matching and Interim Residuals

With exact date-by-date matching, the assets produce precisely the cash needed for each liability payment. No interim excess needs to be carried forward to fund a later liability. The conclusion is about the timing and amounts of cash flows: it does not assert that the assets are free of default, operational, or other risks.

When flows do not match on a date, the residual $C_t$ must be accounted for. For a terminal horizon $H$, a positive residual received at time $t<H$ can accumulate to time $H$. A residual at time $H$ is already at the horizon and is not accumulated.

S_H=\sum_{t=1}^{H} C_t(1+j)^{H-t}

Here $S_H$ is the terminal surplus, and $j$ is a constant effective reinvestment rate per period. A positive $S_H$ means terminal assets exceed the terminal liability; a negative $S_H$ means a shortfall. This formula assumes that every signed residual is carried at the same rate. In particular, it treats a negative residual as a funded amount that accumulates at rate $j$.

If the effective rate differs by period, let $j_k$ be the rate from the end of period $k-1$ to the end of period $k$. A residual received at time $t$ accumulates through periods $t+1$ to $H$:

S_H=\sum_{t=1}^{H} C_t\prod_{k=t+1}^{H}(1+j_k)

For $t=H$, the product has no factors and is defined to equal 1. The order of the rates matters when rates differ, because each cash flow is exposed only to the rates applying after its receipt.

Why the Accumulation Formula Holds

Consider first a single residual $C_t$ received at time $t$, with a constant effective rate $j$. After one period it is $C_t(1+j)$; after two periods it is $C_t(1+j)^2$. Repeating this argument for the $H-t$ periods from time $t$ to time $H$ gives its terminal value:

C_t\underbrace{(1+j)(1+j)\cdots(1+j)}_{H-t\ \text{periods}} =C_t(1+j)^{H-t}

The terminal account value is the sum of the terminal values of the individual residuals. This follows from the additivity of cash balances: accumulating two amounts at the same rate gives the same result as accumulating their sum. Therefore, summing the accumulated value of each dated residual gives $S_H$. For period-specific rates, the corresponding factors are multiplied in chronological order, yielding the product formula above.

Key result: Exact matching makes every dated residual zero, so the residual-based terminal surplus is zero regardless of reinvestment rates. If residuals are nonzero before the horizon, their terminal value generally depends on the rates applied after those residuals arise.

Worked Example: Exact Matching Removes the Need to Reinvest Residuals

A hypothetical fund has liabilities of $110 at the end of year 1 and $121 at the end of year 2. Its assets pay $110 and $121 on those same dates. The annual effective valuation rate is 10%. Determine whether interim reinvestment is needed and verify the present value of the matched cash flows.

Method. First compare the asset and liability cash flows date by date. Then, separately, value the matched cash flows at the stated annual effective rate. This keeps the matching question distinct from the valuation-rate calculation.

At year 1, $A_1=110$ and $L_1=110$, so $C_1=110-110=0$. At year 2, $A_2=121$ and $L_2=121$, so $C_2=121-121=0$. Each asset payment is used to meet the liability due on the same date; there is no residual from either date to reinvest.

At 10%, the discount factor is $v=1/1.10$. The present value of the liabilities is

\frac{110}{1.10}+\frac{121}{(1.10)^2} =100+100 =\$200.00.

The assets have the same dated cash flows, so their present value is also $200.00. This confirms equal present values at the stated rate, while the date-by-date comparison establishes the stronger fact that each payment is matched directly. Check: both years have zero residual, so no choice of reinvestment rate changes the residual-based terminal surplus. The common mistake avoided is treating the first matched payment as an excess that must be reinvested.

Measuring the Effect of a Constant Reinvestment Rate

The constant-rate formula is useful when a portfolio produces a series of early excesses and shortfalls before a terminal liability. The calculation should start by netting assets and liabilities at each date. Accumulating gross asset receipts while forgetting to reflect liabilities paid on those dates can misstate what is actually available to carry forward.

Worked Example: A Partial Match with a Terminal Shortfall

A hypothetical portfolio has a three-year horizon. Its year-end asset and liability cash flows are, respectively, $120 and $100 in year 1; $80 and $50 in year 2; and $0 and $100 in year 3. Calculate the terminal surplus at a constant annual effective reinvestment and financing rate of 5%. Then compare it with the result at 10%.

Method. Find each year’s net residual, then accumulate it from its date to the end of year 3. The common mistake avoided is accumulating early receipts for the wrong number of periods.

The residuals are

C_1=120-100=20,\qquad C_2=80-50=30,\qquad C_3=0-100=-100.

At 5%, the year-1 residual is carried for two periods and the year-2 residual for one period. The year-3 shortfall is already at the horizon:

S_3=20(1.05)^2+30(1.05)-100 =20(1.1025)+31.50-100 =22.05+31.50-100 =-\$46.45.

Thus the terminal shortfall is $46.45. At 10%,

S_3=20(1.10)^2+30(1.10)-100 =24.20+33.00-100 =-\$42.80.

The higher rate reduces the shortfall because the two early residuals are positive and have more value at the horizon. It does not eliminate the shortfall: the accumulated positive residuals are still less than the $100 year-3 liability. Check: at either rate, the year-3 liability is not accumulated beyond year 3, while each earlier residual receives exactly the number of annual accumulation factors implied by its date.

The direction of the effect is not always the same. A higher reinvestment rate increases the terminal value of a positive early residual, but it also increases the magnitude of a negative residual carried forward as financing. With a mixture of positive and negative residuals, the overall effect depends on their amounts and dates.

Changing Rates and the Timing of Exposure

When reinvestment rates vary, each residual has a different exposure period. A cash flow received at the start of the horizon’s final period is exposed to only that final period’s rate. An earlier amount experiences every subsequent period’s rate. Consequently, one annual rate or an unweighted average of the rates generally cannot replace the period-by-period accumulation factors.

Worked Example: Comparing Two Reinvestment Scenarios

A hypothetical portfolio has residuals of $50 at the end of year 1, $25 at the end of year 2, and $-$80$ at the end of year 3. Compare its terminal surplus under two annual rate scenarios. In Scenario A, the effective rates from year 1 to 2 and year 2 to 3 are 4% and 6%. In Scenario B, they are 2% and 3%. Assume any shortfall is financed at the same rate as the positive balances earn.

Method. Apply the rate for each period after each residual’s date. The year-1 amount experiences both rates, the year-2 amount experiences only the second rate, and the year-3 amount is already at the horizon. This avoids applying a full two-year accumulation factor to the year-2 amount.

For Scenario A, the year-1 residual accumulates by $1.04$ and then $1.06$, while the year-2 residual accumulates by $1.06$:

S_3=50(1.04)(1.06)+25(1.06)-80 =55.12+26.50-80 =\$1.62.

For Scenario B,

S_3=50(1.02)(1.03)+25(1.03)-80 =52.53+25.75-80 =-\$1.72.

The terminal results differ by $1.62-(-1.72)=\$3.34$. Scenario A leaves a surplus of $1.62; Scenario B leaves a shortfall of $1.72. Check: in each scenario, the year-1 amount receives two factors, the year-2 amount one factor, and the year-3 amount none. The rates are scenario assumptions, not guaranteed returns established by the cash-flow schedule.

Common Mistakes and Practical Interpretation

Assuming a valuation rate is an earned rate. A rate used to discount cash flows is a valuation assumption. It does not establish the rate at which interim proceeds can actually be reinvested.
Ignoring dates. A residual received earlier has more accumulation periods than a residual received later. Use the time between its receipt and the horizon, not the total length of the horizon.
Accumulating gross cash flows instead of residuals. First record which asset and liability payments occur at each date. The amount available to carry forward is their net difference, subject to the specified financing convention.
Using a single rate for a changing-rate scenario. If period rates differ, multiply the applicable factors for each cash flow’s actual exposure period.
Assuming a higher rate always improves the result. A higher rate helps positive balances accumulate but can also increase the terminal burden of financed shortfalls.

Exact cash-flow matching is therefore a way to remove the need to reinvest interim residuals for the matched liabilities. A portfolio with unmatched interim cash flows instead has a terminal result conditional on how those amounts are carried. Comparing terminal surpluses across plausible rate scenarios makes that dependence visible; it does not turn an assumed reinvestment return into a guaranteed one. These distinctions are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, use asset cash flow minus liability cash flow as the dated residual. Unless a problem says otherwise, rates are effective annual rates and shortfalls are financed at the same rate as positive balances earn.

A fund has asset and liability cash flows of $75 and $75 at year 1, $90 and $90 at year 2, and $40 and $40 at year 3. Are there residuals to reinvest, and what is the terminal residual-based surplus? (Answer: No residuals need reinvestment; terminal surplus is $0.00.)
At the ends of years 1, 2, and 3, a portfolio has residuals of $14, $20, and $-$40$. Find its year-3 terminal surplus at a constant 5% annual effective rate. (Answer: $-\$3.57$.)
Residuals are $25 at year 1, $15 at year 2, and $-$45$ at year 3. The effective rates from year 1 to 2 and year 2 to 3 are 3% and 4%. Find the terminal surplus. (Answer: $-\$2.62$.)
A portfolio has residuals of $40 at year 1, $30 at year 2, and $-$75$ at year 3. Compare its terminal surplus at constant rates of 0% and 5%. (Answer: At 0%, $-\$5.00$; at 5%, $0.60.)
Residuals are $20 at year 1, $30 at year 2, and $-$57.20$ at year 3. Under a constant annual effective rate $j$, determine the nonnegative rate that produces a zero terminal surplus. (Answer: $j=10\%$.)

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