Reinvestment Shortfalls and Target Value Gaps

Introduction

A projected terminal value depends on an accumulation assumption: each cash flow is treated as though it can earn a specified rate until the investment horizon. If actual reinvestment rates differ, the wealth available at that horizon may not equal the projection. The difference is a target value gap. Measuring it requires following each cash flow from its own receipt date, rather than applying one accumulation period to the entire stream.

This analysis is useful when a stated yield or benchmark is being compared with realized reinvestment experience. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus. The central discipline is to mark cash-flow dates and rate intervals on one timeline, then accumulate each receipt only over the time it is actually invested.

Definition: For nonnegative receipts $C_k$ at times $t_k$ and a terminal horizon $T$, the target terminal value at assumed effective annual rate $i$ is the value obtained by accumulating each receipt to $T$ at $i$. The actual terminal wealth uses the actual reinvestment rates. Define the signed gap as target terminal value minus actual terminal wealth. A positive gap is a shortfall; a negative gap means actual wealth exceeds the target.

Accumulate Each Cash Flow Over Its Own Investment Period

Assume the benchmark rate $i$ is an effective annual rate and that fractional-year accumulation uses the factor $(1+i)^s$ over $s$ years. A receipt $C_k$ at time $t_k$, where $0\le t_k\le T$, therefore contributes $C_k(1+i)^{T-t_k}$ to the target at time $T$. The target is the sum of these separate contributions:

V_{\text{target}}(T)=\sum_k C_k(1+i)^{T-t_k}.

Actual rates may change over time. Suppose the rate $r_j$ applies on the interval $[u_{j-1},u_j]$, with fractional-year accumulation at that interval’s effective annual rate. A receipt may arrive inside an interval. In that case, it earns the interval’s rate only for the portion of the interval after the receipt arrives. The duration to use is the length of the overlap between the rate interval and the receipt’s investment period.

Formula: For a receipt at $t_k$, its actual accumulation factor to $T$ is the product of $(1+r_j)$ raised to the time spent in each rate interval after $t_k$. The exponent for interval $j$ is the length of $[u_{j-1},u_j]\cap[t_k,T]$; intervals with zero overlap contribute a factor of 1.

A_k(T)=\prod_{j:\,L_{kj}>0}(1+r_j)^{L_{kj}}, \qquad L_{kj}=\max\!\left(0,\min(u_j,T)-\max(u_{j-1},t_k)\right).

Thus actual terminal wealth and the signed gap are

V_{\text{actual}}(T)=\sum_k C_k A_k(T), \qquad G(T)=V_{\text{target}}(T)-V_{\text{actual}}(T).

The intervals must cover the investment horizon, and their rates must describe the accumulation convention in the problem. If the problem specifies a different convention for fractional periods, use that convention instead of the fractional-year effective-rate rule above. A cash flow received at $T$ has no time to accumulate: both its target and actual accumulation factors are 1.

Why the Overlap Exponent Is Necessary

The overlap rule follows from applying an effective annual rate only during the time the money is invested in the corresponding interval. For example, if an interval lasts from time 0 to time 1 but a receipt arrives at time 0.5, the receipt is invested in that interval for $1-0.5=0.5$ years. Under fractional-year effective accumulation, its factor over that portion is $(1+r)^{0.5}$, not 1 and not $1+r$.

For a receipt at $t_k$, partition the period from $t_k$ to $T$ into its overlaps with the rate intervals. The sum of the overlap lengths is $T-t_k$. Applying each interval’s rate to its own overlap gives the product in the actual accumulation formula. If every interval has the same rate $r$, the product reduces to $(1+r)^{T-t_k}$, as it should.

Mark the timeline.
Record every receipt date, every rate-change date, and the terminal horizon.

Find each receipt’s target contribution.
Accumulate that receipt from its own date to $T$ at the benchmark rate.

Find its actual contribution.
For each rate interval, use only the portion from the later of the receipt date and interval start to the earlier of the horizon and interval end.

Compare and check.
Subtract total actual wealth from total target value. Confirm that each receipt’s accumulation period ends at $T$.

Worked Calculations

Worked Example: Several Receipts and Annual Rate Changes

A fund receives $1,000 at time 0 and $500 at time 1. The terminal horizon is time 3. The target uses an effective annual rate of 10%. Actual effective annual reinvestment rates are 10% from time 0 to 1, 6% from time 1 to 2, and 4% from time 2 to 3. Find the target, actual terminal wealth, and gap.

Method. Calculate the target and actual accumulation for each receipt separately. All receipt and rate-change dates are integer times, so each applicable interval contributes one full year.

For the target, the $1,000 receipt is invested for 3 years and the $500 receipt for 2 years:

V_{\text{target}}(3) =1{,}000(1.10)^3+500(1.10)^2 =1{,}000(1.331)+500(1.21) =1{,}331+605 =\$1{,}936.00.

For actual wealth, the first receipt earns 10%, then 6%, then 4%. The second receipt arrives at time 1 and earns only the 6% and 4% rates:

\begin{aligned} A_0(3)&=(1.10)(1.06)(1.04)=1.21264,\\ A_1(3)&=(1.06)(1.04)=1.1024,\\ V_{\text{actual}}(3)&=1{,}000(1.21264)+500(1.1024)\\ &=1{,}212.64+551.20 =\$1{,}763.84. \end{aligned}

Gap and check. The signed gap is $1{,}936.00-1{,}763.84=\$172.16$, a shortfall. Receipt by receipt, the gaps are $1{,}331.00-1{,}212.64=\$118.36$ and $605.00-551.20=\$53.80$; these sum to $172.16. The common error avoided is applying the 3-year accumulation period to both receipts.

Worked Example: A Receipt Inside a Rate Interval

A $800 receipt arrives at time 0.5, and a further $400 arrives at time 1.5. The horizon is time 2. The target effective annual rate is 10%. The actual effective annual rate is 12% from time 0 to 1 and 4% from time 1 to 2. Use fractional-year effective accumulation and find the gap.

Method. Use the overlap duration for each receipt and rate interval. The $800 receipt overlaps the first rate interval for 0.5 years and the second for 1 year. The $400 receipt overlaps the second interval for 0.5 years.

The target contributions use durations $2-0.5=1.5$ years and $2-1.5=0.5$ years:

\begin{aligned} V_{\text{target}}(2) &=800(1.10)^{1.5}+400(1.10)^{0.5}\\ &=800(1.153689733)+400(1.048808848)\\ &=922.951786+419.523539\\ &=\$1{,}342.48. \end{aligned}

The actual factors reflect the intervals’ overlaps:

\begin{aligned} A_{0.5}(2)&=(1.12)^{0.5}(1.04)^1 =1.058300524(1.04)=1.100632545,\\ A_{1.5}(2)&=(1.04)^{0.5}=1.019803903. \end{aligned}

Therefore,

\begin{aligned} V_{\text{actual}}(2) &=800(1.100632545)+400(1.019803903)\\ &=880.506036+407.921561\\ &=\$1{,}288.43. \end{aligned}

Gap and check. Rounding money to the nearest cent only at the final step, $G(2)=1{,}342.48-1{,}288.43=\$54.05$. The $800 receipt’s actual accumulation includes the half-year from time 0.5 to 1; omitting it would incorrectly treat that interval as if the receipt had not been invested there.

Worked Example: Locating a Shortfall and a Later Partial Recovery

A fund receives $2,000 at time 0 and $1,000 at time 1. The horizon is time 3, and the target effective annual rate is 8%. Actual rates are 8% from time 0 to 1, 3% from time 1 to 2, and 9% from time 2 to 3. Find the terminal gap and identify when it develops.

Method. Track target and actual balances at each integer time. The receipt at time 1 is added to both balances at that time. This shows how the below-benchmark interval creates a gap and how the later above-benchmark interval affects it.

At time 1, both balances are $2{,}000(1.08)=\$2{,}160$. Adding the new $1,000 receipt gives $3,160 in each account. At time 2:

\begin{aligned} V_{\text{target}}(2)&=2{,}160(1.08)+1{,}000(1.08)=\$3{,}412.80,\\ V_{\text{actual}}(2)&=2{,}160(1.03)+1{,}000(1.03)=\$3{,}254.80,\\ G(2)&=3{,}412.80-3{,}254.80=\$158.00. \end{aligned}

The 3% interval creates a $158.00 gap by time 2. Over the final year, the target balance earns 8% and the actual balance earns 9%:

\begin{aligned} V_{\text{target}}(3)&=3{,}412.80(1.08)=\$3{,}685.824,\\ V_{\text{actual}}(3)&=3{,}254.80(1.09)=\$3{,}547.732,\\ G(3)&=3{,}685.824-3{,}547.732=\$138.092. \end{aligned}

Gap and check. To the nearest cent, the final shortfall is $138.09. It is smaller than the $158.00 gap at time 2 because the actual balance earns 9%, above the 8% target rate, during the last year. The final gap is also confirmed directly: the target is $2{,}000(1.08)^3+1{,}000(1.08)^2=\$3{,}685.824$, and actual wealth is $2{,}000(1.08)(1.03)(1.09)+1{,}000(1.03)(1.09)=\$3{,}547.732$.

Interpreting the Gap Carefully

When every receipt is nonnegative and every actual rate over its investment period is below $i$, each receipt’s actual accumulation factor is below its target factor. The total gap is then positive, provided at least one receipt has a positive amount and positive investment duration. However, if some actual rates are above $i$, the total sign cannot be inferred from just one low-rate interval. Later high rates may reduce an existing shortfall or even produce a surplus.

The gap is not generally the sum of rate differences multiplied by the original deposits. Each cash flow has its own investment duration, and rate differences compound through time. The reliable method is to calculate the target and actual terminal contribution for each receipt, then subtract. Keeping intermediate values unrounded also prevents small rounding errors from accumulating across several cash flows.

Common Mistake: Ignoring an Interior Receipt Date

Suppose a receipt arrives at time 0.5, the horizon is time 1, and the actual effective annual rate is 10% throughout $[0,1]$. An incorrect calculation may multiply only over whole rate intervals that start after the receipt date. It then finds no complete interval and assigns an accumulation factor of 1. That misses half a year of investment.

\text{Incorrect factor}=1, \qquad \text{correct factor}=(1.10)^{1-0.5}=(1.10)^{0.5}\approx1.048809.

The correction is to split the rate interval at the receipt date or, equivalently, use the overlap duration $1-0.5=0.5$ in the exponent. Always test the first interval after a receipt, especially when the receipt date does not match a rate-change date.

Key takeaway: A target value gap is a comparison of two terminal values, not a rate difference. Accumulate each receipt from its own date under the target and actual rate paths, use overlap durations for interior dates, and interpret a positive target-minus-actual result as a shortfall.

Check Your Understanding

For each problem, use the stated fractional-year effective accumulation convention where needed. Round monetary answers to the nearest cent.

A $500 receipt arrives at time 0 and is held to time 2. The target rate is 6% effective annually; actual rates are 5% in year 1 and 4% in year 2. Find target value, actual wealth, and gap. (Answer: target $561.80; actual $546.00; gap $15.80.)
A $900 receipt arrives at time 0.25 and is held to time 1. The target rate is 8%; the actual rate is 4% throughout $[0,1]$. Find the target value, actual wealth, and gap. (Answer: target $953.48; actual $926.87; gap $26.61.)
A fund receives $400 at time 0 and $300 at time 1, with horizon time 2. The target rate is 7%; the actual rate is 6% in each year. Find the gap. (Answer: target $778.96; actual $767.44; gap $11.52.)
A $1,000 receipt arrives at time 0.5 and is held to time 1.5. The target rate is 10%. Actual rates are 6% on $[0,1]$ and 12% on $[1,1.5]$. Find the signed gap. (Answer: target $1,100.00; actual approximately $1,089.59; gap approximately $10.41 shortfall.)

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