Matching Performance Measures to Decision Horizons

Why the Decision Horizon Matters

Annual RAROC is a ratio for assessing performance over a period, commonly one year. A business decision, however, may commit capital for several years, produce earnings unevenly, or release capital as risk runs off. Comparing businesses only by one year’s RAROC can therefore miss when value is earned and how long capital remains tied up. The appropriate comparison depends on the decision horizon: a one-year operating decision calls for a period measure, while a multi-year commitment calls for a measure that accounts for the full sequence of projected results.

The key is not to replace RAROC with one universally superior measure. Instead, use annual RAROC to interpret each period’s return relative to the capital supporting it, and use a multi-period value measure when the decision concerns results across several periods. Both measures require consistent definitions of earnings, capital, and the hurdle rate.

Definition: For period $t$, let $P_t$ be the period’s projected risk-adjusted earnings and $K_t$ the economic capital assigned to support the business during that period. Define annual RAROC as $P_t/K_t$, when $K_t>0$. For a capital hurdle rate $h$, period economic profit is $EP_t=P_t-hK_t$. If the period’s earnings and capital are measured on a compatible basis, then $EP_t=K_t(\operatorname{RAROC}_t-h)$.

The earnings measure and hurdle must be compatible. For example, a pre-tax earnings numerator should not be compared with a post-tax hurdle without an explicit adjustment. Capital must also be measured consistently across periods: if it represents capital required to support the coming year, the earnings and capital charge should refer to that same year. Economic capital is an input to these calculations; how it was estimated is a separate modeling question.

From Annual Performance to Multi-Period Value

A multi-period decision requires a sequence of period results, not a single ratio. One common comparison is the present value of economic profits. Assume each $EP_t$ is earned at the end of year $t$, and let $r$ be the effective annual discount rate. For a decision evaluated over $n$ years, define its discounted economic-profit value as:

V_{\mathrm{EP}}=\sum_{t=1}^{n}\frac{EP_t}{(1+r)^t} =\sum_{t=1}^{n}\frac{P_t-hK_t}{(1+r)^t}.

This comparison treats economic profit as the value contribution in each period and discounts later contributions. It does not mean that all firms must use the same $r$ and $h$: the rates should reflect the organization’s stated valuation and capital-charge framework. If the horizon includes years after the stated projection, the analyst must also specify how remaining earnings, capital, and risk are treated. Omitting a material run-off period can distort a comparison.

Derivation: By definition, $\operatorname{RAROC}_t=P_t/K_t$. Multiplying the difference between RAROC and the hurdle by capital gives $K_t(\operatorname{RAROC}_t-h)=K_t(P_t/K_t-h)=P_t-hK_t=EP_t$. Discounting this identity period by period gives $V_{\mathrm{EP}}=\sum_{t=1}^{n}K_t(\operatorname{RAROC}_t-h)/(1+r)^t$. Thus, a period’s value contribution depends on both its return above the hurdle and the amount of capital supporting it; the ratio alone does not give total value.

A present value of economic profits is not automatically interchangeable with a discounted cash-flow valuation. The two approaches require internally consistent definitions of earnings, capital investment, capital release, and discounting. In particular, do not subtract the full capital balance from economic profit and then also charge for that capital in the same valuation unless the framework explicitly calls for both. The period capital charge $hK_t$ already represents the required return on capital in the economic-profit calculation.

Worked Examples

Worked Example: Equal Annual RAROC, Different Capital Profiles

Two hypothetical business lines are evaluated for two years. All amounts are in thousands of dollars. The projected earnings and capital are deterministic inputs for this comparison; no mortality assumption or probability distribution is used, and the stated capital figures already reflect the relevant risk assessment. Both lines use a 10% annual capital hurdle, and the effective annual discount rate is 10%. Earnings occur at each year-end, and $K_t$ supports that year’s earnings.

Line A has earnings of 12 in each year and capital of 100 in each year. Line B has earnings of 6 and 18, and capital of 50 and 150, respectively. Compare their annual RAROC and their two-year discounted economic profit. The method is to calculate each year’s ratio and economic profit first, then discount and sum the economic profits. This avoids treating a ratio as if it were a multi-year value.

Line A, year 1: $\operatorname{RAROC}_1=12/100=0.12=12\%$. Its economic profit is $12-(0.10)(100)=12-10=2$.

Line A, year 2: $\operatorname{RAROC}_2=12/100=12\%$, and $EP_2=12-10=2$.

Line B, year 1: $\operatorname{RAROC}_1=6/50=0.12=12\%$. Its economic profit is $6-(0.10)(50)=6-5=1$.

Line B, year 2: $\operatorname{RAROC}_2=18/150=0.12=12\%$, and $EP_2=18-(0.10)(150)=18-15=3$.

For Line A, the discounted value is $2/1.10+2/(1.10)^2=1.818182+1.652893=3.471074$ thousand dollars. For Line B, it is $1/1.10+3/(1.10)^2=0.909091+2.479339=3.388430$ thousand dollars. The difference, calculated from the unrounded values, is $3.471074-3.388430=0.082645$ thousand dollars. Rounded to the nearest cent, Line A’s discounted economic profit is $3,471.07, Line B’s is $3,388.43, and Line A is ahead by $82.64.

Check: Each line has 12% RAROC in both years, above the 10% hurdle, so every annual economic profit is positive. The present values are also positive. The ranking reflects the amount and timing of capital-supported economic profit, not a difference in annual RAROC.

Common mistake avoided: Concluding that the lines have equal multi-year value because their annual RAROC is the same. RAROC does not measure the dollars of economic profit generated, and the two lines have different capital and earnings profiles.

Worked Example: Discounting Uneven Economic Profits

Two hypothetical strategies have projected annual economic profits, already net of their annual capital charges, in thousands of dollars. Strategy A produces 10 at the end of each of years 1 and 2. Strategy B produces 5 at the end of year 1 and 15 at the end of year 2. Assume deterministic projections, no mortality assumption or probability distribution, and a 10% effective annual discount rate. The decision horizon is exactly two years, with no terminal value. Which strategy has the higher discounted economic-profit value?

The method is to discount each year’s contribution to the decision date and sum. For Strategy A:

V_A=\frac{10}{1.10}+\frac{10}{(1.10)^2} =9.090909+8.264463=17.355372

For Strategy B:

V_B=\frac{5}{1.10}+\frac{15}{(1.10)^2} =4.545455+12.396694=16.942149

The difference using the unrounded values is $17.355372-16.942149=0.413223$ thousand dollars. Rounded to the nearest cent, Strategy A is worth $17,355.37, Strategy B is worth $16,942.15, and A’s advantage is $413.22, or $0.41 thousand. Both strategies have the same undiscounted total economic profit, 20 thousand dollars, but A earns more of it earlier.

Check: Strategy A’s discounted value is greater than Strategy B’s, as expected when equal total contributions are shifted from a later year to an earlier year at a positive discount rate.

Common mistake avoided: Subtracting the individually rounded values in thousands, $17.36-16.94=0.42$, and reporting a $0.42-thousand difference. The correct difference is calculated before rounding: $0.413223\ldots$ thousand, which rounds to $0.41 thousand.

Worked Example: A Weak First Year and a Strong Run-Off Year

A hypothetical run-off business has projected earnings of 8 thousand dollars in year 1 and 18 thousand dollars in year 2. Capital supporting the respective years is 100 and 50 thousand dollars. A comparison business earns 11 thousand dollars each year and requires 100 thousand dollars of capital in each year. Use a 10% annual hurdle and a 10% effective annual discount rate. Earnings are deterministic, occur at year-end, and include all projected expenses and losses; no mortality assumption or probability distribution is used. Assume the stated two-year horizon captures the full value of both businesses.

First calculate the run-off business’s annual RAROC. In year 1, $8/100=8\%$. In year 2, $18/50=36\%$. Its annual economic profits are $8-(0.10)(100)=-2$ thousand and $18-(0.10)(50)=13$ thousand. Its discounted economic-profit value is:

V_{\mathrm{runoff}}=\frac{-2}{1.10}+\frac{13}{(1.10)^2} =-1.818182+10.743802=8.925620

The comparison business has RAROC $11/100=11\%$ in each year and economic profit $11-(0.10)(100)=1$ thousand in each year. Its discounted value is $1/1.10+1/(1.10)^2=0.909091+0.826446=1.735537$ thousand dollars.

The run-off business therefore has the larger two-year discounted economic-profit value, by $8.925620-1.735537=7.190083$ thousand dollars, or $7,190.08 to the nearest cent. Yet its year-1 RAROC is 8%, below the 10% hurdle, while the comparison business earns 11% in year 1. A one-year ranking and a full-horizon value comparison answer different questions.

Check: The run-off business’s first-year economic profit is negative, but its discounted second-year economic profit more than offsets that amount. The total discounted value is positive and exceeds the comparison business’s value.

Common mistake avoided: Rejecting the run-off business solely because its first-year RAROC is below the hurdle. That conclusion would ignore the specified second-year earnings and capital release; it is not a valid full-horizon comparison.

Choosing and Applying the Measure

The period measure and the horizon measure answer related but distinct questions. Annual RAROC asks how much risk-adjusted earnings were generated relative to the capital assigned to one period. Discounted economic profit asks how much value a sequence of period returns above the capital hurdle contributes over a defined horizon. A high RAROC in one period need not imply the greatest total value: the capital base may be small, the result may not persist, or much of the business’s value may arise in other periods.

A disciplined comparison follows a consistent sequence:

Set the decision horizon.
State the periods being compared and whether the horizon includes run-off or a terminal value. A partial horizon should not silently omit material future results.

Align earnings and capital.
For each period, specify the earnings basis and the capital supporting that period. Use a consistent tax and measurement basis across the alternatives.

Calculate period performance.
Compute $\operatorname{RAROC}_t=P_t/K_t$ and $EP_t=P_t-hK_t$ for each period. Check that $EP_t=K_t(\operatorname{RAROC}_t-h)$.

Aggregate for the decision.
If the decision is multi-period, discount each period’s economic profit using the stated rate and timing convention, then sum. Compare unrounded totals before rounding the result.

Discounting later profits less heavily is not a claim that those profits are less important in every respect; it is the consequence of the specified time value of money. Nor does a positive multi-period value guarantee that every year has a positive economic profit. The period-by-period results remain useful for understanding when the business consumes or creates value, while the total measures the result over the chosen horizon.

Common Traps

Ranking a multi-year commitment using only one year. A single-year RAROC does not include future earnings, capital requirements, or run-off unless those are represented in the period being measured.
Adding RAROC percentages across years. Ratios are not dollar contributions and are not additive. Calculate economic profit or another explicitly defined value measure for each period.
Mixing periods or denominators. Pairing current earnings with capital from a different time or scope can make the ratio uninterpretable. State which period the capital supports.
Discounting at an unstated time convention. If results occur at year-end, the year-$t$ discount factor is $1/(1+r)^t$, not $1/(1+r)^{t-1}$.
Rounding too early. Retain unrounded values through aggregation and subtraction, then round the final reported amount as required.
Counting capital twice. In an economic-profit framework, the capital charge is deducted from earnings. Do not also subtract the entire capital balance without a coherent valuation treatment.

Key takeaway: Match the performance measure to the decision. Annual RAROC describes a period’s return on its supporting capital; a multi-period decision needs a complete, consistently defined sequence of results and an explicit method for aggregating their value over time.

Check Your Understanding

For each problem, assume the stated inputs are deterministic projections and use the specified timing and discount conventions.

A business produces economic profits of $4,000 and $6,000 at the ends of years 1 and 2. At an effective annual discount rate of 5%, calculate the two-year discounted value, rounded to the nearest cent. (Answer: $9,251.70.)
A business earns $27,000 in a year and has economic capital of $300,000. Calculate its RAROC. If its hurdle is 8%, calculate its economic profit. (Answer: RAROC = 9%; economic profit = $3,000.)
Two strategies have economic profits in thousands of dollars of A: 8 and 8; B: 4 and 13, paid at the ends of years 1 and 2. At a 10% effective annual discount rate, which has the greater discounted value, and by how much? Round the difference to the nearest cent. (Answer: B, by $495.87.)
A business has earnings of $9,000 and $16,000 in years 1 and 2, with capital of $100,000 and $80,000, respectively. The hurdle and effective annual discount rate are both 10%. Calculate each year’s RAROC and economic profit, then calculate the two-year discounted economic-profit value. (Answer: RAROC = 9% and 20%; economic profit = $-$1,000$ and $8,000; discounted value = $5,702.48.)
At a positive discount rate, two alternatives have the same total economic profit over two years, but one earns more in year 1 and the other earns more in year 2. What information is needed to identify which has the larger discounted value? (Answer: The year-by-year amounts, their payment timing, and the discount rate; discount each amount to the decision date and compare the resulting sums.)

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