Private and Social Appraisal

A community is considering a flood intervention. Paying for construction changes several bank balances. A lower flood probability could preserve buildings, belongings and temporary accommodation resources. An ecological improvement may matter even when it has no agreed monetary price. How should those consequences enter the decision?

This lecture follows one deliberately small example through seven questions. The aim is to make the accounts and assumptions inspectable before interpreting a numerical result.

Preparation and a short refresher

You will use signed cash flows, percentages and finite sums. For a real amount $x_t$ received in year $t$, its present value at an assumed real rate $r$ is $x_t/(1+r)^t$. At a zero rate, present value is the undiscounted amount. The probabilities in this lesson are stated teaching inputs.

Defining the appraisal boundary

Start with the same baseline for each option: no additional public intervention. The comparison covers years zero through five, in constant 2026 Canadian dollars. It includes three owner households, one renter household, a separate landlord, the public authority, suppliers, a lender and an insurer.

The baseline can still contain a flood. It describes the absence of the additional intervention, rather than an assumption that losses disappear. The landlord owns the rental structure; the renter's belongings and temporary accommodation enter the renter's account. Those endpoints must remain separate.

Within a boundary containing both parties, a payment has an equal receipt at the same date. If $G$ is that set of actors and $TR_{g,t}$ is actor $g$'s matched transfer in year $t$, then

$$\sum_{g\in G}TR_{g,t}=0.$$

This is a financial identity. It does not say that constructing the project uses no labour or materials. A transaction crossing the boundary will not have both sides inside this sum.

Private value and resource consequences

For actor $g$, let $\Delta CF_{g,t}(a)$ be the incremental cash flow from action $a$ relative to baseline. Its private present value is

$$P_g(a;r)=\sum_{t=0}^{5}\frac{\Delta CF_{g,t}(a)}{(1+r)^t},\qquad r>-1.$$

A private cash-flow PV describes that account. A supplier's positive receipt is not automatically profit or an additional social benefit. We need a separate resource account for what is constructed, maintained or preserved.

The nature option uses CAD 24,000 of construction resources at year zero and CAD 1,200 of maintenance in each of the next five years. At a 3% real rate, those requirements have a present value of CAD 29,496. Lower assumed flood exposure preserves resources with PV CAD 5,348. The partial monetary score is preserved resources minus required resources: CAD −24,148, after rounding.

Distribution and compensation

The total does not tell us which groups gain or lose. In the base financing arrangement, each owner household and the landlord funds one quarter of the intervention. The renter pays no levy and receives a smaller modeled reduction in expected losses. Rent pass-through is zero in this comparison.

The modeled losses of these five groups would require about CAD 24,456 to compensate at time zero. Their available positive PV is only CAD 309. These gains cannot finance the proposed compensation. That comparison neither supplies an additional subsidy nor establishes agreement by the affected groups.

Appraisal across dates and states

The example allows a first flood in one of five years, or no flood. It excludes repeat floods. The illustrative annual first-event hazard is 2% under baseline and 1.4% under the nature option. Conditional minor and major events have weights 75% and 25%.

Minor restoration, belongings replacement and temporary accommodation require CAD 112,000 in total; major events require CAD 496,000. Their conditional weighted resource burden is $0.75(112{,}000)+0.25(496{,}000)=208{,}000$ CAD. Conditional damage is unchanged by the intervention in this example; the probability weights change.

Let $L(p;r)$ be expected discounted resource loss at hazard $p$ and real rate $r$. Let $B_n$, $C_n$ and $S_n$ denote the nature option's preserved resources, required resources and partial monetary score. The calculation is

$$\begin{aligned} L(p;r)&=208{,}000\sum_{t=1}^{5} \frac{(1-p)^{t-1}p}{(1+r)^t},\\ B_n(r)&=L(0.020;r)-L(0.014;r),\\ C_n(r)&=24{,}000+1{,}200\sum_{t=1}^{5}(1+r)^{-t},\\ S_n(r)&=B_n(r)-C_n(r). \end{aligned}$$

Change the real rate below. The construction cost remains at year zero; the timing of the other entries determines how their PVs change. The choices are the five stated sensitivity settings, not estimated rates or policy prescriptions.

EXPLORE AN ASSUMPTIONIllustrative simulation

What changes with the discount rate?

Compare the same nature option over five years. Only the assumed real discount rate changes.

Real discount rate

0%1.5%3%5%7%

Five-year present value · 2026 CAD

Resources preserved$5,348

Resources required$29,496

$0$35,000

Partial monetary score at 3%

-$24,148

Required resources exceed preserved resources in this calculation. Baseline has a score of zero; the assumed ecological improvement remains a separate consideration.

Read the numerical comparison

Nature option, five-year PV in 2026 CAD

Real rate	Preserved	Required	Partial score
0%	$5,826	$30,000	-$24,174
1.5%	$5,579	$29,739	-$24,160
3%	$5,348	$29,496	-$24,148
5%	$5,062	$29,195	-$24,133
7%	$4,801	$28,920	-$24,120

Choosing within non-monetized constraints

Before ranking monetary scores, state which options meet the decision's conditions. Here the assumptions require no ecological degradation, maintained shelter access and no project-imposed involuntary relocation. Initial resources cannot exceed CAD 35,000, and annual maintenance cannot exceed CAD 1,500. These are conditions of the teaching case, not findings about legal compliance.

Option	Assumed ecological condition	Meets the stated conditions?
Baseline	Maintained	Yes
Barrier	Degraded	No
Nature option	Improved	Yes

The barrier fails the ecological condition even though it meets the budget limits. Baseline and nature remain admissible. Among them, baseline's incremental monetary score is zero and nature's is negative in this partial account. The assumed ecological improvement has no price here; a preference for nature would need a stated justification beyond maximizing this score.

Reconciling the appraisal accounts

A useful check follows each payment to its recipient, then follows each physical consequence to its endpoint. The initial levy and supplier payment balance within the full financial boundary. Construction resources are still counted once. Likewise, an avoided repair should not be entered again as a separate property-value gain without checking whether both entries describe the same consequence.

Keep three results alongside one another: private incidence, the partial resource score, and the conditions or consequences that remain unaggregated. Agreement among spreadsheet totals does not resolve disagreement about which outcomes matter.

Transferring the framework within evidence limits

The conditional result is narrow: baseline has the higher partial monetary score among the admissible options under these five-year assumptions. It does not establish that nature-based interventions generally fail appraisal, or that the monetary account settles the community's overall preference.

The model assumes restoration resources are fully supplied when needed. It does not model household liquidity constraints or repeat floods. Health, biodiversity, cultural meaning and attachment are not assigned invented monetary values. A different horizon, supported costs or an additional verified endpoint could change the comparison; it would require its own calculation and evidence.

The transferable discipline is to state the baseline, classify the entries, examine distribution and explain the limits of the decision rule.

Try it yourself

WORK THROUGH IT

A payment is not a free resource

Four funders each pay CAD 6,000 to the authority. The authority pays CAD 24,000 to the construction supplier. Calculate the matched financial total across those parties. Does that total establish the project's resource cost?

Reveal the worked solution+

The funders' total is CAD −24,000, the authority's net is zero, and the supplier receives CAD 24,000. The matched financial sum is zero. The construction still uses CAD 24,000 of resources in this illustrative example. The financial identity and the resource requirement answer different questions.