Appraisal Reference

Statistics & Cash Flows

Two sets of keys most appraisers never touch, and both do work that matters. The blue statistics functions turn a handful of sales into a trend and tell you whether to trust it. The cash flow keys value an uneven income stream, which is the whole of a discounted cash flow analysis.

The keys

All blue except the clear, which is gold. None of them appear on the key faces.

Σ+

Accumulate a point. Returns the running count, n

gΣ+

Remove a point. Blue Σ− on the same key, for a figure entered in error

fSST

Clear the statistics registers. Gold Σ above SST

Mean. Blue x̄ under the zero. Press x≷y afterwards for the mean of y

Standard deviation. Blue s under the decimal point

Weighted mean. Blue x̄w. Weights go in x, values in y

Estimate y from x. Blue ŷ,r. Then x≷y gives r

Estimate x from y. Blue x̂,r — the same line read backwards

Clear before you start, every time. f Σ is not optional. The statistics registers hold whatever the last problem accumulated, and adding five sales to a set that already has three gives a mean of eight properties you never chose. Nothing on the display warns you — but pressing Σ+ shows n, so glance at it: if the first point does not return 1, the registers were not empty.

One variable: mean and standard deviation

Five sale prices. How they cluster, and how widely.

For a single variable, just enter each figure and press Σ+. No ENTER between them — Σ+ does the storing.

fSST

Clears the statistics registers. Σ is the gold legend above SST — do this first, every time

225000Σ+

First sale accumulated. The display returns 1.00, the running count

264000Σ+

Returns 2.00

270000Σ+

Returns 3.00

312000Σ+

Returns 4.00

336000Σ+

Returns 5.00 — all five in, n = 5

Blue x̄ under the zero key gives the mean

Blue s under the decimal point gives the standard deviation

Mean$281,400.00

Standard deviation$43,391.24

What the pair tells you. A mean of $281,400 with a standard deviation of $43,391 says the sales sit within roughly fifteen percent of each other. Tight enough to reconcile confidently. The same mean with an s of $110,000 would be describing a market you have not yet defined, and the right response is better comparables rather than a more careful average.

Two variables: prediction

The feature the Statistics course sets out. Fit a line through paired sales, then read a value off it.

With two variables the order of entry matters: y first, then x. Here y is the sale price and x is gross living area, because we want to predict price from area.

Sale	Gross living area	Sale price
Sale One	1,450 sf	$225,000
Sale Two	1,680 sf	$264,000
Sale Three	1,820 sf	$270,000
Sale Four	2,050 sf	$312,000
Sale Five	2,340 sf	$336,000

fSST

Clear the registers. Skipping this is the single most common way a prediction goes wrong

225000ENTER1450Σ+

Sale One — price into Y, area into X. Returns 1.00

264000ENTER1680Σ+

Sale Two — price into Y, area into X. Returns 2.00

270000ENTER1820Σ+

Sale Three — price into Y, area into X. Returns 3.00

312000ENTER2050Σ+

Sale Four — price into Y, area into X. Returns 4.00

336000ENTER2340Σ+

Sale Five — price into Y, area into X. Returns 5.00

1950g2

Blue ŷ,r under the two key. Estimates the price for a 1,950 sf subject

x≷y

Swaps X and Y to reveal the correlation coefficient sitting behind the estimate

Predicted price$291,685

Correlation, r0.9890

Read both numbers, in that order. The estimate is $291,685. The r of 0.9890 is what tells you whether to quote it. Squared, it says about 98 percent of the variation in these prices tracks with area alone — which for five sales in one market is believable, and is why the estimate is worth having.

A high r is not a licence. It says the line fits these five points; it does not say area is what drives price, and it cannot see the variable you left out. Five sales from one subdivision will almost always look tight. Run the same keys on sales spread across three neighbourhoods and watch r fall — that fall is the useful part, because it is the calculator telling you the comparables are not comparable.

The same accumulated data also gives you the implied rate of change. The line through these points rises about $125 per square foot, which is a market-derived adjustment rather than a rule of thumb — and it came from the same keystrokes.

For the underlying method rather than the keystrokes, the Statistics course works through what the line means and when a regression is being asked to carry more than it can.

The keys

Blue below the financial keys, with NPV and IRR in gold above them.

gPV

Initial cash flow. Blue CF_o under PV — the cash flow at time zero, today. Negative here because that is money going out, not because the o means “outlay” — it is a date, not a direction

gPMT

Each later cash flow. Blue CF_j under PMT. Year one, year two, and so on in order

gFV

Repeat the last flow. Blue N_j under FV — how many consecutive years it occurs

fPV

Net present value. Gold NPV above PV. Needs a rate in i first

fFV

Internal rate of return. Gold IRR above FV. Needs no rate at all

The whole idea in one line. Direct capitalization takes one year of income and one rate. A cash flow analysis takes every year separately, which is what you need when the income is not level — a lease-up, a step lease, a rehab, or any hold that ends in a sale.

Signs are not decoration here. Money leaving you is negative, money arriving is positive. Enter the purchase price as a positive CF_o and the IRR will be wrong or refused, because you will have described a project that produces income and never cost anything.

N_j: the key that saves the day

Level income for several years, entered once.

A ten-year hold with the same net operating income each year would mean pressing g CF_j ten times. N_j exists so you enter the flow once and then say how many years it repeats.

Take a property bought for $1,000,000, producing $85,000 of net operating income for five years, and sold at the end of year five for $1,150,000 net. Year five therefore carries two things: that year's income and the reversion.

fCLx

Clear the financial registers before anything else. Gold REG above CLx — spoken as f CLx

1000000CHSgPV

CF_o = −1,000,000. CHS makes it negative, because the purchase leaves your pocket

85000gPMT

CF₁ = 85,000. One year of net operating income

4gFV

N_j = 4. Repeats that flow through years one to four, in two keystrokes instead of eight

1235000gPMT

CF₅ = 1,235,000. Year five income of 85,000 plus the 1,150,000 reversion

The discount rate, per period

fPV

Solves for net present value

fFV

Solves for the internal rate of return — and note it needed no rate from you

NPV at 9%$78,041.45

IRR10.91%

Read what those two answers say together. A positive NPV of $78,041 means the stream is worth more than $1,000,000 to a buyer wanting 9 percent. The IRR of 10.91 percent says the same thing from the other side: this stream earns 10.91, which is more than the 9 required. Two numbers, one conclusion — and they must always agree in direction.

N_j counts years, not flows. Entering 85000 g CF_j then 5 g N_j would put five years of income at 85,000 and leave nowhere for the reversion. Four is correct here precisely because year five is different from the four before it. If a flow changes in any year, that year gets its own CF_j.

Where this shows up in appraisal work

Three situations direct capitalization cannot handle.

Situation	Why a single year will not do
A property in lease-up	income climbs for three years before stabilising, so no one year represents it
A step lease	contract rent rises on a schedule you already know
Above or below market rent	the contract runs out partway through the hold and reverts to market

And the honest limit. A discounted cash flow needs a forecast for every year plus a reversion plus a discount rate. Each of those is a judgement, and the arithmetic will process a poor forecast as willingly as a good one. The technique adds precision, not accuracy — the numbers you feed it still have to be defensible one at a time.

Statistics & Cash Flows

The keys

One variable: mean and standard deviation

Two variables: prediction

The keys

Nj: the key that saves the day

Where this shows up in appraisal work

N_j: the key that saves the day