S&P 500 calculator — methodology

Why there is no historical return here

Because this site has no market data source, and a figure invented to fill the box would be worse than the box. Every statement of the form “the index has returned x% a year since 19xx” rests on four choices that are almost never printed beside it:

• The start date. Beginning a measurement just before or just after a crash moves a long-run average by a percentage point or more.
• The end date. Same effect, same size, and it changes every year.
• The currency. A dollar figure and a sterling figure for the same index over the same window are different numbers, and for a UK reader only one of them is about their money.
• Whether dividends were counted. A price index excludes them; a fund receives them. The difference is roughly the yield every year, compounded.

Any of those alone moves a forty-year projection substantially, and a calculator that quotes a single figure is asking a reader to trust four unstated decisions about a number they cannot check. So the rate is an input, the figure the page opens with is described on the page as a round placeholder, and nothing here attributes a return to any index.

The precedent on this site is the way an annuity rate is handled: the rate is the reader’s to supply, and the page says why it will not supply one rather than defaulting to something that reads like a fact.

The rate, composed

sterling  = (1 + index return) × (1 + currency move) − 1
delivered = (1 + sterling) × (1 − charges) − 1

Multiplicative, never additive. A 7% index with a 3% weakening of the dollar is 3.79%, not 4%, and the difference compounds for the whole term. It is the same rule the projection engine applies to a fund’s ongoing charge, where the additive shortcut was measured to overstate a pot by about 2% over forty years.

The second line is applied by the engine rather than by the page, and the delivered rate shown on the calculator is the engine’s own report of what it compounded at. Two implementations of one number agree until the day one of them changes.

Charges are modelled as a single fund-level ongoing charge — a drag on the return rather than a deduction, which is where a tracker investor meets it, and why the deducted column of the schedule is all zeroes. Splitting an ongoing charge from a platform fee, an annual cap and a flat fee is a real subject and belongs to the investment fee calculator.

The required return, and why it is bisected rather than rearranged

“What would the index have to return for this to work?” is answerable exactly, with no market data at all, and it is the more useful question: it turns an assumption nobody can evaluate into a number a reader can hold against their own judgement.

It is solved by running the projection repeatedly and narrowing the range. There is no closed-form inverse to invert that would agree with the schedule: rounding to whole pence every period makes the final value a step function of the rate, and a rearranged annuity formula answers a simpler question and then disagrees with the table printed underneath it. A solver that calls the projection is correct by construction.

Where no rate in the range reaches the target, the answer is that it is out of reach and by how much. The solver never hands back the top of its own search range as though it were the answer — which is the specific lie that is hardest for a reader to notice, and the most damaging on a page about a target somebody is planning around.

The search runs on the sterling rate, because that is what the projection compounds at, and the answer is converted back to what the index itself would have to do. The range searched is the image of the calculator’s own index field, so the answer is always a figure that could be typed back into the box above rather than one the page could not accept.

The spread, which is the honest output

The same plan, the same term and the same charges at seven index returns, and the ratio between the largest pot and the smallest. That ratio says how much of the answer is the assumption rather than the plan, and a calculator that prints a single figure has hidden it.

Nought is one of the seven on purpose. A plan whose pot at no growth at all is already most of the way to its target is a plan that depends mostly on what is paid in, which is a materially different situation from one that needs the market to do the work — and it is not visible from any single-rate answer.

Currency

The index is priced in dollars, so a sterling investor holds two positions: the index and the exchange rate. A strengthening dollar adds to a UK holder’s return and a weakening one takes from it, whatever the index did. That is the part of this index no American calculator covers and every UK holder carries.

The assumption defaults to nought, and that is a decision rather than a convenience. An equity return assumption is at least a claim about something that grows; an exchange rate is a ratio between two economies with no reason to drift in a particular direction for twenty years. Asserting one would be inventing the number on this page that nobody claims to forecast. Nought is also what `NumberInput` substitutes for an empty field, so clearing the box and arriving on a link with the parameter emptied land in the same place.

A currency-hedged share class aims to remove the exposure — the nought row of the table on the calculator. It costs something to run, the amount varies by product and with the gap between the two countries’ interest rates, and this site has no figure for it: put your own in the charges box. No fund is named and neither choice is recommended.

Conventions, stated rather than assumed

• Contributions land at the start of each period, which is what a standing order on payday is.
• The rate is effective, not nominal. The figure you type is what a year compounds to. Dividing an index’s annual return by twelve and compounding it back would inflate an assumption you never made.
• One period governs everything. The frequency setting drives contributions, growth and charges together. Monthly contributions into annually-compounding growth are not expressible, and the page says so rather than quietly picking one.
• Whole pence at every period boundary, so every row of the table adds up exactly rather than approximately, at the cost of a pound or two of drift against a textbook formula over a long term.

What it does not model

• Anything about the index itself. No history, no constituents, no volatility. It has no data.
• The difference between a price index and a total return index. Whatever you type is treated as the return your money earns. The index fund calculator takes that apart, along with the tracking gap.
• Dealing and currency-conversion charges on each purchase. The ETF calculator models a per-order commission.
• Tax of any kind, and no ISA or pension wrapper rules. The stocks and shares ISA calculator prices what a wrapper saves, against figures that carry a verification stamp.
• Sequence of returns. The order returns arrive in changes the outcome, and a constant rate has no order. On an index whose real history is nothing but sequence, this is the largest omission on the page and no amount of arithmetic here repairs it.