Update: the question was updated to make it clear that what is in mind is a foreign currency fund.
There seem to be two main reasons to build up a foreign currency reserve.
- A country has valuable natural resources (typically oil), and a reserve is built up to generate claims on the rest of the world for the time after the resource runs down, and to prevent the domestic currency from getting too strong (which damages the competitiveness of the non-resource industries). Similarly a country might keep its currency undervalued to support export-led growth, which requires building up reserves.
- Build up reserves as a defensive measure, so that it is possible for domestic entities to import goods and service foreign debt. This strategy blends in with export-led growth strategies.
The build up and run down of foreign currency assets is somewhat independent to domestic policy settings, as it a redirection of trade income flows.
For example, if the country spends its rainy day fund, it is presumably to import foreign goods. This means that the extra demand is being met from global productive capacity, and thus it likely has negligible inflationary effect on the domestic economy.
Building up the fund may also not be associated with reducing growth. If it is built up as part of an export-led strategy, growth may be higher (as was the case in Japan and China).
The rest of this answer discusses the issue of rainy day funds in a local currency.
A sub-sovereign does not issue the currency, but has obligations in the currency. It builds up a “rainy day” fund for the same reason an entity in the private sector does - to meet unexpected demands for cash. An increase in the rainy day fund acts as a form of saving, reducing aggregate demand. This will have a mildly deflationary effect. Unwinding the fund is mildly inflationary. This is true of any other entity in the economy that is not the central government, and in most cases, the effect of one sub-sovereign is not measurable.
However, sovereigns cannot really create “rainy day” funds in their own currency. They do keep balances at the central bank, which look like rainy day funds. But since the central bank is owned by the government, this deposit is a internal obligation of the consolidated entity - a debt the consolidated entity owns to itself. As such, the entry is largely cosmetic, and should have no measurable effect on the economy. (For example, such deposits disappear from standard macro models.)
Once again, a sovereign’s “rainy day fund” in its own currency is a debt it owes to itself. Any entity can create infinitely large debts to itself. For example, you can write an IOU to yourself for \$1 billion, and you have a \$1 billion in assets! However, this has not affected your net worth. Your net worth is what matters for economic outcomes, and thus such self-debt has no effect on anything.
About the only way for a central government to save up “money” in its own currency is to have a negative amount of debt - i.e., the private sector issues debt to the government to meet tax obligations. This is an fairly unusual situation, but the rainy day fund is not really money, it consists of private sector debt.