Explain the following:
Ex-post (Actual) saving and Ex-post (Actual) investment.

Actual savings is the actual amount of savings that took place, measured after the fact. Alternatively ex-post savings are those which the households actually save from their income. In short, realised savings of a period, say, a year, are called actual savings (or ex– post savings).
Actual investment is the actual amount of investment that took place, measured after the fact. Alternatively it refers to the actual investment made by all the entrepreneurs in the economy during a given period. In short, the realised investment of a period, say, a year, is called actual investment (or ex-post investment). In Keynesian terminology, investment means real and non–financial investment. The point to be noted is that ex-post investments of firms are always equal to ex-post savings by households at all levels of income as savings finance investments. Ex-post or realised (or actual) saving and investment are necessarily equal and this is brought about by fluctuations in income. Since unplanned investment also gets included in realised investment, realised investment is always equal to realised saving. This equality is proved from the following equations:
Y    (Realised income) = C + S (Realised saving)
Y    (Realised income) = C + I (Realised investment)
C + S = C + I
S (Realised saving) = I (Realised investment)