Theory of Net Expectations

The theory of net expectations explains the dependence of income rates on the urgency of a financial instrument . The theory assumes that instruments with different maturity are substitutes .

If investors expect the next year a certain rate of return on financial instruments with a maturity of one year, then the income rate of the biennial financial instruments of the current year will be equal to the product of the rate of income of the current year to the expected rate of return of the next year. In general, the rate of return of long-term financial instruments is equal to the geometric mean of the rates of income of short-term debt instruments. Or the spot rate (rate on a financial instrument acquired at the moment) of the income of long-term financial instruments is equal to the average geometric spot rate of a short-term financial instrument and forward rates of short-term financial instruments (rate on a financial instrument acquired in the future).

$(1+i_{lt})^{n}=(1+i_{st}^{year1})(1+i_{st}^{year2})\cdots (1+i_{st}^{yearn})$ ${\ displaystyle (1 + i_ {lt}) ^ {n} = (1 + i_ {st} ^ {year1}) (1 + i_ {st} ^ {year2}) \ cdots (1 + i_ {st} ^ {yearn})}$ ${\ displaystyle (1 + i_ {lt}) ^ {n} = (1 + i_ {st} ^ {year1}) (1 + i_ {st} ^ {year2}) \ cdots (1 + i_ {st} ^ {yearn})}$ ,

from where

$(1+i_{lt})={\sqrt[{n}]{\prod _{t=1}^{n}(1+i_{t-1,t})}}$ ${\ displaystyle (1 + i_ {lt}) = {\ sqrt [{n}] {\ prod _ {t = 1} ^ {n} (1 + i_ {t-1, t})}}}$ ${\ displaystyle (1 + i_ {lt}) = {\ sqrt [{n}] {\ prod _ {t = 1} ^ {n} (1 + i_ {t-1, t})}}}$ ,

Where $i_{lt}$ ${\ displaystyle i_ {lt}}$ ${\ displaystyle i_ {lt}}$ - interest rate on long-term financial instrument (long-term), $i_{st}$ ${\ displaystyle i_ {st}}$ ${\ displaystyle i_ {st}}$ - interest rate of short-term financial instrument (short-term), $t$ ${\ displaystyle t}$ $t$ - elapsed time from the current moment.

The formula for the interest rate of a long-term financial instrument takes the following form:

$i_{0,t}={\sqrt[{n}]{\prod _{t=1}^{n}(1+i_{t-1,t})}}-1$ ${\ displaystyle i_ {0, t} = {\ sqrt [{n}] {\ prod _ {t = 1} ^ {n} (1 + i_ {t-1, t})}} - 1}$ ${\ displaystyle i_ {0, t} = {\ sqrt [{n}] {\ prod _ {t = 1} ^ {n} (1 + i_ {t-1, t})}} - 1}$ ,

Where $i_{0,t}$ ${\ displaystyle i_ {0, t}}$ ${\ displaystyle i_ {0, t}}$ - spot rate $t$ ${\ displaystyle t}$ $t$ - urgent financial instrument $i_{0,1}$ ${\ displaystyle i_ {0,1}}$ ${\ displaystyle i_ {0,1}}$ - spot rate of a financial instrument with a repayment period of one year, $i_{1,2}$ ${\ displaystyle i_ {1,2}}$ ${\ displaystyle i_ {1,2}}$ - the forward rate of a financial instrument with a period of one year in a year, counting from the current time, etc.

Theory of Net Expectations

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