Effective interest rate

Effective interest rate ( EPS, EIR, Effective Interest Rat e) - the interest rate (discount rate) at which the discounted value of the cash flow from a financial instrument (asset, liability, investment project, etc.) is equal to some estimate of the current value of this instrument (attachments). The effective interest rate can be determined for any time interval, but usually the annual effective interest rate is implied.

EPS is a compound interest rate that takes into account the time value of money, allowing you to compare different cash flows, instruments, assets, liabilities, projects with each other.

In different situations, different names may apply. For bonds, the concept of yield to maturity (YTM) is used; for investment projects, the internal rate of return (IRR, Internal Rate of Return).

The EPS method is the primary method for measuring financial assets and liabilities in IFRS (see IFRS 9) when they are carried at amortized cost. At the time of initial recognition, the instrument is carried at fair value and, based on it, is determined by EPS. Subsequently, the value of the instrument is determined as the discounted value of the cash flow from the instrument expected after the current moment discounted at this initial EPS

Content

1 Formalized Description
- 1.1 General definition
- 1.2. The simplest special case: interest-bearing instrument with the repayment of the initial amount at the end of the term.
- 1.3 General case of an interest-bearing instrument with the repayment of the original amount over the term
2 See also

Formal Description

General definition

In accordance with the definition, the EPS of a financial instrument with a value of S (at a given point in time) is generally defined as a solution to the r equation

$S=\sum _{i=1}^{n}{\frac {CF_{t_{i}}}{(1+r)^{t_{i}}}}$ ${\ displaystyle S = \ sum _ {i = 1} ^ {n} {\ frac {CF_ {t_ {i}}} {(1 + r) ^ {t_ {i}}}}}$

Where $CF_{t_{i}}$ ${\ displaystyle CF_ {t_ {i}}}$ - payment for the instrument at a time $t_{i}$ ${\ displaystyle t_ {i}}$ (time is counted from the current moment in units of measure r).

If the ESR is determined for a certain base period, then to determine the ESR for a period T containing m base periods (m is not necessarily an integer) in the above equation, in degrees of discount factors, it is also necessary to convert the time to new units, respectively, instead $t_{i}$ ${\ displaystyle t_ {i}}$ need to use $t_{i}/m$ ${\ displaystyle t_ {i} / m}$ . This is equivalent to instead of $1+r$ ${\ displaystyle 1 + r}$ use $(1+R)^{1/m}$ ${\ displaystyle (1 + R) ^ {1 / m}}$ , therefore, we have accrual of compound interest, i.e.

$R=(1+r)^{m}-1$ ${\ displaystyle R = (1 + r) ^ {m} -1}$

The simplest special case: interest-bearing instrument with the repayment of the initial amount at the end of the term.

In the simplest case, when there is an instrument (for example, a bank deposit) with a value S (deposit amount), which is repaid exactly in the same amount at the end of the period for which interest is accrued at the rate q for a fixed period (capitalization period) throughout term of the instrument, the equation for the annual EPS for this base period has the form

$S=\sum _{i=1}^{n}{\frac {q*S}{(1+r)^{i}}}+{\frac {S}{(1+r)^{n}}}=qS{\frac {1-(1+r)^{-n}}{r}}+S(1+r)^{-n}$ ${\ displaystyle S = \ sum _ {i = 1} ^ {n} {\ frac {q * S} {(1 + r) ^ {i}}} + {\ frac {S} {(1 + r) ^ {n}}} = qS {\ frac {1- (1 + r) ^ {- n}} {r}} + S (1 + r) ^ {- n}}$

It follows that q = r , that is, in this case, the EPS for the base period and the nominal rate for the same period are equal to each other (r = q). However, the annual EPS and the annual nominal rate will differ if interest is accrued more often than once a year, since interest calculation for a year depends on the base rate linearly (simple percent), and EPS for an annual period is determined, as indicated above, as complex interest. If m is the number of base periods per year, then the annual nominal rate will be equal to Q = q * m, and the annual EPS will be equal to

$R=(1+r)^{m}-1=(1+Q/m)^{m}$ ${\ displaystyle R = (1 + r) ^ {m} -1 = (1 + Q / m) ^ {m}}$

this ratio links the annual effective and nominal interest rates on the instrument in this simplest case. Accordingly, in such cases there is no need to build cash flows and solve the equation for determining EPS. It is enough to apply the indicated formula.

The general case of an interest-bearing instrument with the repayment of the initial amount during the term

More generally, the initial amount is not paid off at the end of the period, but over the life of the instrument. The cash flow for the instrument is a stream of payments through a fixed period of time (the base period), which are the repayment of the main debt and interest on the remainder of the main debt for the base period. All standard loans and deposits can be attributed to this case, unless there are additional income or expenses on them, taken into account when calculating EPS. At the same time, the schedule of payments does not matter (annuity, differentiated, at the end of the term, etc.), it is only important that the periods of payments (or interest capitalization) are the same, the absence of other cash flows other than repayment of the main debt and interest on its balance. It should be noted that if interest is calculated, for example, on a monthly basis, then formally the months do not have the same duration, however, in a first approximation, this can be neglected in the calculations.

It can be shown that in this general case the EPS for the base period is exactly the nominal rate for the base period. Accordingly, it is possible to determine the annual EPS by the above standard formula for calculating the annual EPS depending on the number of base periods per year.