Interest Rate (i or r)

(noun)

The cost of not having money for one period, or the amount paid on an investment per year.

Related Terms

Examples of Interest Rate (i or r) in the following topics:

  • Single-Period Investment

    • Since the number of periods (n or t) is one, FV=PV(1+i), where i is the interest rate.
    • What is the value of a single-period, $100 investment at a 5% interest rate?
    • PV=100 and i=5% (or .05) so FV=100(1+.05).
    • The number of periods can be represented as either t or n.
    • Interest Rate (i or r) [Note: for all formulas, express interest in it's decimal form, not as a whole number. 7% is .07, 12% is .12, and so on. ]

  • Multi-Period Investment

    • The future value is simply the present value applied to the interest rate compounded one time.
    • Interest Rate (i or r) [Note: for all formulas, express interest in its decimal form, not as a whole number. 7% is .07, 12% is .12, and so on. ]
    • 't' in this equation would simply be 1, simplifying this equation to FV = PV(1+r).
    • Compound interest simple means that the interest from the first period is added to the future present value, and the interest rate the next time around is now being applied to a larger amount.
    • This means that the interest rate of a given period may not be the same percentage as the interest rate over multiple periods (in most situations).

  • Calculating Annuities

    • Annuities can be calculated by knowing four of the five variables: PV, FV, interest rate, payment size, and number of periods.
    • The PV of a perpetuity is the payment size divided by the interest rate:
    • where m is the amount amount, r is the interest, n is the number of periods per year, and t is the number of years.
    • $\displaystyle{P_0 = \frac{P_n}{(1+i)^n} = P \cdot \sum_{k=1}^n \frac{1}{(1+i)^{n+1-k}} = P \cdot \frac{1-(1+i)^{-1}}{i}}$
    • Solving for the interest rate or number of periods is a bit more complicated, so it is better to use Excel or a financial calculator to solve for them.

  • Calculating Present Value

    • Calculating the present value (PV) is a matter of plugging FV, the interest rate, and the number of periods into an equation.
    • But first, you must determine whether the type of interest is simple or compound interest.
    • Interest Rate (Discount Rate): Represented as either i or r.
    • One area where there is often a mistake is in defining the number of periods and the interest rate.
    • The problem may talk about finding the PV 24 months before the FV, but the number of periods must be in years since the interest rate is listed per year.

  • Differences Between Real and Nominal Rates

    • Nominal rate refers to the rate before adjustment for inflation; the real rate is the nominal rate minus inflation: r = R - i or, 1+r = (1+r)(1+E(r)).
    • If the lender is receiving 8% from a loan and inflation is 8%, then the real rate of interest is zero, because nominal interest and inflation are equal.
    • Where r is the real rate, i is the inflation rate, and R is the nominal rate.
    • The real rate can be described more formally by the Fisher equation, which states that the real interest rate is approximately the nominal interest rate minus the inflation rate: 1 + i = (1+r) (1+E(r)), where i = nominal interest rate; r = real interest rate; E(r) = expected inflation rate.
    • Since the future inflation rate can only be estimated, the ex ante and ex post (before and after the fact) real rates may be different; the premium paid to actual inflation may be higher or lower.

  • The Fisher Effect

    • We only discussed nominal interest rates.
    • Investors and savers are concerned about the real interest rate because the real interest rate reflects the true cost of borrowing.
    • However, if the inflation rate or interest rate becomes high, then the approximation loses accuracy as the cross term becomes large.
    • Many financial analysts use nominal interest rates because inflation is low in the United States, averaging 3% per year or less.
    • If investors and the public have higher expectations of inflations ( πe ), then nominal interest rates (i) become greater.

  • International Fisher Effect

    • They set the cross term, $r \times \pi$, to zero.
    • For example, if the expected inflation, $\pi$, is 10% and nominal interest rate, i, equals 5%, subsequently, the real interest rate is approximately -5%.
    • Cross term, $r \times \pi$, is roughly 0.5%.
    • Interest rate differential between Mexico and the United States (iMEX – iUS) ranged between 7% and 16%.
    • Thus, the interest rate differential (iMXN – iUS) is 7%, while the investment period equals one year.

  • Interest Rate Parity Theorem

    • Domestic nominal interest rate in APR equals id while the rate of return is rd.
    • We compute his or her rate of return in U.S. dollar in Equation 25.
    • $F \approx S \left[ 1+\left( i_d - i_f \right) \frac{T}{360} \right]$
    • For example, a U.S. investor can invest in the United States to earn 2% interest or invest in Japan to earn 5%.
    • Although Japan has a greater interest rate, the depreciating yen wipes out any gains from the higher interest rate.

  • Calculating Values for Different Durations of Compounding Periods

    • The EAR can be found through the formula in where i is the nominal interest rate and n is the number of times the interest compounds per year (for continuous compounding, see ).
    • Once the EAR is solved, that becomes the interest rate that is used in any of the capitalization or discounting formulas.
    • For example, if there is 8% interest that compounds quarterly, you plug .08 in for i and 4 in for n.
    • Remember that the units are important: the units on n must be consistent with the units of the interest rate (i).
    • Finding the FV (A(t)) given the PV (Ao), nominal interest rate (r), number of compounding periods per year (n), and number of years (t).

  • Calculating Perpetuities

    • The present value of a perpetuity is simply the payment size divided by the interest rate and there is no future value.
    • The PV is simply the payment size (A) divided by the interest rate (r).
    • Notice that there is no n, or number of periods.
    • The PV of a growing perpetuity is represented as $PVGP \ = \ {A \over ( i - g )}$ .
    • It is essentially the same as in except that the growth rate is subtracted from the interest rate.