Interest Rates: Nominal vs. Effective

Nominal Annual Interest Rate

  • Banks display the nominal annual interest rate, which isn't the actual interest rate due to compounding.
  • Example: A bank advertises a 6% nominal interest rate per annum.
  • This rate doesn't account for the compound interest added throughout the year.

The Misleading Nature of Nominal Rates

  • Banks add interest to loans multiple times a year, making the actual interest owed higher than what the nominal rate suggests.
  • Example: Borrowing 1,000 kroner at a 6% nominal rate implies owing 1,060 kroner after one year. However, due to compounding, the actual amount owed is more.

Effective Annual Interest Rate

  • Lawmakers mandated that banks disclose the effective annual interest rate to provide transparency.
  • The effective rate includes the impact of compounding interest during the year.
  • Banks typically advertise the nominal rate first but must also show the effective rate.

Formula for Converting Nominal to Effective Interest Rate

  • Need to understand how to convert between nominal and effective interest rates.

Notation

  • II = Effective annual interest rate.
  • rr = Nominal interest rate divided into compound periods (periodic interest rate).
  • kk = Number of compounds per year.
  • nn = Total number of interest rate applications during the whole loan period.

Compound Periods

  • Tradition in Denmark: interest added quarterly (every three months).
  • Some banks add interest monthly; some loan companies daily.
  • The more frequent the compounding, the greater the difference between the nominal and effective interest rates.

Periodic Interest Rate

  • To find the periodic interest rate (rr), divide the nominal annual interest rate by the number of terms per year.
  • If the nominal annual rate is 6% and interest is added monthly:
    r=6%12=0.5%r = \frac{6\%}{12} = 0.5\%

Formula for Effective Annual Interest Rate

  • I=(1+r)k−1I = (1 + r)^k - 1

Example Calculation

  • Given a 6% nominal annual interest rate, compounded monthly:
    • r=0.005r = 0.005 (0.5% per month)
    • k=12k = 12 (12 months in a year)
    • I=(1+0.005)12−1=0.0617I = (1 + 0.005)^{12} - 1 = 0.0617 or 6.17%

Significance of the Difference

  • The difference between 6% and 6.17% may seem small, but it can be substantial over long periods or large amounts (e.g., a house loan).
  • Banks advertise the nominal rate because it appears cheaper, even if the margin isn't significant.

Special case

  • The effective rate will always be bigger unless the bank compound once a year, then the nominal and the effective rate will be the same.
  • k=1k = 1 implies I=(1+r)1−1=rI = (1 + r)^1 - 1 = r

Proof of the Formula

  • Initial amount: k0k_0
  • Consider kk terms in one year, where interest is added at each term.

Timeline Representation

  • One year divided into kk equally large periods.
  • Alternatively, consider adding interest only once per year using the effective interest rate.

Compound Interest Formula

  • General formula: k<em>n=k</em>0⋅(1+r)nk<em>n = k</em>0 \cdot (1 + r)^n

Applying the Formula

  • For kk terms: k<em>k=k</em>0⋅(1+r)kk<em>k = k</em>0 \cdot (1 + r)^k
  • For one year with effective interest rate II: k<em>1=k</em>0⋅(1+I)1k<em>1 = k</em>0 \cdot (1 + I)^1

Equating the Amounts

  • The amount after one year should be the same whether calculated using nominal rate with kk compounds or the effective rate once.
  • Therefore, k<em>1=k</em>kk<em>1 = k</em>k

Derivation

  • k<em>0⋅(1+I)=k</em>0⋅(1+r)kk<em>0 \cdot (1 + I) = k</em>0 \cdot (1 + r)^k
  • Divide both sides by k0k_0: 1+I=(1+r)k1 + I = (1 + r)^k
  • Isolate II: I=(1+r)k−1I = (1 + r)^k - 1