FIN 300

Interest Rates and Inflation
Lecture 5b

Topics Covered



  • Annualized Interest Rates
  • Quoted vs Effective Interest Rates
  • Inflation

Annualized Interest Rate


  • Measuring interest rates is dependent on the time frame
  • To get annual rates from shorter periods, interest must be compounded
  • Consider a quarterly interest rate: $r_q = 2\%$;   $r_a = ?$

$\$1$ today   $\rightarrow$ ? in one year

$(1+0.02)$ $\times(1+0.02)$ $\times(1+0.02)$$\times(1+0.02)$$=1.0824$

$(1+r_{q})$ $\times(1+r_{q})$$\times(1+r_{q})$$\times(1+r_{q})$$= 1 + r_a$

$(1+r_{q})$ $\times(1+r_{q})$$\times(1+r_{q})$$\times(1+r_{q})$$= 1 + r_a$


$(1+r_q)^4= 1+r_a$


$r_a=(1+r_q)^4-1$


The General Case:


$r_a=(1+r$$_{\frac{1}{m}}$ $)^m-1$

$m$ is the number of compounds per year
Example:

In annual terms, how much is a daily rate of $0.01\%$?

$r_a=(1+r$$_{\frac{1}{m}}$ $)^m-1$ = $(1+r$$_{\frac{1}{365}}$ $)^{365}-1$



$r_a = (1+0.0001)^{365}-1$ $ =0.037$ $=3.7\%$

Quoted vs Effective Rates

  • Quoted rates understate the true interest rate
  • True rate implied by number of periods compounded
  • Interest on interest makes a difference

A more accurate rate of interest:


Effective Annual Rate (EAR)

Quoted Rate to EAR


  • Quoted Rate: $12\%$, compounded semiannually
  • Every 6 months: interest rate is $6\%$
  • EAR is the annualized return

$EAR = (1+0.06)^2 -1 = 0.124 = 12.4\%$

$EAR = 12.4\% > 12\% =Quoted$ $Rate$


$EAR = \left( 1 + \dfrac{\text{Quoted Rate}}{m}\right)^m -1 $


$m$ is the number of compounding periods


$EAR = \left( 1 + \dfrac{0.12}{2}\right)^2 -1 $


$= 12.4\%$

Inflation


  • Investments usually return currency
  • The value of currency usually decreases
  • Nominal returns are the observed returns in currency
  • Real returns remove the effects of inflation

Fisher Effect:

$1+R = (1+r)\times(1+h)$
Nominal return: R           Real return: r                Inflation: h
$1+R = (1+r)\times(1+h)$

$1+R = 1+r+h+ r\times h$

$R = r+h+ r\times h$

$R \approx r+h$

Nominal returns are approximately equal to real returns plus inflation.

Summary




  • Annualized Interest Rates
  • Quoted Rate to EAR
  • Inflation