Fight Finance

Answer: Good choice. You earned $10. Poor choice. You lost $10.

An APR is a discretely compounding annual rate that compounds multiple times per year.

An APR compounding once per year is an effective annual rate.

An effective rate is also a discretely compounding rate but it compounds only once per period. The time period is not necessarily annual, it can be monthly, daily, two years, or any time.

Therefore answer (c) is incorrect. An effective monthly rate is a monthly rate compounding per month.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

An APR compounding monthly is equal to 12 times the effective monthly rate. There are two steps required to convert an APR compounding monthly to an effective weekly rate:

• Convert the APR into the effective rate that it naturally converts into, an effective monthly rate, by dividing by 12.
• Convert the effective monthly rate into an effective weekly rate by compounding down. Add one and raise it all to the power of the inverse of the number of weeks in a month, all minus one.

The number of weeks per month is about 4, or to be more exact: 52 weeks per year divided by 12 months per year.

Mathematically:

$$r_\text{eff monthly} = \dfrac{r_\text{apr comp monthly}}{12}$$ $$\begin{aligned} r_\text{eff weekly} &= \left( 1 + r_\text{eff monthly} \right)^{1/(\text{number of weeks in a month})} -1 \\ &= \left( 1 + \dfrac{r_\text{apr comp monthly}}{12} \right)^{1/(52/12)} -1 \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

$$\begin{aligned} r_\text{eff monthly} &= \frac{r_\text{apr comp monthly}}{12} \\ &= \frac{0.18}{12} \\ &= 0.015 \\ \end{aligned}$$

$$\begin{aligned} r_\text{eff yearly} &= \left(1+\frac{r_\text{apr comp monthly}}{12}\right)^{12} - 1 \\ &= \left(1+\frac{0.18}{12}\right)^{12} - 1 \\ &= 0.195618171 \\ \end{aligned}$$

$$\begin{aligned} r_\text{eff daily} &= \left(1+\frac{r_\text{apr comp monthly}}{12}\right)^{12/365}-1 \\ &= \left(1+\frac{0.18}{12}\right)^{12/365}-1 \\ &= 0.000489608 \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the coupons are paid annually, by convention (and in some countries by law), we assume that the yield is an APR compounding annually. An APR compounding annually is a special case that is also an effective annual rate.

$$\begin{aligned} r_\text{eff, monthly} =& (1 + r_\text{eff,annual})^{1/12} - 1 \\ =& (1 + 0.1)^{1/12} - 1 \\ =& 0.00797414 \\ \end{aligned}$$

$$r_\text{eff, yearly} = 0.1 \text{ as given}$$

$$\begin{aligned} r_\text{eff, daily} =& (1 + r_\text{eff, yearly})^{1/365} - 1 \\ =& (1 + 0.1)^{1/365} - 1 \\ =& 0.000261158 \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

$$\begin{aligned} r_\text{credit card, eff yrly} &= \left(1 + \frac{r_\text{credit card, apr comp monthly}}{12} \right)^{12} - 1 \\ &= \left(1 + \frac{0.18}{12} \right)^{12} - 1 \\ &= 0.195618171 \\ \end{aligned}$$

$$\begin{aligned} r_\text{bond, eff yrly} &= \left(1 + \frac{r_\text{bond, apr comp 6 monthly}}{2} \right)^{2} - 1 \\ &= \left(1 + \frac{0.06}{2} \right)^{2} - 1 \\ &= 0.0609 \\ \end{aligned}$$

$$\begin{aligned} r_\text{stock, eff yrly} &= \left(1 + \frac{r_\text{stock, apr comp yearly}}{1} \right)^{1} - 1 \\ &= r_\text{stock, apr comp yearly} \\ &= 0.1 \\ \end{aligned}$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

No explanation provided.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Since the assumptions state that there are 30 days per month and therefore 360 days per year, then the annualised percentage rate compounding per day should be:

$$\begin{aligned} r_\text{APR comp daily} &= r_\text{eff daily} \times 360 \\ &= ((1 + r_\text{eff annual})^{1/360}-1) \times 360 \\ &= ((1 + 0.1)^{1/360}-1) \times 360 \\ &= 0.00026478555 \times 360 \\ &= 0.095322798 \\ \end{aligned}$$

Commentary

Notice that the APR's get smaller as the compounding period becomes shorter. The continuously compounded return is the limit when the compounding period is infinitely small. The APR compounding per second is nearly equal to the continuously compounded rate.

Different Return Quotations Equivalent to an Effective Annual Rate of 10%
Quote type	Return (%pa)	Symbol	Formula	Spreadsheet formula
Effective annual rate	10	$r_\text{eff annual}$	$=r_\text{eff annual}$	=0.1
APR compounding per annum	10	$r_\text{apr comp annually}$	$=r_\text{eff annual}$	=0.1
APR compounding semi-annually	9.761769634	$r_\text{apr comp 6mth}$	$=2 \times ((1+r_\text{eff annual})^{1/2}-1)$	=2 * ((1+0.1)^(1/2)-1)
APR compounding quarterly	9.645475634	$r_\text{apr comp quarterly}$	$=4 \times ((1+r_\text{eff annual})^{1/4}-1)$	=4 * ((1+0.1)^(1/4)-1)
APR compounding monthly	9.568968515	$r_\text{apr comp monthly}$	$=12 \times ((1+r_\text{eff annual})^{1/12}-1)$	=12 * ((1+0.1)^(1/12)-1)
APR compounding daily	9.532279763	$r_\text{apr comp daily}$	$=360\times ((1+r_\text{eff annual})^{1/360}-1)$	=360 * ((1+0.1)^(1/360)-1)
APR compounding hourly	9.531070550	$r_\text{apr comp hourly}$	$=360 \times 24 \times ((1+r_\text{eff annual})^{1/(360 \times 24)}-1)$	=360*24 * ((1+0.1)^(1/(360*24))-1)
APR compounding per minute	9.531018861	$r_\text{apr comp per minute}$	$=360 \times 24 \times 60 \times ((1+r_\text{eff annual})^{1/(360 \times 24 \times 60)}-1)$	=360*24*60 * ((1+0.1)^(1/(360*24*60))-1)
APR compounding per second	9.531018227	$r_\text{apr comp per second}$	$=360 \times 24 \times 60 \times 60 \times ((1+r_\text{eff annual})^{1/(360 \times 24 \times 60 \times 60)}-1)$	=360*24*60*60 * ((1+0.1)^(1/(360*24*60*60))-1)
Continuously compounded annual rate	9.531017980	$r_\text{cc annual}$	$=\ln(1+r_\text{eff annual}) = log_e(1+r_\text{eff annual})$	=ln(1+0.1)

 

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The effective monthly rate can easily be found by dividing the APR compounding monthly by 12.

$$r_\text{eff monthly} = \dfrac{r_\text{APR comp monthly}}{12} = \dfrac{0.045}{12} = 0.00375 = 0.375 \%pa$$

Answer: Good choice. You earned $10. Poor choice. You lost $10.

No explanation provided.