# Annualized Returns

To compare returns over different timeframes, we need to annualize them. This means converting daily, weekly, monthly, or quarterly returns into annual figures.

## Non-Annual Compounding

Interest may be paid semiannually, quarterly, monthly, or even daily – interest payments can be made more than once a year. Consequently, the present value formula can be expressed as follows when there are multiple compounding periods in a year:

$$PV=\ {{\rm FV}_N\left(1+\frac{R_S}{m}\right)}^{-mN}$$

Where:

$$m$$ = Number of compounding periods in a year.

$$R_s$$ = Quoted annual interest rate.

$$N$$ = Number of years.

#### Example: Calculating the Present Value of a Lump Sum (More than One Compounding Period)

Jane Doe wants to invest money today and have it become $500,000 in five years. The annual interest rate is 8%, and it’s compounded quarterly. How much should Jane invest right now? Using the formula above: $${FV}_N = 500,000$$. $$R_S = 8\%$$. $$m = 4$$. $$R_s/m = \frac{8\%}{4} = 2\% = 0.02$$. $$N = 5$$. $$mN = 4\times 5=20$$. Therefore, $$PV=\ {{FV}_N\left(1+\frac{R_S}{m}\right)}^{-mN}=\500,000\ \times\left(1.02\right)^{-20}=\336,485.67$$ Using BA II Plus Calculator: • Press the [2nd] button, then the [FV] button to clear the financial registers. The display should show “CLR TVM.” • Enter the future value (FV). This is the amount Jane wants to have in five years, which is$500,000. To do this, type “500000” and press the [FV] button.
• Enter the interest rate (I/Y). This is the annual interest rate, which is 8%. However, since interest is compounded quarterly, we need to divide this by 4. To do this, type “8”, press the [÷] button, type “4”, then press the [ENTER] button, and finally press the [I/Y] button.
• Enter the number of periods (N). This is the number of quarters in five years, which is 5*4 = 20. To do this, type “20” and press the [N] button.
• Compute the present value (PV). To do this, press the [CPT] and then the [PV] buttons. The display should show the amount Jane needs to invest today, approximately \$336,485.49.

## Annualized Returns

To annualize a return for a period shorter than a year, you need to account for how many times that period fits into a year. For example, if you have a weekly return, you would compound it 52 times because there are 52 weeks in a year.

Generally, we can annualize the returns using the following formula:

$${\text{Return}}_{\text{annual}}=\left(1+{\text{Return}}_{\text{period}}\right)^c-1$$

Where:

$${\text{Return}}_{\text{period}}$$ = Quoted return for the period.

$$c$$ = Number of periods in a year.

#### Example: Annualizing Returns

If the monthly return is 0.7%, then the compound annual return is:

\begin{align}{\text{Return}}_{\text{annual}}&=\left(1+{\text{Return}}_{\text{monthly}}\right)^{12}-1\\&=\left(1.007\right)^{12}-1=0.0873=8.73\%\end{align}

For a period of more than one year, for example, a 15-month return of 16% can be annualized as:

\begin{align}{\text{Return}}_{\text{annual}}&=\left(1+{\text{Return}}_{15\ \text{month}}\right)^\frac{12}{15}-1\\&=\left(1.16\right)^\frac{4}{5}-1=12.61\%\end{align}

We may apply the same procedure to convert weekly returns to annual returns for comparison with weekly returns.

$${\text{Return}}_{\text{annual}}=\left(1+{\text{Return}}_{\text{weekly}}\right)^{52}-1$$

For comparison with weekly returns, we can convert annual returns to weekly returns by making $${(\text{Return}}_{\text{weekly}})^{52}$$ the subject of the formula.

#### Example: Comparing Investments by Annualizing Returns

An investor is evaluating the returns of two recently formed bonds. Selected return information on the bonds is presented below:

$$\begin{array}{c|c|c}\text{Bond}&\text{Time Since Issuance}&\text{Return Since Issuance (%)}\\ \hline \text{A}&\text{120 days}&2.50\\ \hline \text{B}&\text{8 months}&6.00\\ \end{array}$$

Which bond has the highest annualized rate of return?

Solution

Bond A:

Time = 120 days.

Return = 2.50% = 0.0250.

$$\text{Annualized Return} = \left(1\ +\ 0.0250\right)^\frac{365}{120}\ -\ 1\ =\ 0.0818 \approx 8.18\%$$

Bond B:

Time = 8 months = 240 days.

Return = 6.00% = 0.0600.

$$\text{Annualized Return} = \left(1\ +\ 0.0600\right)^\frac{365}{240}\ -\ 1\ =\ 0.0972 \approx 9.72\%$$

So, Bond B has the highest annualized rate of return at 9.72%.

## Continuously Compounded Returns

The continuously compounded return is calculated by taking the natural logarithm of one plus the holding period return. For example, if the monthly return is 1.2%, you’d calculate it as ln(1.012), which equals approximately 0.01192.

Generally, continuously compounded from $$t$$ to $$t+1$$ is given by:

$$r_{t,t+1}=\ln{\left(\frac{P_{t+1}}{P_t}\right)=\ln{\left(1+R_{t,t+1}\right)}}$$

Assume now that the investment horizon is from time $$t=0$$ to time $$t=T$$ then the continuously compounded return is given by:

$$r_{0,T}=\ln{\left(\frac{P_T}{P_0}\right)}$$

If we apply the exponential function on both sides of the equation, we have the following:

$$P_T=P_0e^{r_{0,T}}$$

Note that $$\frac{P_T}{P_0}$$ can be written as:

$$\frac{P_T}{P_0}=\left(\frac{P_T}{P_{T-1}}\right)\left(\frac{P_{T-1}}{P_{T-2}}\right)\ldots\left(\frac{P_1}{P_0}\right)$$

If we take natural logarithm on both sides of the above equation:

\begin{align*} \ln{\left(\frac{P_T}{P_0}\right)} &= \ln{\left(\frac{P_T}{P_{T-1}}\right)} + \ln{\left(\frac{P_{T-1}}{P_{T-2}}\right)} + \ldots + \ln{\left(\frac{P_1}{P_0}\right)}\\\Rightarrow r_{0,T} &= r_{T-1,T} + r_{T-2,T-1} + \ldots + r_{0,1} \end{align*}

Therefore, the continuously compounded return to time T is equivalent to the sum of one-period continuously compounded returns.

### Question

The weekly return of an investment that produces an annual compounded return of 23% is closest to:

A. 40%.

B. 92%.

C. 41%.

The correct answer is A.

Recall that:

$${\text{Return}}_{\text{annual}}=\left(1+{\text{Return}}_{\text{weekly}}\right)^{52}-1$$

We can rewrite the above equation as follows:

\begin{align}
\text{Return}_{\text{weekly}} &= \left(1 + \text{Return}_{\text{annual}}\right)^{\frac{1}{52}} – 1 \\
&= \left(1 + 0.23\right)^{\frac{1}{52}} – 1 \\
&\approx 0.40\%
\end{align}

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