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:

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

Annualized Return Calculation

To compare the annualized rate of return for both bonds, you can use the formula for annualizing returns based on different time periods:

Annualized Return = \(\left(1 + \frac{\text{Return Since Issuance}}{100}\right)^\frac{365}{\text{Time Since Issuance}} – 1\)

Let’s calculate the annualized returns for both bonds:

For Bond A:

Time Since Issuance = 120 days.

Return Since Issuance = 2.50%.

Annualized Return for Bond A = \(\left(1 + \frac{2.50}{100}\right)^\frac{365}{120} – 1\).

Annualized Return for Bond A = \(\left(1 + 0.025\right)^{3.0417} – 1\).

Annualized Return for Bond A = 1.079847 – 1 = 0.079847 or 7.98%.

For Bond B:

Time Since Issuance = 8 months = 240 days.

Return Since Issuance = 6.00%.

Annualized Return for Bond B = \(\left(1 + \frac{6.00}{100}\right)^\frac{365}{240} – 1\).

Annualized Return for Bond B = \(\left(1 + 0.06\right)^{1.5208} – 1\).

Annualized Return for Bond B = 1.092751 – 1 = 0.092751 or 9.28%.

Comparing the annualized returns:

Bond A has an annualized return of approximately 7.98%.

Bond B has an annualized return of approximately 9.28%.

Therefore, Bond B has a higher annualized rate of return compared to Bond A.

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. 0.40%.

B. 0.92%.

C. 0.41%.

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}