Another useful feature of log returns relates to limited liability. For many financial assets, including equities and bonds, the most that you can lose is the amount that you've put into them. For example, if you purchase a share of XYZ Corporation for $100, the most you can lose is that $100. This is known as limited liability. Today, limited liability is such a common feature of financial instruments that it is easy to take it for granted, but this was not always the case. Indeed, the widespread adoption of limited liability in the nineteenth century made possible the large publicly traded companies that are so important to our modern economy, and the vast financial markets that accompany them.
That you can lose only your initial investment is equivalent to saying that the minimum possible return on your investment is –100 percent. At the other end of the spectrum, there is no upper limit to the amount you can make in an investment. The maximum possible return is, in theory, infinite. This range for simple returns, – 100 percent to infinity, translates to a range of negative infinity to positive infinity for log returns.
(3.3)
As we will see in the following sections, when it comes to mathematical and computer models in finance, it is often much easier to work with variables that are unbounded, that is, variables that can range from negative infinity to positive infinity.
Graphing Log Returns
Another useful feature of log returns is how they relate to log prices. By rearranging Equation 3.1 and taking logs, it is easy to see that:
where pt is the log of Pt, the price at time t. To calculate log returns, rather than taking the log of one plus the simple return, we can simply calculate the logs of the prices and subtract.
Logarithms are also useful for charting time series that grow exponentially. Many computer applications allow you to chart data on a logarithmic scale. For an asset whose price grows exponentially, a logarithmic scale prevents the compression of data at low levels. Also, by rearranging Equation 3.4, we can easily see that the change in the log price over time is equal to the log return:
It follows that, for an asset whose return is constant, the change in the log price will also be constant over time. On a chart, this constant rate of change over time will translate into a constant slope. Figures 3.1 and 3.2 both show an asset whose price is increasing by 20 percent each year. The y-axis for the first chart shows the price; the y-axis for the second chart displays the log price.
FIGURE 3.1 Normal Prices
FIGURE 3.2 Log Prices
For the chart in Figure 3.1, it is hard to tell if the rate of return is increasing or decreasing over time. For the chart in Figure 3.2, the fact that the line is straight is equivalent to saying that the line has a constant slope. From Equation 3.5 we know that this constant slope is equivalent to a constant rate of return.
In the first chart, the y-axis could just have easily been the actual price (on a log scale), but having the log prices allows us to do something else. Using Equation 3.4, we can easily estimate the log return. Over 10 periods, the log price increases from approximately 4.6 to 6.4. Subtracting and dividing gives us (6.4 – 4.6)/10 = 18 percent. So the log return is 18 percent per period, which – because log returns and simple returns are very close for small values – is very close to the actual simple return of 20 percent.
Continuously Compounded Returns
Another topic related to the idea of log returns is continuously compounded returns. For many financial products, including bonds, mortgages, and credit cards, interest rates are often quoted on an annualized periodic or nominal basis. At each payment date, the amount to be paid is equal to this nominal rate, divided by the number of periods, multiplied by some notional amount. For example, a bond with monthly coupon payments, a nominal rate of 6 %, and a notional value of $1,000, would pay a coupon of $5 each month: (6 % × $1,000)/12 = $5.
How do we compare two instruments with different payment frequencies? Are you better off paying 5 percent on an annual basis or 4.5 percent on a monthly basis? One solution is to turn the nominal rate into an annualized rate:
(3.6)
where n is the number of periods per year for the instrument.
If we hold RAnnual constant as n increases, RNominal gets smaller, but at a decreasing rate. Though the proof is omitted here, using L'Hôpital's rule, we can prove that, at the limit, as n approaches infinity, RNominal converges to the log rate. As n approaches infinity, it is as if the instrument is making infinitesimal payments on a continuous basis. Because of this, when used to define interest rates the log rate is often referred to as the continuously compounded rate, or simply the continuous rate. We can also compare two financial products with different payment periods by comparing their continuous rates.
Sample Problem
Question:
You are presented with two bonds. The first has a nominal rate of 20 percent paid on a semiannual basis. The second has a nominal rate of 19 percent paid on a monthly basis. Calculate the equivalent continuously compounded rate for each bond. Assuming both bonds have the same credit quality and are the same in all other respects, which is the better investment?
Answer:
First we compute the annual yield for both bonds:
Next we convert these annualized returns into continuously compounded returns:
All other things being equal, the first bond is a better investment. We could base this on a comparison of either the annual or the continuously compounded rates.
Discount Factors
Most people have a preference for present income over future income. They would rather have a dollar today than a dollar one year from now. This is why banks charge interest on loans, and why investors expect positive returns on their investments. Even in the absence of inflation, a rational person should prefer a dollar today to a dollar tomorrow. Looked at another way, we should require more than one dollar in the future to replace one dollar today.
In finance we often talk of discounting cash flows or future values. If we are discounting at a fixed rate, R, then the present value and future value are related as follows:
(3.7)
where Vt is the value of the asset at time t and Vt+n is the value of the asset at time t + n. Because R is positive, Vt will necessarily be less than Vt+n. All else being equal, a higher discount rate will lead to a lower present value. Similarly, if the cash flow is further in the future – that is, n is greater – then the present value will also be lower.
Rather than