Price Relatives and Continuously Compounded Returns
If an investor purchases a stock for USD 200 and sells it in four months for USD 230, earning a 15% capital gain, what do you figure the selling price of this stock is relative to the purchase price?
The price relative can be over any time interval. You may have seen this referred to elsewhere as a gross return, since it's really just the holding period return plus 1.
No, be careful. This reflects the gain, which is 15% of the purchase price.
$$ 15 \% \times \mbox{USD}~ 200 = \mbox{USD} ~ 30 $$
But that's not the selling price of the stock.
That's right!
The gain of 15% means that the stock price ended up at 115% of the original price. This value of 115% is called the __price relative__—that is, the ending price of a stock divided by the beginning price—and is found with the following simple ratio of the prices:
$$\displaystyle \frac{S_{t+1}}{S_{t}} = 1 + R_{t{,}t+1}$$.
Here $$R_{t{,}t+1}$$ is the net rate of return from _t_ to _t_ + 1.
No. An increase of 115% would mean that the stock price _gained_ USD 230, not that it rose to USD 230.
So putting that into context, if the price of a stock was EUR 95.50 yesterday, and it closed at EUR 94.75 today, what can you say with certainty about the price relative over the period of this single day?
No, the return is negative in this case, but recall that the price relative is really a gross return.
Right!
There's a negative return plus 1, so the price relative must be less than 1. Or equivalently, the ending price is smaller than the starting price, so the ratio must be less than 1. And it is
$$\displaystyle \frac{S_{t+1}}{S_{t}} = \frac{94.75}{95.50} = 0.9921 $$.
This suggests a return of
$$\displaystyle 0.9921 - 1 = -0.79 \% $$
for the day. There is no limit with prices in some cases, so the price relative can be very high as well. But it's bounded below by zero. So the worst you can do is lose everything, ending up with nothing (zero divided by whatever you started with will leave you with a price relative of zero). Try not to let that happen!
No, a price relative suggests a gain. In this case, there is a loss.
Suppose now that an investor purchases a security for GBP 100 and sells it one year later for GBP 106. No trick here; the holding period return is 6%, with a price relative of 1.06.
But if the GBP 100 had been invested in a 6% APR security, it would have yielded
$$\displaystyle (1 + \frac{0.06}{1})^1 - 1 = 6 \% $$
with annual compounding,
$$\displaystyle (1 + \frac{0.06}{4})^4 - 1 \approx 6.14 \% $$
with quarterly compounding, and
$$\displaystyle (1 + \frac{0.06}{12})^{12} - 1 \approx 6.17 \% $$
with monthly compounding. What do you think the security would yield with _continuous_ compounding?
Not exactly. Notice that the pattern here is an ever-increasing yield with a greater number of compounding periods.
Yes!
The greater the number of compounding periods, the higher the yield. This leads to a limit of infinite compounding periods, which is a __continuously compounded return__. In this case it's close to 6.184%.
Actually, no. There is a limit to this progression. The yield doesn't grow forever.
Now think about this another way. The investor earned a 6% return with that one-year investment. If over the year the investor had earned a continuously compounded return, _r_, of some stated rate, _R_, to yield exactly 6%, do you think the stated rate _R_ needed to provide the 6% continuously compounded yield would be equal to, less than, or greater than 6%?
That's not right.
A rate, _R_, compounded continuously would grow to a larger rate. So if _R_ is greater than 6%, then the continuously compounded rate, _r_, would be larger than 6% as well. So the rate cannot be greater than 6% if the continuously compounded rate, _r_, is equal to 6%.
Exactly!
In fact, using an equation for continuously compounded returns in a single period,
$$\displaystyle FV = PVe^{R} $$,
a little algebra can lead to an explicit solution for _R_:
$$\displaystyle \frac{FV}{PV} = e^R $$,
and so
$$\displaystyle ln \left[ \frac{FV}{PV} \right] = ln(e^R) = R $$.
In this example,
$$\displaystyle R = ln \left[ \frac{FV}{PV} \right] = ln \left[ \frac{106}{100} \right] \approx 5.827 \% $$.
Incorrect. As in the example, the holding period return, _R_, and the continuously compounded return, _r_, are different.
There are a couple of ways to read this equation. Try looking at it carefully to see how you might state the logic of this equation in words. Given a holding period return, _R_, what do you think the equivalent continuously compounded return would be?
No, _R_ is very different from a continuously compounded _r_. In this example, _r_ is 6%, and a continuously compounded _R_ of 6% would lead to a return, _r_, of about 6.184% like earlier.
Actually, no.
The natural log of 6% is -2.81. That's not consistent with this example or the general rule shown.
That's right!
This is what the equation is stating: For a holding period return _R_, the equivalent continuously compounded return is the natural log of 1 + _R_. Notice that the ratio of future value divided by present value is 1.06 here, and that's also 1 +_R_.
The natural log of this provides the equivalent continuously compounded return to the holding period, or _r_, and this is the general rule for its calculation.
Consider also that this ratio of FV to PV is the price relative. How would you state the calculation of _r_, the equivalent continuously compounded return, using the price relative?
To summarize this discussion:
[[summary]]
No, this would lead you to
$$\displaystyle ln(2+R)$$.
Not exactly.
The price relative minus one is the holding period return _R_, not 1 + _R_.
You got it.
The price relative is the gross holding period return, or 1 + _R_. And since for a holding period return, _R_, the equivalent continuously compounded return is the natural log of 1 + _R_, this is the same as saying the natural log of the price relative.
The selling price is 15% of the purchase price
The selling price is 115% of the purchase price
The selling price rose 115% over the holding period
It's negative
It's less than 1
It's greater than 1
Less than 6%
More than 6.17%
Infinity
Equal to 6%
Less than 6%
Greater than 6%
_R_ = _r_
The natural log of _R_
The natural log of 1 + _R_
The natural log of the price relative
The natural log of the price relative plus 1
The natural log of the price relative minus 1
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