# Timing the Taper – are markets overreacting?

This week the markets are again hit by a fresh round of tapering fear. The odd thing about these tapering induced market swings is that the end of QE is certain to happen at some point and must have been priced in already. The only uncertain factor here is the timing. But can the timing of tapering really explain these violent market moves?

To begin with, QE can affect stock prices in two ways:

1. QE pushes down bond yields and so it pushes down the discount rate. This makes the present value of future dividends greater, and thus the stock prices are higher.
2. QE (i.e. lower borrowing costs) will increase demand in the economy and thus increase profits by firms. This increases expected dividends.

Of these, it seems likely that the first effect, via the discount rate, is the one that would be most prone to daily volatility due to tapering timing revisions. But how does the timing of the end to QE affect today’s stock price? And can this explain the volatility in the market today?

Below I do a simplistic back of the envelope analysis which will illustrate that the market has very good reason to be sensitive to the timing of QE tapering. I use a standard dividend discount model calibrated to current data. First we begin with the basic price equation:

$P_t=E\bigg( \sum_{i=1}^{\infty}{\frac{D_{t+i}}{(1+d)^i}} \bigg)$

For simplicity, assume that QE will end abruptly at period N, and that the discount rate then changes from $d_1$ to $d_2$. This gives a price equation as follows:

$P_t=E\bigg( \sum_{i=1}^{N}{\frac{D_{t+i}}{(1+d_1)^i}} + \frac{1}{(1+d_1)^N} \sum_{j=N+1}^{\infty}{\frac{D_{t+j}}{(1+d_2)^j-N}} \bigg)$

Which gives:

$P_t=E\bigg( \sum_{i=1}^{N}{\frac{D_{t+i}}{(1+d_1)^i}} + \frac{P_{t+N}}{(1+d_1)^N} \bigg)$

Now, let us do a rough calibration to the current market by looking at the S&P 500. Today’s price is approximately $1700, the dividend yield is approximately 2%, and thus we have a monthly dividend of approximately$3. For now, we assume a discount rate of 0.5% per month during QE. Once QE ends, this is assumed to increase to 0.8% per month (I will show results for different assumptions later). If QE is expected to end after 12 months, we simply get:

$1700=E\bigg( \sum_{i=1}^{12}{\frac{3}{(1.005)^i}} + \frac{P_{t+12}}{(1.005)^12} \bigg)$

This implies a forecasted price in 12 months of \$1765. We will stick to this implied forecast going forward. Now, if Bernanke makes an announcement that QE will end in 11 months, rather than the expected 12 months, the new price will be:

$P_t=E\bigg( \sum_{i=1}^{11}{\frac{3}{(1.005)^i}} + {\frac{3}{(1.005)^{11}*(1.008)^1}}+ \frac{1765}{((1.005)^{11}*(1.008)^1} \bigg)=1689$

Thus, if markets expected the end of QE in 12 months, and news suddenly revealed it would end in 11 months, the current stock price would fall by approximately 0.49%.

Simulations
I wrote a little program in Matlab that repeats this simulation for an announced end to QE in everything from 1 month up to 12 months compared to an expected end varying between 1 month and 12 months. Some results are plotted in the figure below.

Figure 1: Discount rate during QE = 0.5% per month. Discount rate after QE = 0.8% per month.

The simulations suggest that if the market expects QE to end in for example three months (top left panel) and a surprise announcement by the FED states that it will actually end QE in 2 months, the current stock prices will fall by approximately 0.3%. The fall is fully caused by the revelation that discount rates will increase one month sooner than expected. Remember, here I have assumed that QE keeps the discount rate artificially low at 0.5% per month instead of 0.8% per month. If I change the assumptions so that the end of QE will see a much smaller increase from 0.5% to 0.55% per month, we get the results below:

Figure 2: Discount rate during QE = 0.5% per month. Discount rate after QE = 0.55% per month.

As you see, if QE is assumed to keep monthly discount rates only 5 basis points lower than what they would otherwise be, the current price is still relatively sensitive to the timing of the end to QE. Under this scenario, if we assume the market expects QE to end in 3 months and an announcement today states it will end in 2 months, the current price will fall by 0.1%. If the market expects QE to end in 12 months, and an announcement states it will end in 2 months, the current price will fall by 0.5% (bottom right panel). On the other hand, if the market expects QE to end in 6 months, and an announcement states it will end in 12 months, the current price will increase by 0.3% (top right panel).

Thus, all together, despite the fact that the end to QE is already priced in the market, the market has good reason to be sensitive to its timing. It is easy to say the market is overreacting to news (the reason I started writing this post is because I thought this was the case), but this analysis shows that the current market swings may be completely justified. Any surprises here is guaranteed to induce volatility as markets readjust their expectations of when the FED will begin tapering.

PS: Also note that the day tapering actually begins, the market is practically guaranteed to fall as long as at least one market participant failed to predict the taper to start that day. The simple reason being that no participants will expect taper to begin on a past date (as they would have observed this if it happened), and thus any expectations of tapering to begin on a future date must pull the average expected taper beginning to a future date. This average expectation means that stock prices will never price in an immediate end to QE, no matter how imminent it is. (Of course, the effect of this may be negligible).