Fight Finance

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The Macaulay duration of a fixed-coupon bond will be less than or equal to the bond's maturity, not more. For zero coupon bonds, the duration will equal the maturity since all money is paid at the end. For coupon bonds, the coupons will pull the duration, the average time of payment, slightly lower than the maturity time when the big face value is paid.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The 90-day certificate of deposit (CD) pays no coupons, as is typical of money market securities, so it's equivalent to a zero coupon bond. Therefore its duration would equal its maturity which is 90 days. This duration is far shorter than the other securities whose durations would be years longer.

Note that this money market security's duration is not zero since the final payment in 90 days does not vary with any floating interest rate such as LIBOR or OIS, the final face value payment is fixed and written in the contract.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

The stock's higher duration means that it is likely to be more sensitive to changes in central bank interest rates than the market portfolio. Assuming that central bank monetary policy (interest rate) changes are systematically risky events, then higher sensitivity to these events due to higher duration would imply a higher required return and a higher CAPM beta.

Mathematically, a stock's CAPM beta equals:

###\beta_E = \sigma_{r_E, r_M} . \frac{\sigma_{r_E}}{\sigma_{r_M}} = correl(r_E, r_M) . \frac{sd(r_E)}{sd(r_M)}###

The longer a stock's duration, the more sensitive it is to changes in interest rates, making its standard deviation ##sd(r_E)## higher, which is in the fraction's numerator, making its beta higher.

Also, the longer a stock's duration, the more correlated it is likely to be with the market portfolio, since the stock will react more strongly to surprise increases (or decreases) in interest rates, which affects the market portfolio in the same direction.

While this question specifically asks you to approach the problem from a single-factor CAPM framework, it's interesting to note that the effect of interest rate changes is often thought of as a whole separate factor in multi-factor CAPM's and arbitrage pricing theory (APT) models, which include the market portfolio factor in addition to others such as surprise interest rate changes. For example, here are five factors mentioned by Berry, Burmeister and McElroy (1988):

1. risk of changes in default premiums;
2. risk that the term structure of interest rates may change;
3. risk of unanticipated inflation or deflation;
4. risk that the long-run expected growth rate of profits for the economy will change; and
5. residual market risk, or any remaining risk needed to explain a market index such as the S&P 500.

In the above list, factor 2 concerns interest rates while factor 5 is the market portfolio.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Starting with the CAPM's SML equation we can find the required return from the stock's beta, the market return and the risk free rate:

###\begin{aligned} r_E &= r_f + \beta_E(r_m-r_f) \\ &= 0.05 + 0.5(0.1-0.05) \\ &= 0.075 \\ \end{aligned} ###

Now that we have the total required return on equity, we can find the Macaulay duration using the 'duration of a perpetuity' formula where ##r## is the total required return:

###\begin{aligned} D_\text{Macaulay} &= \frac{1+r}{r-g} \\ &= \frac{1+0.075}{0.075-0.02} \\ &= 19.545455 \text{ years} \end{aligned}###

So the weighted average time when you will receive your dividends is 19.545 years, weighted by the present value of all of the infinite number of dividends which stretch into perpetuity. It's interesting that the dividends and price are not needed in the formula.

Answer: Good choice. You earned $10. Poor choice. You lost $10.

Due to the firm's total required return on equity ##r_E## remaining unchanged, and the market risk premium ##(=MRP = r_M - r_f)## and risk free rate remaining the same, the firm's equity beta ##\beta_E## must also remain unchanged:

##r_E = r_f + \beta_E(\underbrace{r_M - r_f}_{MRP})##

This is confusing because when a firm's duration rises, it becomes more sensitive to changes in required returns, including default-risk-free money market rates set by the central bank which affect government bond yields ##(r_f)##, and these changes affect all (non-zero duration) assets in the country at the same time so you would expect these yield changes to be systematically risky events. Macro-economic events such as changes in monetary policy are commonly thought of as being systematic events that are not easily diversifiable, at least within the one country.

But this is a sort of trick question where it's impossible for the stock's equity beta to rise when its required return is steady. Perhaps the increase in the stock's systematic risk with respect to changes in monetary policy (interest rates) was offset by a decrease in some other systematic risk factor such as inflation or some other factor which makes the overall CAPM beta unchanged.

Another view point is that the ordinary 'static' CAPM assumes that the risk free rate is constant and doesn't change, which is an ideal assumption that's not true in real life. Government bond yields change every day. Therefore the static CAPM is incompatible with changes in the risk free rate.