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An Efficient Market Story

An investment increases in value, as there is an expected return resulting from a compensated risk. In an efficient market and when properly measured, this relationship between expected return and risk is expected to be proportional, where it is assumed that the current valuations within the market are already correct for bonds, equities, commodities, or any other asset based on their expected future valuations. It is possible to systematically identify factors in order to target a certain level of risk, although this will result in a reduction of diversification.

Expected Return And Risk

An investment can be defined as an asset with an expected return following from a discount rate applied to the future value of the asset. This results in a present value which is usually expected to increase over time until the future value is realized. However, it should be noted that this does not guarantee that the present value will increase over time, as the discount rate is not necessarily fixed over time and may change with future information, so there is an associated likelihood or probability for a distribution of outcomes. The variability in the discount rate, which is equivalent to the uncertainty of the expected return, is associated with the risk of the asset. From the perspective of an investor, this risk or uncertainty of the expected return can also be viewed as the uncertainty of future consumption, where there is also a consideration for a temporal component from the short-term to the long-term.

Mathematical definition of the pricing of assets based on the present value, future value, and discount rate:
\[\begin{gather*} P_{t} = \frac{E[F]_{t+\tau}}{R_{t,\tau}} \end{gather*}\]

This approach is valid even when viewing a seemingly safe asset with a fixed return. For example, consider a principal amount of R1,000 invested in a guaranteed bond with an annual interest rate of 12%, maturity of 1 year, and ...zero coupon re-paid at the end of the period... , where any future consumption is perfectly aligned with the maturity and it is assumed that there is no future information which will have an effect on the discount rate. Thus, the future value of this bond is R1,120 in 1 year, as the discount rate will be equal to the interest rate of 12% (obviously, the present value is R1,000 in this case).

Decomposition of the discount rate for a safe asset with a fixed return:
\[\begin{gather*} P_{t} = \frac{E[F]_{t+\tau}}{R_{t,\tau}} \text{ with } R_{t,\tau} = R_{int} \rightarrow P_{t} = \frac{E[F]_{t+\tau}}{R_{int}} \end{gather*}\]

Consider a more realistic representation of a bond, where there is the possibility of future information and, initially, this future information is only associated with future changes in interest rates. Essentially, as a result, there will be a component of the discount rate related to the expectation for future changes in interest rates. To demonstrate this, the previous example is still valid if, for example, a bond with a maturity of 1 month also has an annual interest rate of 12% (around 0.95% per month after accounting for effects from compounding) without any expected future changes in interest rates (with bonds of other maturities also being equivalently balanced within the market). In other words, the expected return of continuously re-investing in a bond with a maturity of 1 month (or any other maturity) will be identical to the return of a bond with a maturity of 1 year after 1 year.

Instead, if the future interest rate of bonds is expected to change, the present value of the bonds must also change to accommodate this. For example, if the expected annual interest rate of a bond with a maturity of 1 month is expected to be 12% for 6 months and then 10% for the next 6 months (around 0.80% per month after accounting for effects from compounding), then the effective annual interest rate from continuously re-investing in a bond with a maturity of 1 month will be around 11%. Thus, without any adjustments, it would obviously be more favourable to invest in the bond with an annual interest rate of 12% and maturity of 1 year. However, since the market is efficient and recognizes this disparity, the present value of the bond with an annual interest rate of 12% and maturity of 1 year will adjust to emulate the equivalence of continuously re-investing in a bond with a maturity of 1 month. This adjustment can be accounted for through a component of the discount rate related to the expectation for future changes in interest rates. In this case, as the future value is still fixed at R1,120, the discount rate will become around 11% and equal to the effective annual interest rate from re-investing in a bond with a maturity of 1 month. Through the calculation, this then leads to a present value of R1,009 instead of R1,000.

Any good ideas to indicate annualized versus monthly rates?
Equivalence of long-term interest rates to short-term interest rates with perfect balance without premiums:
\[\begin{gather*} (1 + R_{int,lng})_{\tau} = E[(1 + R_{int,shrt})_{1} (1 + R_{int,shrt})_{2} \cdots (1 + R_{int,shrt})_{\tau-1} (1 + R_{int,shrt})_{\tau}] \text{ if } R_{int,lng} = R_{int,shrt} \\[5px] \therefore (1 + (R_{int,lng} + \lambda_{mat}))_{\tau} = E[(1 + R_{int,shrt})_{1} (1 + R_{int,shrt})_{2} \cdots (1 + R_{int,shrt})_{\tau}] \text{ if } R_{int,lng} \neq R_{int,shrt} \end{gather*}\]
Decomposition of the discount rate with consideration for effects from varying maturities:
\[\begin{gather*} P_{t} = \frac{E[F]_{t+\tau}}{R_{t,\tau}} \text{ with } R_{t,\tau} = R_{int} + \lambda_{mat} \rightarrow P_{t} = \frac{E[F]_{t+\tau}}{R_{int} + \lambda_{mat}} \end{gather*}\]
Example of a yield curve showing the effective annual interest rate of bonds with different maturities:
...

However, this pricing would not be expected to be balanced without additional adjustments, as there is a risk due to the uncertainty associated with the expectation for future changes in interest rates. This effect can be seen as a risk premium based on the term of the bond, where this term premium is the compensation which an investor requires for bearing the risk that interest rates may unexpectedly change over the life of the bond. The magnitude and sign of the term premium is related to the uncertainty in estimating the effective annual interest rate from re-investing in a bond with a shorter maturity relative to the ...rigidity... of holding a bond with a longer maturity.

However, it may be better to think of risk through the uncertainty around future consumption. An investor has future consumption at a certain horizon (whether this horizon is 1 day for transactions or 100 years for an endowment), such that there will be an average duration at which the investors in the market have an implied future consumption. As a result, any deviation in term from this average duration would be expected to be compensated with a risk premium, where the magnitude of this risk premium would depend on the obscurity of the deviation - more common deviations expect a lower risk premium (with the most common deviation having no risk premium), while less common deviations expect a higher risk premium (if the investor is able to bear the risk).

Equivalence of long-term interest rates to short-term interest rates with perfect balance and a term premium:
\[\begin{gather*} (1 + ((R_{int,lng} + \lambda_{mat}) + \lambda_{trm}))_{\tau} = E[(1 + R_{int,shrt})_{1} (1 + R_{int,shrt})_{2} \cdots (1 + R_{int,shrt})_{\tau-1} (1 + R_{int,shrt})_{\tau}] \end{gather*}\]
Decomposition of the discount rate with consideration for effects from the term premium:
\[\begin{gather*} P_{t} = \frac{E[F]_{t+\tau}}{R_{t,\tau}} \text{ with } R_{t,\tau} = R_{int} + \lambda_{mat} + \lambda_{trm} \rightarrow P_{t} = \frac{E[F]_{t+\tau}}{R_{int} + \lambda_{mat} + \lambda_{trm}} \end{gather*}\]

When comparing bonds with different maturities, the relative changes for balance and any risk premiums will depend on the maturities being compared. So, for a generalization, the discount rate can be defined relative to a common baseline, such as the interest rate of a bond with a maturity of 1 month. This can also be viewed with the reasonable assumption that the interest rate of a bond with a maturity of 1 month is the universal discount rate for a safe asset without significant risk. Obviously, this assumes that bonds of other maturities are being equivalently balanced within the market, as is expected in an efficient market with the transfer of information into prices as assets are traded, such that the effective annual interest rate for any bond is equal before risk premiums are considered. It should be noted that, any risk premium is deterministically unobservable in the present and can only be quantified with hindsight (although it can be estimated in the present based on valuations and historical inferences if these are assumed to be similar in the future).

Decomposition of the discount rate relative to a common base assumed to be the risk-free return:
\[\begin{gather*} P_{t} = \frac{E[F]_{t+\tau}}{R_{t,\tau}} \text{ with } R_{t,\tau} = R_{int,rsk} + (R_{int,trm} - R_{int,rsk}) + \lambda_{mat} + \lambda_{trm} = R_{int,rsk} + \lambda_{trm} \rightarrow P_{t} = \frac{E[F]_{t+\tau}}{R_{int,rsk} + \lambda_{trm}} \end{gather*}\]

With additional analysis, the discount rate can be further segmented into components for ...real...inflation... . This can be seen in a comparison of nominal bonds against inflation-protected bonds. Essentially, without the risk of inflation, a nominal bond will be priced such that its return is identical to an inflation-protected bond for a given maturity. However, this pricing would not be expected to be balanced without additional adjustments, as there is a risk due to the uncertainty associated with the expectation for future inflation. This effect can be seen as a risk premium based on the inflation-protection of the bond, where this inflation premium is the compensation which an investor requires for bearing the risk that bearing the risk that inflation may unexpectedly change over the life of the bond. As before, the magnitude and sign of the inflation premium is related to the uncertainty in estimating the effective annual interest rate from a bond with inflation-protection relative to the ...rigidity... of holding a bond without inflation-protection.

Decomposition of the discount rate with consideration for effects from inflation protection: \gamma_{ips} + \gamma_{inf} likelihood of default. Could have a graph of the term premium over time if it is available from French? aggregated discount rate

Efficient Markets

diversification, . From the perspective of the average investor, there should be no preference for any individual security over any other security, as the current valuations already reflect the consensus of preferences of the market. When a portfolio is less diversified, it introduces idiosyncratic characteristics, as information is being excluded. An idiosyncratic characteristic can be seen as a risk which is not expected to be compensated. This risk is not compensated, because, when aggregated with other information, it is negated without further effects. For example, consider Company A and Company B making up the supply each with a market share of 50% in an industry with fixed demand - it can be assumed that both companies have capacity to increase or decrease production, while their competition has resulted in a stable price for the supply. If an event occurs and Company A (or Company B) is no longer able to maintain its market share, then this market share will be absorbed by Company B (or Company A). Obviously, this event will affect the future earnings of each individual company and, subsequently, the present value of each individual company. However, in aggregate, the total future earnings of both companies is still the same and only the distribution of earnings between the companies has changed, so the total present value for both of the companies will not actually change. So, as the present value of each individual company has changed, choosing only one of the companies before the event would have been a bet based on luck, as there was no initial reason to have a preference for either of the companies - in other words, choosing only one of the companies introduced risk with an equal chance of being a winning or losing bet, where the outcome was only decided by luck and should not have been expected to go either way. Example of the considerations for idiosyncratic characteristics which are uncompensated: R_{div} = (R_1 + I_1) + (R_2 + I_2) \text{ where } I_1 = - I_2 \text{ or } - I_1 = I_2 \rightarrow R_{div} = R_1 + R_2

Active Management

Active management can be defined as stock picking and market timing. These activities are essentially bets that individual securities or groups of securities are mispriced and overvalued or undervalued. However, if it is accepted that the market is efficient, picking stocks will only result in an aggregated discount rate for the portfolio which is arbitrarily selected, as well as increased uncompensated risk due to ...higher... idiosyncratic characteristics from a reduction in diversification. In a similar way, it should be expected (although not guaranteed) for a positive return every day and, thus, trying to time the market is irrational, as it involves missing this positive return and going against the consensus of the market that current valuations are correct. In an efficient market, active management, as defined as stock picking and market timing, can only be seen as nonsensical and foolish.

arithmetic of active management

Factor Investing

It is possible to systematically identify factors in order to target a certain level of risk. This involves considering the fundamentals of a security in relation to other similar securities.

To emphasize, this does not mean that a security is overvalued or undervalued. It means that any outperformance or underperformance is expected relative to the current level of risk of the security - there is a reason for the security to have its current valuation and an investor will be compensated for this reason if they are willing to accept it while other investors avoid it.

As certain securities are being deliberately excluded, this will inevitably reduce diversification. However, as long as extreme concentration is avoided, this tends to be of minimal concern. It is estimated that ... .

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