Category: Finance
A New Study of the Central Bank’s B/S
Stock Market Reactions on Taper & the Increase in Federal Rate
Factors affecting stocks valuation are liquidity of the market, Prosperity of the overall market, investors’ preference, etc.
In simply the DCF model, the impacts of those factors would be reflected in the discounted rate. As we consider separating the interest rate into a risk-free rate and the premium for a certain firm, the premium is idiosyncratic. For example, with low liquidity of the capital market, investors would expect a liquidity premium; worse economic conditions and low investors’ preferences would increase the risk premium.
Therefore, we could predict that an increase in the federal fund rate, as what the Fed is doing to face the hyper-inflation, and quantitive tightening would have the following impacts. Firstly, the quantitive tightening (QT) or TAPER means the Fed would actively decrease its balance sheet by sell-out/stoping re-issuing those MBS or Government debts. This conduction would decrease the amount of money available in the market, and thus result in higher costs of borrowing money. The liquidity premium would increase. Secondly, if there is less supply of money, then the higher cost of using money, the interest rate, would increase. Both the increase in the interest rate and the premium would increase the discount rate for a certain company. Applying the higher discounted rate to the DCF model for that firm would end up with a lower valuation.
Conclusively, an increase in the Federal Fund Rate and QT/TAPER would generally result in a lower valuation of firms.
Bond Price Approximation
The Price-Yield Curve descripts the relationship between a bond’s price and yield, and they are normally negatively correlated.
We here consider the Bond’s price as a function of the bond’s yield, \(P(Y)\), and study the approximation of the bond’s price.
We firstly apply the Taylor Expansion at \( (Y_0,P_0 \),
$$ P(Y)\approx P(Y_0)+\frac{dP}{dY}(Y-Y_0)+\frac{d^2P}{dY^2}\frac{(Y-Y_0)^2}{2!}+O((Y-Y_0)^3) $$
$$ P(Y)-P_0 \approx +\frac{dP}{dY}(Y-Y_0)+\frac{d^2P}{dY^2}\frac{(Y-Y_0)^2}{2!}+O((Y-Y_0)^3) $$
$$ \triangle P \approx \frac{dP}{dY}\triangle Y+\frac{d^2P}{dY^2}\frac{(\triangle Y)^2}{2!}+O((\triangle Y)^3) $$
Devided by P from both side, then the LHS means the percentage change of Price.
$$\frac{ \triangle P}{P} \approx \frac{dP}{dY}\triangle Y \frac{1}{P}+\frac{d^2P}{dY^2}\frac{(\triangle Y)^2}{2!}\frac{1}{P}+O((\triangle Y)^3) $$
By definition, the Modified Duration \( D=\frac{\triangle P / P}{\triangle Y} = \frac{dP}{dY}\frac{1}{P}\), and convexity \( C=\frac{d^2 P}{d Y^2}\frac{1}{P} \). We replace them into the expansion function. and drop the last term.
$$\% \triangle P \approx D\cdot \triangle Y+\frac{C}{2}\cdot (\triangle Y)^2$$
Finally, we get the approximated bond price curve. Second order Taylor Expansion is applied, and the Duration and Convexity, two important properties of bonds are included. For more accurate approximation, more terms need to be expanded.
Modigliani-Miller (M&M) Theorem
M&M theorem (Modigliani and Miller, 1958) is used to value a firm. It states that a firm’s value is based on its ability to earn revenue plus its risk of underlying assets. The way a firm finances its operations should not affect its value.
At its most basic level, the theorem argues that, with certain assumptions in place, it is irrelevant whether a company finances its growth by borrowing, by issuing stock shares, or by reinvesting its profits.
Assumptions are 1. the markets are completely efficient; 2. there are no costs of bankruptcy or agency dynamics and no taxes.
However, there are of course taxes and costs in the reality, and the assumptions do not hold. Therefore, the M&M theorem implies that firms are more valuable if financed by debts than financed by equities. The reason is the tax shield effects of debts.
Mathematic Example
Consider two companies, same risks, same expected cash flow before interest, \( Y\).
- Co1, has debt with market value of \(D_1\). Total market value \(V_1=E_1+D_1\).
- Co2, has no debt. \(V_2=E_2\), market value of equity.
We can invest,
- Investment A: We own a fraction \(a\) (e.g. 6%) of shares in Co1z. They worth \(aE_1\) (e.g. $6,000). Expected cash flow from investment A, \(y_A\), is:
$$(y_A=\underbrace{a}_{SharesOwned} \times \underbrace{(Y-R_D D_1)}_{Co1’s EarningAfterInterest}$$
Co1 needs to pay an interest rate of its debt, \(R_D D_1\). Purchasing Co1 means only purchasing the equity of Co1, which is EV-Debts.
- Investment B: We sell the shares of Co1. We receive amount \(aE_1\) ($6,000). Then, we use this to buy shares in Co2, which is ungeared.
To Produce the same gearing and risk as investment A, we borrow a further amount worth \(aD_1\) (‘home-made gearing’ or ‘artificially gearing’), at interest rate \(R_D\), and buy more shares in Co2.
E.G. if gearing of Co1 is \( \frac{D_1}{D_1+E_1}=0.25\), then to get same gear for our investment B, we borrow \(6,000\times \frac{0.25}{0.75}=2,000\).
Expected cash flow from investment B, \( y_B\), is:
$$ y_B=\frac{aE_1+aD_1}{E_2}Y-R_D\times aD_1 = a\frac{V_1}{V_2}Y-R_D\times aD_1$$
In equilibrium, \( y_A=y_B\). Otherwise, arbitrage opportunity emerges. Therefore, we get \(V_1=V_2\).
In conclusion, gearing does not affect value (Equity plus Debt).
Implication
Since expected net cash flow (\(Y\)) and company value are the same for each company, the cost of equity for ungeared Co2 and WACC for geared Co1 must be equal. So WACC must be constant with respect to gearing.
Let the cost of equity for a company with no debt be \(R_{ungeared}\).
$$ R_{ungeared}=R_D\frac{D}{V}+R_E\frac{E}{V}=WACC$$
$$ R_{ungeared}=R_D\frac{D}{D+E}+R_E\frac{E}{D+E}=WACC$$
WACC is constant w.r.t. \(\frac{D}{V}\).
$$ R_{ungeared}(D+E)=R_DD+R_EE$$
$$ R_EE=R_{ungeared}(D+E)-R_DD $$
$$ R_E=R_{ungeared}+(R_{ungeared}-R_D)\frac{D}{E} $$
So we get a linear relation between cost of equity and D/E, assuming a constant cost of debt, assuming a constant cost of debt (and assuming \(R_{ungeared}>R_D\)). Implicly, changes in gearing structure (or how to finance the business) do not affect the WACC for a company.
Violation of the constant debt assumption of course would make WACC unconstant.
MM Theory & Beta
In the CAPM, expected returns on assets differ because their betas differ.
So we can write,
$$ \beta_{ungeared}=\beta_{debt}\frac{D}{D+E}+\beta_{geared}\frac{E}{D+E} $$
$$\beta_{geared}=\beta_{ungeared}+(\beta_{ungeared}-\beta_{debt})\frac{D}{E}$$
, where \(\beta_{ungeared}\) denotes asset beta, and \( \beta_{geared}\) denotes actual beta of shares of a geared company (estimated from the market data).
Calculations are similar to those as above.
If assume \(\beta_{debt}=0\), then
$$ \beta_{ungeared}=\beta_{geared} \frac{E}{V}$$
$$ \beta_{geared}=\beta_{ungeared}(1+\frac{D}{E}) $$
A Modigliani-Miller Theorem for Open-Market Operations
Monetary policy determines the composition of the government’s portfolio. Fiscal policy (the size of the deficit on the current account) determines the path of net government indebtedness. Wallace showed that alternative paths of the government’s portfolio consistent with a single path of fiscal policy can be irrelevant. The irrelevance means that both the equilibrium consumption allocation and the path of the price level are independent of the path of the government’s portfolio.
Typos are there. See the original paper issued by Fed in 1979.
Reference
Modigliani, F. and Miller, M.H., 1958. The cost of capital, corporation finance and the theory of investment. The American economic review, 48(3), pp.261-297.
Wallace, N., 1981. A Modigliani-Miller theorem for open-market operations. The American Economic Review, 71(3), pp.267-274.
The Fed’s Balance Sheet
Here we would try to understand the different components of the Fed’s balance, for the further study of policy impacts on the economy and the cash in advance model.
The fed’s balance sheet (H.4.1 report) is weekly updated every Thursday, and it presents the assets and liabilities of Federal Reserve Banks.
Assets
Fed’s assets are how the Fed uses money. Most of the money is spent on securities, unamortized premiums and discounts, repurchase agreements, and loans. For example, if the fed buys assets (MBS) or bonds (T-securities) by printing money, then those purchased investments are the assets of the Fed. The followings are some typical assets of the Fed that occupy a significant proportion.
- Treasury Securities: Treasury Securities account for the highest proportion of the fed’s assets. They are consisted of the T-bills, T-notes, and T-bonds by maturity. Treasutry secuirities account for 64% of total assets, (5,533,219 million dollors).
- Mortgage-backed Securities: MBS are securitisations of a basket of home loans. They occupy about 29% of total assets (2,527,824 million dollars).
- Loans: Those are money borrowed by other commerical banks through repo or discount window. The discount window is simplised as that Fed lends money to those borrowers and chages for an interest rate, federal discount rate. Loans have 48,317 million dollars balance.
(Total assets are 8,574,871 million dollars.)
Liabilities
The liabilities of the fed represents how the money comes from, consisting of Federal Reserve Notes, Reverse repurchase agreements, and deposits.
- Federal Reserve Notes: Paper currency (Federal Reserve notes) outstanding net of the quantities held by Reserve Banks. In short, they are net of money oustanding. The balance of it is 2,158,089 million dollars, which is about 25% of total liabilities of the Fed. From historial data, we can find that the money growth at relatively constant rate except during Aprial, 2020. The growth rate jumps.
- Reverse repurchase agreements: Fed sells securities to a counterparty subject to an agreement to purchase back the securities at a later date. The Fed uses Repo or Reserve Repo to conduct open market operations. By Repo, the Fed can release liquidity into the financial amrket. In contrary, the Fed tightens liquidity by revsere repo.
- The deposits, which is also call reserves, are amount of money deposided by commercial banks. The reserve ratio is proportion required in order to keep financial banks operating less risky, and it is also an important tool of the monetary policy. When reserve ratio decrese, then the money supply woul increase. The Board was reduced to zero on 26 March 2020.
Fed’s Tools
- Securities held outright: Treasury security and MBS together consist of the Securities held outright. The increase in those is due to the QE by Federal Reserve. Since the financial crisis of 2008, the Fed started to use QE to boost the economy, through which it purchased a huge number of financial assets (especially long-term assets). That account increased dramatically as well during the pandemic.
- Repos and Reverse Repos: Those transactions are between cash and Treasury securities (mostly short term ones). These open market operations support effective monetary policy implementation and smooth market functioning by helping maintain the federal funds rate within target ranges. Repo: the desk purchases securities from the counterparty subject to an agreement to resell the securities at a later date. Each repo is similar to a loan collateralised by securities, and temporarily increase the supply of reserve banlances in the banking system. Reverse repo is the opposite. The desk sells securities and reduce supply of money.
- Reserve rate as I have discussed before.
Federal Fund Rate
The following content is directly copied from the FRED. Links would be attached in the reference.
The federal funds rate is the interest rate at which depository institutions trade federal funds (balances held at Federal Reserve Banks) with each other overnight. When a depository institution has surplus balances in its reserve account, it lends to other banks in need of larger balances. In simpler terms, a bank with excess cash, which is often referred to as liquidity, will lend to another bank that needs to quickly raise liquidity. The rate that the borrowing institution pays to the lending institution is determined between the two banks; the weighted average rate for all of these types of negotiations is called the effective federal funds rate. The effective federal funds rate is essentially determined by the market but is influenced by the Federal Reserve through open market operations to reach the federal funds rate target.
The Federal Open Market Committee (FOMC) meets eight times a year to determine the federal funds target rate. As previously stated, this rate influences the effective federal funds rate through open market operations or by buying and selling of government bonds (government debt). More specifically, the Federal Reserve decreases liquidity by selling government bonds, thereby raising the federal funds rate because banks have less liquidity to trade with other banks. Similarly, the Federal Reserve can increase liquidity by buying government bonds, decreasing the federal funds rate because banks have excess liquidity for trade. Whether the Federal Reserve wants to buy or sell bonds depends on the state of the economy. If the FOMC believes the economy is growing too fast and inflation pressures are inconsistent with the dual mandate of the Federal Reserve, the Committee may set a higher federal funds rate target to temper economic activity. In the opposing scenario, the FOMC may set a lower federal funds rate target to spur greater economic activity. Therefore, the FOMC must observe the current state of the economy to determine the best course of monetary policy that will maximize economic growth while adhering to the dual mandate set forth by Congress. In making its monetary policy decisions, the FOMC considers a wealth of economic data, such as: trends in prices and wages, employment, consumer spending and income, business investments, and foreign exchange markets.
The federal funds rate is the central interest rate in the U.S. financial market. It influences other interest rates such as the prime rate, which is the rate banks charge their customers with higher credit ratings. Additionally, the federal funds rate indirectly influences longer-term interest rates such as mortgages, loans, and savings, all of which are very important to consumer wealth and confidence.
Key Takeaway: The Fed can adjust the effective federal fund rate by Open Market Operation, QE, etc.
Reference
Fed’s statement until 10 Nov 202`1: https://fred.stlouisfed.org/release/tables?rid=20&eid=1194154#snid=1194156
A good explanation of Repo: https://www.bankrate.com/banking/federal-reserve/why-the-fed-pumps-billions-into-repo-market/
https://www.newyorkfed.org/markets/domestic-market-operations/monetary-policy-implementation/repo-reverse-repo-agreementsMarket Efficiency And Anomaly
Background: Efficient Market Hypothesis
The efficient market hypothesis generally states that the asset price has already reflected all the information (Fama, 1998).
This implies the stock price depends on all fundamentals and includes all information existing in the market. Any small deviations might be quickly diminishing with new information coming out. Therefore, the EMH rules out the profitable investment opportunity in the market, because the price is already the right price.
Is the market efficient or not?
The price might be incorrect in the real market. Actually, the anomaly, which is the deviation of the strategic price from the expected true price, seemingly does exist. Then an issue arises:
Whether the investor could beat the market?
To address this issue, we apply an asset pricing model such as CAPM to calculate the expected benchmark return we should get subject to some factors.
The anomaly can be expressed as:
\alpha=r_{real}-r_{benchmark}
This equation shows that alpha (anomaly), which is the alpha in CAPM, is the real-world return less the benchmark return.
Then, the efficiency can be examined by testing the existence of an anomaly. If an anomaly exists, it means the market is not quite efficient. Therefore, investment strategies could be applied to earn the abnormal return and beat the market. This test can be conducted by constructing a regression model as, \(r_{i,t} =\alpha+r_{f,t}+\beta \times (r_{m,t}-r_{f,t})\) and do joint hypothesis testing.
Here are some relative literature introducing that the market is inefficient. Schwert (2003) stated that the market is not efficient, indicated by the existence of anomaly; Jensen (1978) proofed the inefficiency after adjusting the inconsistent data and missed techniques; Latif et al. (2011) also demonstrated different types of anomalies generated from technical, fundamental and behavioural aspects.
Given that the market is inefficient, the question now is how investors could earn the abnormal profit.
How could investors earn the abnormal profit?
The strategy is to construct a financial portfolio earning the abnormal return (anomaly), and in that strategy, investors select stocks based on their certain characteristics. (two examples below).
The momentum strategy: stocks that performed well in the previous period would also outperform others in the following period (Jegadeesh & Titman, 1993). The contrarian effect is the opposite.
The value investing strategy: stocks with a high P/E ratio or P/B ratio are overvalued, vice versa. Stocks with such as low P/E ratio, Dividend Yield, P/B ratio, etc. are generally under-priced. Thus, buying those cheap stocks could result in a profitable opportunity.
There are numerous investment strategies as we discussed above, and investors could simply apply one strategy or even combine some of the strategies to construct a portfolio to beat the market.
How to test ‘beat the market’?
Applying back-testing to simulate the portfolio’s performance
back-testing is the methodology by which people can see the performance of a certain strategy by some historical data (Campbell, 2005). Here is an example of back-testing below:
Firstly, investors could sort stocks by strategy. If they apply the momentum strategy, they simply sort stocks’ historical return \( (r_{t=1,i}) \) from high to low. Then, investors would make the quartile cut-off, selecting stocks in the first quartile (best-perform stocks) and keeping them into the next period. This periodical return would be the strategic return. After that, we sort the second period return \( (r_{t=2,i}) \), select a first quartile stock again, and keep them into the next period again and again.
Under this procedure, we could use the historical data to track strategy periodical performance. Then, we could examine the existence of an anomaly by subtracting strategic return from the benchmark return.
Reference
Campbell, S.D., 2005. A review of backtesting and backtesting procedures.
Fama, E.F., 1998. Market efficiency, long-term returns, and behavioral finance. Journal of financial economics, 49(3), pp.283-306.
Fama, E.F., 1998. Market efficiency, long-term returns, and behavioral finance. Journal of financial eco
Jegadeesh, N. and Titman, S., 1993. Returns to buying winners and selling losers: Implications for stock market efficiency. The Journal of finance, 48(1), pp.65-91.
Jensen, M.C., 1978. Some anomalous evidence regarding market efficiency. Journal of financial economics, 6(2/3), pp.95-101.
Latif, M., Arshad, S., Fatima, M. and Farooq, S., 2011. Market efficiency, market anomalies, causes, evidences, and some behavioral aspects of market anomalies. Research Journal of Finance and Accounting, 2(9), pp.1-13.
Lux, T. and Marchesi, M., 2000. Volatility clustering in financial markets: a microsimulation of interacting agents. International journal of theoretical and applied finance, 3(04), pp.675-702.
Schwert, G.W., 2003. Anomalies and market efficiency. Handbook of the Economics of Finance, 1, pp.939-974.