Financial Economics - Prof. Varaldo 2023/2024, Appunti di Economia Finanziaria

Scarica Financial Economics - Prof. Varaldo 2023/2024 e più Appunti in PDF di Economia Finanziaria solo su Docsity! FINANCIAL ECONOMICS Prof. Varaldo II period – 6 CFU 1 INTRODUCTION TO FINANCIAL ECONOMICS Financial economics studies the behavior of the financial markets in correlation with the real economy to identify the best asset allocation to increase the wealth = Monetary assets + Bond + Equity + Commodities 𝐶𝑃𝐼 Central concept in financial economics is the investment which is the current commitment of money or other resources in expectation of reaping a future benefit that justify both the time the money is tide as well as the risk of the investment (on value of money and on future payment) → reason is the enter temporal preference for future higher consumption. Therefore we can recognize as basic concept for the investment: ➢ Pure rate of interest: Exchange between future consumption and current consumption; ➢ Pure time value of money: quantification willingness to pay the difference for borrowed funds or desire to receive a surplus on their savings → compensation to keep money in the market for time period; ➢ Inflation; ➢ Uncertainty: under it the investor will demand higher interest rate. Market indexes In order to have a benchmark in financial markets exist several indexes for securities. They hey are used to: ➢ Measure return and volatility of the financial markets; ➢ Benchmark to the performance of professional money manager; ➢ Represent market portfolio of risky asset; When constructing an index it's necessary to look at the sample (which securities are considered with which characteristic), weight of members (which is the influence assigned to each member of the index) and computational procedure (arithmetic average, change of the basic index or geometric average). ➢ Dow Jones industrial average: oldest one, composed of 30 large corporations in NYSE, computed as arithmetic mean of current stock prices adjusted for stock splits(balance number shares prices) and changes in the sample → its purchase represents returns on portfolio folio that invest one share in each of the stock → price index: overvalue more priced Criticized because it's not widely representative (small sample of industrial firms) and because of a downward bias from fast growing companies was stock split. ➢ S&P500: 500 firms, market value weighted index, so it is computed by calculating the total market value of the firms: their daily growth is there growth of index → rate of return of an investor holding a portfolio of all 500 (value of free float) → market value index: overvalue bigger companies. I can also find a series of indexes for the bond market. Usually Between stock index there is a good correlation, while If we consider bond: ➢ High correlation with investment grade bonds which confirms that the overriding determinate of return are the interest rates → same source; ➢ High yield bond has weaker relationship because of strong characteristic; Debt market Bond is a debt instrument that promises a fixed income stream to the holder. According to maturity I can define: ➢ bill or paper: lower than one year ➢ notes: under seven years ➢ bonds: higher time period The characteristic of the bond are: ➢ par value: value paid at maturity → It is not the market value; ➢ coupon: income received over the life defined by a coupon rate; ➢ term to maturity: years before the maturity, usually a single one, except for the serial obligation bonds; 4 • Yield to call: For bonds issued with I call provision is necessary to estimate differently the return, the Yield To Call, We used the discounted cash flow method and adjust it to obtain the current market price: Where: 𝒏𝒄𝒂𝒍𝒍 is the number of years to the first call date and 𝑷𝒄𝒂𝒍𝒍 is the call price of the bond which includes a small premium → more dates necessary to compute the worst yield → YTM of call; • Realized/horizon yield: if interested to anticipate the selling prior to its maturity The investor might be interested to compute the expected rate of return for a holding period lower than the maturity→ YTM for selling before: Bond yields coupon rates and prices Analyzing the way prices are defined we can see that a bond value is inversely related to its yield to maturity and the relationship is convex meaning that decreases in yields have bigger impact on prices than increases in yields of equal magnitude. Since interest rates can fluctuate substantially this is the main loss for fixed income investments. However comparing bonds it is possible to notice differences in sensitivity for interest variation: A. Long term bonds more sensitive: we discount more distant cashflows. However sensitivity increases at decreasing rate as maturity increases; B. Lower coupon bonds more sensitive: higher fraction of value attached to more distant payment; C. Lower YTM more sensitive: lower yield increases the present value of all bond payments but more for more distant payments. Duration We need a measure of the average maturity of the bond (how long to be repaid for price) from promised cash flows, its balance maturity date: the duration. Ceteris paribus: • duration increases when coupon rate lower: more weights on early payment: repayment time up; • duration increases with maturity: distribute more the weights: repayment time up; • duration increases with lower YTM: lower yield means lower average return greater weight of the more distant payments: repayment time up. Duration is the weighted average of times to each coupon or principle payment, is the proportion of the total value divided by the price. • Measured in unit of time; • ZCB have a duration equals to its maturity. An extremely useful property of the duration statistic is that it can be considered as the elasticity coefficient of price respect to yield changes (small change in the yield to maturity), so the relative variation of it can be: ➔ If periods differ from annual, we have to divide the interest for the frequency of payments. We could also use the modified duration (D/1+i) to adjust the former equation as a product between the modified duration and the change in bond yield (multiplying by actual price get the price variation). ➔ Duration is fundamental for bond portfolio management depending on direction of i changes. 5 Convexity As a measure of sensitivity duration is a key tool but is not a good approximation for bigger changes in yield: we need higher level approximation. Indeed, the tradeoff between bond price and the yield to maturity is not a straight line but it's curved, is a convex function, so linear approximations: • Overestimate price decline; • Underestimate price increase → more conservative. Thanks to convexity we approximate better bond prices and also how modified duration will change with yield curve shift: Observing the equation: • This is the rate of change of the slope of the price yield curve as a fraction of bond price; • Degree of convexity inversely related with yield → larger at lower interest → larger gains than loses for same variation → good characteristics for bond portfolio; • Positive for non-callable bond (negative for callable bonds, higher with put): when yield decline price increase at slower rate; • Same maturity but lower coupon rate means larger convexity; • Same coupon rate but longer maturity means larger convexity. With annual payments: Duration in portfolio Investors are often concerned with the effects of interest rate on portfolio, and they need a date that immunize the portfolio from the interest rate risk. In actual practice this is the duration of the portfolio. • Duration of the portfolio is the weighted average of the payment dates for all the cash flow across the entire collection of bonds; • The duration is the weighted average of the duration statistics for each of the bonds with weights equal to the value proportion on the total portfolio: ➔ Stock has not deterministic time period so we cannot compute duration → could theoretically approximate it finding the average number of years necessary to pay back P → DDM. 6 INTEREST RATES Given that who issued a bond is theoretically subject to default risk we must distinguish between the promised yield to maturity and the expected yield, indeed in order to compensate for the possibility of default bonds must offer a premium respect to the riskless bond: the difference is just called the yield spread or credit spread. Pointwise comparison: • traditional yield spread: comparison between two similar bonds in characteristic considering only a single point in the term structure rather than the entire shape of the curve. • interpolated spread: often is impossible to find two bonds with similar coupons and maturities so the spread is the difference between the YTM, and a fitted yield for the same maturity on an appropriate reference yield curve (benchmark curve). Entire shape of the yield: • z-spread: spread that makes the price of a security equal to the present value of its cash flow when added to the yield at each point on the spot rate curve → cash flow discounted at the benchmark rate plus the spread → used to compute relationship treasury-corporate. • option adjusted spread: if a bond has embedded options, we average the spread over the treasury spot rate given the potential interest rate paths. In order to estimate it: i. compute implied forward rates given a probability distribution for treasury's spot rates and the Monte Carlo simulation to generate interest rate paths. ii. develop rules to determine when option exercised. iii. determine cash flows from the paths. iv. using an assumed spread compute to the present value for the paths. v. compare the average present value computed with the market price of the bond: If they are equal, they assumed the spread is the option adjusted spread otherwise repeat. N.B. we can have also spread duration to compute the sensitivity to changes in the option adjusted spread. I = (RFR + 𝝅) + RP → yield to maturity express the expected return to the bond: macroeconomic factors (real risk-free rate as an expression of monetary policies) + issue characteristic (risk premium as determined by the quality of the issue and which is the liquidity). Yield curve The yield curve/term structure of interest rates summarize the relationship between the yield and the maturity for a sample of bonds compatible except for the maturity. Not all the curves have the same shape signs the behavior overtime of the rate is fluid: it represent expectation for future interest rate of the market. We can see that the curve might have different shapes: • Rising: interest rates are at low or moderate level; • Declining: rates are high; • Flat: short term and long-term issues shares the same rate; • Humped: extremely high rates near in the future are expected to decline. ➢ Par yield: YTM for coupon bearing bonds; ➢ Spot yield: expected return from ZCB( F-P/P) → if used curves are upward sloping the spot is higher. Theories of the term structure We have different theories to explain the reasons under different shapes in the curve: 1. Expectation hypothesis: shape of curve results from interest rate expectation of market participants, Therefore long-term interest rate are a geometric mean of current and future one year interest rates (forward) expected to prevail. The equilibrium long term yield is the rate the long-term bond investor would expect to earn through successive investment in short term bonds: a. Rising: expect rising short terms rates; b. Declining: expect falling short term rates; 9 quantity of loans is connected to amount that banks can borrow is linked to their loans to non-financial corporations and households; • APP: asset purchases program (QE), buy of sovereign debt, securitization, guaranteed bonds to strengthen credit normalization; • DEPO: a negative rate on deposits put costs for bank in deposits into the ECB. N.B. during covid coordination between CB was a strong response. N.B. the reason why investors invests in negative interest rates are: interest rate cost less than deposit rates and the investment on solid bonds guarantee certainty return with lower cost, or I expect decrease of interests so there are positive returns on prices. High yield bonds Low rated bonds, under BBB, became notorious as vehicles for hostile takeover attempts, characterized by low liquidity and high volatility: destined to institutional investors. A corporate might be high bond as a consequence of: • Leverage buy-out • First issue of low graded company • Downgrading In this case the objective is to borrow money to lower interests than bank and be listed, acquired releveraged or became investment grade. Empirically we can see that spread of high yield bonds versus treasuries: • widened with ri-leveraged phases (with more debt I need higher returns); • widened when default rate increase (when there are more defaults, I need higher returns); • widened when banking reduces the lending (when it is more difficult to obtain funds from banks, I need higher returns). High yield bond might also be subordinated to the debt of the issuer structurally or by a contractual covenant. In addition, in case of covenants the parts might contract other aspects in favor of the investor: limitation on additional leverage/restricted payments/transaction with affiliates/dividends (guarantees on the ability to repay), assets sales, merger/consolidation/change of control, or from the bank some corporate indicators. Bond portfolio management Including a bond to the portfolio must combine two different approaches: 1. Top-down: get the asset class evaluation by general analysis: a. Macro-context b. Monetary situation c. Relative value analysis with other asset classes 2. Bottom-up: determines the quantity to invest in a specific asset/sector: a. Fundamental analysis: get from all the business information the intrinsic value (economic fundamental value of a business) of the security also compared to the market evaluation → anticipate movement (especially long run). b. Technical analysis: use the pattern of historical financial data to analyze the price trends or volume trends to anticipate movement → bubbles (especially short run). c. Sector analysis. Going into details on fixed-income portfolios we can see that they produce less volatility and more stable returns than other classes. In addition, during the recent decades with higher volatility in interest rates (movement of prices) in the last decades we get increasingly attractive returns to bond investors: capital gain from those shifts to be especially attractive. 10 In this context, is important to analyze the credit quality especially in the high-yield segment: knowing the likelihood of quality change (e.g. downgrade or default) and quantifying correctly is critical to the success in the risky-segment → balancing correctly return with risk. The strategies on bond-portfolios variate accordingly to two characteristics: credit quality and duration (interest rate sensitivity). Then the main families for bond strategies are: 1. Passive management: divided in: a. Buy and hold: according to necessity of client I build a specific portfolio and hold it until maturity: I target specific characteristics (quality, coupon… ) to get the desired return and risk. When the bond mature the bond manager should reinvest the funds: it happens at managed regular intervals. A modified version accept the possibility of trade into more desirable position if big occasions arises; b. Indexing strategy: the construction of the portfolio should match as closely as possible a specific index with full replication or with a stratified sampling, that is to reproduce the main characteristics but maintaining a more cost-effective portfolio. The ability here is to match as good as possible the main features to reduce the tracking error; 2. Active management strategies: construct portfolios whose risk-adjusted returns outperform a specific benchmark. The portfolios differs for the relative active weight assigned to more macro top-down views (behavior of different interests) or bottom-up issuers/issues. Important elements: a. Interest rates anticipation: I alter the portfolio duration to preserve capital anticipating interest rates increase or achieve capital gain when I anticipate decline. In case of interest increase, I switch ZCB to intermediate maturity high coupon bond to shorten the duration (shorten when rising interests, lower than benchmark, moving between countries also). Another way to shorten maturities is to use a “cushion bond” that is a high-yield long term bond with coupon rate above the market level and that because of call features has a price lower than normal given market yields: sold at premium and get before P (D down). For rate decline we increase the D because in this way the greater is the positive price volatility: exploit potential for capital gain. Given that sensitivity is critical (more prices return) the manager should focus on high grades bond (more sensitive) and without call risk (call used to have better interest). Portfolio managers can distinguish expectation for different maturities and therefore for certain bucket/ranges of maturities play their bet in accordance with the expectations. b. Credit allocation: the allocation between different issuers is critical. For example in case of expectation of a recovery if we have low or negative yields for governments, we might underweight it and have a positive allocation in corporate bonds. However, bond markets rather than being a stand-alone asset class have an important role in a broader- meaning portfolio construction: using the safest segment we can appreciate the diversification role correlating it with the equity market. ➔ Remember flying to quality 11 EQUITY In order to evaluate the profitability, the efficiency and the risk related to the investment into a specific firm is necessary to look at the documents that shows in a standardize and formal way the resources and how are financed also in terms of time dynamics: 1. balance sheets; 2. income statements: profitability of firm indicating how this profitability is produced; 3. cash flow statement: shows the effect on the cash flows of income flows and changes in the items of balance sheet, dividing it in different categories operating/investing/financing. The analysis is conducted via ratios which consider relative to the size and main characteristics of the firm and aim to estimate future performances. The value must then be put in perspective with the industry level or the entire economy using average performances, subset of competitors or composite indicators, and in time perspective. • Vertical analysis: relationship between items for a single year; • Horizontal analysis: absolute time change variation. Main topic: 1. Liquidity: convert assets into cash with short notice a. Current ratio: pay tempestive way current liabilities. b. Quick ratio: remove inventories from current ratio, eliminate the most rigid element current asset. c. Cash ratio: fresh money to repay the liabilities → conservative form of liquidity ratio. d. Receivables turnover: quality of accounts receivable, that is how efficiently a company collect receivables from client, the ratio measures the times that receivables are converted to cash during a certain time period. e. Inventory turnover: Number of times the inventory has been replaced, that is we measure the excess/lack of demand relative to the supply: inverted*365 = days to sell inventory. → PROF USES ALSO SALES INSTEAD OF COGS 2. Operating performances: we measure the quality of management, in terms of a. Efficiency: how are used the assets/capital, how many dollars of sales we create from an asset dollar. The measures are the total asset turnover, representing the effectiveness on the use of total/fixed assets: → net assets = minus depreciation of fixed assets → fixed=Long term (Net sales= sales- discounts-returns…) b. Profitability: percentage of sales or investment converted into profit. ➢ Gross profit margin: profitability on sales (gross profit = net sales - cost of goods sold). 14 • Relative valuation technique: how much I’m paying for dollar values of earnings/asset value/CF obtained by comparing the relative price to significant variables (sales, book value, earning, cash flow ratios) , using information about how the market is valuing stocks (EXPECTATION) but no guidance as to whether valuation are appropriate (relative to current market but not intrinsic). Both methods try to evaluate if the business can be purchased at a significant discount to its fundamental intrinsic value. Dividend discount model The value of a share is the present value of all future dividends, that is the value of common stock as a sum of actualized dividend: If the firm grow at constant and infinite rate the equation can be developed as: Instead looking at the value at some future period N.B. the required rate of return is higher than the growth, not the “growth firms” which reinvest the high percentage of returns, for them we use a multistep DDM (evaluate the years of supernormal growth, temporary, and then use a sustainable rate). g: the historical growth rate, where the retention rate is how much of the net income invested is reinvested into the company (what guarantee the growth). It can be expressed alternatively as geometric mean or reinvestment rate of return on equity: k: When valuing equity we look at the risk in the equity investment, given by specific characteristics of the firms (as financial leverage): from specific level of risk we have consequent required rate of return . The required rate of return/cost of capital or equity is a function of Risk-free rate of return, market rate of return and the risk coefficient for holding the bond: → CAPM like formula Considering the value of the stock we can transform it into the useful ratio of P/E, used to explain the reason why P represents the return on the title. According to the DDM P/E is affected by the cost of equity, the growth rate of dividends and the payout ratio (how much of earn is distributed). Free cash flow to equity or firm When we do not have dividends or earnings, we cannot use the former formula and in this case, we evaluate the stock using the cash flow disposable without the capital expenditure, that is we consider the excess cash after meeting the operating and reinvestment need: we want to see the cash generated after accounting for outflows to maintain the capital asset or support operation → exclude from income statement non-cash expenses but include investment. We define: • FCFE= net income + depreciation - capital expenditure – variation in working capital – principal debt repayment + new debt issues → income - old debt expenses – gross variation in capital • FCFF = EBIT (1- tax rate) + depreciation expense – capital spending – Δ in working capital – Δ in other assets (after tax operating income) → to the firms mean that we consider the cash going to the firm before debt cost. N.B. Or the FCFF can be also replaced by the operating free cash flow to the firm. 15 In an analogous way as the DDM (objective is to compare them) we can use a model for constant growth: The expected growth rates of free cash flows can be computed as: • Return on invested capital (ROIC) is the amount of money a company makes that is above the average cost it pays for its debt and equity capital → use of capital to generate profits • Weighted average cost of capital: sum of cost of equity and debt (k and i) weighted for the proportion of equity and debt on total capital. 𝑊𝐴𝐶𝐶 = 𝑊𝐶 ∗ 𝑘 + 𝑊𝐷 ∗ 𝑖 Equity portfolio management strategies One way to distinguish between active/passive strategy is to decompose the actual return in: • passive strategy: I get only the expected return: buy-and-hold, track an index (approximatively given commission), match market performances → strong pro are lower cost of management. The basic techniques for passive management are: A. Full replication: ➢ replicate all securities in the same proportion as index → close tracking ➢ increase transaction costs → dividend reinvestment B. Sampling: ➢ representative sample of stocks of index → fewer so lower commission → easier to reinvest dividends ➢ lower tracking → tracking error C. Quadratic optimization: ➢ use historical information on price changes and correlations to find the portfolio composition that minimize the tracking error ➢ failure of the method is given by change in historical correlations between securities. Whereas the tracking error are the standard deviation of the differential between the returns of the portfolio and of the benchmark normally annualized: → goal: minimize it (minimize return volatility portfolio-benchmark) Sometimes, active and passive might be complemented to cover the entire market: if the portfolio overweight some sectors or stock types (to find alpha) the manager might want to invest some remaining funds to “fill the holes” passively. • active strategy: I try to outperform the benchmark (selected) with the alpha: risk-adjusted the portfolio to seek alpha → outperform the benchmark in terms of net return (also offset the transaction costs, moving between securities) and given higher risk, taking care of it in the needed performance. Top-down: country/asset→ sector→ individual securities Bottom-Up: selection of securities without initial market analysis → equities purchased at discount (valuation model). When is actuated an active strategy is important to understand the timing and direction (forecast) of the broad equity market, in order to correctly shifts between securities, and of the sectors, move across sectors, in order to find also undervalued stocks. A similar method is to use a business-cycle analysis to understand when is best to invest in specific titles: ➢ late/decline: consumer staples (necessary goods) → low risk ➢ recession/bottom: consumer durable → low risk ➢ early/rise: capital goods or consumer discretionary → high risk ➢ mid/peak: basic industries → medium risk 16 Asset class allocation correlated also to the offensive/defensive position of the cycle. According to some analyst, there might be some anormal trends in stock prices concentrated in specific periods of the calendar January/weekend. Then, three technical strategies used as guidance in investment are: • Contrarian investment strategy: according to this theory the optimal timing on an investment is when the investors are too pessimistic about the stock. Indeed, using the idea that the stock will eventually come back to its long-run average level it is possible to exploit a performance that is greatly different from the historical average before coming back to normal pattern. A case of abnormal performance are given by the overreaction hypothesis: overbought or oversell as a consequence of cognitive bias. In this case historical information are a useful tool to rebalance the mix to take advantage of the changing market conditions. • Price momentum strategy: using some technical indicator I can assume that recent trend in past prices will continue I try to capitalize this continuance of existing market trend (post 2008). • Earnings momentum strategy: in this case momentum measurable as the difference between earning per share (EPS) actual and expected, that is if the earnings are accelerating than prior periods: earnings variation might indicate anticipate the variation in prices. • Factor based investment strategy: if the manager believe that exist some characteristics in stocks (size, r volatility, …) that produce higher risk-adjusted returns than the benchmark, he can achieve both high risk-premia and better diversification than traditional passive index fund. We can also assess how active a manager strategy is by looking at the portfolio holdings compared to those in the benchmark, that is by looking at the average difference in weight between the portfolios: RISK AND RETURNS As we have seen interest rate directly determine the expected return in the fixed income market: the promised return however might be affected by movement in the purchasing power. For this reason we need to distinguish between the promised/nominal interest rate and real interest rate, the growth rate of the purchasing power: r = i−π 1+π (or approximated as difference). N.B. in Italy destroyed much of purchasing power inflation historically. ➔ Even if there are more rates, they move together so we can talk of single representative rate (S=D). The in order to compute the net real return we have to multiply the return by (1-t) to contemplate the possible effect of income taxation. For securities an important measure of return is the computation of the return as percentage of initial price of the stock if the share is held for the entire period: Risk There is evidence, taking as an example Italy, that especially in the stock market there is considerable uncertainty about the performances of shares’ in the futures: using a portfolio with the best 3 titles of the past 3 years we have poor performances → past returns gives biased analysis alone → best in one year is not in the next year. ITA: BOT 3M has not been always a risk-free investment, as we can see from the high volatility in sovereign debt crisis. To characterize the uncertainty in investment return we use standard deviation: The problem of this characterization that the distance, volatility from the average result does not express the direction of the deviation. 19 N.B. For dollar and euro exchange the quotation is EUR/USD (not consistent with mathematical representation) the method is uncertain/certain or $/€: how many dollars I need to buy 1 €: ➢ Price of 1$ is 1/E € ➢ Price of 1 € is E $ Macroeconomic reasons for currency variation or risk: • Interest differential: different returns on deposit; • Net exports which brings in and out the reserves; • Politic specific situations that bring uncertainty. In order to get the forward exchange we need to consider the spot exchange plus the interest rate differential. N.B. currency swap curve is also a benchmark for evaluation of the expectation on exchanges. Credit default swap The CDS is an option-like arrangement (initial premium) where one party swap/offset its credit risk with that of another investor: the lender buy the CDS from another investor who agrees to reimburse him (settlement payment, pay par , physical or cash, or the difference with the post-default value) in case of default but obtain an ongoing premium payment until maturity (usually quarterly). This instrument is used for: • Bankruptcy; • Failure to pay in a time → default; • Default rating downgrade; • Restructuring. The CDS premium approximately: risky par yield – treasury par yield (same maturity) → it increases dramatically with falling credit ratings. Given that the protection buyer pays that risk, what is actually doing is transfer the credit spread/risk premium to the protection seller → insurance policy CDS buyers against loss of principal (naked CDS, not owing the object of the CDS, is betting). Futures contracts The position of many company might heavily depend on the actual decision on prices of the something in the future, which in case of high volatility might represent a strong business risk. In this cases the subject might hedge the risk of price volatility with a forward contract (only settlement period differently to swap): • Forward contract: right and full obligation to conduct a transaction involving another security or commodity: ➢ predeterminate date T ➢ predetermined price 𝐹0,𝑇 → locked-in for future events ➢ predetermined buyer (long position) predetermined seller (short position) • Forward intrinsic value: price at maturity -locked price → (computed everyday). Futures formalize and standardize the forward contracting because both party are in a central exchange market should meet some “standard” features, ( daily settlement price at end of day, volumes, delivery…) : we eliminate the flexibility (customization) in forward contract, but with standardization we make it easier to be traded, giving more liquidity as trader prefer to concentrate on small set of contract and instead of paying the entire contract amount as guarantee, should only met a security deposit “margin” (1% or 2%). Clearing house: institution who facilitate the exchange of payment, standing between two clearing firms. Aiming to reduce costs, settlement/operational risk → neutral, both long/short. The long position profit from price increase (because the purchase is a lower cost) while the profit of the short position is specular (sell at higher revenue than the cost): Profit to long= (spot price – original future price)* tick * n° contracts Profit to short= (original future price – spot price)* tick * n° contracts = -profit to long Where: tick= unitary value in the contract (e.g. quantity of corn protected) 20 The process by which profits, and losses are accrued (accumulated) is called “marking to market”, done with reference to the margin (deposit on the account) daily → less cost (only margin) and daily profits → room for speculation. N.B. Future contract is a zero-sum game → aggregate zero profit in market → no effect on prices. • Contago= future prices higher than actual spot, so forward curve slope up but concavely (differential with spot reducing) → weak demand so market amplify contago; • Backward= future prices lower than actual spot, so forward curve slopes down but convexely (differential with spot reducing). If we fear that we could face a big loss given to adverse variation of the underlying, we could use a future/forward with an opposite direction to the price variation and get an aggregate position that is approximately null. Interest rate future In this case the underlying instrument pays interest: the contract regards the future delivery of any interest- bearing asset, locking in the price. In case of bond with fixed rate the question is how to transform future value today in future value, the conversion rate, for the exchange/future value: 𝐸𝑥𝑐ℎ𝑎𝑛𝑔𝑒 𝑣𝑎𝑙𝑢𝑒 = ( p5min 100 ∗ 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑓𝑢𝑡𝑢𝑟𝑒 ∗ 𝑛°𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠 ∗ 𝐹𝐶) + 𝑖 𝐹𝐶 = 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 = AV (1 + isemiann) 𝑔 182.5 - c ∗ 180 − h + 1 180 ∗ 1 100 h= commercial days → get proportion of working days g=days between delivery and coupon date c= gross coupon AV= actual value of the coupon flows and nominal amount of the underlying p5min= price 5 min before closing Option The option are contracts similar to the forwards/futures (both over the counters and on exchanges) except that: • Forward does not need any up front (=in advance) payment while option does; • Option allows but not require the future settlement payment → predetermined price and date. If the investor sell= put option while if buy= call option, the main characteristics of the contract are: • Exercise/strike price: X is the call buyer will pay to or they put buyer will receive in the transaction; • Option premium: 𝐶0,𝑇 is the price that the option buyer must pay to the seller at date 0 to acquire the option → pay the right; • European option can only be exercised at maturity, American any time up to the expiration date; • Intrinsic value: value that the buyer could extract if she exercised the option immediately (payoff) now (what the possibility value now); • Time value is difference between the option premium and the intrinsic component → difference price and value if expiring now. An investor exercise X if he has advantage to S, given this if this isn’t advantageous the condition is “out of the money”, if is equal “at the money” if advantageous “in the money” (cover partially premium, profit break even or net profit, which can be unlimited for call) → profit =payoff- premium N.B. Put and Call are contrary in terms of payoff and profit (i decide to sell at X with put and decide to buy at X with call) while if the investor is the writer (short) of the option, the payoff is the contrary because I depend in other people X decision. 21 Option strategies Many variety of payoff patterns can be achieved by combining put and calls with various exercise prices. We distinguish (bull=up, bear=down): • spread: only calls or only puts some are purchased and some written: ➢ Bull spread: buy a call and sell an otherwise identical call with higher strike price → bet on low increasing prices→ three regions: limit downside loss and diminish cost, bet on low variation, limit upside →bullish as the payoff increase with prices; ➢ Bear spread: is a position in which I write a call and buy an otherwise identical call with higher strike price→ bet on low decreasing prices → specular to bullish spread; ➢ Box spread: use option to create synthetic long forward at one price and short forward at a different price combining bull + bear spread→ to borrow or lend money RF (PV intrinsic value box – net premium paid); ➢ Ratio spread: buying m calls at one strike and selling n calls (m>n) at a different strike (higher) (same time to maturity and asset) → small downside moves with unlimited risk; • collar: purchase of a protective put and sale (write) a call option with higher strike price both with the same underlying asset and same expiration date: create a synthetic short position with potentially no cost if done correctly; • protective put: I protect myself against stock value drop: unwilling to bear losses beyond some given level, purchase a put + stock→ combine their payoff together: form of portfolio insurance. ➔ protective call if I’ve short position on the security Hedge with options: - not limit upside - costly because of premium - not forced to exercise • covered call: purchase of stock coupled WRITING (selling) of a call on that stock, out of the money, to finance myself, but I need to be “covered” by the stock in the portfolio by also buying the stock: stock price go up I benefit from it and from gaining the premium, this until the person does not exercise the call. Non directional positions: do not care of direction of the stock but only of how much it moves → bet/diversify on volatility. • Straddle: long straddle established by buying both a call and a put with the same exercise price and the same expiration date, useful where there is a lot of variability, but we don't know the direction: bets on volatility. The main disadvantage of this strategy is that the purchase of both put and call: the value of portfolio must exceed the initial double option payment in order to clear a profit; 24 • Capped call option at 3.5%; • Capped put option at 0.5%; • Combine them with a swap; Instruments to invest in volatility: • Option on volatility: combining options to achieve exposure only to volatility without directional price is the oldest way. We use liquid instruments, but it might requires expensive hedging; • Volatility/variance swap: (actually forward) the two parties agrees to exchange cash flows based on the realized volatility of the underlying asset, computed in unit of variance or standard deviation (volatility): 𝑷𝒂𝒚𝒐𝒇𝒇 = 𝑵𝒐𝒕𝒊𝒐𝒏𝒂𝒍 𝑨𝒎𝒐𝒖𝒏𝒕 ∗ (𝑽𝒐𝒍𝒂𝒕𝒊𝒍𝒊𝒕𝒚 – 𝑽𝒐𝒍𝒂𝒕𝒊𝒍𝒊𝒕𝒚 𝑺𝒕𝒓𝒊𝒌𝒆) • VIX future: VIX represents the estimated forward volatility ( weighted average of implied volatility for a range of strikes) usually on S&P. Differently to variance and volatility swap this is not related to the realized volatility but to the implied volatility → useful to have an estimation of volatility. Model to volatility: • Black Scholes derivates formula: heteroskedasticity in equity return assumption; • GARCH most accepted model in short time frames: mean reversion end necessity that volatility does not change its regime; • Clustering (if now low likely to be low in near future) and mean reversion characteristic (if increase a lot in a day it will bounce back soon). MARKOWITZ, CMT AND CAPM Assumption: • The objective of an investor is to maximize from the total return from investments for a given level of risk; • Portfolio not only collection of investment but considered in their relationship; • Risk aversion: same expected return but prefer the lower risk → evidence from insurance or different yield with classification; N.B. Not all investors are risk averse: small amounts risk is smaller but with large share of losses. • Risk is the uncertainty of future outcome or the probability of adverse outcome. Markowitz theory Markowitz theory is based on some assumption on investor behavior: • all investment are alternative (comparable)depending on probability distribution of potential return; • rational egoism of utility and the diminishing marginal utility of wealth; • risk is estimated with variability of returns (VAR/STD, range of returns, semi-variance (only above a benchmark of the mean) ) → prefer VAR/STD because of their simplicity and usage in models; • only factor for investment decision are expected return and variance: setting risk prefer higher return, setting return prefer lower risk. From this we can define a portfolio efficient if there is no other asset or portfolio of assets that is a pareto improvement for risk/return with constant risk/return. Important measures: E(r)= probability - pondered average of potential return = weighted average of individual investment expected returns Individual asset variance: 25 Portfolio covariance and correlation: N.B. We are equally interested in upward and downward volatility. Portfolio STD: Looking at the STD of the portfolio formula if we add another asset in the portfolio: • modify covariance of returns → strongest variation since the relative weight is greater than the unique weight of the asset alone (more factor more is this true) → what focus on; • modify the weighted sum of variances. In order to reduce the risk in the portfolio and therefore improve its performance, exploiting this relationship we can exploit diversification, that is we can find the right types of assets so that in terms of risk we are better off: • Equal returns and variability: not perfectly positively correlated assets (we spread out variability) , but finding negative covariance (reduce volatility of results) term in order to offset standard deviation and leading risk to zero → risk free, constant in time return → maximum benefits of diversification give an equal return is given by negatively correlated assets; • In reality the return and the standard deviation of assets are not constant: even if perfectly negatively correlated no 0 risk of portfolio if we keep the weights constant. However, we can see that modifying the weights with the same correlation between the two assets we can have different performances in terms of expected returns and portfolio risk → lower is the correlation the best I exploit benefits of diversification. If we generalize, if stocks returns can be described by a single market model the number of correlations required reduces to the number of assets. Efficient frontier The efficient frontier represent that set of portfolios with the maximum rate of return for any given level of risk or the minimum risk for every level of return → pareto efficiency for portfolios. Therefore, to find the portfolios on the efficient frontier the problem that investor needs to solve is to find the weights of investments that minimize the portfolio risk producing a desired level of expected return with weights of investments equal to 1 → called the mean variance optimization because we minimize the variance given the mean. The tradeoff between return and risk is described by its utility curve which therefore give us the optimal portfolio for the investors: tangency point between the two curves. The higher is the slope of the indifference curve the more risk averse the individual is: to counterbalance the increase of risk he ask for much higher returns. Capital market line Sharpe extended the Markowitz efficient frontier result in order to derive the optimal portfolio investment strategy, we get the so-called Capital Market Theory, this approach depends on the existence of a risk-free asset to build the market portfolio. The assumption are: • Markowitz efficiency of investors; • Infinite ability to borrow or lend at risk free rate; • Homogeneous expectation: same probability distribution; 26 • Single horizon of investment; • Investment infinitely divisible → any fractional share; • No transaction or taxes cost; • Total anticipation of inflation and changing interest rates → infinite predictivity → no EMH; • Equilibrium of capital markets → correct pricing with risk. Firstly, we introduce a risk-free asset in the portfolio, it has by definition perfectly known return therefore has standard deviation and its covariance and correlation with the assets are zero. Therefore, a portfolio with a risk-free asset has: Substituting for the objective return we get the allocation of funds between the two choices. Therefore, solving together the two equations: This formula tell us that for the risky portfolio M the return are given by the risk-free rate plus a compensation for the number of risk units. This linear relationship risk-return is called capital allocation line, the slope is the “return to variability ratio” or Sharpe ratio and is the excess of rate of return of the portfolio to the RFR normalized on its standard deviation. This represent a line connecting a portfolio on the Efficient Frontier. Whenever the portfolio contain all risk assets had in the marketplace this linear relationship is called the capital market line: the market level of additional returns from RFR as a function of the level of risk. Considering the CML together with the efficient frontier we get that the optimal portfolio is the one at which the Sharpe ratio, equals the slope the of the Markowitz efficient frontier, the one with the highest CAL (best Sharpe ratio). Therefore, by incorporating all the relevant information about the universe of securities, they can purchase the optimal portfolio, which is the market one → passive strategy is efficient. ➢ Portfolios above the CML does not exist → better Sharpe ratio than the efficient portfolio; ➢ Portfolios under the CML → under the EF → inefficient portfolios. Given the possibility, by assumption, to borrow/lend at RFR, according to the risk taking/risk averse position of the investor is possible to move from the optimal portfolio on the CML: • Increase return by borrowing at RFR → risk taker than optimal allocation; • Decrease risk by lending at RFR → risk averse than optimal allocation. Theory of separation the step for the managers are two and separate: 1. Find the optimal risky portfolio by maximizing the CAL; 2. Move according to investor risk taste on the CPM → financing decision: we are still buying efficient portfolios in terms of slope but moving according to risk taste to increase the exposure relative to M → indifference curves. Therefore, the only relevant portfolio is the market portfolio M, this means that the only important decision for the individual asset is to determine the correlation with M → covariance is the relevant element (how the asset variate respect to M) → reproduce market volatility so positive correlation. CML risk Given that the CML contains all the risky assets on the markets, it represent a completely diversified portfolio, so the unique risk of individual asset (e.g. firm performances) is diversified: we reduce the standard deviation of the total portfolio, because when we add securities we expect the average covariance to decline, but not to go to zero given imperfect correlation. The one risk that remain in the portfolio is the systematic risk, which is the risk attributable to market wide sources, macroeconomic variables. Where: ➢ 𝑹𝒎 returns for the aggregate stock market; 29 a. RF portfolio → net zero exposure To replice the RF portfolio we take a short position to negative weights portfolios and long position to positively weights ones. b. absence of arbitrage (𝑟𝑓 = 𝑟𝑝) → portfolio risk free: Return to portfolio equal return to the factor (same risk) , which, given the 0 exposure to the factor , gives 0 identity, if we substitute for the weights: ➔ risk premium in portfolio is the ratio between exposure to factor-risk and E returns. With a single factor I can derive the risk premium and the risk free rate knowing b and the expected returns for the single security, from the system (without arbitrage opportunity = RF mispricing we know that for the same factor there is the same lambdas): ➔ they share only risk premium (reward for same factor) and RF. Step to find the arbitrage opportunity (one factor): • Use the betas and expected returns of the 2 portfolios to compute the lambdas implied in the price (no arbitrage condition) → joint portfolio; • Estimated the beta of a third market portfolio; • Compute the expected return of portfolio E using the pricing equation of APT: returns of synthetic portfolio of the first two that replicate the third (pricing equation of portfolio 1 and 2 but exposure of three) → lambdas from 1 and 2 and exposure from 3 (weights from there) → short the one with lower E(r) and long the one with higher E(r). Multifactor model I can generalize to a K factor model, where I can use K+1 diversified portfolio to build the risk-free portfolio, solving the system. That is that we can: 1. find the lambdas of market; 2. obtain the weights by replicating a mispriced portfolio → b replica portfolio is the weighted average of underlying betas; 3. short sell the one with lower expected return and long the other. In this case the cash flows are: 30 a. 0 initial CF given the short/long b. positive CF from the expected values c. 0 exposure to the factors Empirical studies To identify and measure factors and risk premium there are three approaches: 1. Factor analysis: determine betas and lambdas jointly, In this case we determine the number of factors, but we capture linear combination of economic forces which might variate according to the sample; 2. Returns to K well diversified portfolios to find a representation of the risk premium; 3. Use a set of macroeconomic factors: time series regression of returns and factors to estimate betas and then cross-sectional regression of the stocks return on betas to estimate Lambda, here factors must explain large portion of returns variability and must be interpretable a robust overtime → focus here is on choices. Chen, Roll and Ross used as f (unanticipated variation): • Industrial production → not significant • Inflation (also expected) • Risk premium → confidence • Term structure of rates → slope is time horizon • Return to equally and value weighted NYSE portfolios → not significant N.B. Even if this struggle with the definition of Lambda the risk premium might be negative because with inflation the exposition is negative and the contribute to the return is positive. • BARRA: multi factor model that measure the overall risk associated with the security, relative to the market : ➢ confidence as the default spread ➢ time horizon as the slope of the term structure rates ➢ inflation risk as unexpected inflation ➢ business cycle risk as monthly industrial production ➢ market timing risk as residual of the S&P 500 regression on the previous factors BLACK & LITTERMAN MODEL This is a model that combine market equilibrium theory with investors opinions using the Bayes theorem to calculate conditional probability: it is forward looking because investors look at the future. • Start from neutral position: compute the equilibrium returns, the one that equalize supply and demand (market portfolio) using reverse optimization using the matrix of variance, the risk aversion factor and the market capitalization of asset classes as weights in the market portfolio. Then from the variance matrix times a proportionality constant we compute the variance of the mean estimate of return • Take additional input from investors: they can express the view as absolute (expected return on a single asset) or a relative view (differential between two asset classes), the views are independent and it's not mandatory to express them. Given K views on n assets they can be represented as a linear combination of expected returns, which is normally distributed (Q vector of views and omega matrix of uncertainties) Banca Aletti Use B&L modern improving it who is the collaboration with the center of econometric analysis of the bias Business School in London. 31 • Returns as VAR model • Volatility with GARCH model → volatility estimation techniques • Modify calculation of equilibrium return: doesn't using CAPM assumption but historical average return and prospective returns • risk define the allocation constraints of assets analyzing systemic risk and liquidity risk • use monthly indicator • incorporating specific views → directionality and confidence • principle of homogeneity mutual exclusivity diversification and availability of information • tested the product before selling them Liquidity focus Liquidity is the ability to exchange a cash flow with other agents of the financial system: CB liquidity concept (aggregate), funding liquidity (debt/credit), market liquidity (trade). In normal times liquidity flow easily between the level of the demand (agents) of the aggregate supply (CB) and of the intermediaries (redistribution in the system), however when illiquidity creeps into the system it creates a vicious circle: • propagate because of interlinkages among banks • banks may seek liquidity through fire sales • fire sales determines impact on balance sheet ➔ need for liquidity injection → CB This problem and risk is quantified by the liquidity risk indicator of the Banca Aletti. EMH AND BEHAVIORAL FINANCE Efficiency market hypothesis: return reflect both price and risk, therefore prices are an image of “all possible information quickly and accurately”, otherwise competitive pressure would lead the price to the equilibrium level: on aggregate investing reduce the asymmetries. All new information, still not reflected in prices, will determine unpredictable price variation: stock prices follow a random walk which is the result of incorporating the information: new equilibrium price which is static → no competitive advantage → zero autocorrelation in time. N.B. investor objective maximize marginal utility. Fama hypothesis start from definitions → differ on interpretation of “all available information” → not future or total info: • Weak efficiency: assets prices reflect public information (historical prices and returns, traded volumes, technical information) from data provider with no cost → technical info from data (statistical analysis of prices) → no info from trend: already incorporated; • Semistrong efficiency: the price of an asset better reflect all the information available both public and private (also data about the company and workers) related to a listed company → firm data is not an opportunity for the investors; • Strong efficiency: all the most important info about an asset, even confidential, are available to investors → no present info to exploit for abnormal return. Other results of the theory: • Technical analysis useless: past info already incorporated, like most of fundamental analysis: only if unknown insight are brought up; • Portfolio management is on the construction of efficiently diversified portfolio depending on risk taking → In addition, to outperform buy and hold only strategy with filter of 0.5% but not if transaction cost are considered (not beneficial active management )→ active management and arbitrage should not let to aggregate profits → more information of some agents is a profit opportunity (Stiglitz); • Normally distributed returns (proved initially but not on effective annual return). This theory started to seem to be appliable from the 70s after a strong speculative phase after oil shock where the poor technical analysis didn't get the price trends.