Finance Notes PDF

Chapter 2: Overview of the Financial System =========================================== **Securities (AKA Financial Instruments)**: claims on the borrower's future income or assets - Securities are assets for the person who buys them and liabilities for the person/firm who sells them **Meth...

Chapter 2: Overview of the Financial System =========================================== **Securities (AKA Financial Instruments)**: claims on the borrower's future income or assets - Securities are assets for the person who buys them and liabilities for the person/firm who sells them **Methods of raising funds:** 1. Debt Instruments 2. Equities **Debt Instrument:** a contractual agreement by the borrower to pay the holder of the instrument fixed dollar amounts at regular intervals (interest payments) until a specified date (the maturity date) when a final payment is made - **Maturity:** the time (term) to that instrument's expiration date - Short term if its maturity is less than a year - Long term if its maturity is ten years or longer - Intermediate term if its maturity is between 1 and 10 years **Equities:** claims to a share in the net income (income after expenses and taxes) and the assets of a business - If you own 1 share of common stock of a company that has issued 100 shares, you are entitled to 1% of the firm's net income and 1% of the firm's assets - Make periodic payments (dividends) to their shareholders - Considered long-term securities because they have no maturity date *Disadvantage of holding a company's equities rather than debt:* the equity holder is a **residual claimant** -- the corporation must pay all its debt holders before the equity holders *Advantages of holding equities*: equity holders benefit directly from increases in the corporation's profitability or asset value because equities confer ownership rights on the equity holders **Primary Markets**: new issues of a security are sold to initial buyers by the company or government agency borrowing the funds **Secondary Markets:** securities have been previously issued (and are thus secondhand) can be resold - Ex. New York and American stock exchanges -- previously issued stocks are traded - **Brokers**: agents of investors who match buyers with sellers of securities - **Dealers**: link buyers and sellers by buying and selling securities at stated prices - **Purpose of secondary markets**: - Make financial instruments more **liquid** - Determine the price of the securities in the primary market - Can be organized into **exchanges** and **over-the-counter** markets **Underwriting Securities (what investment banks do):** guarantees a price for a corporation's securities and then sells them to the public **Exchanges:** buyers and sellers of securities (or their agents or brokers) meet in one central location to conduct trades - NY and American stock exchanges and the Chicago board of trade for commodities **Over-the-counter (OTC) Market:** dealers at different locations who have an inventory of securities stand ready to buy and sell securities "over the counter" to anyone who comes to them and is willing to accept their prices - The US government bond market is set up as an OTC market **Money Market:** a financial instrument in which only short-term debt instruments (maturity of less than 1 year) are traded - Usually more widely traded than longer-term securities, making them *more liquid* - Short-term securities tend to be less volatile making them also safer investments **Capital Market:** the market in which longer-term debt and equity instruments are traded - Often held by financial intermediaries such as insurance companies and pension funds, which have little uncertainty about the amount of funds they will have available in the future. Money Market Instruments ------------------------ **US Treasury Bills** - Short-term debt instruments of the US government - Issued in 3, 6, and 12 month maturities to finance the deficits of the federal government - Pay a set amount off at maturity and have no interest payments - Effectively pay interest by initially selling at a discount (a price lower than the set amount paid at maturity) - The most liquid of all money market instruments because they are the most actively traded - Safest of all money market instruments because there is no possibility of **default** - **Default**: a situation in which the party issuing the debt instrument is unable to make interest payments or pay off the amount owed when the instrument matures - Federal government can always pay off its debt instruments because it can raise taxes or issue **currency** to pay it off - Treasury bills are held mainly by banks, although small amounts are held by households, corporations, and other financial intermediaries **Negotiable Bank Certificates of Deposit** (CD) - **Definition**: a debt instrument sold by a bank to depositors that pays annual interest of a given amount and at maturity pays back the original purchase price - *Negotiable*: can be sold to others; can be redeemed from the bank before maturity without paying a substantial penalty **Commercial Paper**: short-term debt instrument issued by large banks and well-known corporations **Banker's Acceptances:** a bank draft (a promise of payment similar to a check) issued by a firm, payable at some future date, and guaranteed for a fee by the bank that stamps it "accepted" - Created in the course of carrying out international trade and have been in use for hundreds of years - Firm issuing the instrument is required to posit the required funds into its account to cover the draft - If the firm fails to do so, the bank's guarantee means that it is obligated to make good on the draft - *Advantage*: the draft is more likely to be accepted when purchasing goods abroad because the foreign exporter knows that even if the company purchasing the goods goes bankrupt, the bank draft will still be paid off **Repurchase agreements:** - Short-term loans (usually with a maturity of less than 2 weeks) in which treasury bills serve as collateral **Federal (Fed) Funds**: typically overnight loans between banks of their deposits at the federal reserve - Loans are not made by the federal government, or by the federal reserve, but rather by banks to other banks - **Federal funds rate:** a closely watched barometer of the tightness of credit market conditions in the banking system and the stance of monetary policy - When it is high, it indicates that the banks are strapped for funds - When it is low, banks' credit needs are low **Eurodollars**: US Dollars deposited in foreign banks outside the US or in foreign branches of US banks - American banks can borrow these deposits from other banks or from their own foreign branches when they need funds Capital Market Instruments -------------------------- Captial market instrumetns have far wider price fluctuations than money market instruments and are considered to be fairly risky investments **Stocks**: equity claims on the net income and assets of a corporation - Individuals hold around ½ of the value of stocks; the rest are held by pension funds, mutual funds, and insurance companies **Mortgages:** loans to individuals or firms to purchase housing, land, or other real structures, that in turn serve as collateral for the loans - The mortgage market is the *largest debt market in the US* **Corporate Bonds:** long-term bonds issued by corporations with very strong credit ratings - Sends the holder an interest payment twice a year and pays off the face value when the bond matures - *Convertible bonds*: have an additional feature of allowing the holder to convert them into a specified number of shares of stock at any time up to the maturity date - Makes them a more desirable purchase than bonds without it and allows corporation to reduce its interest payments because these bonds can increase in value fi the price of the stock appreciates sufficiently **US Government Securities** - Long-term debt instruments - Issued by US Government to finance the deficits of the federal government - Most widely traded bonds in the US; most liquid security traded in the capital market - Held by federal reserve, banks, households, and foreigners **US Government Agency Securities** - State and local bonds; AKA *municipal bonds* - Long-term instruments issued by state and local governments to finance expenditures on schools, roads, and other large programs - Their interest payments are exempt from federal income tax and generally from state taxes in the issuing state - Commercial banks are the biggest buyer (own over ½ of total amount outstanding) **Consumer and Bank Commercial Loans** - Loans to consumers and businesses made principally by banks, but in case of consumer loans, also by finance companies - Often no secondary markets in these loans, which makes them the *least liquid* of the capital market instruments Financial Intermediaries ------------------------ **Financial intermediation:** the process of indirect finance using financial intermediaries - primary route for moving funds from lenders to borrowers **Commercial Banks:** - raise funds primarily by issuing checkable deposits (deposits on which checks can be written), savings deposits (deposits that are payable on demand but do not allow their owner to write checks), and time deposits (deposits with fixed terms to maturity) - they use these funds to make commercial, consumer, and mortgage loans and to buy US government securities and municipal bonds **Savings and Loan Associations:** - obtain funds primarily through savings deposits (called shares) and time and checkable deposits - usually used for mortgage loans - 2^nd^ largest group of financial intermediaries after commercial banks **Mutual Savings Banks** - Raise funds by accepting deposits (shares) and use them to primarily to make mortgage loans - Differ from S&L Associations because of their corporate structure - Always structured as "mutuals" -- they function as cooperatives - The depositors own the bank - Located primarily in NY state and New England **Credit Unions** - Organized around union members, employees of a particular firm, and so forth - Raise funds from deposits (shares) and primarily make consumer loans - Allowed to issue checkable deposits and can make mortgage loans in addition to consumer loans Contractual Savings Institutions -------------------------------- **Life Insurance Companies** - Insure people against financial hazards following a death and sell annuities (annual income payments upon retirement) - Raise funds from premiums that they charge to keep their policies in force and use them mainly to buy corporate bonds and mortgages - Also purchase stocks but are restricted in the amount that they can hold **Fire and Casualty Insurance Companies** - Similar to life insurance companies but have a greater possibility of loss of funds if major disasters occur - They use their funds to buy *more liquid assets* than life insurance companies do (because they have a greater possibility of major losses) **Pension funds and Government Retirement Funds** - Provide retirement income in the form of annuities to employees who are covered by a pension plan - Largest asset holdings are corporate bonds and stocks Investment Intermediaries ------------------------- **Finance Companies** - Raise funds by selling commercial paper (a short-term debt instrument) and by issuing stocks and bonds - Lend funds to consumers who make purchases of such items as furniture, automobiles, and home improvements - Some companies are organized by a parent corporation to help sell its product **Mutual Funds** - Raise funds by selling shares to many individuals and use the proceeds to purchase diversified portfolios of stocks and bonds - Allow shareholders to pool their resources so that they can take advantage of lower transactions costs when buying large blocks of stocks or bonds - Allow shareholders to hold more diversified portfolios than they otherwise would - Can sell (redeem) shares at any time, but the value of these shares will be determined by the value of the mutual fund's holdings of securities - Investments in mutual funds can be risky **Money Market Mutual Funds** - Have the characteristics of a mutual fund but also function to some extent as a depository institution because they offer deposit-type accounts - Sell shares to raise funds that are then used to buy money market instruments that are both safe and liquid - Interest on these assets is then paid out to shareholders - Key feature of money market mutual funds: shareholders can write checks against the value of their shareholdings - Restrictions on the use of check-writing: checks frequently cannot be written for amounts less than a set minimum and a substantial amount of money is required to initially open an account Regulation of the Financial System ---------------------------------- **Regulatory Agency** **Subject of Regulation** **Nature of Regulation** ----------------------------------------------- ----------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------- Securities and Exchange Commission (SEC) Organized exchanges and financial markets Requires disclosure of information, restricts insider trading Commodities Futures Trading Commission (CFTC) Futures market exchanges Regulates procedures for trading in futures markets Office of the Comptroller of the Currency Federally chartered commercial banks Charters and examines the books of federally chartered commercial banks and imposes restrictions on assets they can hold National Credit Union Administration (NCUA) Federally chartered credit unions Charters and examines the books of federally chartered credit unions and imposes restrictions on assets they can hold State banking and insurance companies State-chartered depository institutions Charter and examine the books of state chartered banks and insurance companies, impose restrictions on assets they can hold, impose restrictions on branching Federal deposit insurance corporation (FDIC) Commercial banks, mutual savings banks, savings and loan associations Provides insurance of up to \$100k for each depositor at a bank, examines the books of insured banks, and imposes restrictions on assets they can hold Federal reserve system All depository institutions Examines the books of commercial banks that are members of the system, sets reserve requirements for all banks Office of Thrift Supervision Savings and Loan associations Examines the books of savings and loan associations, imposes restrictions on assets they can hold The government has issued 6 different types of regulations to protect the public and economy from financial panics: 1. Regulations on who is allowed to set up a financial intermediary a. Regulated by state banking and insurance commissions, as well as the office of the comptroller of currency b. Individuals or groups must obtain a charter from a state or the federal government 2. Stringent reporting requirements for financial intermediaries c. Bookkeeping must follow strict principles and are subject to periodic inspection d. Must make certain information available to the public 3. Restrictions on what financial intermediaries are allowed to do and what assets they can hold e. Restricted in engaging in certain risky activities to ensure financial intermediaries can meet their obligations f. Restricted from holding certain risky assets or at least a large quantity of risky assets 4. Government can insure people providing funds to a financial intermediary from any financial loss if the financial intermediary should fail g. FDIC insures each depositor at a commercial bank or mutual savings bank up to losses of \$100k h. All commercial and mutual savings banks make contributions into the FDIC which are used to pay off depositors in the case of a bank's failure 5. Restrictions on unbridled competition between financial intermediaries i. Restrictions on the opening of additional locations (branches) j. Restrictions on interest rates that can be paid on deposits i. Banks are prohibited from paying interest on checking accounts 6. The FED setting reserve requirements -- the fraction of deposits that depository institutions must keep in accounts with the FED Chapter 3: Understanding Interest Rates ======================================= Amortization ------------ **Amortization:** refers to the process of spreading out a loan or intangible asset's cost over a specified period, through periodic payments or accounting adjustments. - In the context of loans, it involves paying off debt over time in equal installments, which include both interest and principal. - In the context of intangible assets, it reflects the gradual expenses of the asset's cost over its useful life **Loan Amortization** 1. Interest: A cost for borrowing money, based on the outstanding balance. 2. Principal: The repayment of the actual loan amount. \ [\$\$FP = \\frac{{P \\times r \\times \\left( 1 + r \\right)}\^{n}}{\\left( 1 + r \\right)\^{n} - 1},\\ \\ \\ \\ where\$\$]{.math.display}\ \ [*FP* = *Fixed* *Payment* *Amount*]{.math.display}\ \ [*P* = *Loan* *Principal* (amount borrowed)]{.math.display}\ \ [*r* = *Periodic* *Interest* *Rate* (annual rate divided by number of payments per year) ]{.math.display}\ \ [*n* = *Total* *Number* *of* *Payments* *over* *the* *loan*^′^s life]{.math.display}\ \ [\$\$FP = \\frac{10,000 \\times 0.005 \\times \\left( 1 + 0.005 \\right)\^{60}}{\\left( 1 + 0.005 \\right)\^{60} - 1} \\approx 193.01\$\$]{.math.display}\ **Amortization Schedule** - outlines each payment, breaking it into: - **Interest Payment (**[**I**~**t**~]{.math.inline}**):** Based on the remaining loan balance. - **Principal Payment (**[**P**~**t**~]{.math.inline}**):** The remaining portion of the fixed-payment amount after interest. - For month [*t*]{.math.inline}, the interest and principal are calculated as: \ [*I*~*t*~ = *r* \* *B*~*t* − 1~]{.math.display}\ \ [*P*~*t*~ = *FP* − *I*~*t*~]{.math.display}\ \ [*B*~*t*~ = *B*~*t* − 1~ − *P*~*t*~]{.math.display}\ Where [*B*~*t* − 1~]{.math.inline} is the balance at the beginning of month [*t*]{.math.inline}. **Loan-to-Value Ratio (LTV):** Measures the remaining loan balance relative to the asset's value \ [\$\$LTV = \\frac{B\_{t}}{V}\$\$]{.math.display}\ Where [*V*]{.math.inline} is the value of the asset. Amortization reduces [*B*~*t*~]{.math.inline} over time, lowering LTV. **Debt Service Coverage Ratio (DSCR):** used in evaluating loan repayment capacity \ [\$\$DSCR = \\frac{\\text{Net\\ Operating\\ Income\\ }\\left( \\text{NOI} \\right)}{Total\\ Debt\\ Services\\ (TDS)}\\ \$\$]{.math.display}\ Where TDS includes all FP Payments. Proper amortization ensures manageable DSCR. **Intangible Asset Amortization** **Key Financial Ratios** 1. **Return on Assets (ROA):** Reflects profitability relative to asset amortization \ [\$\$ROA = \\frac{\\text{Net\\ Income}}{\\text{Total\\ Assets}}\$\$]{.math.display}\ 2. **Earnings Before Interest, Taxes, Depreciation, and Amortization (EBITDA)** \ [*EBITDA* = *Net* *Income* + *Interest* + *Taxes* + *Depreciation* *and* *Amortization*]{.math.display}\ Credit Market Instruments ------------------------- 4 types of Credit Market Instruments 1. **Simple Loan:** Provides the borrower with an amount of funds (principal) that must be repaid to the lender at the maturity date along with an additional amount known as an *interest* payment 2. **Fixed-payment Loan:** Provides a borrower with an amount of funds that is to be repaid by making the same payment every month, consisting of part of the principal and interest for a set number of years 3. **Coupon Bond**: pays the owner of the bond a fixed interest payment (coupon payment) every year until the maturity date, when a specified final amount (face value or par value) is repaid; identified by three pieces of information: a. The corporation or government agency that issues the bond b. The maturity date of the bond c. The **coupon rate**: the dollar amount of the yearly coupon payment expressed as a percentage of the face value of the bond - Treasury bonds and notes and corporate bonds are examples of coupon bonds - Ex. A coupon bond with \$1000 face value pays \$100 per year for 10 years, then repay the face value of \$1000; the coupon rate here is 10% - **Consol (AKA Perpetual Bond):** no maturity date and no repayment of principal that makes fixed coupon payments of \$*C* forever. a. Relatively rare in the American capital market today 4. **Discount Bond (AKA zero-coupon bond)**: bought at a price below its face value (at a discount) and the face value is repaid at the maturity date d. Does not make any interest payments (unlike a coupon bond); it just pays off the face value. e. Ex. a discount bond with a face value of \$1000 is bought for \$900 and then in a year, the face value of \$1000 is repaid Because some of these instruments above make interest payments (simple loan and coupon bonds) and some don't (fixed-payment loans and discount bonds), we use the concept of *present value* to provide a procedure for measuring interest rates on different instruments. Present Value ------------- **Present Value**: based on the commonsense notion that a dollar paid to you one year form now is less valuable to you than a dollar today **Simple Interest Rate:** the measure of the cost of borrowing funds based on interest payments on a simple loan. \ [\$\$i = \\frac{\\text{Total\\ Interest\\ Payments}}{\\text{Initial\\ Loan\\ Value}}\$\$]{.math.display}\ Ex. a simple loan of \$100 today, with interest payments totaling \$10 over the next year, then repaid in full (\$100) next year, has interest rate, \ [\$\$i = \\frac{10}{100} = 0.1 = 10\\%\$\$]{.math.display}\ If you are the loaner, you will return *\$A* when loaning out all your money each year starting with *I*, the initial loan value \ [*A* = *I* × (1+*i*)^*n*^]{.math.display}\ Where I is the initial loan value, i is the simple interest rate, and n is the number of years you consecutively loan out all the money. **General Formula for Present Value (PV):** [\$PV = \\frac{\\text{FV}}{\\left( 1 + r \\right)\^{n}},\\ where\\ \$]{.math.inline} \ [*FV* = *Future* *Value*]{.math.display}\ \ [*r* = *discount* *rate*]{.math.display}\ \ [*n* = *number* *of* *periods* (*years*, *months*, *etc*.)]{.math.display}\ **Example 1: Single Cash Flow** \ [\$\$PV = \\frac{\\\$ 1000}{\\left( 1 + 0.05 \\right)\^{3}} \\approx \\\$ 863.84\$\$]{.math.display}\ **Formula for Present Value of a Stream of Cash Flows:** \ [\$\$PV = \\sum\_{t = 1}\^{n}\\frac{CF\_{t}}{\\left( 1 + r \\right)\^{t}},\\ \\ \\ \\ \\ \\ where\$\$]{.math.display}\ \ [*CF*~*t*~ = Cash Flow at time t]{.math.display}\ \ [*r* = *discount* *rate*]{.math.display}\ \ [*n* = *Total* *number* *of* *periods*]{.math.display}\ **Example 2: Cash Flow Stream** - - - \ [\$\$PV = \\frac{\\\$ 100}{\\left( 1 + 0.1 \\right)\^{1}} + \\frac{\\\$ 200}{\\left( 1 + 0.1 \\right)\^{2}} + \\frac{\\\$ 300}{\\left( 1 + 0.1 \\right)\^{3}} = \\\$ 481.79\$\$]{.math.display}\ **Yield to Maturity:** the interest rate that equates the present value of payments received from a debt instrument with its value today. **Calculating Yield to Maturity for the Four Types of Credit Market Instruments** **Simple Loan.** For simple loans, YTM = the simple interest rate \ [\$\$YTM = \\left( \\frac{\\text{FV}}{\\text{PV}} \\right)\^{\\frac{1}{n}} - 1\$\$]{.math.display}\ **Fixed-Payment Loan** YTM is calculated by solving for *r* in the equation, \ [\$\$FV = \\ \\sum\_{t = 1}\^{n}\\frac{\\text{FP}}{\\left( 1 + r \\right)\^{t}},\\ \\ \\ \\ \\ where\\ \$\$]{.math.display}\ \ [*FV* = *Loan* *Face* *Value* ]{.math.display}\ \ [*FP* = *Fixed* *Payment* ]{.math.display}\ \ [ *r* = *YTM* (unknown) ]{.math.display}\ \ [*n* = *number* *of* *payments*]{.math.display}\ \ [\$\$1000 = \\frac{300}{\\left( 1 + r \\right)\^{1}} + \\frac{300}{\\left( 1 + r \\right)\^{2}} + \\frac{300}{\\left( 1 + r \\right)\^{3}} + \\frac{300}{\\left( 1 + r \\right)\^{4}}\$\$]{.math.display}\ \ [\$\$P = \\sum\_{t = 1}\^{n}{\\frac{C}{\\left( 1 + r \\right)\^{t}} + \\frac{\\text{FV}}{\\left( 1 + r \\right)\^{n}},\\ \\ \\ \\ where\\ }\$\$]{.math.display}\ \ [*P* = *Price* *of* *the* *Bond*  ]{.math.display}\ \ [*C* = *Coupon* *Payment* (*annual* *or* *semi* − *annual*) ]{.math.display}\ \ [*FV* = *Face* *Value* *of* *the* *Bond* ]{.math.display}\ \ [ *r* = *YTM* (unknown)]{.math.display}\ \ [*n* = *number* *of* *periods*]{.math.display}\ 1. When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate. 2. The price of a coupon and the yield to maturity are negatively related, that is, as the yield to maturity rises, the price of the bond falls. If the yield to maturity falls, the price of the bond rises. 3. The yield to maturity is greater than the coupon rate when the bond price is below its face value. - - - - \ [\$\$950 = \\ \\sum\_{t = 1}\^{10}{\\frac{50}{\\left( 1 + r \\right)\^{t}} + \\frac{1000}{\\left( 1 + r \\right)\^{10}}} \\approx 5.5\\%\$\$]{.math.display}\ \ [\$\$YTM = \\left( \\frac{\\text{FV}}{P} \\right)\^{\\frac{1}{n}} - 1\$\$]{.math.display}\ \ [*FV* = *Face* *Value*  ]{.math.display}\ \ [*P* = *Price*  ]{.math.display}\ \ [*n* = *Time* *to* *maturity*]{.math.display}\ \ [\$\$YTM = \\left( \\frac{1000}{800} \\right)\^{\\frac{1}{5}} - 1 \\approx 4.56\\%\$\$]{.math.display}\ **Long Term Bond YTM** can be estimated using the formula: \ [\$\$YTM \\approx \\frac{\\frac{C + \\frac{FV - P}{n}}{FV + P}}{2}\$\$]{.math.display}\ **Current Yield**: the yearly coupon payment divided by the price of the security; an approximation of the yield to maturity on coupon bonds that is often reported because in contrast to the yield to maturity, it is easily calculated. \ [\$\$i\_{C} = \\frac{C}{P\_{b}}\$\$]{.math.display}\ - Equals the coupon rate when the bond value is at par - The closer the bond price is to par, the closer the current yield is to the YTM **Yield on a Discount Basis (AKA Discount Yield):** [\$i\_{\\text{db}} = \\frac{F - P\_{d}}{F}\*\\frac{360}{\\text{days\\ to\\ maturity}}\$]{.math.inline}, Where [*i*~db~ = yield on a discount basis]{.math.inline} [*F*= face value of the discount bond]{.math.inline} [*P*~*d*~= purchase price of the discount bond]{.math.inline} **Distinction Between Interest Rates and Returns** **Return:** How well a person does by holding a bond or any other security over a particular time period. **Rate of Return:** the payments to the owner plus the change in its value, expressed as a fraction of its purchase price. **Rate of Return for a Coupon Bond:** - Face Value = \$1,000 - Purchase Price = \$1,000 - Coupon Rate = 10% - Time to Maturity = 10 years - Sold for \$1,200 after 1 year (after receiving \$100 coupon) \ [\$\$\\mathrm{\\text{Rate\\ of\\ Return\\ }} = \\frac{\\\$ 100 + \\\$ 200}{\\\$ 1000} = \\frac{\\\$ 300}{\\\$ 1000} = 30\\%\\ \$\$]{.math.display}\ **The return on a bond will not necessarily equal the interest rate on that bond.** **General Formula for Rate of Return (RoR) with Income:** \ [\$\$\\mathrm{\\text{RoR}} = \\frac{\\mathrm{Final\\ Value\\ + \\ Income\\ } - \\mathrm{\\text{\\ Initial\\ Value}}}{\\mathrm{\\text{Initial\\ Value}}} = i\_{C} + g\$\$]{.math.display}\ Where: - [RoR= return from holding the bond]{.math.inline} - [\$i\_{c}\\mathrm{= \\ }\\frac{\\mathrm{\\text{Income}}}{\\mathrm{\\text{Initial\\ Value}}} = \\mathrm{\\text{current\\ yield}}\$]{.math.inline} - [*g*= *Rate* *of* *Capital* *Gain* (*below*)]{.math.inline} \ [\$\$g = \\frac{\\mathrm{\\text{Final\\ Value\\ }} - \\mathrm{\\text{\\ Initial\\ Value}}}{\\mathrm{\\text{Initial\\ Value}}} = \\frac{\\\$ 1200 - \\\$ 1000}{\\\$ 1000} = 20\\%\$\$]{.math.display}\ \ [\$\$CAGR = \\left( \\frac{\\mathrm{\\text{Final\\ Value}}}{\\mathrm{\\text{Initial\\ Value}}} \\right)\^{\\frac{1}{n}} - 1\$\$]{.math.display}\ 1. The only bond whose return equals the initial yield to maturity is one whose time to maturity is the same as the holding period. 2. A rise in interest rates is associated with a fall in bond prices, resulting in capital losses on bonds whose terms to maturity are longer than the holding period. 3. The more distant a bond's maturity, the greater the size of the price change associated with an interest-rate change. 4. The more distant a bond's maturity, the lower the rate of return that occurs because of the increase in interest rate. 5. Even though a bond has a substantial interest rate, its return can turn out to be negative if interest rates rise. **Maturity and the Volatility of Bond Returns: Interest-Rate Risk** Prices and returns for long-term bonds are more volatile than those for shorter-term bonds. **Interest-Rate Risk:** the riskiness of an asset's return that results from interest rate changes. **Reinvestment Risk:** the risk that an investor will not be able to reinvest cash flows (such as interest payments, dividends, or principal repayments) from an investment at a rate equal to or higher than the investment\'s original rate of return. **Reinvestment Risk with Bonds when Holding Period \> Time to Maturity** If, at the end of the first year, interest rates are - *20*%: Reinvesting his money (now total \$1100) into a one-year bond will leave him with [\$1100 × (1+0.20) = \$1320]{.math.inline} and his two-year return will be [\$\\frac{\\\$ 1320 - \\\$ 1000}{\\\$ 1000} = 32\\%\$]{.math.inline} - Which is an annual rate of [14.9%]{.math.inline}. - *5%*: Reinvesting his money into a one-year bond will leave him with [\$1100 × (1+0.05) = \$1155]{.math.inline} and his two-year return will be [\$\\frac{\\\$ 1155 - \\\$ 1000}{\\\$ 1000} = 15.5\\%\$]{.math.inline} - Which is an annual rate of [7.2%]{.math.inline}. **Mitigating Reinvestment Risk** 1. Use zero-coupon bonds (do not pay coupons, so cannot reinvest the coupons) 2. **Laddering**: Spreading investments across various maturities to minimize the impact of rate changes. 3. Invest in Non-Callable Bonds. Avoid callable bonds, which are often called when rates fall. **Calculating Duration to Measure Interest-Rate Risk** **Duration**: the average lifetime of a debt security's stream of payments **Effective Maturity:** the term to maturity that accurately measures interest-rate risk. **Interest-Rate Risk Differences by Coupon Rate** - Bonds with lower coupon rates are more sensitive to changes in interest rates compared to those with higher coupon rates. - Example: - Bond A: 10 years to maturity, 2% annual coupon. - Bond B: 10 years to maturity, 8% annual coupon. - If market rates rise by 1%, Bond A will experience a larger price drop than Bond B because Bond A has fewer interim cash flows to offset the interest rate impact. **Exercise: Rate of Capital Gain on 10-yr Zero-Coupon Bond** - Interest-Rates rise from current 10% to 20% next year - Face Value of \$1000, which it pays at the end of ten years' time \ [\$\$\\frac{\\\$ 1000}{\\left( 1 + 0.10 \\right)\^{10}} = \\\$ 385.54\$\$]{.math.display}\ \ [\$\$\\frac{\\\$ 1000}{\\left( 1 + 0.20 \\right)\^{9}} = \\\$ 193.81\$\$]{.math.display}\ \ [\$\$g = \\frac{P\_{t + 1} - P\_{t}}{P\_{t}} = \\frac{\\\$ 193.81 - 385.54}{\\\$ 385.54} = - 0.497 = - 49.7\\%\$\$]{.math.display}\ **Calculating Duration** \ [\$\$D = \\sum\_{t = 1}\^{N}{t\\frac{CP\_{t}}{\\left( 1 + i \\right)\^{t}}/\\sum\_{t = 1}\^{N}\\frac{CP\_{t}}{\\left( 1 + i \\right)\^{t}}\\text{\\ \\ }}\$\$]{.math.display}\ where [*D* = duration]{.math.inline} [*t* = years until cash payment is made]{.math.inline} [*CP*~*t*~ = *cash* *payment* (*interest* *plus* *principal*) *at* *time* *t*]{.math.inline} [*i* = interest rate]{.math.inline} [*N* = years to maturity of the security]{.math.inline} **Table 3.** Calculating Duration on a \$1000, Ten-Year 10% Coupon Bond When Its Interest Rate is 10% **Year** **Cash Payments (Zero-Coupon Bonds)** **Present Value (PV) of Cash Payments (i=10%)** **Weights (% of total PV = PV/\$1000)** **Weighted Maturity (1 x 4)** ---------- --------------------------------------- ------------------------------------------------- ----------------------------------------- ------------------------------- 1 100 90.91 9.091 0.09091 2 100 82.64 8.264 0.16528 3 100 75.13 7.513 0.22539 4 100 68.30 6.830 0.27320 5 100 62.09 6.209 0.31045 6 100 56.44 5.644 0.33864 7 100 51.32 5.132 0.35924 8 100 46.65 4.665 0.37320 9 100 42.31 4.231 0.38169 10 100 38.55 3.855 0.38550 10 1000 385.54 38.554 3.8550 Total 1000.00 100.00 6.75850 **Relationship Between Interest Rate Risk and Duration** 1. **Higher Duration = Greater Interest Rate Risk**: - Bonds with higher duration experience larger price swings when interest rates change. - Example: - Bond A: Duration = 3 years. - Bond B: Duration = 10 years. If interest rates rise by 1%, Bond A's price might fall by \~3%, while Bond B's price might fall by \~10%. 2. **Factors That Increase Duration (and Interest Rate Risk)**: - **Lower Coupon Rates**: Bonds with lower coupons rely more on the distant principal repayment, increasing duration. - **Longer Maturity**: More distant cash flows are affected more by interest rate changes, increasing duration. - **Lower Yield to Maturity**: A lower yield reduces the discounting effect, making distant cash flows more significant and increasing duration. 3. **Mathematical Link**: The price change of a bond due to interest rate changes can be approximated using modified duration: \ [*ΔP* ≈  − *Duration* × *Δr* × *P*]{.math.display}\ - [*ΔP* = change in bond price]{.math.inline} - [*Δr* = change in interest rates]{.math.inline} - [*P* = current bond price]{.math.inline} **Practical Implications** 1. **For Investors**: - Bonds with higher duration are riskier in terms of price volatility but may offer higher returns if interest rates fall. - Bonds with lower duration are more stable and less sensitive to rate changes. 2. **Portfolio Management**: - To manage interest rate risk, investors adjust portfolio duration. - Shorten duration in rising rate environments to reduce risk. - Lengthen duration in falling rate environments to maximize gains. 3. **Zero-Coupon Bonds**: - Zero-coupon bonds have the highest duration for a given maturity, as all cash flows occur at the end. **The Distinction Between Real and Nominal Interest Rates** **Nominal Interest Rate:** interest rate that makes no allowance for inflation **Real Interest Rate:** the interest rate that is adjusted for expected changes in price level so that it more accurately reflects the true cost of borrowing. \ [*i*~*R*~ = *i* − *π*^*e*^]{.math.display}\ Where: - [*i*~*R*~ = real interest rate]{.math.inline} - [*i* = nominal interest rate]{.math.inline} - [*π*^*e*^ = expected rate of inflation]{.math.inline} Chapter 4: Portfolio Choice =========================== **Asset:** a piece of property that is a store of value (e.g., money, bonds, stocks, art, land, houses, farm equipment, and manufacturing machinery) **Factors when considering buying/holding an asset:** 1. **Wealth:** the total resources owned by the individual, including all assets. 2. **Expected Return:** the return expected over the next period 3. **Risk:** the degree of uncertainty associated with the return 4. **Liquidity:** the ease and speed with which an asset can be turned into cash **Wealth Elasticity of Demand:** similar to the concept of income elasticity of demand; measures how much, with everything else unchanged, the quantity demanded of an asset changes in percentage terms in response to a percentage change in wealth. \ [\$\$\\frac{\\mathrm{\\%\\ Change\\ in\\ quantity\\ demanded}}{\\mathrm{\\%\\ change\\ in\\ wealth}} = \\mathrm{\\text{wealth\\ elasticity\\ of\\ demand}}\$\$]{.math.display}\ **Necessity:** an asset is a necessity if there is only so much that people want to hold, so that as wealth grows, the percentage increase in the quantity demanded of the asset is less than the percentage increase. - *The wealth elasticity of a necessity is 1*. Because the quantity demanded of a necessity does not grow proportionally with wealth, the amount of this asset that people want to hold relative to their wealth falls as wealth grows. - Currency and checking account deposits are necessities **Luxury:** as wealth grows, the quantity demanded of this asset grows more proportionally, and the amount that people hold relative to their wealth grows. - *The wealth elasticity of a luxury is greater than 1.* - Common stocks and municipal bonds are luxury assets ***Key Point 1 (Wealth):** Holding everything else constant, an increase in wealth raises the quantity demanded of an asset, and the increase in the quantity demanded is greater if the asset is a luxury than if it is a necessity.* ***Key Point 2 (Expected Returns):*** *an increase in an asset's expected return relative to that of an alternative asset, holding everything else unchanged, raises the quantity demanded of that asset.* ***Key Point 3 (Risk):** holding everything else unchanged, if an asset's risk rises relative to that of alternative assets, its quantity demanded will fall.* ***Key Point 4 (Liquidity):** the more liquid an asset is relative to alternative assets, holding everything else unchanged, the more desirable it is, and the greater will be the quantity demanded.* **Theory of Portfolio Choice** The **theory of portfolio choice** states that, holding all the other factors constant: 1. The quantity demanded of an asset is usually positively related to wealth, with the response being greater if the asset is a luxury rather than a necessity. 2. The quantity demanded of an asset is positively related to the risk of its returns relative to alternative assets. 3. The quantity demanded of an asset is negatively related to the risk of its returns relative to alternative assets. 4. The quantity demanded of an asset is positively related to its liquidity relative to alternative assets. **Risk** \ [*Asset* *Risk* = *Systematic* *Risk* + Nonsystematic Risk]{.math.display}\ **Systematic Risk:** risk that cannot be eliminated through diversification. - **Beta:** a measure of the sensitivity of an asset's return to changes in the value of the entire market of assets - Used to measure the systematic risk of an asset \ [\$\$\\beta = \\frac{\\mathrm{\\Delta}P\_{\\text{asset}}}{\\mathrm{\\Delta}P\_{\\text{market}}}\$\$]{.math.display}\ - Ex. a 1% rise in the value of the market portfolio leads to a 2% rise in the value of the asset beta for the asset is 2.0 when the value of the market fluctuates by a certain amount, the value of the asset changes by twice as much - ***The greater an asset's beta, the greater the asset's systematic risk and the less desirable the asset.*** **Nonsystematic Risk:** the risk unique to an asset that can be diversified away. - Related to the part of an asset's return that does not vary with returns on other assets, therefore the risk is unique to that asset - With many assets in a portfolio, nonsystematic risk becomes less import because when the nonsystematic risk of one asset goes up, it is likely that the nonsystematic risk of another asset goes down nonsystematic risks cancel out in a well-diversified portfolio - ***The risk of a well-diversified portfolio is due solely to the systematic risk of assets in the portfolio*** **Capital Asset Pricing Model (CAPM)** **Capital Asset Pricing Model** provides an explanation for the magnitude of an asset's risk premium. **Risk Premium:** the difference between the asset's expected return and the risk-free interest rate (the interest rate on a security that has no possibility of a default) - Essentially measures the expected payout for taking on a risky investment \ [Risk Premium = *R*^*e*^ − *R*~*f*~ = *β*(*R*~*m*~^*e*^ − *R*~*f*~)]{.math.display}\ Where: - [*E*(*R*~*i*~) = *Expected* *return* *of* *the* *investment* (*or* *security*)]{.math.inline} - [*R*~*f*~ = *risk* − *free* *interest* *rate*]{.math.inline} - [*β* = beta of the asset]{.math.inline} - [*E*(*R*~*m*~) = expected return for the market portfolio]{.math.inline} THEN \ [*E*(*R*~*i*~) = *R*~*f*~ + *β*~*i*~  × (*E*(*R*~*m*~)−*R*~*f*~)]{.math.display}\ Where: - [*E*(*R*~*i*~) = *Expected* *Return* *of* *the* *investment* (*or* *security*)]{.math.inline} - [*R*~*f*~ = *Risk* − *Free* *interest* *rate*]{.math.inline} - [*β*~*i*~ = Beta of the investment]{.math.inline} - A beta of greater than 1 means the investment's returns move in line with the market - A beta greater than 1 indicates higher risk and potential reward compared to the market - A beta less than 1 indicates lower risk and return relative to the market - [*E*(*R*~*m*~) = Expected return of the market portfolio]{.math.inline} - [*E*(*R*~*m*~) − *R*~*f*~ = Market risk premium]{.math.inline} - [The additional return investors expect for taking on market risk]{.math.inline} **Key Assumptions of CAPM:** 1. **Risk-averse investors:** Investors prefer higher returns for taking on additional risk. 2. **Efficient markets:** All information is available, and securities are fairly priced. 3. **Diversified portfolios:** Investors hold diversified portfolios, eliminating unsystematic risk (company-specific risk). 4. **Single-period model:** CAPM considers a single time horizon. 5. **No taxes or transaction costs:** These factors are ignored in the model for simplicity. **How CAPM Works:** CAPM assumes that the return of an investment depends on two components: 1. **Risk-free return (**[**R**~**f**~]{.math.inline}**):** The return an investor would receive for taking no risk, such as investing in government bonds. 2. **Risk premium (**[**β**~**i**~ **\*** **(E**(**R**~**m**~)**−R**~**f**~**)**]{.math.inline}**):** Compensation for taking on the additional risk of investing in a particular security compared to the market. The model shows that the expected return of a security is directly proportional to its beta, meaning that riskier investments (higher beta) should provide higher returns. **Applications of CAPM:** 1. **Estimating Cost of Equity:** CAPM is commonly used in corporate finance to calculate the cost of equity, which is a key input in valuation models like discounted cash flow (DCF). 2. **Portfolio Management:** CAPM helps investors decide whether a security offers a fair return given its risk. 3. **Security Pricing:** CAPM can help determine if a stock is overvalued or undervalued compared to its expected return. **Limitations of CAPM:** 1. **Simplistic Assumptions:** CAPM assumes no taxes, transaction costs, or restrictions on borrowing, which are unrealistic in the real world. It also **assumes that there is only one source of systematic risk.** 2. **Market Portfolio:** The true market portfolio is theoretical and difficult to replicate in practice. 3. **Beta Limitations:** Beta is based on historical data and may not accurately reflect future risk. 4. **Single-factor Model:** CAPM only considers market risk and ignores other potential factors like size, value, or momentum, which are included in multi-factor models like the Fama-French model. Despite its limitations, CAPM remains a foundational tool in finance for understanding the relationship between risk and return. **Arbitrage Pricing Theory** The **Arbitrage Pricing Theory (APT)** is an asset pricing model that determines the expected return of a financial asset based on multiple macroeconomic factors or systematic risks. It is an alternative to the **Capital Asset Pricing Model (CAPM)**, offering a more flexible framework for estimating returns by considering multiple sources of risk. **APT Formula:** The expected return of a financial asset under APT is expressed as: \ [\$\$E\\left( R\_{i} \\right) = R\_{f} + \\sum\_{i = 1}\^{n}{\\beta\_{i}F\_{i}}\$\$]{.math.display}\ Where: - [*E*(*R*~*i*~) = Expected return of the asset]{.math.inline} - [*R*~*f*~ = *Risk* − *free* *rate* (*e*.*g*., *return* *on* *government* *bonds*)]{.math.inline} - [*β*~*i*~ = *Sensitivities* (*factor* *loadings*) *of* *the* *asset* *to* *the* *respective* *risk* *factors*]{.math.inline} - [*F*~*i*~ = *Risk* *factors* *affecting* *the* *asset*′*s* *return*]{.math.inline} - I.e., [*inflation*, *interest* *rates*, *GDP* *growth*, *or* *oil* *prices*]{.math.inline} **Key Concepts of APT:** 1. **Multi-Factor Model:** - Unlike CAPM, which relies on a single factor (market risk), APT uses multiple factors to explain returns. - These factors can include economic variables, industry-specific influences, or even company-specific attributes. 2. **No Arbitrage Principle:** - APT is based on the idea that arbitrage opportunities (risk-free profits) should not exist in efficient markets. - If mispricing occurs, investors would exploit the price differences, bringing the asset prices back to equilibrium. 3. **Linear Relationship:** - APT assumes a linear relationship between an asset's expected return and the various risk factors. **Calculating Gains from Diversification** One way to calculate risk is by calculating the expected return and variance / standard deviation of the returns: \ [\$\$E\\left( R\_{i} \\right) = \\sum\_{i = 1}\^{n}{p\_{i}R\_{i}}\$\$]{.math.display}\ Where: - [*E*(*R*~*i*~) = Expected Return]{.math.inline} - [*p*~*i*~ = probability of getting the realization *R*~*i*~]{.math.inline} - [*R*~*i*~ = realization of the return]{.math.inline} The variance of the return [*σ*^2^]{.math.inline} equals: \ [\$\$\\sigma\^{2} = \\sum\_{i = 1}\^{n}{p\_{i}\\left( R\_{i} - E(R\_{i}) \\right)\^{2}}\$\$]{.math.display}\ The standard deviation, the more commonly cited measure of risk, is the square root of the variance: \ [\$\$\\sigma = \\sqrt{}\\sigma\^{2}\$\$]{.math.display}\ The **return on the portfolio (**[**R**~**p**~]{.math.inline}**)** is the weighted average of the returns on the individual assets, with the weights [*x*~*i*~]{.math.inline} reflecting the proportion of the portfolio invested in each asset: \ [\$\$R\_{p} = \\sum\_{i = 1}\^{n}{w\_{i}R\_{i}}\$\$]{.math.display}\ Where: - [*R*~*p*~ = Return on portfolio]{.math.inline} - [*w*~*i*~ = the weight of the *i* − *th* *asset* *in* *the* *portfolio* (*percentage* *of* *total* *portfolio*)]{.math.inline} - [*R*~*i*~ = Return on the *i* − *th* *asset*]{.math.inline} The **expected return on the portfolio (**[**E(R**~**p**~**)**]{.math.inline}**)** is \ [\$\$E\\left( R\_{p} \\right) = \\sum\_{i = 1}\^{n}{w\_{i}\*E(R\_{i})}\$\$]{.math.display}\ Where: - [*E*(*R*~*p*~) = Expected Return on portfolio]{.math.inline} - [*w*~*i*~ = the weight of the *i* − *th* *asset* *in* *the* *portfolio* (*percentage* *of* *total* *portfolio*)]{.math.inline} - [*E*(*R*~*i*~) = Expected Return on the *i* − *th* *asset*]{.math.inline} Calculating the standard deviation for the portfolio's return involves **covariance of returns between two assets**, which measures the degree to which the returns on different assets move together. \ [\$\$\\sigma\_{\\text{ab}} = \\sum\_{i = 1}\^{n}{p\_{i}(R\_{\\text{ai}} - E\\left( R\_{a} \\right))(R\_{\\text{bi}} - E\\left( R\_{b} \\right))}\$\$]{.math.display}\ Where: - [*σ*~ab~ = the covariance between returns on two assets A and B]{.math.inline} - [*p*~*i*~ = probability of the *i* − *th* *outcome*]{.math.inline} - [*R*~ai~ = Return of Asset A in the *i* − *th* *outcome*]{.math.inline} - [*R*~bi~ = Return of Asset B in the *i* − *th* *outcome*]{.math.inline} - [*E*(*R*~*a*~), *E*(*R*~*b*~) = Expected Returns of Assets A and B]{.math.inline}