MAT 483 Mathematical Models in Finance and Investments Fall 2008

Financial Markets Main ideas: Major types of financial instruments How are they… Initiated Traded Priced Quoted Some specific focus on the nature of interest rates

Some general vocabulary Market: a place where goods are bought and sold This can be a very wide definition, covering many types of transactions How do markets evolve? What types of markets are there?

Some general vocabulary Barter Market: Most basic, exchange of goods and services Most common accepted form of commerce prior to the invention of currency 1626: Peter Minuet trades beads, knives and kettles for... Issues: Equality of value Often a necessity to bring goods to a central place

Some general vocabulary Cash or Spot Market: Purchase items with a currency, with delivery either immediate or delivered within a short amount of time Most common type of market we know today Goods and services can range from very simple to very complex financial instruments

Some general vocabulary Futures / Forward Markets: Agree to terms at present time, but currency exchange and delivery is many months or more into the future The term “futures” is most commonly used when the asset being purchased is a standardized contract The term “forward” is most commonly used when the asset being purchased is non-standardized; amount, quality shape and form need negotiated between the buyer and seller The evolution of futures markets have given many industries an enhanced stability for risk management, budgeting, planning and confidence in production inputs and outputs

Some general vocabulary “Underlying” assets traded can be very diverse: Tangible items: commodities, animals, production inputs, metals, stocks, bonds, etc quoted in terms of the price of the item Intangible items: Stock indexes, currency exchange rates, interest rates The evolution of futures markets have given many industries an enhanced stability for risk management, budgeting, planning and confidence in production inputs and outputs

Futures Markets Chicago Mercantile Exchange Largest U.S. Futures Exchange 20 S. Wacker Drive, Chicago, IL Agricultural Products: Beef, Dairy, Hogs, Lumber, Fertilizer Financial Products: Equity Index Futures (S&P 500, NASDAQ), Interest Rate Futures (T- Bill), Foreign Currency Futures (Euro)

Futures Markets 141 West Jackson Boulevard (Jackson and LaSalle) Ceres, the Roman goddess of agriculture Agricultural Products: Corn, Soybeans, Wheat, Oats Financial Products: Equity Index Futures (Dow), Interest Rate Futures (Treasury Notes), Metal Futures (Gold, Silver)

Futures Markets See sample printed copies of corn and lean hog futures contracts…

More vocabulary Financial Market: A place where financial assets are bought and sold – most common is a stock exchange like NYSE, but doesn’t need to be a physical location (National Association of Securities Dealers Automated Quotation System - NASDAQ) Primary Market: Transaction between the creator of the security (issuer) and the first owner Secondary Market: Owners sell the security to a new owner, typically in a financial market Derivative Securities: Securities whose prices are dependent or “derived” from another security’s price – options, futures, etc.

Interest rates: T-Bills Start with Treasury Bills or “T-Bills” – one of the most common fixed income securities in U.S. Primary market: issued by New York Fed every week for short term US financing – face amounts start at $10,000 Maturities up to 1 year are offered but not the same every week Do NOT pay coupons – hence a “discount” or “zero-coupon” security – Repayment of face is only cash flow Secondary Market extremely active – very liquid investments

Spot and Forward Rates Spot rates: rates derived from the prices of interest rate securities – usually zero-coupon securities like CD’s, money market securities, T-Bills, etc. Forward rates: rates derived from spot rates that are implied for periods of time in the future Example: If a one year CD yields 5.50%, a two-year CD yields 5.80% and a three-year CD yields 6.20%, then what does it imply about a one-year rate one year from today?

Spot and Forward Rates The facts imply there is some rate, f, that will be effective one year from today for a one-year period that satisfies: ( ) * (1 + f) = ( ) 2 f =.061 or 6.10%

Spot and Forward Rates The facts imply there is some rate, f, that will be effective two years from today for a one-year period that satisfies: ( ) 2 * (1 + f) = ( ) 3 f =.070 or 7.00% Note that these rates are “implied” – but may or may not come true – they are driven by the expectations of the market today in how they price the spot instruments

Spot and Forward Rates Note that derivation of spot and forward rates is dependent upon the set of assets in the marketplace. Example: Price of Asset 1= $89.60; cash flow = $100 in 2 years Price of Asset 2= $104.58; cash flow = $7 in 1 year, $100 in 2 years Price of Asset 3= $96.42; cash flow = $4 in 1 year, $4 in 2 years, $100 in 3 years

Spot and Forward Rates Note that derivation of spot and forward rates is dependent upon the set of assets in the marketplace. Example: Price of Asset 1= $89.60; cash flow = $100 in 2 years Price of Asset 2= $96.25; cash flow = $7 in 1 year, $100 in 2 years Price of Asset 3= $91.53; cash flow = $4 in 1 year, $4 in 2 years, $100 in 3 years 1-Year spot rate = 5.40% 2-Year spot rate = 5.64% 3-Year spot rate = 5.92%

Spot and Forward Rates There is a general relationship between the spot curve and the forward curve dependent upon the characteristics of the spot curve…. If the spot curve is increasing, forward rates are greater than spot rates If the spot curve is level, forward rates are equal to spot rates If the spot curve is decreasing, forward rates are less than spot rates

Other interest rate ideas Yield to Maturity In general the “effective yield”, or “yield to maturity” of a fixed income instrument is the interest rate that discounts the entire set of cash flows to the current time to get the current price Since most bonds have coupons and then return the principal at maturity, there are many cash flows to consider Generally an annuity discount factor used on the coupons

Other interest rate ideas Equivalent Taxable Yield Earnings on some investments are deemed to exempt from federal income tax, such as debt securities issued by states and local municipalities; these are often called municipal bonds or “muni” bonds In order to compare alternative investment choices, investors must calculate the equivalent taxable yield on muni bonds Equivalent Taxable Yield = Tax-Free Yield / (1 – Tax Rate) Also, can calculate the tax rate where the investor becomes indifferent between taxable and tax-free yields Tax Cutoff Bracket = 1 – (Tax-Free Yield / Taxable Yield)

Other interest rate ideas Example: Investor has a tax rate of 32% and a Muni bond yields 6% Equivalent Taxable Yield = Tax-Free Yield / (1 – Tax Rate) =.06 / (1 –.32) =.06 /.68 =.0882 = 8.82%

Other interest rate ideas Example: A muni bond has a yield of 5.25% versus a taxable investment of 7.00% Tax Cutoff Bracket = 1 – (.0525/.0700) = 1 –.75 =.25 = 25% If the tax rate is less than 25%, say 10%, then the Equivalent Taxable Yield on the muni bond would be (.0525 /.90) = 5.83% and the taxable investment would be preferable If the tax rate is greater than 25%, say 40%, then the Equivalent Taxable Yield on the muni bond would be (.0525 /.60) = 8.75% and the muni bond would be preferable