MAT 483Mathematical Models in Finance and Investments

Fall 2010

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 valueOften 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

More vocabularyFinancial 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.

Some general vocabulary

Futures / Forward Markets: Agree to terms at present time, but currency exchange and delivery is in the future, often several months

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 MarketsChicago Mercantile ExchangeCMEGroupLargest U.S. Futures Exchange20 S. Wacker Drive, Chicago, ILwww.cmegroup.com

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)

Combined with CMEGroup in 2008

Futures MarketsSee sample printed copies of lean hog and lumber futures

contracts…

Interest rates: T-BillsStart 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 RatesSpot 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 RatesThe facts imply there is some rate, f, that will be effective one

year from today for a one-year period that satisfies:

(1+.0550) * (1 + f) = (1 + .0580)2

f = .061 or 6.10%

Spot and Forward RatesThe facts imply there is some rate, f, that will be effective

two years from today for a one-year period that satisfies:

(1+.0580)2 * (1 + f) = (1 + .0620)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 RatesNote 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

Spot and Forward RatesNote 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 = 5.26%

2-Year spot rate = 5.64%

3-Year spot rate = 5.92%

Spot and Forward RatesThere 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 ideasYield 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 ideasEquivalent 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 ideasExample: 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 ideasExample: 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