Ch 1: BASIC CONCEPTS IN FINANCE

• Finance is the study of how resources are valued and allocated in time.

• Outcomes of financial decisions are spread out over time and not known with certainty in advance

• Three key concepts in finance are : Time value of money Asset Valuation (stocks, bonds, derivatives,...) Risk management

1.1: Interest and return

• Income almost never matches consumption desires exactly. Either one will need to borrow to purchase more than one can afford or save excess income.

• Costs / benefits of financial decisions are spread over time. So one needs to compare values of cashflows which mature at different times.

Time value of money: 1ZAR in the hand today is worth more than the expectation of 1ZAR in the future. Why? • Opportunity cost: To give up consumption of your 1ZAR today, you would expect to be rewarded with a greater amount in the future; the promise of consumption at a higher level in the future motivates one to save. The desire to receive surplus on savings leads to an interest rate called the pure time value of money. • Inflation: Prices of goods rarely stay the same over time. The purchasing power of 1ZAR now is (usually) greater than 1ZAR later. Investors expect a higher rate of return to compensate for inflation. • Uncertainty: One may not receive the expected sum - this is referred to as investment or credit risk.

• Opportunity cost: Pure time value of money give rise to pure rate of interest.

• Inflation: The rate of interest on top of the rate of inflation is the nominal riskfree rate.

• Uncertainty: The excess amt added to the nominal risk-free interest rate is the risk premium.

1.1.1: Interest Borrowing is not free: borrower pays premium for sum of money from lender. The cost of borrowing is interest. • Interest rates are not necessarily fixed - they vary at different times - different rates may be charged for different durations of lending • Magnitude of interest depends on - economic factors [inflation; growth rate of economy, money supply, trade deficits,...] - credit rating of borrower [government as a borrower usually pays the lowest interest rate (treasuries/gilts); non-investment grade bonds (”junk bonds”) have highest premiums; the difference or spread reflects default probabilities].

Types of interest For simplicity, assume interest rates are constant. • PT denotes the future value at time T of P0 invested at time 0:

PT = F V (P0). • P0 is the present value of PT : P0 = P V (PT ). Computing present values is referred to as discounting and the interest used is referred to as the discount rate. Computing future values is referred to as compounding. PV and FV make it possible to compare different cash flows at different times.

• Simple interest : Invest P0 at (annual) rate r for time T years. Then PT = P0(1 + rT ). • Discretely compounded interest: Invest P0 at (annual) rate r, compounded n times annually for 1 year. Then r P1 = P0(1 + )n. n Invest P0 at (annual) rate r, compounded n times annually for time T years. Then r PT = P0(1 + )nT . n

• Continuously compounded interest: Invest P0 at (annual) rate r, compounded continuously for time T years. Then PT = P0erT . x Here we are using lim (1 + n )n = ex. n→∞

For compound interest, the expectation of receiving an amount K at a future date has a value today of: P V (K) = Ke−rT .

• What is the present value of an investment that will pay R1000 per year for 5 years given an annual rate of interest of 8.5% per annum (p.a.)?

PV

1000 = n n=1 1.085 1 − 1.085−5 = 1000 0.085 = 3940.64.

5

The investment is less than R4000 - it is worth no more than R3940.64 at the given interest rate.

• Which is better: an investment which offers 8.5% or one which offers 8.4% continuously compounded? Finding the continuously compounded rate rc which corresponds to 0.85% semi-annually: 0.085 2 ) 2 rc = 0.0832

erc = (1 + ⇒

i.e. rc = 8.32%. Thus, the continuous rate of 8.4% is better.

1.1.2: Returns

• Returns are similar to interest rates. The main difference is that interest rates are promised and predictable returns on deposits, while returns on other assets (e.g. stocks) are generally uncertain.

• Shares are riskier investments than deposits. ⇒ expected return on a share should be greater than the interest offered by a bank. NB: returns can be negative; interest rates must be positive (?!)

Returns

• Roughly, the return on an investment is measured by: return = final value + interim cashflows - initial value initial value

• Mean rates of return are computed for investments spread over several years. Annual rates of return vary. Suppose the return for year 1 of an investment is r1 = −30% and for year 2 is r2 = 42.857%. The arithmetic mean return is r = (r1 + r2)/2 = 6.42857%. ¯

But it would be wrong to assume that an initial wealth of P0 = R100 will become P2 = W0 ∗ (1 + r)2 = R113.27. ¯ In fact P2 = P0(1 + r1)(1 + r2) = 100 ∗ 0.7 ∗ 1.42857 = R100. The geometric mean return satisfies (1 + rg )2 = (1 + r1)(1 + r2) = 1, ¯ i.e. for this example rg = 0%. ¯

Generalising, (1 + rg )n = (1 + r1)(1 + r2)...(1 + rn) ¯ and Pn = P0(1 + rg )n. ¯

• Shares with greater expected return are considered riskier than shares with lower expected return: If 2 shares had the same risk and one of them had greater return than the other, then everyone would buy the share with greater return; this would push up its price and reduce its return.

• Risk is measured by volatility: this is the standard deviation of returns.

• Returns may be measured as both continuously and discretely compounded

• Returns on bonds are referred to as yields.

A fundamental relationship in finance:

E[return] = f (risk), where f is an increasing function. The risk premium E represents a composite of uncertainties such as business risk, financial risk, liquidity risk, exchange rate risk, credit risk, country risk, operational risk, regulatory risk, ... See also The 100 risks in financial services by H.-U. Doerig, (2001): Operational Risks in Financial Services, Credit Suisse Group.

1.2: Markets and instruments

• Securities are contracts for future delivery of goods or money, e.g. shares, bonds, derivatives.

• Shares, bonds, currencies, interest rates and indexes are examples of primary or underlying instruments.

• A derivative is an instrument whose value depends on the value of some underlying asset - forward, contracts, futures, options and swaps are examples of derivatives.

• There is a distinction between primary and secondary markets. Securities are issued for the first time on a primary market and are then traded a secondary market the latter provides liquidity.

• Borrowing takes place in fixed-income markets; the money-market is for very shortterm debt (≤1 yr ).

• There is a distinction between the spot market and the forward market: most transactions are spot transactions (pay now and receive goods now); to hedge or speculate on future values, it is possible to sell goods for delivery in the future by means of forward and future contracts.

• Equity refers to stocks or shares which represent ownership of a small piece of a company. Shareholders own a corporation, while directors (are supposed to) act in the shareholder’s best interest; public limited companies are listed on a stock exchange - the ownership of such companies are easily transferred, the shareholders share the profits but have limited liability - at most they can lose is there investment.

• Most shares pay regular dividends - the amount varies depending on profitability and opportunities for growth - a share may be bought cum- or ex-dividend; on the ex-dividend date, the share price decreases by the amount of the dividend

• Occasionally a company may announce a stock split: after a 4-for-1 stock split, the single stock priced at R1 000 is converted to four shares each valued at R250.

• You can sell a share which you don’t own! (in the hope that it can be obtained later at a cheaper price) - a broker borrows the share from a client, you sell it and later buy it in the market to return it to the broker who, in turn, returns it to the client. Any dividends that were issued in the interim are paid to the original owner.

• Commodities are raw materials such as metals, oil, agricultural products, etc .... These are often traded by speculators who have no need for the material but who are betting on future price movements - this sort of trading is done (indirectly) in the futures market and contracts are closed before delivery date.

• Currencies are traded on forex markets.

• Indices: an index tracks the changes in a hypothetical portfolio of instruments. A typical index consists of a weighted sum of a basket of representative stocks. These representatives and their weights are not necessarily fixed. E.g. S&P500, DJIA, FTSA100, DAX-30, NIKKEI225. The most popular SA indices include – ALSI (All share index). This consists of all shares on the JSE (bar about 100, these being, for example, pyramids or debentures) – TOPI (Top 40 listed companies index). Until June 2002 called the ALSI40. – INDI25 (Top 25 listed industrial companies index) – FINI15 (Top 15 listed financial companies index) – RESI20 (Top 20 listed resources companies index) Main SA derivatives indices are the TOPI, INDI25.

• Fixed income securities include bonds, notes, bills. These are debt instruments and promises to pay a certain rate of interest which may be fixed or floating. E.g. a 10-yr, %5 semi-annual coupon bond with a face value of ZAR 1m promises to pay R25 000 every 6 months for 10 yrs and a lump-sum of ZAR 1m at the end of the 10-yr period.

LINKS to explore http://www.jse.co.za http://www.bondexchange.co.za http://www.safex.co.za http://www.satrix.co.za History & development of the JSE http://www.jse.co.za/informational/ historyofjse/history.htm Understanding Financial Markets & Instruments by Braam van den Berg, http://www.eagletraders.com/books/afm/ afm0.htm See Chapter 1, Introduction to the Financial Markets and Chapter 2, The Equity Market for notes on the SA Markets and Market instruments.

Ch 2: Derivatives

• A derivative security is a financial contract whose values is derived from an underlying variable, such as a stock price, the level of an index or an interest rate.

• Examples: - a stock option’s value depends on the value of the underlying stock; - the price of a zero-coupon bond depends on the prevailing interest rate - the profit/loss made by buying oil futures depends on the changes in the spot price for oil.

Reasons for using derivatives

• Hedging and speculation: derivatives are tools for transferring risk - hedgers wish to reduce risk while speculators want to take on more risk in anticipation of greater rewards

• Arbitrageurs seek low risk profits by entering into off-setting positions in different markets or instruments.

2.1: Forwards and futures

• A forward contract is an agreement to buy or sell an asset S (the underlying) at a certain date T (delivery date, maturity) for a certain price F (forward price , delivery price) - The party who agrees to buy the asset is said to have a long position ; the party who agrees to deliver the asset is said to have the short position. - If the forward price is chosen carefully, then the contract has no value initially, i.e. it costs nothing for either party to enter into the contract. The contract value changes in time depending on conditions in the market.

• If the spot price of the asset at time T is S(T ) then the party in the long position has agreed to pay (and the party in the short position has agreed to accept) a price F for what is worth S(T ). The payoff to the holder is therefore S(T ) − F, which may be positive or negative; similarly, the payoff to the seller is F − S(T ).

• Payoffs cancel out: one party’s gain is the other party’s loss ⇒ Forwards are a zero-sum game.

[ see payoff diagram ]

• A future’s contract is very similar to a forward contract. However, interim profits and losses are paid throughout the life of the contract rather than just at maturity.

• Example - agriculture forwards: Mrs Mkhize is a farmer in Northern Kwa-Zulu Natal and one of her specialities is potatoes. She knows that her potatoes will be ready for sale and delivery in three months’ time. Due to the good rains during the season, she expects to harvest three tons of potatoes. She also knows that the harvest countrywide will be a good one and is worried that she won’t be able to sell all her potatoes, or that she will be forced to sell them at a discounted price and suffer a loss. The Roar Food Company in Johannesburg produces potato chips. The company expects an influx of tourists to South Africa due to a reduction in the currency and the sporting events taking place during the summer. The company has budgeted a huge increase in production in three months’ time, and is scared that there will not be sufficient potatoes available in the market, or that the demand would increase, thus pushing up prices.

Mrs Mkhize and the Roar Food Company close a contract whereby Mrs Mkhize undertakes to supply the company with three tons of potatoes in three months’ time. The Roar Food company undertakes to buy three tons of potatoes from Mrs Mkhize at R1 000 per ton on delivery of the potatoes. The market price of potatoes at the closing of the contract is R950 per ton. Both parties are using the forward contract, a derivative as a hedging tool. The market price of potatoes at the closing of the contract has no direct effect on the contract except that it acts as a guideline to the determination of the contract price (R1000). At the date of delivery determined in the contract (called the close-out date), Mrs Mkhize has an obligation to supply three tons of potatoes and the Roar Food Company has an obligation to take delivery of the potatoes and pay Mrs Mkhize R3000 (R1 000 x 3).

If the market price of potatoes on the day of delivery (the close-out day of the contract) is R1050, then Mrs Mkhize could have sold her potatoes in the market at R1 050 (assuming the demand is high enough). She loses while Roar gains. On the other hand, if market price drops to R850, then she saves R450 through the contract while Roar forks out the same amount extra for the hedge.

• Example - Forex forwards: A local Biotech company knows that it will need to buy equipment to the value of Eu. 1 million from a German manufacturer in 3 months time. To hedge against forex risk, the company looks into the possibility a forward contract with a suitable bank (an SA Reserve Bank approved forex dealer) . The following ZAR/Euro exchange rates are quoted: Spot 90-day forward Bid 7.1789 7.2590 Offer 7.200 7.2622

There are two options for the company enter forward contract to buy Euros at 7.2622 ZAR per Euro or buy Euros at the prevailing rate in 3 months time

How are forward rates calculated? The bank determines the forward rate in the contract in the following manner: The bank takes on the foreign exchange exposure and has to cover this risk. It must have 1 million Euros in three months’ time to give to the company. It is an approved forex dealer, so the bank can buy Euros now and place it on deposit in a German bank, to ensure that it has the right amount in three months’ time. The bank, however, does not have to buy the full amount now, as it will receive interest (say the European deposit rate is 4%) for the three months on the deposit at the US bank. The bank therefore only buys: 1 000 000 1 + (i ∗ t/365) where i denotes short-term deposit rate in the European Union and t denotes the term of deposit.

The bank thus buys: 1 000 000 = 990 126.87 1 + (0.04 ∗ 91/365) To buy 990 126.87 Euros now, the bank must pay rand at the current exchange rate - the bank thus pays: 990 126.87 ∗ 7.2 = R7 128 913.46. The bank must borrow this ZAR amount to purchase Euros at the current shortterm interest rate in SA for 3 months (or at the cost of capital of the bank if it uses internal funds). If the current short-term rate in SA is 7.5%, the bank will have to pay: 91 = R133 300.92 R7 128 913.46 ∗ 7.5% ∗ 365 interest to its lender in 3 months time.

This interest paid by the bank will be borne by the company and will be discounted in the forward rate. The total cost to the bank for this transaction is thus: The cost of the Euro loan R7 128 913.46 plus the interest on the loan R133 300.92, i.e. R7 262 214.38. The forward rate is then calculated by the total cost divided by the amount in Euros: 7 262 214.38 ≈ R7.2622. 1 000 000 This gives 3 month forward rate. R 30 and 60 day forward rates can be obtained similarly.

2.2: Options

• An european call option gives the holder the right to buy an asset S (the underlying) for an agreed amount K (the strike or exercise price) on a specified future date T (the maturity or expiry date). - The party who undertakes to deliver the asset is called the writer of the option. - The holder of the option does not have to exercise the option. Hence, the payoff to the holder is never negative (ignoring the cost of the contract). Thus the payoff is

max{S(T ) − K, 0}.

[ insert payoff diagram for call option]

• Since the payoff is never negative, the contract comes with a price ! i.e. option contracts, unlike forwards, have a price associated with them.

• The purchaser (holder) pays the writer a premium up front to enter into the contract.

• For a call option the option price represents a premium paid for the risk that the spot price of the underlying will rise above the agreed upon strike price at maturity.

[ insert profit diagram for call option]

• A european put option gives the holder the right to sell an asset at a future date at an agreed upon price.

• An american call (put) option gives the holder the right to buy (sell) an asset for an agreed amount. However, the option can be exercised at any time, not just at maturity.

• An option expires out of the money if - S(T ) < K for a call and - S(T ) > K for a put option.

[ insert payoff and profit diagrams for put option]

• The buyer of an option stands to lose at most the premium paid up front. This happens if the option expires out of the money and option is not exercised.

• Comment: a forward contract can be constructed using options. A forward contract with delivery price K and expiry T is equivalent to a portfolio which is long a call option and short a put option which both have strike price K and expiry T .

• Using options to hedge: An investor owns 1000 shares of ABC, valued at R100 per share. If the share price drops to R90 in the next quarter the investor will suffer a loss of R10 000. To hedge against this risk, the investor purchase a put option to sell 1000 shares at R95 per share in 4 months time. Now if the price drops to R90 per share, the loss thus becomes limited to R5 000 plus the premium for the put. If the price rises instead to R104 per share, then the investor gains R4 000 minus the premium of the put.

• Using options for leverage: Joe believes strongly that the shares of firm GenX, which manufactures power generators, will rise rapidly in the next quarter. He is willing to speculate R10 000 on this view. GenX shares are currently trading at R50 per share, so Joe could buy 200 shares. If the share price rises to R60 in 4 months time, then Joe will make a profit of R2 000. If the price drops to R40 per share, then he will lose R2 000.

Joe decides instead to buy call options. Call options to buy 100 GenX shares, striking at R53, maturing in 4 months time and which are priced at R200 are available. Joe can buy 50 of these with his R10 000; he will be able to purchase 5 000 shares if the options mature in the money. If the share price rises to R60, then Joe will make a profit of 5000 ∗ (60 − 53) − R10 000 = R25 000. If the share price drops below R53, then he loses his entire R10 000.

Derivative Markets (international, 2002)

DERIVATIVE TYPES

amounts (billions of US$) 270 100 31 075 16 031 8 236 6 809 204 393 13 573 163 749 27 071 5 145 1 176 3 968 1 693 288 1 406 27 793

OTC contracts: 1. FOREX contracts Forwards & forex swaps Currency swaps Options 2. Interest rate contracts FRA’s Interest rate swaps Options 3. Equity-linked contracts Forwards and swap Options 4. Commodity contracts Gold Other commodities 5. Other OTC contracts

DERIVATIVE TYPES

amounts (billions of US$) 58 281.4

Exchange traded contracts: 1. Futures Interest rate Currency Equity index 2. Options Interest rate Currency Equity index TOTAL

20 696.9 19 860.3 109.7 726.8 37 584.5 32 794.9 63.1 4 726.5 328 381.4

http://www.bis.org/publ/quarterly.htm

Myths, legends and tales of woe ... Famous case studies: * LTCM, * Metallgesellschaft, * Barings (Nick Leeson - Rogue trader). http://www.erisk.com/Learning/CaseStudies.asp http://www.erisk.com/Learning/CaseStudies/ WheelofMisfortune.asp

BOOKS: Inventing money by Nicholas Dunbar (LTCM - more math), When Genius Failed by Roger Lowenstein (LTCM - more people), Fooled by Randomness by Nassim Taleb (general) Liar’s Poker by Michael Lewis (bond trading and Wall Street in the 80’s)

SOME Look-up LINKS: http://www.riskglossary.com http://www.optionsxpress.com/ http://www.investopedia.com/ http://www.investorwords.com/ http://global-derivatives.com/maths/a-e.php http://global-derivatives.com/options/o-types.php http://www.wikipedia.org/

http://www.in-the-money.com/glossarynet/keyindex.