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BASIC MATHS FOR FINANCE

BASIC MATHS FOR FINANCE. MTH 105. FINANCIAL MARKETS. What is a Financial market: A financial market is a mechanism that allows people to trade financial security. Transactions occur either: either in an Exchange; a building where securities are traded

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BASIC MATHS FOR FINANCE

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  1. BASIC MATHS FOR FINANCE MTH 105

  2. FINANCIAL MARKETS • What is a Financial market: • A financial market is a mechanism that allows people to trade financial security. • Transactions occur either: • either in an Exchange; a building where securities are traded • Over the counter; electronically or telephone

  3. FINANCIAL MARKETS • Types of Financial Markets: • Capital Markets: Long term securities (> 1yr) Stock Markets Bond Markets • Commodity Markets • Money Markets: Short term (< 1yr)

  4. FINANCIAL MARKETS • Derivative markets • Foreign Exchange Markets

  5. FINANCIAL MARKETS • Capital Markets: are for trading securities with original maturity that is greater than 1yr (Stocks and Bonds) • Stocks: -represent ownership -shareholders receive dividends -they can also decide to sell their shares NB: Stocks are risky assets(uncertain future return) High risk, High return

  6. FINANCIAL MARKETS • Bonds: represents a debt owned by the issuer to the investor -Bonds obligate the issuer to pay a specific amount at a given date, generally with periodic interest payments NB: Bonds issued by the Government are risk free : Low risk, Low return

  7. FINANCIAL MARKETS STOCKS BONDS • Risky (uncertain future returns) • No maturity date • Have control over the firm (vote, firm’s activities, etc) • Receives dividends (not guaranteed) • Receiving capital invested is not guaranteed • Risk-free (future return is certain) • Maturity date • Have no control over the firm • Receives interest payments (guaranteed) • Receives capital invested

  8. FINANCIAL MARKETS • Capital Market Participants: 1.Government: Issues bonds to finance projects such as schools etc. or pay off its debt. NB: Government never issues stock

  9. FINANCIAL MARKETS 2.Companies: Issue both stocks and bonds to fund investment projects 3.Investors/Households: Purchasers of capital market securities

  10. A SIMPLE MARKET MODEL • Assume that only two assets are traded in the financial market: • Stocks: Risky assets [uncertain future returns] • Bonds: Risk-free [future returns are certain]

  11. A SIMPLE MARKET MODEL • The time scale to be used: t=0; represents today t=1; represents the future (tomorrow, 1yr time etc)

  12. A SIMPLE MARKET MODEL • For Stocks let: S(t): the price of a share at time t S(0): the price of a share today (certain) S(1): the price of a share in future (uncertain) Ks: Return on Stocks Ks = S(1) – S(0) (uncertain) S(0)

  13. A SIMPLE MARKET MODEL • For Bonds let: A(t): the price of a bond at time t A(0): the price of a bond today (certain) A(1): the price of a bond in future (certain) KA = A(1) – A(0) (certain) A(0)

  14. A SIMPLE MARKET MODEL • For a Portfolio let: x: number of shares held by an investor y : number of bonds held by an investor Portfolio (x shares and y bonds)

  15. A SIMPLE MARKET MODEL • The total wealth of an investor holding x shares and y bonds at time t is given by: V(t) = x S(t) + y A(t) NB: S(t), A(t) ; price per share and bond

  16. A SIMPLE MARKET MODEL • At t=0 V(0) = x S(0) + y A(0) • At t=1 V(1) = x S(1) + y A(1)

  17. A SIMPLE MARKET MODEL • Return on a Portfolio (x, y) Kv = V(1) – V(0) uncertain V(0)

  18. A SIMPLE MARKET MODEL • Examples: • If S(0) = GH 50 and, S(1) = GH 52 with probability p, GH 48 with probability 1 − p, for a certain 0 < p < 1. Find the return on this stock or find Ks.

  19. A SIMPLE MARKET MODEL • If A(0) = 100 and A(1) = 110 Ghana cedis. What will be the return on an investment in this bond? Or find KA.

  20. A SIMPLE MARKET MODEL • If S(0) = GH 50, A(0) = GH100, A(1) = GH110 S(1) = GH 52 with probability p, GH 48 with probability 1 − p, for a certain 0 < p < 1. For a portfolio of x = 20 shares and y = 10 bonds find; V(0): value of the portfolio at t=0 V(1): value of the portfolio at t=1 Kv: return on this portfolio

  21. A SIMPLE MARKET MODEL • If S(0) = GH 40, A(0) = GH 50, A(1) = GH 70 S(1) = GH 50 with probability p, GH 35 with probability 1 − p, For a portfolio of x = 10 shares and y = 200 bonds find; Ks, KA, and Kv

  22. A SIMPLE MARKET MODEL • Let A(0) = 90, A(1) = 100, S(0) = 25 dollars S(1) = 30 with probability p 20 with probability 1−p where 0 <p<1. For a portfolio with x = 10 shares and y = 15 bonds calculate Ks, KA and Kv

  23. A SIMPLE MARKET MODEL • Assumption 1: Randomness The future stock price S(1) is a random variable with at least two different values. The future price A(1) of the risk-free security is a known number.

  24. A SIMPLE MARKET MODEL • Assumption 2: Positivity of Prices All stock and bond prices are strictly positive, A(t) > 0 andS(t) > 0 for t =0 ,1.

  25. A SIMPLE MARKET MODEL • Assumption 3: Divisibility, Liquidity and Short Selling An investor may hold any number of x shares and y bonds, whether integer or fractional, negative, positive

  26. A SIMPLE MARKET MODEL • Divisibility: The fact that one can hold a fraction of a share or bond is referred to as divisibility. • Liquidity: It means that any asset can be bought or sold on demand at the market price in arbitrary quantities.

  27. A SIMPLE MARKET MODEL + x : long position in shares (buying shares) + y : long position in bonds (buying bonds) Long Position; If the number of securities of a particular kind held in a portfolio is positive, we say that the investor has a long position.

  28. A SIMPLE MARKET MODEL - x : short position in shares (short selling shares) - y : short position in bonds (issuing bonds) Short Position; If the number of securities of a particular kind held in a portfolio is negative, we say that the investor has a short position.

  29. A SIMPLE MARKET MODEL • Short Position in Stocks: Short selling: • Borrow stocks (-x) • Sell stocks • Use the proceeds to make some other investments NB: Owner of stocks keeps all rights to it

  30. A SIMPLE MARKET MODEL • Closing the short position: Bonds: Paying interest and Face value Shares: Repurchase the stock and return to the owner

  31. A SIMPLE MARKET MODEL • Assumption 4: Solvency The wealth of an investor must be non-negative at all times, V (t) ≥ 0 for t =0 ,1. - Admissible Portfolio: A portfolio satisfying this condition is called admissible

  32. A SIMPLE MARKET MODEL • Assumption 5: Discrete Unit Prices The future price S(1) of a share is a random variable taking only finitely many values.

  33. A SIMPLE MARKET MODEL • Assumption 6: No-Arbitrage Principle • We shall assume that the market does not allow for risk-free profits with no initial investment • If the initial value of an admissible portfolio is zero, V (0) = 0, then V (1) =0 • There is no admissible portfolio with initial value V (0) = 0 such that V (1) > 0

  34. A SIMPLE MARKET MODEL - If a portfolio violating this principle did exist, we would say that an arbitrage opportunity was available.

  35. A SIMPLE MARKET MODEL • Suppose that investor A in Accra offers to buy 100 shares at GH 2 per share while investor B in Kumasi sells 100 shares at GH1per share. If this were the case, the investors would, in effect, be handing out free money. An investor with no initial capital could realise a profit of 200 − 100 = 100 Ghana cedis by taking simultaneously a long position (thus buying from investor B)and a short position (selling to investor A).

  36. BINOMIAL MODEL • BINOMIAL MODEL: • The choice of stock and bond prices in a binomial model is constrained by the No-Arbitrage Principle.

  37. BINOMIAL MODEL • Suppose that: S(0)=A(0) S(1)= S u when stocks go up S d when stocks go down • Then: Sd <A(1) <S u, or else an arbitrage opportunity would arise. (Insert Diagram).

  38. BINOMIAL MODEL • Show that an arbitrage opportunity would arise when A(1) ≤ Sd A(1) ≥ Su.

  39. BINOMIAL MODEL • Example: If S(0) = A(0) = GH 100, S(1) = 125 with probability p 105 with probability 1-p Proof that an arbitrage opportunity will arise if A(1)=GH 90

  40. BINOMIAL MODEL • Assignment If S(0) = A(0) = GH 100, S(1) = 125 with probability p 105 with probability 1-p Proof that an arbitrage opportunity will arise if A(1)=GH 130

  41. RISK AND EXPECTED RETURN • Risk • The uncertainty associated with receiving future returns • Expected Return The weighted average of all possible returns from a portfolio

  42. RISK AND EXPECTED RETURN • Example: Given that; S(0)=25, A(0)=90, A(1)=100 S(1) = 30 with probability 0.6 20 with probability 0.4 if x=5 shares and y=10 bonds, find the expected return and risk on: • Stocks • Bonds • Portfolio

  43. RISK AND EXPECTED RETURN • Given the choice between two assets or portfolios with the same expected return, any investor would obviously prefer that involving lower risk. • Similarly, if the risk levels were the same, any investor would opt for higher return.

  44. OPTIONS • An option is a financial derivative which gives the holder the right (but not the obligation) to buy (call option) or sell (put option) an asset on a specific pre-determined future date and price. • Call Option: Gives the holder the right to buy an asset on a fixed future date and price. • Put Option: Gives the holder the right to sell an asset on a fixed future date and price.

  45. OPTIONS • TERMS: • Strike price: the agreed upon price to buy or sell an asset in future • Delivery date: the agreed upon future date to buy or sell an asset • Exercising the option: buying or selling an asset at a pre-determined price and date

  46. OPTIONS • 2 TYPES OF OPTIONS • Call Option: gives the holder the right to buy an asset at a pre-determined date and price. • A call option is exercised only when the market price S(1) is above the strike price. • If the stock price falls below the strike price, the option will be worthless

  47. OPTIONS • Example: Let A(0) = 100, A(1) = 110, S(0) = 100 GH and S(1) = 120 with probability p, 80 with probability 1 − p, where 0 < p < 1. Strike Price: GH 100

  48. OPTIONS • Hence at t=1, the value of a call option is: C(1) = S(1) – Strike price = 120 - 100 = 20 with prob. p 80 - 100 0 with prob. 1-p

  49. OPTIONS • At t=0, the value of a call option is found in 2 steps: Step 1 (replicating the option) - Construct a portfolio of x shares and y bonds such that the value of the investment at time 1 is the same as that of the option, xS(1) + yA(1) = C(1) 120x + 110y = 20 80x + 110y = 0 solve for x and y

  50. OPTIONS x = 1 2, y= −4 11 To replicate the option we need to buy 1̸ 2 a share of stock and take a short position of − 4̸ 11 in bonds.

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