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
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3. FINANCIAL MARKETS Types of Financial Markets:
Capital Markets: Long term securities (> 1yr)
Stock Markets
Bond Markets
Commodity Markets
Money Markets: Short term (< 1yr)
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4. FINANCIAL MARKETS Derivative markets Foreign Exchange Markets
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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
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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
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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
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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
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9. FINANCIAL MARKETS
2.Companies: Issue both stocks and bonds to fund investment projects
3.Investors/Households: Purchasers of capital market securities
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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]
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11. A SIMPLE MARKET MODEL The time scale to be used: t=0; represents today
t=1; represents the future (tomorrow, 1yr time etc)
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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)
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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)
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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)
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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
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16. A SIMPLE MARKET MODEL At t=0 V(0) = x S(0) + y A(0) At t=1
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17. A SIMPLE MARKET MODEL Return on a Portfolio (x, y)
Kv = V(1) – V(0) uncertain
V(0)
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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.
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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.
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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
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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
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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
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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.
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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.
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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
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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.
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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.
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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.
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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
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30. A SIMPLE MARKET MODEL Closing the short position:
Bonds: Paying interest and Face value
Shares: Repurchase the stock and return to the owner
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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
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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.
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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
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34. A SIMPLE MARKET MODEL
- If a portfolio violating this principle did exist, we would say that an arbitrage opportunity was available.
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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).
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36. BINOMIAL MODEL BINOMIAL MODEL:
The choice of stock and bond prices in a binomial model is constrained by the No-Arbitrage Principle.
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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).
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38. BINOMIAL MODEL
Show that an arbitrage opportunity would arise when A(1) ≤ Sd
A(1) ≥ Su.
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39. BINOMIAL MODEL Example: If S(0) = A(0) = GH 100,
S(1) = with probability p
with probability 1-p
Proof that an arbitrage opportunity will arise if
A(1)=GH 90
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40. BINOMIAL MODEL Assignment If S(0) = A(0) = GH 100,
S(1) = with probability p
with probability 1-p
Proof that an arbitrage opportunity will arise if
A(1)=GH 130
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41. RISK AND EXPECTED RETURN
The uncertainty associated with receiving future returns
Expected Return
The weighted average of all possible returns from a portfolio
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42. RISK AND EXPECTED RETURN
Example:
Given that; S(0)=25, A(0)=90, A(1)=100
S(1) = with probability 0.6
with probability 0.4
if x=5 shares and y=10 bonds, find the expected return and risk on:
Stocks
Bonds
Portfolio
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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.
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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.
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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
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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
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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
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48. OPTIONS Hence at t=1, the value of a call option is:
C(1) = S(1) – Strike price
= = with prob. p
with prob. 1-p
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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 = solve for x and y
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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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51. OPTIONS Step 2: (Valuing the Option)
Compute the time 0 value of the investment in stock and bonds. It will be shown that it must be equal to the option price,
xS(0) + yA(0) = C(0)
xS(0) + yA(0) =1̸2 × 100 −4̸11 × 100 = GH
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52. OPTIONS
Put Option: gives the holder the right to sell an asset at a pre-determined date and time
A put option is exercised only when the market price S(1) is below the strike price.
If the stock price rises above the strike price, the option will be worthless
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53. OPTIONS Put Option: 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
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54. OPTIONS Hence at t=1, the value of put option is:
P(1) = Strike price - S(1)
= = with prob. p
with prob. 1-p
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55. 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
- For a call option with strike price GH 100, find C(0)
- For a put option with strike price GH 100, find P(0)
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56. OPTIONS Solution: C(0) 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 portfolio at t = 1 is the same as that of the option.
At t=1; V(1) = C(1)
xS(1) + yA(1) = C(1)
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57. OPTIONS Step 2: (Valuing or Pricing the Option)
At t=0, the value of the portfolio in stock and bonds must be equal to the value of the option.
At t=0;
V(0) = C(0)
xS(0) + yA(0) = C(0)
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58. OPTIONS Assignment Let A(0) = 100, A(1) = 110, S(0) = 100 GH and
S(1) = 120 with probability p,
80 with probability 1 − p
Compute the value of a call option C(0) if:
Strike price = GH 90
Strike price = GH 110
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59. OPTIONS Portfolio (x,y,z)
In a market in which options are available, it is possible to invest in a portfolio (x, y, z) consisting of x shares of stock, y bonds and z options.
At t=0 and t=1, the value of such a portfolio is:
V (0) = xS(0) + yA(0) + zC(0)
V (1) = xS(1) + yA(1) + zC(1)
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60. FORWARD CONTRACT
Forward contract is a financial derivative which obligates the holder to buy (long forward contract) or sell (short forward contract) an asset on specific pre-determined future date and at a fixed price.
Long forward: Obligates the holder to buy an asset on a fixed future date and price.
Short forward: Obligates the holder to sell an asset on a fixed future date and price
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61. At t=1, the value of a Long Forward Contract: LF(1) = S(1) – Forward price At t=1, the value of a Short Forward Contract: SF(1) = Forward price-S(1) At t=0. F(0)=0 no value
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62. SIMILARITIES OPTION FORWARD CONTRACT Right to buy (call option)
Right to sell
(put option)
Fixed price
(strike price)
Obligation to buy
(long forward contract)
Obligation to sell
(short forward contract)
Fixed price
(forward price)
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63. SIMILARITIES
In both option and forward contracts, agreement is made today, but transaction takes place in future.
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64. DIFFERENCES Not compulsory to buy or sell assets Down payment required
OPTIONS
FORWARD CONTRACT
Not compulsory to buy or sell assets
Down payment required
(Premium)
Compulsory to buy or sell assets
No down payment is required
(no premium)
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65. FORWARD CONTRACT Portfolio (x,y,z)
It is possible to invest in a portfolio (x, y, z) consisting of x shares of stock, y bonds and z future contracts.
At t=0
V (0) = xS(0) + yA(0) no value for F(0)
At t=1
V (1) = xS(1) + yA(1) + zF(1)
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