Financial Mathematics I Notes

Transcription

1 Financial Mathematics I Notes Contents... 3 Introduction to interest... 3 Simple Interest... 4 Practical Applications of Simple Interest in Discount Securities... 4 Simple Discount... 5 Compound Interest... 5 Discount Value... 5 Accumulation Factors... 5 Force of Interest... 6 Equivalent statements... 7 Describing Present Values and Accumulations for Constant Force of Interest Cash Flow Model Flat Rates of Interest Equation of Value Propositions (sufficient not necessary) Linear interpolation Bisection: Loan Loan Schedule Checking Bonds Three principal asset types Property: real estate Comparison Real estate investment trust (REITs) Ordinary shares (common stocks): listed businesses Comparison Debt (fixed interest): loans, marketable securities and deposits Types Other assets

2 Differences between property and shares Similarities (property and shares) Why shares have a higher return than fixed interest securities Differences between convertible loan stock and ordinary shares Similarities between convertible loan stock and ordinary shares Differences between options and issuing prospectus that gives people the option to subscribe for its ordinary shares at a fixed price at any time in the future Function of clearing house in exchange traded derivatives markets Difference between buying a call option and selling a put option over the same underlying asset Difference between purchasing a call option and entering into a open long futures and forward position in the same underlying asset Factors that influence the time value of option prices Counterparty risk Novation clearing Outright purchase: Forward European When bonds are redeemed at later date

3 Introduction to interest An incentive, reward for depositing money. The amount of interest paid depends on 1. The amount of deposit 2. The interest rate offered 3. Time that money is left on deposit is the effective rate of interest per time unit iff an amount of at time becomes at time o Proof by induction for all using o Effective rate Compound rate of interest. Converse is not true. Eg 6% p.a. compounded half-yearly does not imply the effective rate is 6% p.a. (The effective rate is 3% per half yearly). But 6% p.a. effective implies 6% p.a. compounded yearly. We can also say 6% is the nominal rate of interest convertible half yearly. o is a special case of and is used when the interest is constant over time. is the effective rate of interest per time unit iff an amount of at time becomes at time o Proof by induction for all using where o is a special case of the nominal interest rate, when the effective interest rate for the period is. is the nominal rate of interest per time unit convertible p th ly iff is the effective rate of interest for time units of.. o Definition of and gives which also means knowing the effective rate,, for one time unit allows you to deduce the effective rate for multiples of this time unit through compounding. But the converse is not true. Eg Effective rate of 6% pa does not imply per half year. Need to assume constant force of interest for this to be true. o is a special case of and is used when interest is constant over time and interest rate cover factions of a unit which are reciprocals of integers. is the nominal rate of interest per time unit if the effective rate of interest is for the period of length starting at. 3

4 Simple Interest Interest earned is Since when, investors earning simple interest would be advised to consider closing simple interest accounts at regular intervals and opening new accounts using both principal and accumulated interest from their old accounts thereby achieving a form of do-it-yourself compounding. Simple interest would result in unnecessary administrative work for banks and competitive pressure. If interest payments are the same,. Note if interest payments are the same,. If, Simple discount pay larger interest as it is based on, accumulation which is larger than, principle which is what simple interest is proportional on. Practical Applications of Simple Interest in Discount Securities Discount securities are fixed interest investments which do not pay interest; investors earn interest by purchasing these securities for less than the amount (discount) at which they will be redeemed and subsequently receiving the full redemption value (face value) at a later date. Features of discount securities o Fixed known maturity (redemption) date o Known amount payable on maturity (face value) o No interest payments Treasury Notes (issued by Australian Government with Federal government guarantee), Bank Accepted Bills of Exchange (Guaranteed by and payable at the accepting bank normally regarded as negligible default risk) and Promissory Notes (companies and institutions redemption depends on financial strength of issuer). They differ by the certainty at which the face value will be paid on maturity. Assume equivalence of Settlement date (date buyer pays the seller for the security for security purchased) and Transaction Date (date transaction is agreed). Days to Maturity: count either the settlement or maturity date buy not both. Use 365 days even leap year of 29 days in February. 4

5 Simple Discount Compound Interest Interest earned is using Binomial Theorem Assumptions using proof by induction for all using o Interest rate does not depend on amount of principal o Compounding occurs at unit time intervals and so ] o Rate of interest are constant over time. [Introduce ]. [Introduce Discount Value For, the discount value at of 1 due in is o Special case which gives present value o Known as discount value (due at or after ) at for due in time. o If due at or before accumulation. Accumulation Factors is the amount returned at time for an investment of at. 5