Financial Mathematics

Financial Operations and Financial Quantities

• Financial operation is every agreement concerning the exchange of money at different time dates.
• Simple financial operations have only two cash flows.
• Complex financial operations have several cash flows.
• Financial calculus is based on two types of processes: accumulation and discount.

Accumulation

• Principal value C is paid at time to.
• Final value M is cashed at time T.

Discount

• Present/discounted value C is cashed at time to.
• Nominal value M is paid at time T.
• Accumulation shows today's €1,000 worth in 1 month (€1,100).
• From agent 2's point of view, it is a discount operation.
• €100 is the price of money an economic agent to pay for getting €1,000 now instead of 1 month.
• Works under the assumption that the market is efficient (= fair).
• Basic principle of financial calculus: €1 today is better than €1 tomorrow.

Accumulation Process

• Given C, to, and T, the value of M needs to be found.

Discount Process

• Given M, to, and T, the value of C needs to be found.
• A fair financial operation means the engagement value into the operation must be less than or equal to the right-hand-side value.

Terminology

• C = starting value (time 0).
• M = accumulated value (time T).
• Financial equivalence (~) exists between C (€1000 available now) and M (€1100 available at time T) because money has a time dimension.

Financial quantities

• Accumulation factor provides the capital value of €1 available today at some future time T.
• Money available today x accumulation factor = value of money at time T.
• If two operations have the same accumulation factor, then they are fair.

Discount Factor

• Tells how much €1 available at future time (t=T) is worth now (t=0).
• Knowing the discount factor allows to know if the offered money today in exchange of a higher amount given in the future is fair or not.
• Discount is the inverse operation of accumulation, in this case we say that the accumulation and discount factors are conjugate.

Interest and Interest Rate

• Interest rate is the interest (reward) an entity gets for each unit of invested money.
• Interest I, can also be called discount (suffered to have money available instead of having them in the future).
• Interest rates are usually expressed by time unit interval (day, month, year).

Note

• i (to, T) or i(0,T - to) are different in general.
• Spot interest rate represents the interest on a unit of capital invested today after one year.
• Forward interest rate is the interest of a unit of capital invested for one year from one year from now.
• Two interest rates are usually different, even if they refer to financial operation of the same length.

Discount Rate

• Is the ratio between the suffered discount to have now money instead of money available in T, and the money available in T
• If i increases, also d rate increases.

Common Financial Laws

• A financial law is a rule that describes how interests are accumulated or accrued.
• It is needed to compare interest rates with different time dimensions (ex: monthly vs annual).
• Financial law can be mathematically represented as a two-variable function, one variable being time, and the other interest rate.
• f (t; i) is the value of €1 after t units of time when invested at an interest rate for unit of time given by i.
• f(1; i) is the interest accumulated after 1 unit of time based on an interest of €1.
• Any function f (t; i) can be used as a financial law if certain conditions are satisfied.

Conditions

1. f(0; i) = 1 has to be 1.
2. f must be a monotonically increasing function with respect to t.

• 10% is always the interest rate.
• Interest rate (i), accumulation factor (f), discount factor (φ) and discount rate (d) are numerically the same for the two operations.
• i, f, g, and d, have a time dimension.
• So, that the 4 financial quantities very important, changes the convenience of those financial quantities.
• The biannual interest rate, not annual

Simple Accumulation Rule (Simple Interest Rate)

• The interest of the financial operation is proportional to the capital (C) and to investment time (t).
• The interest rate does not participate in the accumulation of forward interest rate.
• Interest does not generate further interest: each year interests only depends on initial conditions.
• Stopping the operation and restarting it immediately after brings extra rewards.
• The accrued capital M evolves as: t = 0 → M(0) = C, t = 1 → M(1) = C(1 + i) = C + iC etc.

Compound Accumulation Rule (Compound Discount)

• Interest has to be computed using the simple accumulation rule.
• I₂ = M(1)i > 0,1(= iC), this means
• the cash in would be €1,1 which is the capital at period 1.
• Start a new investment of €1,1, immediately after at an interest rate of 10% according to rule.
• the money comes back at t= 0 → M(2) = C(1 + i) → COMPOUND ACCUMULATION RULE

Example

• Under the simple accumulation rule, M(10) = 100(1 + 10(0,1)) = €200.
• Under the compound accumulation rule, M(10) = 100(1 + 0,1)10 = €259,4
• MC – MS = €259,4 – €200 = €59,4 which is the extra interest generated by accumulated interest.

Financial Law

• Simple accumulation fs(t; i) = fs(t) = 1 + ist, is =simple interest rate.
• Compound accumulation fc(t; i) = fc(t) = (1 + ic)t, ic compound interest rate.
• Between times 0 and 1, simple accumulation is preferable because the interest accumulates faster.
• After time period of 1.1 compound accumulation would be preferable or 1 would just equal.
• Compound accumulation process includes the term I₁i is the interest rate accumulated over the past interest rate.
• Where at each period, the interest rate yielded up is added to the principle becoming new capital invested.

Continuation Accumulation Factor

• Using compound accumulation there is no advantage in doing intermediate capitalization.
• Simple accumulation does not apply because: fs(t; i) = fs(t₁ + t2; i) = 1 + (t₁ + t₂)i.
• Under simple accumulation process, money at period end is: C*(1+SUM(i(t))

Complex Financial Operations

• Any kind of operations that involves several investments made regularly to get the final amount + interest.
• Annuity is a sequence of payments with fixed frequency (annual or monthly fixed payments in the bank account).
• If money inflows are not in fixed time frequencies, it is not an annuity.
• Annuity class with a constant amount are called a certain annuity.
• A financial derivative that flows money is a contingent annuity.

Terminology

• Time between two consecutive payments (month) for a monthly annuity is the Payment period.
• Time from the beginning of the first payment period time and the end of the last payment period is the Term.
• Payments are made at the end of the payment period for an Ordinary Annuity.
• Payments are made at the beginning of the payment period is an Annuity Due.
• When the term of annuity is not to begin until some time period in the future, is a Deferred Annuity.
• Anual Rent refers to the sum of payments made in 1 year.

Accumulated Value

• The present value of geometric series/sums needs discounting.
• Installments over $1 need multiplying by installemt value.

Discount factor

• The future payments need to be discounted, so that the discount factor, which is between 0 and 1.
• Value of a deferred annuity to the present value discounted for m periods.
• Ordinary deferred annuity, The formula is: a n i

Financial Laws

• Financial laws are (in the more general form) functions of 3 variables and depend on a parameter.
• Described by, F(C; to; T; β) = M, where, F establishes a financial equivalence between C available in to and M available in T.
• For "F" to be a financial law, F must satisfy: 3 principles.
• F must be homogeneous of 1st degree with respect to C
• F(C; to; T; β) = CF (1; to; T; β) — f(x) = xf(1) If f(t) is the accumulation factor that describes a financial law (generic).
• The speed in which interest accumulates is maximum at the beginning, and declines over time.
• Force of interest for compound accumulation is: d = ln (1 i), where d is independent with time.
• One - variable financial law is decomposable only where if its force of interest is constant ( it doesnt depend on time ) meaning a the compound accumulation rule.

Theorem

• Any financial law that is identified in a unique way by either a factor f(t) of by its force of interest.
• The compound is the only one - variable financial law with a constant force of interest

Interest rate and market of riskless bond

• A government issues zero coupon bond is A type of legal contact it can issues the bond directly and the party that lends money to the government.
• Its zero because there's no intermediate payments there is only. Face value is the amount of money that the government promises at maturity, that the that party who loans the money to that to party and it knows that that will happen
• A ZCB( Zero Coupon Bond) with face value FV and maturity T is a financial instrument classified for income and non fixed income instruments it can be non fixed if they are sold before maturity . Pure-discount bond: is 1 ZCB with unitary face value also denoted by V It gives the market the term structure of interest rates

Term structure of interest rates

• Market discount factor is V = P/FV= (price you pay in time 0)/face value. For t years the annual interest rate compounded per period is equal to i(0)(0,t} = (1/√(0)(0,t)) − 1
• A forward contract has this definition it is 1 contract where 2 parties agree to exchange and you also need face value maturity
• At the current time agreed before exchange . This is known as forward price and it known as forward discount .

Yield, the measure of what to expect on the future value of the market

• To amortize debt with it will need to be re-payed with terminologies described with a description based on R ( installement paid at the beginning) , Amount S ( total debit ) and time or year based on the component the period can will will need to be known . Depending on the type of payment plan :
• a) Global Final Refund (payment include 0 installments , its interest is 12% then with the amortization quota and a chart can then solve)
• c) Pure Mortgage or Amortization , A amortization installment chart is needed for the time year amount and can then be calculated .. To solve this amount the elementary final closure conditions must be fulfilled . A calculation needed from A to M must be obtained .

Internal Rate of Return (IRR)

• The IRR is compounded rate (usually denoted by y) that makes a sum of discounted values of a cash load Financial operations equals 0 with formulas .
• If it's 0 then one cash equation must used Adjusted Present Value is the difference in operation between different types of investments .. The ex-ante return can also be evaluated here. If you are also an ex inter investements you must have to check value of the rewards in the market.
• The general adjusted present value is that the cash flows must to be recognized and with that the bank loans as well.
• There needs to show and conclude the connections , market operations must used and compared etc.

Cash flow

• The financial average leveraged in connected with that if the internal rate equals i is higher then amount invested in the in can or can't depend on certain things Financial will depend the cost debt and if theres other factors involved
• The final decision is it can you need the the amount if you have if need debt can really need the leverage

4. Generalized adjusted value the pressence the rate the different types things The oppurtuinty costs of equality or it is not more easy to pay to single interest depends The NPV is very easy 297.52$