Financial Mathematics: Annuities

• These notes cover Financial Mathematics, Semester 2, for first-year TM & IGE students, academic year 2024/2025

Annuities

• Annuities involve a series of monetary flows received or paid at equal time intervals.
• The term "annuity" is typically applied to annual periodicities; other periodicities are referred to as "semi-annual," "quarterly," or "monthly".
• The analysis of annuities determines the present or future value of a series of flows, considering the timing of the first flow, the periodicity, the number of flows, and the amount of each flow.
• Annuities are "constant" when the amounts are equal.
• Or, annuities are "variable" when the amounts change from period to period.
• Annuities can be received or paid at the beginning or end of a period.
• Annuities are "certain" if the period is constant or "random" if the period varies.

Annuities at the End of the Period

• The "future value" (Vn) of a series of constant annuities at the end of the period is the sum of the annuities immediately after the last annuity payment.

Future Value (Vn)

• Vn is the value acquired from a series of annuities.
• 'a' represents the constant annuity at the end of each period.
• 'n' is the number of periods or annuities
• 'i' is the interest rate per capitalization period.
• Vn = a + a(1+i) + a(1+i)² + ... + a(1+i)n-2 + a(1+i)n-1 can be simplified to Vn = a [1 + (1+i) + (1+i)² + ...+ (1+i)n-2 + (1+i)n-1 ].
• This is a geometric series with a first term of 1 and a common ratio of q = (1+i) with 'n' terms.

The Formula

• Vn = a * [(1+i)^n - 1] / [(1+i) - 1] which simplifies to Vn = a * [(1+i)^n - 1] / i.
• Example: Placing 5,000€ each year for 5 years with annual capitalization at 6% requires calculating the future value at the last payment and one year after.
• Solution: Given a = 5000, i = 0.06, and n = 5.
• The value acquired at the 5th payment = 5000 * (1.06^5 - 1) / 0.06 = 28,185.46 €.
• One year later = 28,185.46 * 1.06 = 29,876.27 €.
• The interest acquired is 29,876.27 - (5 * 5000) = 4,876.27 €.
• "Present value" is the current sum of a series of constant annuities at the end of the period, discounted to the original date.
• Recalling that the present value of a sum Ak is the amount placed that, after interest, produces Ak.
• Vo represents the present value per the series of annuities.
• 'n' is the number of periods for the annuities.
• 'a' is the constant annuity at the end of each period.
• 'i' is the interest rate per capitalization period.
• Vo = a(1+i)^-1 + a(1+i)^-2 + ... + a(1+i)^-n+1 + a(1+i)^-n can be written Vo = a [(1+i)^-1 + (1+i)^-2 + ... + (1+i)^-n+1 + (1+i)^-n].
• Vo = a (1+i)^-1 [1 + (1+i)^-1 + ... + (1+i)^-n+2 + (1+i)^-n+1] represents a geometric series of first term 1 and common ratio q = (1+i)^(-1) with 'n' terms.
• The formula becomes Vo = a (1+i)^-1 * [1 - (1+i)^-n] / [1 - (1+i)^-1], simplifying to Vo = a * [1 - (1+i)^-n] / i.
• Example: To find the present value at a 6% discount rate of a series of constant annuities of 1500 euros paid at the end of each year for 7 years.
• Solution: VA = 1500 * [1 - 1.06^-7] / 0.06 ≈ 8373.57.
• Exercice: Calculating the present value of a series of monthly payments of 1,000 € for 24 months at a discount rate of 0.5% per month, one month before the first payment.
• Value = 1000 * [1 - 1.005^-24] / 0.005 = 22,562.87 €.
• Application Question 1: Determine the amount one must borrow at a 3% monthly rate to repay 230 Euros for the next three months at the end of each period.
• Solution: Calculate the present value of these three sums of money: VA = 230(1+3%)^-1 + 230(1+3%)^-2 + 230(1+3%)^-3 = 650.58 Euro.
• The future value (Vn) is considered when flows are paid at the beginning of the period, altering the calculations.
• With payments at the beginning, the graph shifts so Vn = a(1+i) + a(1+i)² + ... + a(1+i)n-1 + a(1+i)n Which is a (1+i) [1 + (1+i) + (1+i)² +.....+ (1+i)n-2 + (1+i)n-1].

With Formulas

• Vn = a (1+i) * [(1+i)^n - 1] / [(1+i) - 1], resulting in Vn = a (1+i) * [(1+i)^n - 1] / i.
• The value is now calculated in present tense.
• Vo = a + a(1+i)^-1 + a(1+i)^-2 + ... + a(1+i)^-n+1 = a [1 + (1+i)^-1 + (1+i)^-2 + ... + (1+i)^-n+1] forms a geometric series. The simplified form is Vo = a * [1 - (1+i)^-n] / [1 - (1+i)^-1].
• Now: Vo = a * (1+i) * [1 - (1+i)^-n] / i.
• Example 1: In depositing an amount of money on the first of each month from January 1, 2002, to January 1, 2003, with the goal of accumulating $1000 by January 1, 2003, with a monthly interest rate of 0.005, what must be the value of the money deposited each month?
• 1. 005 * [1.005^13 - 1]$ or $74.27 deposited each month
• Example 2: Calculate the amount needed to pay on the first of January of each year to reimburse a loan of 90,000 DH with a rate of 7% over 8 years with a = 14086 DH

Variable Annuities.

• Variable Annuities refer to annuities where the payments are not constant.

The Acquired Value

• Vn = a_n + a_(n-1)(1+i) + ... + a_2(1+i)^(n-2) + a_1(1+i)^(n-1). Which can be written like= ∑ a_p (1+i)^(n-p) from p=1 to n.
• The Actual Value = Vo = a_1(1+i)^(-1) + a_2(1+i)^(-2) + ... + a_(n-1)(1+i)^(-n+1) + a_n(1+i)^(-n). Which can be written like ∑ a_p (1+i)^(-p) from p=1 to n
• For example, calculate the actual and acquired value of a series of 7 annual payments made at the end of the period. The payments were are of 1000 DH, 800 DH,900DH, 1200DH, 1000DH,700DH, and 600DH with an interest rate of 8%.
• Value Acquired with payments at the beginning of the period is now calculated with Vn = a_n(1+i) + a_(n-1)(1+i)^2 + ... + a_2(1+i)^(n-1) + a_1(1+i)^n
• Vo = ∑ a_p (1+i)^(n-p+1) where P is from 1 to n
• Value Acquired with the actual value is now calculated with = a_1 + a_2(1+i)^(-1) +...+ a_(n-1)(1+i)^(-n+2) + a_n(1+i)^(-n+1).
• Vo = ∑ a_p (1+i)^(-p+1) where P is from 1 to n
• An example will look like one calculating the actual and acquired values of a series of 6 annual investments made at the beginning of the periods with payments of resp 1200DH, 1000DH, 900DH, 1000DH, 1500DH, and 1000DH and having a total interest rate of 7%.

Individual Loans:

• Individual loans are defined as a loan agreement between a single lender and a single borrower.
• These sorts of loans are contractually set to be fixed at the time of the agreement (amortization processes)
• To calculate them, you use the amount of the loan C_0
• You need the Total time of the load T and the Rate of the loan i.
• the repayment terms can be: repayment, constant amortization, and Constant Annuities.

Amortization Tables

• To build an amortization table, you need to find the capital which is C_0
• The fixed interest rate is i, their need to be a capital amortization and a length of the loan which is marked as T

Breakdown

• Period, Capital at the beginning of the period , Interest period( I=C.i), amortization rate, and annuity at the end of the period ( a=I+m)
• C_0 = the Capital that remains at the beginning of the first year of the loan, that is equal to the amount
• I_p= interest of the pth period
• A_p is the Annuity of the pth period
• In the given table, the annuity term can be expressed with terms dealing with certain term such as Semesterly, monthly or trimestrial.
• Amortization is comprised of two things, the capital which is it plus the amortized capital m_t
• Its calulated using this equation a_t= i_t + m_t
• Interest paid at the end of each period are calculated against the starting nominal rate
• With I_t = C_(t-1)*i

Properties of a loan

• You need the remaining capital
• The remainign capital after the kth payments is the beginning rate less the kth levels of amortization
• C_0 - ∑ m_t where t=1
• Amortiation is used to repay the debts that are equal to the capital that is borrowed which is the sum total of m_p is equal to C_0
• the capital remainign is equal to zero therefore the debt not remibursed is = m_n and called C(n-1)= m_n

Additional Details:

• The amount of interest paid after the kth annumities is described with I_k= ∑ I_t where t=1 through k
• The cost of the loan is the sum of interest fees over the period which is I= ∑ I_t where t=1 through n
• The last depreciation and annuities ae interlinked through a formula with a_n = m_n+(C_(n-1)*i)

Loan Amortiation

• Loan Amortization is when a loan has set fees/fees for the end of the period
• "The repayment of the Capital is done at the end of the agreement, interest period is always set, the capital is the same during that period, and payments are always equal by the amount of it"

Details for one to create that table:

• Period, Capital at the beginning of the period , Interest period( I=C.i), amortization rate, and annuity at the end of the period ( a=I+m)
• Let us say how capital is 100,000 at a period of 4 years wiht at a period of 6%. You can calculate each period with these set parameters.

Amortieraztion by constants

• a loan who is rembursed witb constant amortization or sereis equals
• "Each period there is a constant capital payed, the capital decreases normally, annuities are the sum of the amortization and capital"

To create that amortization table we need this equation

• m= C_0/T
• let us say we have a capital of C0=100,000. We can deduce m

Annuity Reimbursement

• It is where a loan is reimbursed with constant annuities that comprises of interest rates and a fraction of amortized capital where C_i can be deducted through certain formulas