📊Actuarial Mathematics Unit 3 – Financial Mathematics and Interest

Key Concepts and Definitions

• Interest represents the cost of borrowing money or the return on invested funds over a specified period
• Principal refers to the initial amount of money borrowed or invested before interest accrues
• Accumulation function $A(t)$ determines the future value of an investment or loan at time $t$
• Discount function $v(t)$ calculates the present value of a future amount at time $t$
  • Discount factor is the reciprocal of the accumulation factor, expressed as $v(t) = \frac{1}{A(t)}$
• Effective rate of interest $i$ measures the actual rate of return on an investment over a specific compounding period
• Nominal rate of interest $r$ represents the stated annual interest rate without considering the effect of compounding
• Force of interest $\delta(t)$ represents the instantaneous rate of interest at a specific point in time $t$

Time Value of Money

• Money has a time value due to its potential to earn interest over time
• A dollar today is worth more than a dollar in the future because of the opportunity cost of foregone interest
• Present value (PV) is the current worth of a future sum of money or stream of cash flows given a specified rate of return
  • Calculated by discounting the future value using the appropriate discount factor
• Future value (FV) represents the amount to which an investment or loan will grow over a specified period at a given interest rate
  • Determined by applying the accumulation function to the present value
• Compounding frequency affects the growth of an investment, with more frequent compounding leading to higher future values
• Discounting is the process of finding the present value of a future amount using the discount function

Simple Interest vs. Compound Interest

• Simple interest is calculated only on the principal amount, disregarding the effect of interest on interest
  • Formula for simple interest: $I = P \times r \times t$, where $I$ is interest, $P$ is principal, $r$ is annual interest rate, and $t$ is time in years
• Compound interest is calculated on both the principal and the accumulated interest from previous periods
  • Formula for compound interest: $A = P(1 + i)^n$, where $A$ is the future value, $P$ is principal, $i$ is the periodic interest rate, and $n$ is the number of compounding periods
• Compound interest leads to exponential growth due to the effect of interest on interest
• The difference between simple and compound interest becomes more significant as the time horizon and interest rate increase
• Continuous compounding assumes an infinite number of compounding periods, resulting in the highest possible future value for a given interest rate

Interest Rates and Conversion

• Interest rates can be expressed in various forms, such as effective rates, nominal rates, and force of interest
• Effective annual rate $i$ represents the actual rate of return earned over a year, considering the effect of compounding
  • Formula for effective annual rate: $i = (1 + \frac{r}{m})^m - 1$, where $r$ is the nominal annual rate and $m$ is the number of compounding periods per year
• Nominal annual rate $r$ is the stated annual interest rate, which does not account for the impact of compounding
  • Formula for nominal annual rate: $r = m[(1 + i)^{\frac{1}{m}} - 1]$, where $i$ is the effective annual rate and $m$ is the number of compounding periods per year
• Force of interest $\delta(t)$ measures the instantaneous rate of interest at a specific point in time
  • Related to the effective annual rate by the formula: $\delta = \ln(1 + i)$, where $i$ is the effective annual rate
• Interest rate conversions allow for the comparison and transformation of rates between different compounding frequencies
  • Converting between nominal and effective rates using the formulas mentioned above
  • Determining equivalent rates for different compounding periods (annual, semi-annual, quarterly, monthly, daily)

Present and Future Value Calculations

• Present value calculations determine the current worth of a future sum or series of cash flows
  • Formula for present value of a single amount: $PV = \frac{FV}{(1 + i)^n}$, where $FV$ is the future value, $i$ is the periodic interest rate, and $n$ is the number of periods
  • Formula for present value of an annuity: $PV = PMT \times \frac{1 - (1 + i)^{-n}}{i}$, where $PMT$ is the periodic payment, $i$ is the periodic interest rate, and $n$ is the number of periods
• Future value calculations determine the value of a current sum or series of cash flows at a future date
  • Formula for future value of a single amount: $FV = PV(1 + i)^n$, where $PV$ is the present value, $i$ is the periodic interest rate, and $n$ is the number of periods
  • Formula for future value of an annuity: $FV = PMT \times \frac{(1 + i)^n - 1}{i}$, where $PMT$ is the periodic payment, $i$ is the periodic interest rate, and $n$ is the number of periods
• Net present value (NPV) is used to evaluate the profitability of an investment by discounting all future cash flows to the present
  • Calculated by summing the present values of all cash inflows and outflows
• Internal rate of return (IRR) is the discount rate that makes the net present value of an investment equal to zero
  • Represents the expected rate of return on an investment
• Perpetuities are annuities that continue indefinitely, with the present value formula: $PV = \frac{PMT}{i}$, where $PMT$ is the periodic payment and $i$ is the periodic interest rate

Annuities and Cash Flows

• An annuity is a series of equal payments made at regular intervals for a specified period
• Ordinary annuity assumes payments occur at the end of each period
  • Present value of an ordinary annuity: $PV = PMT \times \frac{1 - (1 + i)^{-n}}{i}$
  • Future value of an ordinary annuity: $FV = PMT \times \frac{(1 + i)^n - 1}{i}$
• Annuity due assumes payments occur at the beginning of each period
  • Present value of an annuity due: $PV = PMT \times \frac{1 - (1 + i)^{-n}}{i} \times (1 + i)$
  • Future value of an annuity due: $FV = PMT \times \frac{(1 + i)^n - 1}{i} \times (1 + i)$
• Perpetuity is an annuity that continues indefinitely, with a fixed payment amount
  • Present value of a perpetuity: $PV = \frac{PMT}{i}$
• Uneven cash flows can be analyzed using present value and future value calculations for each individual cash flow
  • Net present value (NPV) is the sum of the present values of all cash inflows and outflows
  • Internal rate of return (IRR) is the discount rate that makes the NPV equal to zero
• Deferred annuities have a waiting period before the first payment is made
  • Calculated by discounting the present value of the annuity to the start of the deferral period

Loan Amortization

• Amortization is the process of gradually reducing a loan balance through a series of equal payments
• Each payment consists of an interest portion and a principal portion
  • Interest portion is calculated by multiplying the loan balance by the periodic interest rate
  • Principal portion is the remainder of the payment after subtracting the interest portion
• Amortization schedule shows the breakdown of each payment into interest and principal, as well as the remaining loan balance
• Sinking fund is a method of accumulating funds to repay a loan or meet a future obligation
  • Regular contributions are made to the sinking fund, which earns interest over time
• Amortization and sinking fund calculations involve the application of annuity formulas
  • Loan payment formula: $PMT = \frac{PV \times i}{1 - (1 + i)^{-n}}$, where $PV$ is the loan amount, $i$ is the periodic interest rate, and $n$ is the number of payments
  • Sinking fund formula: $PMT = \frac{FV \times i}{(1 + i)^n - 1}$, where $FV$ is the future value (loan amount), $i$ is the periodic interest rate, and $n$ is the number of contributions

Applications in Actuarial Practice

• Actuaries use financial mathematics principles to assess and manage risks in insurance, pensions, and other financial products
• Life insurance and annuity valuation involve the application of present value calculations
  • Pricing insurance premiums based on the expected present value of future benefits and expenses
  • Determining reserves required to meet future policy obligations
• Pension plan valuation and funding use actuarial assumptions and financial mathematics techniques
  • Calculating the present value of future pension benefits
  • Determining the required contributions to fund the pension plan over time
• Investment and asset management strategies employ the concepts of time value of money and risk-return analysis
  • Evaluating the performance of investment portfolios using measures like NPV and IRR
  • Assessing the impact of interest rates and inflation on investment returns
• Risk management and financial planning rely on the understanding of interest rates, annuities, and cash flow analysis
  • Analyzing the financial impact of different risk scenarios
  • Developing strategies to mitigate potential losses and optimize returns
• Actuarial models and simulations incorporate financial mathematics principles to project future outcomes and assess risk
  • Stochastic modeling techniques to generate a range of possible scenarios
  • Sensitivity analysis to determine the impact of changes in assumptions on financial results