Money today is worth more than money tomorrow. This simple concept forms the basis of discounting, a crucial tool in finance and actuarial science. By understanding discounting, we can compare cash flows across time and make informed financial decisions.

Inflation, the silent eroder of purchasing power, adds another layer of complexity to financial planning. Actuaries must account for inflation when pricing insurance products, reserving for future liabilities, and valuing pensions to ensure long-term financial stability and fairness.

Time value of money

• Fundamental concept in finance recognizing that money available now is worth more than the same amount in the future due to its potential earning capacity
• Enables comparison of cash flows occurring at different times by discounting future values to their present value equivalents

Concept of discounting

• Process of determining the present value of a future sum of money or stream of cash flows given a specified rate of return
• Discounting adjusts for the time value of money to reflect opportunity cost and risk associated with future cash flows
• Discount rate represents the rate of return that could be earned on an investment in the financial markets with similar risk
• Formula for discounting a future value (FV) to its present value (PV) is: $PV = \frac{FV}{(1+r)^n}$, where $r$ is the discount rate per period and $n$ is the number of periods

Compound vs simple interest

• Simple interest calculated on the principal amount only, ignoring any interest accumulated in previous periods (e.g., $Interest = Principal \times Rate \times Time$)
• Compound interest calculated on the principal and the interest accumulated in previous periods, leading to exponential growth (e.g., $FV = PV(1+r)^n$)
• Compounding can occur annually, semi-annually, quarterly, monthly, or continuously
• Compound interest results in higher future values compared to simple interest for the same initial principal, rate, and time

Nominal vs effective interest rates

• Nominal interest rate stated rate before adjusting for compounding frequency (e.g., 6% per annum compounded monthly)
• Effective interest rate (EIR) actual rate earned after accounting for compounding frequency, calculated as: $EIR = (1 + \frac{r}{m})^m - 1$, where $r$ is the nominal rate and $m$ is the number of compounding periods per year
• EIR allows for fair comparison of interest rates with different compounding frequencies
• Higher compounding frequency leads to a higher EIR for the same nominal rate

Inflation

• Sustained increase in the general price level of goods and services in an economy over time, resulting in a reduction of purchasing power per unit of currency
• Important consideration in long-term financial planning and actuarial valuations

Measuring inflation

• Inflation rate percentage change in a price index (such as CPI) over a given period, typically one year
• Calculated as: $Inflation Rate = \frac{CPI_1 - CPI_0}{CPI_0} \times 100%$, where $CPI_1$ is the price index at the end of the period and $CPI_0$ is the price index at the beginning of the period
• Historical inflation rates used to estimate future inflation for planning purposes
• Central banks often aim to maintain a stable, low inflation rate (e.g., 2% per annum) to promote economic stability

Consumer Price Index (CPI)

• Measures the average change in prices paid by urban consumers for a representative basket of goods and services
• Basket includes items such as food, housing, transportation, healthcare, and education, weighted by their importance in a typical consumer's budget
• CPI is the most widely used measure of inflation and is often used to adjust wages, pensions, and social security benefits
• Limitations of CPI include the fixed basket not reflecting changing consumer preferences, and the difficulty in accounting for quality improvements in goods and services

Real vs nominal values

• Nominal value expressed in current dollars without adjusting for inflation (e.g., nominal wages, nominal interest rates)
• Real value adjusted for inflation to reflect the actual purchasing power (e.g., real wages, real interest rates)
• Real value calculated by dividing the nominal value by a price index (such as CPI) and multiplying by 100
• Importance of distinguishing between real and nominal values when making long-term financial decisions or comparisons across different time periods

Discounted cash flow (DCF)

• Valuation method used to estimate the present value of future cash flows by discounting them at a required rate of return
• Widely used in investment analysis, capital budgeting, and actuarial valuations

Present value (PV) of lump sums

• Discounting a single future cash flow to its equivalent value today
• Formula: $PV = \frac{FV}{(1+r)^n}$, where $FV$ is the future value, $r$ is the discount rate per period, and $n$ is the number of periods
• Allows for comparison of lump sum cash flows occurring at different times
• Higher discount rates and longer time horizons result in lower present values

Present value of annuities

• Annuity series of equal cash flows occurring at regular intervals for a fixed period
• Present value of an ordinary annuity (payments occur at the end of each period) calculated using the formula: $PV = PMT \times \frac{1 - (1+r)^{-n}}{r}$, where $PMT$ is the periodic payment, $r$ is the discount rate per period, and $n$ is the number of periods
• Present value of an annuity due (payments occur at the beginning of each period) is the PV of an ordinary annuity multiplied by $(1+r)$
• Annuities commonly encountered in actuarial applications include insurance premiums, pension payments, and bond coupon payments

Net present value (NPV)

• Difference between the present value of cash inflows and the present value of cash outflows over a project's life
• Formula: $NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$, where $CF_t$ is the cash flow at time $t$, $r$ is the discount rate per period, and $n$ is the total number of periods
• NPV used to evaluate the profitability of investment projects, with a positive NPV indicating a profitable project
• Limitations of NPV include the sensitivity to the choice of discount rate and the assumption that future cash flows are known with certainty

Internal rate of return (IRR)

• Discount rate that makes the net present value of a project's cash flows equal to zero
• Calculated by solving the equation: $0 = \sum_{t=0}^{n} \frac{CF_t}{(1+IRR)^t}$, where $CF_t$ is the cash flow at time $t$ and $n$ is the total number of periods
• IRR used to compare the profitability of different investment projects, with a higher IRR indicating a more attractive project
• Limitations of IRR include the possibility of multiple solutions for non-conventional cash flows and the assumption that cash flows can be reinvested at the IRR

Inflation-adjusted cash flows

• Incorporation of inflation into the analysis of future cash flows to account for the erosion of purchasing power over time
• Essential for long-term financial planning and actuarial valuations

Real vs nominal cash flows

• Nominal cash flows expressed in current dollars without adjusting for inflation
• Real cash flows adjusted for inflation to reflect constant purchasing power
• Real cash flow calculated by dividing the nominal cash flow by a price index (such as CPI) and multiplying by 100
• Importance of using real cash flows when evaluating long-term projects or comparing cash flows across different time periods

Constant vs current dollars

• Constant dollars cash flows expressed in terms of the purchasing power of a particular base year (e.g., 2021 dollars)
• Current dollars cash flows expressed in terms of the actual dollar amounts at each future time period, incorporating the effects of inflation
• Converting current dollars to constant dollars involves dividing the current dollar amount by the cumulative inflation factor since the base year
• Constant dollars allow for easier comparison of cash flows across different time periods by removing the distorting effects of inflation

Inflation premium in discount rates

• Discount rates used in DCF analysis can be expressed in nominal or real terms
• Nominal discount rate includes an inflation premium to compensate investors for the erosion of purchasing power over time
• Real discount rate removes the inflation premium to reflect the true return after adjusting for inflation
• Relationship between nominal and real discount rates given by the Fisher equation: $(1 + nominal rate) = (1 + real rate) \times (1 + inflation rate)$
• Consistency between the type of cash flows (nominal or real) and the type of discount rate used is crucial for accurate DCF analysis

Actuarial applications

• Discounting and inflation adjustments are fundamental concepts in various actuarial applications involving long-term cash flows and valuations

Pricing insurance products

• Insurers use discounted cash flow techniques to determine the present value of future claims and expenses when pricing insurance products
• Discount rates reflect the insurer's investment returns and risk profile
• Inflation assumptions incorporated to account for expected increases in claims costs over time
• Accurate pricing ensures the insurer's long-term financial stability and ability to meet policyholder obligations

Reserving for future liabilities

• Actuaries estimate the present value of future claim payments and expenses to determine the appropriate level of reserves an insurer should hold
• Discounting used to reflect the time value of money, as reserves are invested and earn returns until needed to pay claims
• Inflation adjustments made to account for expected increases in claim costs over time
• Adequate reserving is critical for the insurer's solvency and ability to meet future obligations

Pension valuation

• Actuaries use discounted cash flow analysis to estimate the present value of future pension obligations
• Discount rates based on high-quality corporate bond yields or government bond yields, reflecting the long-term nature of pension liabilities
• Inflation assumptions incorporated to account for expected increases in salaries and cost of living adjustments for pension benefits
• Accurate pension valuation is essential for determining the required contributions and ensuring the long-term sustainability of the pension plan

Adjusting for inflation in contracts

• Many long-term contracts, such as insurance policies or leases, include provisions for inflation adjustments to maintain the real value of payments over time
• Actuaries help design and price these contracts by incorporating appropriate inflation assumptions and adjustment mechanisms
• Common methods include indexing payments to a specific inflation measure (e.g., CPI) or applying a fixed annual increase rate
• Inflation adjustments protect the interests of both parties to the contract and ensure the long-term fairness and sustainability of the arrangement