Time Value of Money: Concepts and Interest Rates

The Time Value of Money

• Individuals often save for the future or borrow for current needs, requiring calculation of investment amounts or borrowing costs.
• Investment analysts evaluate transactions with present and future cash flows, determining the worth of securities based on future cash flow streams.
• Understanding the mathematics behind time value of money problems is crucial for accurate task completion.
• Money possesses time value, with individuals valuing earlier receipt more highly.
• A smaller amount of money now can equate to a larger future amount.
• The time value of money involves relationships of equivalence between cash flows across different dates.
• Mastering associated concepts and techniques is vital for investment analysts.

Interest Rates

• Interest rates can be seen as required rates of investment return.
• Interest rates include discount rates
• Interest rates show opportunity costs
• An interest rate reflects the relationship between differently dated cash flows.
• Compensation for receiving money in the future is required, expressed as a rate of return.
• Opportunity cost is the value investors forgo by choosing a particular action.
• Interest rates are determined by supply and demand in the marketplace.
• Investor's view: interest rate (r) = real risk-free rate + premiums for distinct risks.
• r = Real risk-free interest rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium
• Real risk-free interest rate: single-period rate for a risk-free security with no expected inflation.
• This rate reflects individual preferences for current versus future real consumption.
• The Inflation premium compensates for expected inflation and reflects the average inflation rate overthe maturity of the debt.
• Inflation reduces currency's purchasing power.
• A nominal risk-free interest rate totals the real risk-free rate plus inflation premium.
• Many countries have governmental short-term debt offering a window into the nominal risk-free rate.
• A 90-day US Treasury bill (T-bill) interest rate represents the nominal risk-free interest rate
• US T-bills boast high liquidity, minimal transaction costs and are backed by the US government's credit.
• Default risk premium compensates investors for potential borrower default.
• Liquidity premium compensates for potential loss from converting investments to cash quickly.
• US T-bills carry no liquidity premium due to ease of sale.
• Maturity premium compensates for market value sensitivity to interest rate changes with debt extension.

Future Value of a Single Cash Flow

• Time value is associated with lump-sum investments.
• An amount invested today as present value (PV) that earns a rate of return converts to a future value (FV) after N periods.
• PV represents investment's present value
• FVN represents investment's future value N periods from today
• r denotes rate of interest per period
• With N = 1, future value is FV₁ = PV(1 + r)
• With annual compounding, interest is credited annually, becoming part of the investment base.
• Simple interest is the rate times the principal.
• Principal is the initially invested funds.
• Compounding is when interest earns interest, growing over time.
• Formula for compounding over multiple periods: FVN = PV(1 + r)N
• Key point: stated interest rate (r) and compounding periods (N) must align in time units.
• A time line aids in ensuring compatibility between time units and interest rate per period.
• Time indexes on the timeline show periods from today,
• Amount available for investment today is indexed as t = 0.
• Present value and future value are separated in time, and this has important consequences like we can only add money if indexed at the same time

Non-Annual Compounding (Future Value)

• Some investments pay interest more than once yearly, through monthly compounding.
• Rather than listing the periodic rate, financial institutions often quote a specified annual interest rate
• Formula with more than one compounding period yearly: FVN = PV (1 + rs/m)^mN
• r_s is the stated annual interest rate
• m is the number of compounding periods per year
• N is the number of years
• The periodic rate, r_s/m, and compounding periods, mN, must be compatible.

Continuous Compounding

• Discrete compounding credits interest after a discrete period.
• Continuous compounding means crediting after an infinite number of periods.
• Formula for the future value with continuous compounding: FVN = PVe^(rsN)
• e ≈ 2.7182818 is a transcendental number, and r_sN is the power
• The Effective annual rate (EAR) result is the EAR for an 8 percent stated annual interest rate with semi-annual compounding, the EAR is 8.16 percent

A Series of Cash Flows

• Annuity: A finite set of level sequential cash flows.
• Ordinary annuity: A first cash flow occurring one period from now (indexed at t = 1).
• Annuity due: A first cash flow occurring immediately (indexed at t = 0).
• Perpetuity: A perpetual annuity; never-ending sequential cash flows starting one period from now

Equal Cash Flows—Ordinary Annuity

• The future value of each $1,000 deposit can be determined.
• The arrows extend on the timeline from the payment date to t = 5.
• For instance, the first $1,000 deposit made at t = 1 will compound over four periods.
• Formula for the future value of an ordinary annuity

Unequal Cash Flows

• In many cases, cash flow streams are unequal, precluding the simple use of the future value annuity factor.
• Often an investor would have a savings plan which involves unequal cash payments depending on the month of the year, or lower savings during a planned vacation.
• One can always calculate the future value of a series of unequal cash flows by compounding the cash flows one at a time.
• The most direct approach to getting the future value at a certain time is to calculate the future value of all payments as of the time and then add the individual future values

Present Value of a Single Cash Flow

• The time-value enables discounting future value to present value
• Formula: PV = FVN(1 + r)^-N
• Present value factor, (1 + r)^-N, is the reciprocal of the future value factor, (1 + r)^N.

Non-Annual Compounding (Present Value)

• Interest is frequently paid semi-annually, quarterly, monthly, or even daily.
• The present value formula can be adjusted for interest paid more than once a year.
• r_s is the stated rate and corresponds to the periodic rate multiplied by compounding periods each year.
• The formula for PV is: PV = [FV_N] / [1 + (r_s/m)]^(mN)
• m = frequency of compounding periods per year
• r_s = the quoted annual interest rate
• N = number of years

Present Value of a Series of Equal and Unequal Cash Flows

• The present value, PV, of an ordinary annuity formula: PV = A/{[(1+r)^1] + [(1+r)^2] + [(1+r)^3] +…+ [(1+r)^N]}
• where A = the annuity amount
• r = the interest rate per period corresponding to the frequency of annuity payments
• N = number of annuity payments

Present Values Indexed at Times Other than t = 0

• Analysts frequently need to find present values indexed at times other than now.
• An annuity or perpetuity beginning sometime in the future can be expressed in present value terms one period prior to the first payment.
• That present value can then be discounted back to today's present value.

Solving for Interest Rates and Growth Rates

• Formula to solve for growth rates: g= [(FVN/PV)^1/N] − 1

Solving for the Number of Periods

• Equation to solve for the number of periods is: N = ln(FV/PV) / ln(1+r)

Present and Future Value Equivalence and the Additivity Principle

• Finding present and future values involves moving amounts of money to different points across the timeline.
• Operations can occur because values are corresponding measure are distributed as time progresses.
• The cash flow additivity principle involves indexed amounts at the same point in time, this is one of the most important concepts in time value of money
• This principle has been mentioned and applied previously
• A lump sum can actually generate an annuity if a lump sum is placed in the appropriate account that receives a stated rate of interest.
• Amortized loans are an example of how lump sums can generate annuities.