Time Value of Money & Interest Rates

The Time Value of Money

• Money has varying values at different times; present value is worth more than future value.
• Companies fund investments by raising capital and taking on liabilities, such as bank loans.
• A key principle is that the same amount of money has different values depending on the time.
• Preferences exist for immediate consumption over delayed consumption.
• Purchasing power decreases over time due to inflation.
• The present value is certain, but future cash inflows are less so.
• Inflation is a general increase in prices, eroding money's purchasing power.
• The nominal interest rate is 6% in the example given, while the real interest rate is effectively zero due to a matching inflation rate
• The real interest rate can be calculated with the formulas: 1 + real interest rate = (1 + nominal interest rate)/(1 + inflation rate) or real interest rate ≈ nominal interest rate - inflation rate (when rates are close to zero).

Effective Annual Interest Rates

• Interest rates are quoted for various intervals, so comparing them requires using the Effective Annual Rate (EAR).
• If $100 is borrowed at 1% monthly, the repayment after 12 months is $112.68, equating to an interest rate of 12.68%.
• Annual Percentage Rates (APRs) are helpful for comparing interest rates, but can be misleading if rates are simply annualized by multiplying the rate per period by the number of periods in a year.
• Effective rate = (1 + APR/m)^(m*t) - 1 = (1 + EAR)^t - 1
  • m is the compounding frequency within a year.
  • t is the period for which the interest rate is calculated, expressed in years
• You can calculate the 1 day, week, month, quarter, semi-annual, annual or every 2 years based on the rate, expressed in term of 1 year (t = 1/365, 1/52, 1/12, 1/4, 1/2, 1, 2)

Future Values and Compound Interest

• Investing $100 at 6% annually yields $6 interest in the first year, making the investment worth $106.
• This grows to $112.36 after two years.
• For n years, the investment grows to $100 × (1.06)^n.
• Compound interest earns interest on interest, unlike simple interest, which is calculated only on the original investment.
• Compounding determines the future value of a sum of money.
• Future Value (FVn) with compound interest is calculated as: FVn = C0(1 + i)^n
  • C0 is the initial investment.
  • i is the interest rate per period.
  • n is the number of periods.

Present Values

• Calculating present value involves discounting the future value at the interest rate over the number of periods.
• Discounting determines the present value of a future sum.
• Present values are calculated with compound interest.
• PV = Cn / (1 + i)^n
  • Cn is payment in the future
  • i is the interest rate per period.
  • n is the number of periods.
• With compound interest, future value increases over time while present values decline as future cash payments are delayed.

Multiple Cash Flows, Annuities, and Perpetuities

• Accumulating savings with 8% annual interest, putting aside $1,200 this year, $1,400 next year, and $1,000 one year later, results in a total cash amount at the end of three years.

• The value of an annuity can be determined by calculating the present value of each cash flow and summing them.

• For ordinary annuities, payments occur at the end of each period.

• PV0 of annuity due = PV0 of ordinary annuity × (1 + i)

• A growing ordinary annuity involves a series of payments at fixed intervals that increase: PV0 =( C/(i – g)) (1 − (1 + g)^n/(1 + i)^n)

• In contrast, growing annuity due payments are made at the beginning of each period.

• To calculate the future value of an annuity saving $3,000 at the end of each year with 8% interest, add up the future values of the four payments to be $13,518.

• FVn = Present value of annuity × (1 + i)^n

• Valuing cash flows requires an interest rate consistent with the payment frequency

• A perpetuity is a stream of equal payments at regular intervals that continues forever.

• An Ordinary Perpetuity is a perpetuity whose payments occur at the end of each period: PV=C/i

• A Perpetuity Due is a perpetuity whose payments occur at the beginning of each period: PV=(1+i)*C/i

• Growing ordinary perpetuity: A perpetuity can also take the form of a stream of payments growing at a constant rate that occurs at regular intervals in time and that continues forever.

• Growing perpetuity due: A perpetuity can also take the form of a stream of payments growing at a constant rate that occurs at regular intervals in time and the start of the period and that continues forever.

Notes on Time Value of Money

• The value of money is relative, measuring the goods and services it can buy.
• $1,000 today or 30 years ago had more value than $1,000 today.
• $1,000 today has more value than $1,000 twenty years from now.
• The timing of monetary flow affects its value; future flow's value is lower than present flow's value.
• Purchasing power decreases over time due to inflation.
• Present value is more certain than future cash inflow, which is subject to inflation and default risks.
• Lending money necessitates compensation; the interest rate is the cost per unit of capital and time.
• Requiring a 6% annual rate on a €1,000 deposit shows indifference between holding €1,000 today and receiving €1,060 in one year.
• Two types of interest rates exist with future cash flows: real (in consumption terms) and nominal (in monetary terms).
• Nominal interest rate and real interest rate can be approcimated as: Nominal interest rate ≈ real interest rate - inflation