Zerbe and Bellas, Chapter 9

Discounting

Why discount the future?

  1. Inflation
  2. Investment
  3. Uncertainty about future
  4. Impatience
  5. Time pattern of benefits and costs

Projects with Lives of One Year

  1. Future Value Analysis (compounding): FV = X(1 + r)
  2. Present Value Analysis (discounting): PV = Y/(1 + r)
  3. Discounting is the opposite of compounding: PV = FV/(1 + r)
  4. NPV = PV(benefits) – PV(costs)

Multiple Years

  1. Future Value over Multiple Years: FV= X(1 + r)t
  2. Present Value over Multiple Years: PV = Y/(1 + r)t 
  3. The present value for a stream of values is: PV=ΣY/(1 + r)t 
  4. Net Present Value of a Project: PV=ΣNB/(1 + r)t

Short cut formulas

  1. Present value of an annuity of $A for T years at r percent: PV = A((1-(1 + r)-T)/r)
  2. Present value of a perpetuity (i.e., an annuity that continues indefinitely): PV = A/i
  3. Rule of 72: # of years for a value to double = 72/i

Real interest rates

  1. Loanable funds theory
  2. Fisher equation: i = r + m + r*m, (the size of r*m falls as r, m falls)
  3. Consumer price index: http://www.bls.gov (problems: commodity substitution effect, new goods, quality improvements, discount stores)
  4. GDP deflator: http://www.bea.gov
  5. Use real dollars and r instead of i in NPV formulas
  6. Estimates of expected inflation: TIPs, forecasts (Livingston Survey)
  7. Exercise: use data from the Federal Reserve and the BLS to find the real Aaa corporate bond interest rate using CPI

More on interest rates

  1. Current interest rates
  2. Liquidity preference: yield curve
  3. Default risk
  4. After tax rate: i* = i(1 - tax rate)

What discount rate to use?

  1. Social rate of time preference
  2. Opportunity cost approach:
    -- federal opportunity cost of capital
    -- private opportunity cost of capital (real, pre-tax rate of return on corporate investments
  3. CBO
  4. OMB

Long-lived projects

  1. Differential discounting: r = f(t)
  2. Ramsey equation: SDR = p + ge, where p = the pure rate of time preference, g = the growth in per capita consumption, e = the absolute value of the rate at which the marginal value of consumption declines as per capita consumption increases
  3. the long-term discount rate should be equal to the growth rate, at least
  4. discounting example.xls

Alternative decision rules

  1. Benefit-cost ratio: BCR = ΣB/(1 + r)t / ΣC/(1 + r)t
  2. Internal rate of return: ΣNB/(1 + IRR)t = 0
  3. Payback period
  4. See page 234 ...

 

Comparing Projects with Different Time Frames

  1. If project A spans n times the number of years as project B, then assume that project B is repeated n times and compare the NPV of n repeated project Bs to the NPV of (one) project A.
  2. For example, if project A lasts 30 years and project B lasts 15 years, compare the NPV of project A to the NPV of 2 back-to-back project B’s, where the latter is computed: NPV = x + x/(1+i)15, where, x = NPV of one 15-year project B.