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Compounding and Discounting -A Presentation to DVC Field Trip. Tony Wu PG&E 4/15/2008. Compounding and Discounting are used to calculate:. How much is one dollar now worth at t years later? How much is one dollar at t years later worth now?. Interest. The time value of money

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Compounding and discounting a presentation to dvc field trip

Compounding and Discounting-A Presentation to DVC Field Trip

Tony Wu

PG&E

4/15/2008


Compounding and discounting are used to calculate
Compounding and Discounting are used to calculate:

  • How much is one dollar now worth at t years later?

  • How much is one dollar at t years later worth now?


Interest
Interest

  • The time value of money

    • In your bank account, $1 now (the present value) becomes $1.06 one year later (the future value)

    • The interest is $0.06

    • The interest rate is $0.06/$1 = 6% per year

  • When do you pay interests?

    • Student loans, Credit Cards, Mortgage

  • When do you receive interests?

    • Saving accounts


Compound interest
Compound Interest

  • Present Value, V0

  • Interest rate, r per year, calculate the interest once per year

  • One year later

    • V1=V0 (1+r)

  • Two years later

    • V2=V1 (1+r) = V0 (1+r)^2

  • N years later

    • VN = V0 (1+r)^N


Compounding at various intervals
Compounding at Various Intervals

  • Present Value, V0; Nominal rate, r per year

  • One and two years later

    • If compounding quarterly

      • V1 = V0 [ 1 + (r/4) ]^4

      • V2 = V0 [ 1 + (r/4) ]^8

    • If compounding monthly

      • V1 = V0 [ 1 + (r/12) ]^12

      • V2 = V0 [ 1 + (r/12) ]^24

    • If compounding daily

      • V1 = V0 [ 1 + (r/365) ]^365

      • V2 = V0 [ 1 + (r/365) ]^730


Effective interest rate
Effective interest rate

  • Nominal rate, r per year

  • Compounding m times per year

  • Effective rate, r’ per year

    • 1 + r’ = [ 1 + (r / m) ] ^ m

  • Example, the Credit Card APR (annual percentage rate) is a nominal rate and compounds monthly

    • If the APR is 15%

    • Then the effective APR is

      • 1 + r’ = [ 1 + (15% / 12) ] ^ 12

      • r’ = 16.1%


  • Continuous compounding
    Continuous Compounding

    • Nominal rate, r per year

    • Compounding at infinite small time intervals

    • Effective rate, r’ per year


    Continuous compounding1
    Continuous Compounding

    • Present Value, V0

    • Nominal rate, r per year

    • One year later

      • V1 = V0 exp (r)

    • Two year later

      • V2 = V1 exp (r) = V0 exp (2r)

    • t year later (t is a real number)

      • Vt = V0 exp (rt)


    Discounting
    Discounting

    • How much is Vt dollar at t years later worth now?

    • Nominal rate, r per year

    • Discount factor, Dt, is the factor by which the value at t years later must be multiplied to obtain an equivalent present value

    • Compounding m periods per year

      • V0 = Vt [ 1 + (r / m) ] ^ (- m t) = Dt Vt

      • Dt = [ 1 + (r / m) ] ^ (- m t)

  • Continuous Compounding

    • V0 = Vt exp (- r t) = Dt Vt

    • Dt = exp (- r t)


  • Compounding and discounting with variable interest rate
    Compounding and Discounting with variable interest rate

    • Present Value, V0

    • Variable interest rate, r(u) at time u

    • Continuous compounding

    • Value at time t, Vt

    • Discount Factor


    Application of discounting
    Application of Discounting

    • Present Value of a future cash flow

      • At time t, t=1,2, … N; Payment Vt; Discount factor Dt; then the PV is

      • Could be used to compare two projects


    Unit pv and levelization calculation
    Unit PV and levelization calculation

    • At time t, t=1,2, … N; Payment Vt per unit; Volume at t, Ut unit; Discount factor Dt;

      • The Unit PV is

      • The levelized PV is


    Either pv is validated implicating different views of the value of project in the future
    Either PV is validated, implicating different views of the value of project in the future

    • A Project receives $1 million 1 year later, D1=0.75, and $3 million 2 years later, D2 = 0.5

      • If we use the unit PV method, then V0U=(1X0.75+3X0.5)/2=1.125 million

        • It means that the average unit present value is 1.125 million

      • If we use the levelized PV method, then V0L=(1X0.75+3X0.5)/(0.75+0.5) = 1.8 million

        • It is equivalent to receive a flat future cash flow

        • V1 = 1.8 million

        • V2 = 1.8 million


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