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# Compounding and Discounting -A Presentation to DVC Field Trip - PowerPoint PPT Presentation

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

Tony Wu

PG&E

4/15/2008

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

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

• 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

• 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

• 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

• 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%

• Nominal rate, r per year

• Compounding at infinite small time intervals

• Effective rate, r’ per year

• 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)

• 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)

• Present Value, V0

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

• Continuous compounding

• Value at time t, Vt

• Discount Factor

• 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

• 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

• 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