Compounding and Discounting -A Presentation to DVC Field Trip Tony Wu PG&E4/15/2008

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 –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 Present Value, V 0 Interest rate, r per year, calculate the interest once per year One year later –V 1 =V 0 (1+r) Two years later –V 2 =V 1 (1+r) = V 0 (1+r)^2 N years later –V N = V 0 (1+r)^N

Compounding at Various Intervals Present Value, V 0 ; Nominal rate, r per year One and two years later –If compounding quarterly V 1 = V 0 [ 1 + (r/4) ]^4 V 2 = V 0 [ 1 + (r/4) ]^8 –If compounding monthly V 1 = V 0 [ 1 + (r/12) ]^12 V 2 = V 0 [ 1 + (r/12) ]^24 –If compounding daily V 1 = V 0 [ 1 + (r/365) ]^365 V 2 = V 0 [ 1 + (r/365) ]^730

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 Nominal rate, r per year Compounding at infinite small time intervals Effective rate, r’ per year

Continuous Compounding Present Value, V 0 Nominal rate, r per year One year later –V 1 = V 0 exp (r) Two year later –V 2 = V 1 exp (r) = V 0 exp (2r) t year later (t is a real number) –V t = V 0 exp (rt)

Discounting How much is V t dollar at t years later worth now? Nominal rate, r per year Discount factor, D t, 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 V 0 = V t [ 1 + (r / m) ] ^ (- m t) = D t V t D t = [ 1 + (r / m) ] ^ (- m t) Continuous Compounding V 0 = V t exp (- r t) = D t V t D t = exp (- r t)

Compounding and Discounting with variable interest rate Present Value, V 0 Variable interest rate, r(u) at time u Continuous compounding Value at time t, V t Discount Factor

Application of Discounting Present Value of a future cash flow –At time t, t=1,2, … N; Payment V t ; Discount factor D t ; then the PV is –Could be used to compare two projects

Unit PV and levelization calculation At time t, t=1,2, … N; Payment V t per unit; Volume at t, U t unit; Discount factor D t ; –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, D 1 =0.75, and $3 million 2 years later, D 2 = 0.5 –If we use the unit PV method, then V 0 U =(1X0.75+3X0.5)/2=1.125 million It means that the average unit present value is million –If we use the levelized PV method, then V 0 L =(1X0.75+3X0.5)/( ) = 1.8 million It is equivalent to receive a flat future cash flow V1 = 1.8 million V2 = 1.8 million