Modeling Notes on TVM

Cell Reference (Cell Address)  Excel formulas use cell reference, e.g. =B2+B3 –Goal is to create formulas that can be copied easily –Why? Saves time Reduces mistakes –Relative reference A1, B3, etc. Changes automatically when copied –Absolute reference $A1 –Column reference will not change when copied A$1 –Row reference will not change when copied $A$1 –Entire reference will not change when copied

Range Names  Range names –Same as $A$1  Excel Command –Insert Range Name Create Define –Very important when you need to change/delete a range name  Applications –Useful when the value of an input variable do not vary over time –Avoid assigning names to time varying parameters Since range name functions like $A$1, it does not change when copied

Input Variables for Computing FV of Single Cash Flow  Input –Single Cash Flow Today (PV) Does not change overtime, by definition –Discount Rate (r) Constant discount rate –Number of periods (T) Does not change overtime

FV of Single CF (Time Line Model)  Time line model –Current Time period (t) Increase by 1 in each period –cumulative balance and annual interest –General TVM formula: FV T = CF t * (1+r) (T-t) r is the constant discount rate –Absolute reference T is the ending period –Absolute reference t and cash flows change each time period –Relative reference  Output variable (decision variable) –Total FV

FV of Single CF (Time Line) Example

FV of Single Cash Flow (Formula and Excel Function Models)  Models (continued) –Single FV Formula model FV T = CF t * (1+r) (T-t) –Excel Function FV FV(r,t,PMT,PV) Inflow versus outflow assumptions –Use minus sign to change display if desired

PV of Single Cash Flow  Input –Same characteristics as FV of single CF  Model –Time line model Time period and Cash flows –Same characteristics as FV of single CF General TVM formula: PV 0 = FV t / (1+r) t –r is the constant discount rate –t and cash flows change each time period Relative reference –Single PV Formula model PV 0 = FV t / (1+r) t –Excel Function PV PV(r,t,PMT,FV)

Annuity Models  There are five variables in an annuity problem –PV: present value –FV: future value of an annuity –PMT: periodic payment –N: number of time periods –r: interest rate  PV and FV of annuities can be solved using any of the 3 methods: –The time line method using the basic TVM formula –The annuity formula –Excel TVM functions: PV() and FV()  PMT can be solved using –the annuity formula –Excel TVM function, PMT()  N and r need can only be solved using Excel TVM functions –Solution is obtained by trial-and-error (iterative method) –NPER() and RATE()

NPV Models  Input –Constant Discount Rate Enter once –Cash Flows Time varying  Model –Time line Similar to basic PV model –Excel Function NPV(discount rate, cash flow range) Excel assumes that cash flow starts in year 1 Excel NPV function = PV of CF1 + PV of CF2 +…+ PV of CF T Excel’s assumptions differ from standard textbook definition of NPV NPV = CF0 + PV of CF1 + PV of CF2 +…+ PV of CF T Textbook NPV = CF0 + Excel’s NPV function –Note: CF0 is usually negative, representing initial costs of a project

Constant versus Time Varying Interest Rate –Time constant (constant discount rate) Enter once Use absolute reference or range name –Time varying (general discount rate) (1+r 1_2 ) = (1+r 1 ) * (1+r 2 ) If r 1 = r 2, this will result in (1+r) 2 Many solutions –Remember: goal is to create formulas that can be easily copied