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: FVT = CFt * (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

Interest 10%

Year Account Interest Total inbalance earned account

beg. year during year end of year0 1,000.00 100.00 1,100.00 <-- =C8+B81 1,100.00 110.00 1,210.002 1,210.00 121.00 1,331.003 1,331.00 133.10 1,464.104 1,464.10 146.41 1,610.515 1,610.51 161.05 1,771.566 1,771.56 177.16 1,948.727 1,948.72 194.87 2,143.598 2,143.59 214.36 2,357.959 2,357.95 235.79 2,593.74

10 2,593.74=D8

=$B$3*B8

FV of Single Cash Flow (Formula and Excel Function Models)

Models (continued)– Single FV Formula model

• FVT = CFt * (1+r) (T-t)

– Excel Function FV• FV(r,t,PMT,PV)

• Inflow versus outflow assumptions

– Use minus sign to change display if desired

SIMPLE FUTURE VALUE

Input VariablesSingle CF today $1,000Interest Rate 10%Term 10 yearsFrequency 1 per year

FormulaFuture Value 2,593.74$

Excel FunctionFuture Value ($2,593.74)

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: PV0 = FVt / (1+r)t

– r is the constant discount rate– t and cash flows change each time period

• Relative reference

– Single PV Formula model• PV0 = FVt / (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 CFT

• Excel’s assumptions differ from standard textbook definition of NPV• NPV = CF0 + PV of CF1 + PV of CF2 +…+ PV of CFT

• 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+r1_2) = (1+r1) * (1+r2)

• If r1 = r2, this will result in (1+r)2

• Many solutions– Remember: goal is to create formulas that can be easily

copied