Class Notes And Derivation Of Tvm Equations 6j565d

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Time Value of Money Class Notes and derivation of TVM equations This material covers types of cash flows, their characteristics, their valuation procedure and derivation of some of the elementary equations which form basis for the values given in the annuity tables. Asymmetric time of Cash flow arrivals differ in their value due to Inflation and Opportunity Cost. A way of making cash flows (occurring at different points in time) comparable is to adjust their value using the risk free interest rate. This is denoted by r, and it is normally approximated by rate of return on a government backed treasury instrument. This rate normally s for opportunity costs and includes within itself the inflation observed over the period. A timeline is a visual depiction of occurrence of CF over the investment/financing horizon. Since time has only two dimensions, the value of CF occurring on the timeline has to be adjusted in either of these two directions in order to compare it with other cash flows of similar risk class. If the present cash flow has to be compared with a cash flow in future it can be achieved in either of the following ways, 1. Compounding of the present CF to FV 2. Discounting of the future CF to PV 3. Compounding of the present CF and discounting of future CF to a common point in time. THE CONCEPT OF TIME VALUE AND MECHANISM OF DISCOUNTING AND COMPOUNDING MAKES CASH FLOW A TIME TRAVELER WHERE YOU CAN TAKE THEM TO ANY TIME AND COMPARE, CETERIS PARABUS. The basic equation of discounting entails C/(1+r)^T And the basic equation for compounding is given as C*(1+r)^T Where C is the cash flow at time T, r is the risk free rate of return, T is the unit of time period. Using this basic equation different cash flows could be discounted and compounded. In order to simplify the time value analysis of complex financial opportunities, some procedures have been developed to quicken the discounting/compounding procedure by recognizing basic types of cash flows. Different types of Cash flows could be described through the following chart depicting their characteristics and application of different PV and FV table’s values. In this chart g represents the per period growth factor in the cash flow where as r & k and n & T have been used interchangeably respectively;

Class notes and derivation of TVM equations. Anand© 2011. All rights reserved.

2

Cash Flow (Observed/Expected)

Single CF

Multiple CF

Future Value

Present Value

Fluctuating CF (Finite without growth...why?)

Constant C.F. =C*FVFk,n

=C*PVFk,n

Finite CF series: Growing Annuity

Future Value

Present Value

=c*((1+r)^n-(1+g)^n)/(rg)

=c/(r-g)*(1((1+g)/(1+r))^n)

Class notes and derivation of TVM equations.

Future Value...Only if you know CF or their patterns in blocks/phases.

No growth serial growth factor/Static

With Serial growth factor

Infinite CF Series: Growing Perpetuity

Future Value... Indeterminable...why?

Infinite Series: Perpetuity

Finite CF Series: Annuity

Present Value

Future Value

Present Value

Present Value

=C/(r-g)

=C*FVIFAk,n

=C*PVIFAk,n

=c/r

Anand© 2011. All rights reserved.

Present Value...Only if you know CF or their patterns in blocks/phases.

Future value... How?..in a future time before infinity.

3 The annuity tables give us values of PVIFA and FVIFA for a series of combinations of interest rates and periods (k,n). An important thing to while applying those table values for PVIFA and FVIFA to different problem sets is that they are calculated assuming an Ordinary Annuity of the following type (Salary, Interest on bond, Cash Flow at the end of the period):

Time

0

CF

1

2

3

c

c

c

…

n-1

n

c

c

Where the PV would be calculated at time zero and future value would be calculated at time n. But in case the problem has the following type of cash flow described as Annuity Due (Insurance , rent, Cash Flow at the beginning of the period), Time

0

1

2

3

CF

c

c

c

c

…

n-1

n

c

The above mentioned factors are required to be adjusted for one period of excess discounting (PVIFA)/ one period of under compounding (FVIFA) in the in the ordinary annuity compared to the Annuity Due. Therefore in case of application of PVIFA for getting Present value of Annuity Due, since the PVIFA value is discounted for an additional period (from 1 to 0) compared to ordinary annuity CF pattern, we would need to remove effect of that excess discounting by compounding PVIFA by one period. This can be easily achieved by multiplying PVIFA with (1+r). =c*PVIFAk,n*(1+r) In case of Future value of Annuity Due as well, the FVIFA value would be required to be compounded for one more period (from n-1 to n) by multiplying FVIFA with (1+r). This is because the ordinary annuity compounds’ CF only until the last CF where as in case of Annuity Due we need to compound the CF until one period after the last CF. =c*FVIFAk,n*(1+r) TVM formula derivation Future value after 1 period

PV0*(1+r) = FV1

… (1)

Future Value after 2 periods

FV1*(1+r) = FV2

… (2)

Substituting (1) into (2) we have PV0*(1+r) *(1+r) = FV2 Or, PV0*(1+r)2= FV2

… (3)

PV0*(1+r)n= FVn

… (4)

Generalizing for n number of years we have Future value after n years of a single CF at rate r

Class notes and derivation of TVM equations. Anand© 2011. All rights reserved.

4 Dividing both sides by (1+r)n we have

PV0*(1+r)n/(1+r)n = FVn*1/(1+r)n

Present Value of a single CF after n years at rate r

PV0= FVn*1/(1+r)n

Future Value of Ordinary Annuity (FVA) FVAn=C*(1+r)n-1+ C*(1+r)n-2+ …+C*(1+r)+C

… (5)

… (6)

(Notice that the above series is compounding till n-1 periods only) Multiplying both sides by (1+r) we get FVAn*(1+r)=C*(1+r)n+ C*(1+r)n-1+ …+C*(1+r)2+C(1+r)

… (7)

Subtracting (6) from (7) we have

FVAnr=C*[(1+r)n-1]

… (8)

Dividing both the sides by r we get

FVAnr/r=C*{(1+r)n-1}/r

… (9)

Future Value of Annuity cash flows C for n years at rate r

FVAn=C*[{(1+r)n-1}/r]

… (10)

The second term on the RHS of the equation in also given as FVIAF for a given r and n. Present Value of Ordinary Annuity (PVA)

PVAn=C*(1+r)-1+ C*(1+r)-2+ …+C*(1+r)-n

… (11)

(Notice that the above series is discounting till 0 period) Multiplying both sides by (1+r) we get

PVAn(1+r)=C+ C*(1+r)-1+ …+C*(1+r)-n+1

… (12)

Subtracting (11) from (12) we get

PVAnr=C*{1-(1+r)-n}

… (13)

Or, PVAnr=C*{1-1/(1+r)n}

… (14)

PVAnr*(1/r)=C*{1-1/(1+r)n}*1/r

… (15)

Dividing both sides by r will give

Future Value of Annuity cash flows C for n years at rate r

PVAn=C*{1-1/(1+r)n}*1/r

… (16)

In some books (Eg. PC) an alternative derivation of similar nature could be found… taking from (13)… We can rewrite the equation as

PVAnr=C*[{(1+r)n-1}/(1+r)n]

…(14.1)

Dividing both sides by r will give

PVAn=C*[{(1+r)n-1}/r(1+r)n]

…(15.1)

PVAn=C*[{1-(1/(1+r)n}/r]

…(16.1)

In either case, the second term on the RHS of the equation (16) or (16.1) is given by the annuity table as PVIAF for a given r and n. If you need to calculate Present or future value of annuity due using PVIFA or FVIFA tables, you need to multiply it by (1+r) to adjust its value for one period. Present value of a perpetuity

PV0= C*(1+r)-1+ C*(1+r)-2+ C*(1+r)-3+ …

Class notes and derivation of TVM equations. Anand© 2011. All rights reserved.

… (17)

5 Multiplying both sides of (17) with (1+r) we get PV0*(1+r)= C+ C*(1+r)-1+ C*(1+r)-2+ …

… (18)

Subtracting (17) from (18) we get

PV0r= C

… (19)

Dividing both sides by r we get

PV0r*(1/r)= C*(1/r)

… (20)

Present value of a perpetuity at rate r

PV0= C/r

… (21)

……………………………………………………………………….. Suggestions to update/improve/correct the above document for any deficiency is welcomed and would be appreciated by the author. Anand © 2011. All rights reserved. Email: [email protected]

Class notes and derivation of TVM equations. Anand© 2011. All rights reserved.