class notes and derivation of tvm equations

8/10/2019 Class Notes and Derivation of TVM Equations

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

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 valueusing the risk free interest rate. This is denoted by r, and it is normally approximated by rate of return ona government backed treasury instrument. This rate normally accounts for opportunity costs andincludes 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 ineither 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 thefollowing ways,

1. Compounding of the present CF to FV2. Discounting of the future CF to PV3. 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 CASHFLOW 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 havebeen developed to quicken the discounting/compounding procedure by recognizing basic types of cashflows. Different types of Cash flows could be described through the following chart depicting theircharacteristics and applica tion of different PV and FV tables values . In this chart g represents the perperiod growth factor in the cash flow where as r & k and n & T have been used interchangeablyrespectively;


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

Single CF

Future Value

=C*FVFk,n

Present Value

=C*PVFk,n

Cash Flow(Observed/Expected)

Multiple CF

Constant C.F.

With Serial growth factor

Finite CF series: GrowingAnnuity

Future Value

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

Present Value

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

Infinite CF Series:Growing Perpetuity

Future Value...Indeterminable...why? Present Value

=C/(r-g)

No growth serial growthfactor/Static

Finite CF Series: Annuity

Future Value

=C*FVIFAk,n

Present Value

=C*PVIFAk,n

Infinite Series:Perpetuity

Present Value

=c/r

Future value... How?..ina future time before

infinity.

Fluctuating CF (Fiwithout growth...wh

Future Value...Only ifyou know CF or their

patterns inblocks/phases.

Present Value...Oyou know CF o

patterns inblocks/phas


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The annuity tables give us values of PVIFA and FVIFA for a series of combinations of interest rates andperiods (k,n). An important thing to remember while applying those table values for PVIFA and FVIFA todifferent 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 1 2 3 n-1 n

CF c c c 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 (Insurancepremium, rent, Cash Flow at the beginning of the period),

Time 0 1 2 3 n-1 n

CF c c c c 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 valueis discounted for an additional period (from 1 to 0) compared to ordinary annuity CF pattern, we wouldneed to remove effect of that excess discounting by compounding PVIFA by one period. This can beeasily 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 forone more period (from n-1 to n) by multiplying FVIFA with (1+r). This is because the ordinary annuitycompounds CF only until the last CF where as in case of Annuity Due we need to compound the CF untilone period after the last CF. =c*FVIFAk,n*(1+r)

TVM formula derivation

Future value after 1 period PV 0*(1+r) = FV1 (1)

Future Value after 2 periods FV 1*(1+r) = FV2 (2)

Substituting (1) into (2) we have

PV0*(1+r) *(1+r) = FV 2

Or, PV0*(1+r)2= FV2 (3)

Generalizing for n number of years we have

Future value after n years of a single CF at rate r PV0*(1+r)n= FVn (4)


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Dividing both sides by (1+r) n we have PV 0*(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 (5)

Future Value of Ordinary Annuity (FVA) FVA n=C*(1+r) n-1+ C*(1+r) n-2+ +C*(1+r)+C (6)

(Notice that the above series is compounding till n-1 periods only)

Multiplying both sides by (1+r) we get FVA n*(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 FVA nr=C*[(1+r)n-1] (8)

Dividing both the sides by r we get FVA nr/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) PVA n=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 PVA n(1+r)=C+ C*(1+r)-1+ +C*(1+r)-n+1 (12)

Subtracting (11) from (12) we get PVA nr=C*{1-(1+r)-n} (13)

Or, PVA nr=C*{1-1/(1+r)n} (14)

Dividing both sides by r will give PVA nr*(1/r)=C*{1-1/(1+r) n}*1/r (15)

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 PVA nr=C*[{(1+r)n-1}/(1+r) n] (14.1)

Dividing both sides by r will give PVA n=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 asPVIAF 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 tomultiply it by (1+r) to adjust its value for one period.

Present value of a perpetuity PV 0= C*(1+r)-1+ C*(1+r) -2+ C*(1+r) -3+ (17)


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Multiplying both sides of (17) with (1+r) we get PV 0*(1+r)= C+ C*(1+r)-1+ C*(1+r) -2+ (18)

Subtracting (17) from (18) we get PV 0r= C (19)

Dividing both sides by r we get PV 0r*(1/r)= C*(1/r) (20)

Present value of a perpetuity at rate r PV0= C/r (21)

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Suggestions to update/improve/correct the above document for any deficiency is welcomed and wouldbe appreciated by the author. Anand 2011. All rights reserved.

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class notes and derivation of tvm equations

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