Additive property of NPV rule

One of the important attractions of NPV rule is its additive property. It can be stated as

NPV(X+Y)=NPV(X)+NPV(Y)

Let us look at a one period case of cash flow stream X. A person invests X0 now and expects to receive X1 one period later. The NPV of this investment would be: NPV=-X0+PVX1

Under equilibrium, the market forces would ensure that the NPV is zero. Hence, PVX1=X0. We know that, r=(X1-X0)/X0

The Capital Asset Pricing Model states that: r=rf+β.(rm-rf)

As X0 is known with certainty, we can rewrite β as {covariance(X1,rm)/Variance(rm)}/X0. Defining {covariance(X1,rm)/Variance(rm)} as λ(X1),

r=rf+ λ(X1).(rm-rf)/X0

PVX1=X1/(1+r)

PVX1=X1/(1+rf+ λ(X1)/X0)
That is,

PVX1. (1+rf+ λ(X1)/X0)=X1

Noting that PV=X0, we can simplify this expression as,

PVX1={X1- λ(X1)}/(1+rf) ……….. (1)

NPVX1=PVX1-X0={X1- λ(X1)}/(1+rf)-X0

Let us now look at two different cash flow streams, X and Y.

NPV(X)= ={X1- λ(X1)}/(1+rf)-X0

NPV(Y)= ={Y1- λ(Y1)}/(1+rf)-Y0

It is evident that NPV(X+Y)=NPV(X)+NPV(Y)

If you recall, we tried to capture the risk premium using σ as the measure of risk in one of our earlier sessions. Once you learn the Capital Asset Pricing Model, you would know that the measure of risk is not σ but β. Now look at expression (1).

PVX1={X1- λ(X1)}/(1+rf)

There are two components on the RHS:
{X1}/(1+rf)

{ λ(X1)}/(1+rf)

The second term is the adjustment for risk. The expected cash flow is reduced to account for risk.

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[ 1 ]. You will learn this later in the course. We have dropped the expectation operator for convenience of writing. r=return from the investment; rm=return from the market; rf=risk-free rate; β=covariance(r,rm)/variance(rm).
[ 2 ]. If you recall, we derived it as