Discounting Future Cash Flows with Changing Rates and Risk

# Answer by Attack68 (score 2)


# Why in calculating value of an asset in a changing discount rate environment, a discount rate in time t, should impact future cashflows?












In the page 2 of this link (which is written by famous professor Damodaran), an equation is presented for calculating the value of an asset (2twh equation, alternative and more general version) which its risk and consequently its discount rate change over time. Although, I can understand the justification for changing discount rate over time, but I have hard time understanding why a changed discount rate in time t (rt), in addition of impacting present value of its current cashflows (E(CFt)) should impact future cashflows ( E(CFt+1), E(CFt+2) and etc). in the other words, I expect the correct equation should be something like this (My expected equation)

In which r1 and r2 and …are different. For an intuitive explanation, take an extreme situation in which risk and r1 is so great in t1 such that discounting every cashflow with it, result in a number near zero.(Imagine a firm that its cashflows in t come from a very risky project) and due to changing discount rate over time, rt+1 is something very low .( Imagine the firm in t+1, has sold its risky asset and has invested all its assets in free risk government bonds)? if we calculate the E(CFt+1) which has no risk, based on suggested Formula of the link, its calculated value get near zero; which, in my opinion, in the first glance, is something against intuition and logic.

Take another example:

Imagine there are two separate assets: A and B which asset A provides a very risky cashflow in a year from now(t+1) and asset B which provides a risk free cashflow two years from now(t+2). ِDo you accept that : asset B must be discounted with free risk and not with Hight discount of asset A in t+1 (take the discount of asset B very Hight such that discounting every cashflow with it, result in a number near zero)?

Do you accept that packing asset A and asset B is equivalent of the asset I have assumed in my example( a asset with very risky cashflow in t+1 and free risk cash flow in t+2)? if you accept these two proposition, the Damodaran formula must not hold, because sum of value of asset A and asset B must be more than zero. ( because it is assumed the discount rate of asset A is very high, if Damodaran formula be used the packed asset has zero value) In the other words, if we name the packed asset C, my expected equation implies Asset C value= Asset A value+ Asset B value And Domodaran equation implies Asset C value ≅ 0 ≠ Asset A value+ Asset B and value of asset B vanishes when combine with asset A.

Can you tell me why I am wrong and which aspects I have ignored?

Edit to response Kurt G comment: The discount rate, reflects two different things: risk free rate and risk premium (Take CAPM formula, for example). If risk free rate component (interest rate of bonds, inflation etc.) changed over time, there is no question that it impacts future cashflows. Assuming rf for risk free component and rp for risk premium. The more general form of my expected equation is In the other words, in the first version of my expected equation, for simplicity, I have implicitly assumed that the risk free rate is constant and it is rp (risk premium) that is changing over time. The Domodaran equation is saying rp1, in addition to impacting E(cf1) impacts E(cf2) ,…and E(cfn) and my expected equation says rp1 just impacts E(cf1). So my question, as my examples emphasize, relates to changing risk premium not overall interest rate.(regarding my previous example, in the case of changing risk free(interest rate) environment still Asset C value= Asset A value+ Asset B value)

## Answer by Attack68 (score 2)

https://quant.stackexchange.com/a/80678

We know that as of September 2024, the FED is cutting interest rates. The rate for this year is 5%, while the rate for next year is likely to be 4%.

What is the PV of a receivable 1\$ cashflow in a US account in 1 year. If rates this year, $r_{1y}$ are 5% then it is $\frac{1}{1+5\%}=\frac{1}{1+r_{1y}}$.

What is the PV of a receivable 1\$ in a US account in 2 years. If $r_{1y}$ and $r_{1y1y}$ are 5% and 4% respectively it is: $\frac{1}{(1+5\%)(1+4\%)}=\frac{1}{(1+r_{1y})(1+r_{1y1y})}$

There is nothing more complicated to this than rational expectation.

Why is the answer not $\frac{1}{(1+r_{1y1y})^2}$? Becuase that doesn't replicate the evolution of the money. It is simply an errorneous formula, in the same way that it also isn't $\frac{1}{(1+100)}$ or $sin(2.5\pi)$, which are also made up formulas.