Investment under uncertainty, widget example

cdsharm75 · October 8th, 2023, 1:01 am

Folks, have gotten stuck with a rather simple example in this book (Investment under uncertainty, Dixit and Pindyck).
I found some slides online with a similar example, so will use that to ask the question.

slide location: https://web.mit.edu/rpindyck/www/Course ... Slides.pdf

My confusion stems from the calculation of the asset value in the last slide attached - "Analogy to financial options ". In all the slides, please note the time index used for summation.

First two slides - make sense to me.

: first two slides

The summation starts at t=0 for the standard NPV rule and, as expected, at t=1 for the "waiting value".

However, in the following slide, I'm not sure why the time index includes 0 for a value that's being calculated "next year" (or at time t=1; shouldn't the indexing start at t=1)?

1) Why does the value labeled as V1 (meaning calc. at time t=1) include the price from time t = 0?
2) Why is the price at time t=0 not the original 100 (refer to the price tree from the first slide)?

My suspicion is that the author means that the entire price tree itself shifts so that "time 0 next year" starts with a price of 150. In that case...V1 doesn't literally refer to the time index....it just means value in year 1 when the entire price profile for time [0, infinity] = 150 (or 50).....but I'm not completely sure.

Would appreciate some guidance, particularly from folks who are familiar with the book. Thanks!

Alan · October 8th, 2023, 4:06 pm

To value an American-style option you start by imagining you are at expiration and value the payoff, at that time -- conditional on the underlying security price. Here 'expiration' means 'next year' (in the language of the first slide). So, when you are in the next year, cash flows for that year are indexed with t=0 and not discounted because they are 'immediate'. That is what is going on in the 2nd slides.

Then, you work your way backwards in time, until you reach "today" -- which is "now" in the language of the first slides. Presumably that is what comes next (I am not familiar with the book).

If you are unfamiliar with it, you might want to read up on the valuation algorithm for American call options in a binomial model, for example. Then compare ...

cdsharm75 · October 8th, 2023, 6:31 pm

Thanks Alan. He's talking about European options btw. I've found the first chapter online....I'll add that as well for clarity (not always possible to get full context from just the slides). Appreciate your time!

cdsharm75 · October 8th, 2023, 6:36 pm

Page 18-20 has a discussion of the original example (widget factory, Real options) and analogy with the financial call option.

katastrofa · October 8th, 2023, 10:51 pm

Is this example explicitly about pricing options or rather about evaluating the opportunity cost of delaying the action (in which case you start from t=0, because you’re making the decision now)?

cdsharm75 · October 9th, 2023, 2:13 am

Thanks Katastrofa. The former; its applying options theory to capital projects; Real Options. t = 0 works fine in the first slide, when the decision has to be made today. Also, in the second slide of the first set, it makes sense to start the valuation at t= 1, as we're delaying by a year.

It's in the second slide set where I'm missing it. It says the value of the asset "next year" would involve the t index starting at zero.....and that's what's not making sense to me. It'll be easier to get the full context with the attached DixitPindyck1994.pdf attached.....that describes in more detail what's going on and would also clarify where the confusion is stemming from. Appreciate the time!

katastrofa · October 9th, 2023, 12:32 pm

IMHO, it’s pretty straightforward (perhaps apart from the time notation). The new price at t=1 is $150 and that’s what you get then without discounting, hence 150$/(1+r)^0 + 150$/r (discounted profit from the following years).

cdsharm75 · October 10th, 2023, 12:38 am

Thanks Katastrofa - but at t = 0, the price isn't 150........it's 100 right? So I'm thinking of it as 100/(1+r)^0 + 150/r. That's where I'm tripping up. Appreciate your time!

Marsden · October 10th, 2023, 1:11 am

Everything in the "V1 =" equation is as of time 1. There are a lot of sloppy things about this example, but the relevant bit of sloppiness here is that t is used both as the absolute time in the first slide and in all of the summations as the index. In the second slide "NPV = (.5) ..." equation, which is calculating a value at time 0, the summation index begins at 1 (the time 0 widget is accounted for in the earlier term), which agrees with the time of production of the widget of each particular term of the sum.

In the third and fourth slides, however, values V1 and F1 are taken at time 1, so the first term of the summations in each of the equations is the value at time 1 of the widget produced at time 1. However, the indices of the summations begin with t=0 ... which seems to be the source of your confusion.

I suggest that in each of the V1 and F1 equations, you have the indices of the summations begin with t=1 -- agreeing with the time of production of the widget of each term of the sum -- and then just use "t-1" in place of "t" in the denominator of the summed terms.

Does that make sense?

katastrofa · October 10th, 2023, 9:46 am

Thanks Katastrofa - but at t = 0, the price isn't 150........it's 100 right? So I'm thinking of it as 100/(1+r)^0 + 150/r. That's where I'm tripping up. Appreciate your time!

You’re in the scenario in which you didn’t invest immediately, but waited a year to see if the price goes up or down - and invested when it was $160. So your annual cash flows start to count from then - on the first year you get $150 without discount.

cdsharm75 · October 12th, 2023, 12:48 am

@Marsden and @Katastrofa - folks, truly appreciate your patience!

It's super clear now - this is the worst case of brain fog I've had! I was reading it literally and getting stuck on the time index ; @Marsden...yes sir as you describe it - "in the third and fourth slides....the indices of the summation.....begin with t=0"... that was definitely the source of the confusion - it's not written out very clearly....and I should have taken a little more time to think through it. Good for now - thanks again!!

katastrofa · October 12th, 2023, 10:32 am

The path dependence can lead to the following kind of sequence o events. When the firm first arrives on the scene and contemplates investment the current profit is in the intermediate range between the two thresholds Therefore the firm decides to wait. Then profit rises past the upper threshold so the firm invests. Finally, profit falls back to its old intermediate level, but that does not take it down to the lower threshold where abandonment would occur. Thus the underlying cause (current profitability) has been restored to its old level, but its effect (investment) has not.

Similar effects have long been known in physics and other sciences. The most familiar example comes from electromagnetism. Take an iron bar an( loop an insulated wire around it. Pass an electric current through the wire the iron will become magnetized. Now switch the current off. The magnetism is not completely lost; some residual effect remains. The cause (the current was temporary, but it leaves a longer-lasting effect (the magnetized bar).

Brain fog? My brain is evaporating!

cdsharm75 · October 13th, 2023, 11:06 pm

@katastrofa - LOL isn't the easiest read.....I'm going to take a break from the book for a little bit:)

Investment under uncertainty, widget example

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