Mathematical brainteaser by Investment Manager
I was given a verbal mathematical-centered quiz by my investment manager, who told me to think about it over the weekend and let him know my answer by Monday. The question is as follows:
'A CFO received a very sizeable investment opportunity that he is considering. In order for the investment to be a success, there are 2 things that MUST be achieved. It does not matter what those 2 things are, but for the sake of giving an example, he used a smartphone manufacturer: the first thing that has to be achieved is they have a design ready to go (lets call this x1); the second thing is they receive their chips from their suppliers (lets call this x2). The core issue here is timing and the time value of money. There are 2 scenarios:
Scenario 1: They only start x1 (design process) once they have received x2 (chips) in order to ensure that there is no loss of money / time / resources in case the supplier falls through and is unable to deliver the chips (or delivers them very late). Exit occurs in 5 years.
Scenario 2: They start the design process (entailing costs) in HOPES of receiving the chips on time, meaning x2 beings after x1. If this goes well, exit occurs in 5 years. There is no difference in returns or IRR or anything of the sort; the investment horizon and returns are the exact same. BUT, if the supplier is delayed, the investment horizon is extended by an additional 3 years (8 total), and additional costs are entailed during this period.
Which scenario would I choose?'
He noted that there is indeed a correct answer, which is mathematically driven. He also mentioned NPV, and that there is a formula that would (for scenario 2) essentially provide a cut-off time where it is no longer worth waiting for the the chips to be delivered; i.e. that the cost is outgrowing the profit that was forecasted.
Can anyone shed some light on this?! I would greatly appreciate it! Thanks
I think you need to assign probabilities to all three events (delivery, late delivery and no delivery) and then do an expected value for each scenario. Like say 50% for on-time, 40% for late delivery and 10% for no delivery. This would impact how long you're willing to wait. If the chance of late delivery was only 5% and no delivery 0%, logically you'd be willing to wait longer since the value from success would be higher.It's unclear to me what an extra three years would do. Does the salvage/exit value change if the project is extended? Is there even one to begin with or is it just project cash flows?Algebraically you're going to get an answer for the delay time in terms of the discount rate. I think you can make some assumptions like assuming nominal and consistent margins, constant yearly volume and upfront design costs to play around in excel so you can atleast visualize the numbers.Numbers aside, scenario 1 looks to be the best. No upfront risky CAPEX. Agree at a certain point if you wait long enough the value will be lower than the expected value of the gamble.I'm at a restaurant so can't bust out the pen and paper but I'm down to crack this with you by the end of today.
Worked a bit longer on this and came to some conclusions:
In scenario 1, the NPV for a late delivery is the same as the NPV for an early delivery just multiplied by a discount factor of the delay (1/(1+r)^d) where d is the delay time. The horizon is still 5 years just shifted forward since you're only drawing down capital when chips arrive.
In scenario 1, the NPV for no delivery is zero because you put no money down.
The only advantage of choosing scenario 2 over scenario 1 is the added benefit of the 8 year horizon vs. 5 year horizon to the NPV. Since even when the chips are on time, between both scenarios the returns are the same, despite Scenario 2's headstart on development (this doesn't really make sense). So if there's no headstart benefit, the benefit must come from these 3 extra years that then must outweigh the loss on development if the chips are never received for it to be worth it. You mention additional costs for these three extra years but surely there will also be additonal revenues?
For scenario 2 the investment horizon begins as soon as development is started. You're not gauranteed 5 years of cash flows like in scenario 1 (assuming 0 days DIO) because if there's a 4 year delay, you're only getting 4 years of cash flows. You're money is at work Day 1 unlike scenario 1.
To take one variable out of the equation it would be helpful to know what the average delay is. It would also be helpful to know what the maximum development time is. Theoretically you could receive the chips before finishing development and vice-versa. But I think you can have these both at the same variable because you're solving for what delay time is the time you'd shut down development.
Using these assumptions and assuming a yearly profit of 'P' each year the project is running (after chips received and development finished), and a delay time of 'd', development costs of 'D' per year, along with the annuity formula you can solve this. I did it on a piece of paper it doesn't look pretty...
...but I still don't get how the returns are the same between both scenarios when the chips arrive on time since Scenario 2 has a headstart...
I'll give it a shot from an econ major's POV:
I think there's a few things missing here that in the real-world would significantly impact your decision-making. In scenario 1, the time it takes to develop the design is not specified. If there's any risk of the design taking a long time for example, it would surely be better to receive up-to-date chips after the design is completed so their value doesn't depreciate as technology improves.
If however we assume some time horizon T that it takes to develop in both scenario 1 and 2, and that technology doesn't become outdated in time T, and holding all else constant (ie; the nominal value of the revenue from the phone in both scenario is equal, just the time value varies), then it basically becomes a question of whether R&D is more expensive than the expected delay:
In scenario 1, suppose we incur the chip costs (C) at time 0. We expect to receive the chips at time t=E[Delay], and start the design process/incur the R&D cost (R) then. We will then have inventory ready to sell at time t=T+E[Delay]. If, for simplicity's sake, we assume we can clear all inventory at nominal price P (for total inventory) instantaneously once x1 and x2 are achieved, then the expected NPV of scenario 1 is: (-C)+(-R)/(1+r)^(E[Delay])+P/(1+r)^(T+E[Delay]).
Now for scenario 2: We incur R&D costs immediately, and after period T, we then incur chip costs which arrive in E[Delay] years. Following similar logic to above, we can see that the expected NPV in this scenario is thus: (-R)+(-C)/(1+r)^(T)+P/(1+r)^(T+E[Delay]).
The key difference between the two scenarios is the timing and order of the costs. If R+C/(1+r)^T>C+R/(1+r)^(E[Delay]), then scenario 1 is optimal in expectation. (Intuition being if the cost of paying for research now and chips later is higher than the cost of paying for chips now and research later, then it's better to delay research)
Actually, it might make more sense to assume that chip costs are incurred upon delivery, while R&D costs are incurred upfront. That doesn't really change the argument, just when the cash flows occur. I'll leave it as an exercise to the reader to solve for that scenario :)
Also its possible you meant that R&D and waiting for delivery can go on simultaneously in scenario 2. If that's the case, and assuming there's no lost revenue from inferior designs or whatever, its much better than waiting for chips then designing for a current period perspective since you can realize revenues (profits) sooner (assuming there's somewhat substantial margins to offset the earlier costs)
Not sure where the 5/8 year exit comes in. Tweak the assumptions for when revenue is realized, and maybe you can make an argument after plugging in delay values for which scenario would give you a higher exit value. (ie; in Scenario 2, even if chips aren't delivered, you can still sell IP and patent rights hopefully, while in scenario 1 you're just a dude waiting for chips to arrive with nothing to show for it)
I came up with something somewhat similar to this:
C0 = cost at time 0
AC = additional cost
Scenario 1 would be relatively straight forward: S1 = -C0 + (CF1 / (1+r) ^1)... + (CF5 / (1+r) ^5)
Scenario 2 factors in additional cost: S2 = -C0 - AD + (CF1 / (1+r) ^1)... + (CF8 / (1+r) ^8)
I would love to hear your opinion!
Aren’t your worried the interviewer peruses the forum once in awhile?
When I posted this, I had already come up with a formula (very similar to BasicUsername's; see my reply to his comment). In any case, it is not an interview question, but rather my investment manager trying to understand my train of thought. I very much enjoy these theoretical exercises; I was asking predominantly to see what other people's solving would be.
FWIW if I were the CFO I'd make a DCF with following option/scenario matrix:
Assume cashflows:
Then probability weight Scenarios A, B, C, run and NPV at x discount rate for each Scen, and have prob weighted NPV for Option 1 and Option 2.
Now just get some clarification on my key inputs around disc rate, 4 cash flows, and delay. If NPV of 1a=2a and you have a discount rate, only need either the cash flows or the delay to goal seek the other variable.
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