Help on this IB interview question?

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G-izzo

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Hi, I came across this technical question in the book “running with the bulls” and had no idea how to solve it so I figured this was the best place to ask. Forgive me if this is not the right place
Q: Say you were a computer company and you jut bought $5mil in equipment. Would you rather pay $5mil today or $1.5mil over the next five years? With continuous compounding , is it less or more than twently percent?
The answer is apparently 16.6%.
I barely know how to solve for i for discrete compounding. How do you do it for an annuity when using continuous compounding?
Anyone?
Thanks
 
If I assume
1) PV=5
2) 5 times PMT=1.5 (END)
3) continuous compounding
I get i = 14.18%
 
Way i did it:
Set CFo to -5,000,000 and C01 to 1,500,000 with F01 (freq.) at 5.00. calculated IRR to be 15.24%
Then 2ND e^x 0.1524 minus 1 to get 0.1646 or 16.46
So basically, 16.46% is the rate of return you would require to be indifferent between paying today compared to paying 1.5 million within five years. From the looks of it, 16.46% means you pay more through PMT so i would choose paying 5 million today. (this is atleast what i would of done in the interview ;)… not sure if its 100% right).
 
This seems like a semi-continuous problem. You pay $1.5 million at 5 discrete points (t = 0, 1, 2, 3, 4) but you are continuously discounting.
You want to therefore solve:
5 = 1.5 * (1 + e^(-r) + e^(-2r) + e^(-3r) + e^(-4r))
I get r = 22.8564%. If payments are due at the end of the year, i = 14.1831%.
This problem definitely needs better wording to be clearly understood.
 
Antticfa wrote:So basically, 16.46% is the rate of return you would require to be indifferent between paying today compared to paying 1.5 million within five years. From the looks of it, 16.46% means you pay more through PMT so i would choose paying 5 million today. (this is atleast what i would of done in the interview ;)… not sure if its 100% right).
Click to expand...
You understand the definition of TVM, but then you completely ignore it when making a decision.
Let’s assume 16.46% is correct. If the actual discount rate were lower than 16.46%, the present value of the stream of payments would be higher than $5 million, and thus you’d want to pay $5 million today. If the discount rate were higher than 16.46%, the present value of the stream of payments would be lower than $5 million, and thus you’d want to pay over time.
 
thanks for the clarification aaron!
btw, Is there a way to calculate “5 = 1.5 * (1 + e^(-r) + e^(-2r) + e^(-3r) + e^(-4r))” using the TVM or CF functions in the BA II Plus calculator?
 
thanks for the clarification aaron!
btw, Is there a way to calculate “5 = 1.5 * (1 + e^(-r) + e^(-2r) + e^(-3r) + e^(-4r))” using the TVM or CF functions in the BA II Plus calculator?
 
There is no way direct way to solve that particular function with the calculator, since it would require something like Newton’s Method to solve (this is what the calculator does to solve for interest rates in the TVM/CF worksheets). The only way to do it manually is trial and error; I just pasted it into Wolfram Alpha.
However, since the function is an annuity that is just discounted continuously, you can solve for it with TVM and convert the interest rate. Note that it’s an annuity due as written (the first term is 1.5 * 1, so no discounting – payment is made at t=0).
PV = 5
PMT = -1.5
N = 5
FV = 0
Solve for i = 15.24
Now, to convert just use ln(1 + i) = ln(1.1524) which is approximately 14.18% as I suggested before.
 
Thanks alot guys I really appreciate it. I remember that I got pretty much the same answer as you guys. It’s probably a bad question like you guys said.
 
I also recommend practicing Antticfa’s method (using the CF function on BAII+). aaron’s works too, but its best ot know both methods.

Hit CF, then clear work (2nd CE|C)
CFo: -5,000,000 (enter) (cash outlay)
C01: 1,500,000 (enter) (first cash flow)
F01: 5 (enter) (frequency: how many periods for first cf)
Then hit IRR, CPT to get 15.2382371.
Hit “=” to leave the IRR calculator
Divide by 100 to get 0.1523…, the periodic rate
Since 1+i = e^r, r=ln(1+i) so
ln(0.1523…+1)=14.18%
as has been said, the question really needs to elaborate on due/ordinary.
 
And if it’s a due annuity, we get r=22.856% by simply changing CFo=-3,500,000 and F01=4. All that said, I’m not sure where the 16.6% comes from.
 
How can an annually counponded rate be greater than continously counponded? I can’t understand that!
 
Well 10% on a dollar once over a year will yield $1.10. If you semi-annually compounded the dollar to end at $1.10, you’d require a lower interest rate as your second period will be compounding on the principle PLUS the first periods compounded amount. Take that to infinity and you can see why continuous will be the lower rate.
 
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