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CBR, always, but specifically for mutually exclusive projects: Select the project with the highest NPV.
IRR is misleading as a capital budgeting rule, because if you have project A that offers an IRR of 100%, it is not necessary better than project B that offers an IRR of 50%. If you have only either A or B,
Ex: A project with no IRR: +$1, –$0.01, +$1,000 . Take this project regarless of the economy-wide cost of capital (interest rate).
Ex: A project with multiple IRRs: can happen if there are multiple sign changes — and Excel won’t tell you which one it has given you.
Be very suspicious: Don’t use IRR if multiple sign changes in $C$.
Advice: Calculate NPV instead.
IRR is only guaranteed to be “safe” for minors if there is one negative cash flow upfront, all positive cash flows thereafter, or vice-versa.
Because IRR is so common, you have to understand these problems.
Just because a project gives you back all your money sooner does not make it better. Which project would you rather have?
A: $100, $101, $2.
B: $100, $40, $40, $1,000,000.
What are the payback periods?
That was really all we covered for the payback rule.
Intuition:
Make up examples, as always.
Imagine inflation is 50\%. So, $100 today buys the same as $150 next year (as $225 the year after). Equivalently, imagine each Apple costs $1 today. So, it will cost $1.50 next year (and $2.25 thereafter).
Imagine that typical projects that cost $10 today deliver $18 next year. This is the nominal opportunity cost of capital in the economy. Orchards offer nominal rates of return of 80%.
Now we have our own project, an apple farm:
Real: Give up 100 apples today. Get 120 apples next year. Our real apple rate of return is 20a%. At 20% real, we end up even. \(NPV = -100a + 120a/(1+20a\%) = 0\)
Let’s translate this. 100 apples cost $100 today. 120 apples next year cost $180 next year.
Nominal: Give up $100 today. Get $180 next year. At 80% nominal, we end up even. \(NPV = -\$100 + \$180/(1+r_N) = 0\) At $r_N=80\%$, this is a 0 NPV project — which we already know it is.
Ergo: If the inflation rate is 50\%, then 80\% nominal is 20\% real.
(Fisher is) not an intuitive “1+” formula, but this is what it is.
If numbers are small, then $r+\pi\approx n$. Here, $20\% + 50\% \approx 80\%$…uggh, not so small.
On exams, other than this one-period “1+” simplest formula, I would not ask you to work inflation conversions in multi-period or risky settings.
Simple rule you must remember: If you use nominal dollars in the numerator (cash flows), you must use nominal rates of return (say, from Treasuries) at the bottom of the PV formula, too. See previous page.
Almost all financiers work with nominal rates all the time. (Of course, everyone really cares only about real returns. In high inflation countries, financiers may “index” contracts.)
Modeling “variable cash flows” is difficult. We have always taken cash flows as “given” to us. Modeling cash flows is usually beyond the domain of finance. Sure won’t be on exams.
Comparing projects: See above for mutually exclusive projects.
We didn’t cover this, except to say that managers often assume that in some long-run state, nominal cash flows of a business may tend to grow with inflation. (They use this mostly in terminal value growth rates.) They can then use nominal discount rates to discount nominal cash flows, presumed to be growing at the inflation rate.
PS: This presumption is not always true. Gold tends to increase more with inflation. Salaries often lag behind. Economics determines this. It’s beyond finance. More details won’t be on exams.
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With zero volatility, the arithmetic and geometric average rates of return are identical.
+1\% and –1\% (almost) leaves you unchanged. The bigger the delta, the higher the volatility, and the worse you end up being off.
Gaining 100\% and losing 100\% does not leave you with unchanged. Instead, it leaves you with nothing. $100 becomes $200 and then becomes $0 (at –100\%).
With higher volatility, your arithmetic average looks much better than your geometric average.
This matters greatly when you compare investing in bonds vs. stocks.
You typically care about geo returns.
This is important to understand.
This sucks…not just in this course, but in real life. Except for banks and simple loans, where EAR tends to be a holding rate of return already properly compounded, and plain nominal interest rates are divided by 12 (in monthly cases), you will just have to ask.
If this instills the fear of G’ in you, then all the better. Interest quotes are a minefield. Don’t get cheated.
My big “schtick” is BEWARE. It is that you know to beware. It is not that you should know all the conventions out there.
Compare two runners. A covers 42.2 km in 300 minutes. B covers 100m in 40 seconds. Who is faster?
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The standard metric for comparing (interest) rates across different terms is to “annualize them.” The wiggle is that the average rate accumulates interest over time, too.
You have to think about what you are doing. No way around it. No magic “bullets” or “hints.”
Understanding “1+” or not “1+” has to become second nature. Knowing that 1.5x is 50\% RoR must become second nature.
I don’t have a funnel. See also my posted note from last week.
YTM is the internal rate of return.
IRR is always solving for unknowns — and typically only done numerically.
I will not ask you to solve anything on an exam where having a financial calculator or Excel will give you an advantage. This greatly limits what I could ever ask on exams.
It’s often a little easier if all payments are identical, because then \(NPV = \frac{C}{r}\cdot\left( 1 - \frac{1}{(1+r)^T} \right)\) Nevertheless, you still cannot solve this in closed-form for $r$ (i.e., IRR). (You can solve it in closed-form for $C$.)
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What yield curve shifts? Chances are this is way beyond what I would ever ask on an exam.
The default part was basically our payoff tables…I think.
We did not mix term structure scenarios with risk scenarios. The two work very well together, but it’s simply more time-intensive and cluttered. No way I could ask this on an exam.
The thing to remember is that you obtain the principal reduction by subtracting the interest ($r\cdot NPV$) from the loan payment ($C$).
That is, if your remaining principal is $200, and your fixed loan payment is $25, and the original contracted (not prevailing) interest rate is $5\%$, then you are paying off $$200\cdot5\%= $10$ in interest.
Thus, you are reducing your principal by $15.
Thus, you will have remaining principal of $185 next period.
We’ve done nada else on this.
We’ve only done the very simplest version of this. These can be difficult problems.
We’ve thought of a “buy” as the equivalent of lease with a final sale. In both cases, we got use of the asset for a fixed amount of time.
The problems can become tricky, because they can involve different time frames as well as different upfront payments.
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Our approach was
Example, prevailing CoC = r = 1\% per month:
A has $5,000 payment upfront. Then 24 monthly payments of $1,000 each. Then life ends.
B has $2,000 payment upfront. Then 36 payments of $2,000 each. Then $30,000 resale. Then life ends.
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A:
B:
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We are not done. The remaining problem is that A is also cheaper because you only get to have the asset for 2 years; while B gives it to you for 3 years.
What are monthly “rents”?
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The world is a total mess in how it describes interest rates.
The typical US mortgage quotes the annual not-effective rate, but simply the nominal rate divided by 12.
In addition, almost all mortgages have upfront payments — not just points, but more.
The APR takes into account upfront payments (good), but is inconsistent on whether it uses an effective interest rate or a nominal interest rate (bad).
Understand the widespread heterogeneity in what you are being quoted: don’t let yourself be cheated by others quoting you what they find better for them rather than for you. Don’t let yourself be “baited-and-switched.”
For exams, just state which one you are using and why.
On exams, I am not asking you anything about forward rates.
I want you to know that they are implied in the term structure and you could lock them in. That’s it.
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Turn it around \(IRR = \frac{\mbox{Annual Dividends or Earnings}}{ \mbox{Price} } \;+\; \mbox{Growth of D or E}\)
If the dividend-price ratio is 1.5\% per year, and you believe (uggh) that the dividends will grow by 2\% per year as is forever, then you have told me that you expect the stock to give you 3.5\% per year in the future. The stock is “priced as if.”
We assumed PCM: no x-costs, no taxes, no opinions, no market power.
you only need to recite this, so you know what you are doing so far.
not yet covered what s-show happens if this does not hold.
Was taught in statistics.
You must be able to calculate simple risk metrics, either from probability tables or from data (which is like an equal-weighted probability table, albeit with a small-sample correction in Excel).
Example: What is the risk of a portfolio that earns either –30% or +50% with equal probability?
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Calculate the mean: +10%.
Calculate the squared deviations from the mean: $(-30\% - 10\%)^2= 40\%^2 = 0.16$ and $(50\% - 10\%)^2= 40\%^2 = 0.16$.
Calculate the probability-weighted average: Var=$\frac12\cdot 0.16 + \frac12\cdot 0.16 = 0.16$.
The risk is usually the standard deviation, which is just the squareroot of the variance: $\sqrt{0.16}= 0.40 = 40\%$.
Incidentally, for two equally likely outcomes, the standard deviation is just the half the distance between the two outcomes.
Payoff tables. See sample midterm. shows understanding of risk, reward, promised rates, and contingent claims (bonds and stocks).
Example: 20% prevailing interest rate. Project worth $50 or $100, equally likely. Raise $50 in debt.
| Bond | Stock | |
|---|---|---|
| P1=$50 | ||
| P1=$100 | ||
| EP1= | ||
| P0= |
Forecasting: very difficult in the real world. Economics, not finance.
Mechanics is easy: just stick this into a PV table in Excel.
=sumproduct(b2:b20, c2:c20)See above on lease calculations.
All just PV tables.
Rule of 72 (72/(100*r)) is obsolete.
It doesn’t emphasize intuition…it has no intuition afaics.
Don’t be afraid. Just use \(t=\frac{\log(2)}{\log(1+r)}\) and get it right. Oh, and this also works for tripling, quadrupling, etc.
All: Payments always begin next period.
Growth (in GP) always begins after the initial $C$.
See above for growth/inflation: often managers presume that $g$ will happen to be inflation. Then they pray.
Not sure what “right formulas across contexts” means.