10.15: The First and Second Derivatives Illustrated
- Page ID
- 88592
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- We have just demonstrated that changes in longer-term discount rates affect present value (i.e., prices) more than in the case of short-term rates and cash flows. In our example, the rate of changes in present value factors increased in time, e.g., from 26%, to 101%, to 303%.
- However, the rate of increase is decreasing (i.e., “deceleration” or second derivative), e.g., from 126%, to 71%, to 38%.
- In Calculus, we would say here that the first derivative is positive while the second derivative is negative. If we think of the price of the cash flows as distance covered as a function of time (“speed”), then we can think of the “First Derivative.” We can think of the “Second Derivative” here as the change in the rate of speed (“deceleration”) Were the second derivative positive, we would be speaking of “acceleration.” We observe these characteristics in this diagram:
Imagine a payment due in the future; its present value would be determined by its rate – a short-term rate (and PVF!) for a short-term payment and so on. The rate of change in present value was noted in the first derivative column, while the “rate of change in the rate of change” in the present value was noted in the second derivative column.
In other words, as time increases, the differences in present values – for 5% versus 10% – increase, but the rate of increase decreases! Remember: the price of a financial asset today is its present value!
The Take-away (again): If rates change, prices (i.e., present values) could change dramatically!!
Interest (lit. “bite”). Exodus, 22:24: “Bite” means “interest,” since it is like the bite of a snake, which bites a small wound on one’s foot, which he does not feel, but suddenly it swells and blows up as far as his head. So, with “interest,” one does not feel (it) and it is not noticeable until the interest increases and causes him to lose much money.
-Rashi’s commentary on the Bible.
(Translated by Ben Isaiah and Sharfman, Brooklyn, NY: S. S. & R. Publishing Company, 1949)
On Work
Success in business requires training and discipline and hard work. But if you’re not frightened by these things, the opportunities are just as great today as they ever were.
-David Rockefeller
Economist and Banker
If you train hard, you’ll not only be hard, you’ll be hard to beat.
-Herschel Walker
American professional football player
and Politician
What doesn’t kill me makes me stronger.
-Friedrich Nietzsche
German Philosopher
Whenever heaven is about to confer a great responsibility on any man, it will exercise his mind with suffering, subject his sinews and bones to hard work, expose his body to hunger, put him in poverty, place obstacles in the path of his deeds, so as to stimulate his mind, harden his nature, and improve wherever he is incompetent.
-Meng Tzu, Third Century B.C.E.
Quoted in The Coddled Mind, Chapter One, by Jonathan Haidt
Ben Heh-Heh used to say: “According to the effort is the reward.”
-Mishnah, Avoth, 5:26
No pain, no gain.
-Somebody
When the goin’ gets tough, the tough get goin’.
-Somebody Else