Charts, Trends & Dow Theory

Question 1

(a) (5 marks)

• Dow uptrend definition: a series of successively higher highs (HH) and higher lows (HL) — each rally peak exceeds the prior peak and each reaction trough is above the prior trough. (2)
• Highs in $P$ (odd positions): $118, 132, 150$. Check $118 < 132 < 150$ ✓ — strictly increasing, so HH holds. (1.5)
• Lows in $P$ (even positions): $100, 108, 121$. Check $100 < 108 < 121$ ✓ — strictly increasing, so HL holds. (1.5)
• Conclusion: both HH and HL satisfied ⇒ confirmed uptrend.

(b) (8 marks)

def classify_trend(pivots):
    prices = [p for _, p in pivots]
    highs = prices[1::2]   # positions 1,3,5,... are highs
    lows  = prices[0::2]   # positions 0,2,4,... are lows
 
    def strictly_increasing(xs):
        return all(a < b for a, b in zip(xs, xs[1:]))
    def strictly_decreasing(xs):
        return all(a > b for a, b in zip(xs, xs[1:]))
 
    hh = strictly_increasing(highs)
    hl = strictly_increasing(lows)
    lh = strictly_decreasing(highs)
    ll = strictly_decreasing(lows)
 
    if hh and hl:
        return "uptrend"
    if lh and ll:
        return "downtrend"
    return "range"

Marks: correct high/low separation (2), increasing/decreasing helpers (2), uptrend logic (2), downtrend logic (1), range fallback (1).

(c) (6 marks)

• Leg 1: $100\to118$: $\frac{18}{100}=18.0\%$. (1)
• Leg 2: $108\to132$: $\frac{24}{108}=22.22\%$. (1)
• Leg 3: $121\to150$: $\frac{29}{121}=23.97\%$. (1)
• $18.0\% < 22.22\% < 23.97\%$ ⇒ momentum (% gain per leg) is accelerating. (1)
• Dow confirmation: a healthy accelerating/continuing uptrend should show volume expanding on the up-legs (rallies) and contracting on pullbacks; rising volume confirms the primary uptrend. (2)

(d) (3 marks)

• Slope $m = \frac{108-100}{9-0} = \frac{8}{9} \approx 0.8889$. Intercept $b=100$. Line: $p(t)=\frac{8}{9}t + 100$. (1.5)
• At $t=18$: $p(18)=\frac{8}{9}(18)+100 = 16+100 = 116$. (1)
• Actual low $121 > 116$ ⇒ price sits above the trendline; the low is holding above projected support — bullish, uptrend intact (lows rising faster than the base trendline). (0.5)

Question 2

(a) (4 marks)

• Linear pixels for $40\to80$: $(80-40)\cdot c = 40c$. (1)
• Linear pixels for $80\to160$: $(160-80)\cdot c = 80c$. (1)
• Ratio $= \frac{80c}{40c}=2$. (1)
• Both are $+100\%$ moves yet the second occupies twice the vertical space — equal-percentage moves are visually unequal on a linear scale. (1)

(b) (5 marks)

• Pixel positions: $y=d\log_{10}p$. (1)
• Move 1: $d(\log_{10}80-\log_{10}40)=d\log_{10}2$. (1.5)
• Move 2: $d(\log_{10}160-\log_{10}80)=d\log_{10}2$. (1.5)
• Equal, each $=d\log_{10}2 \approx 0.30103\,d$ pixels. (1)

(c) (5 marks)

• Displacement $=d(\log_{10}(p(1+r))-\log_{10}p)$ (2)
• $=d\log_{10}\!\frac{p(1+r)}{p}=d\log_{10}(1+r)$. (2)
• Independent of $p$ ⇒ constant-percentage moves have constant vertical size on a log axis. (1)

(d) (6 marks)

import math
def to_log_pixels(prices, d):
    return [d * math.log10(p) for p in prices]
 
y = to_log_pixels([40, 80, 160], d=100)
gap1 = y[1] - y[0]
gap2 = y[2] - y[1]
assert abs(gap1 - gap2) < 1e-9

Marks: function correct (3), gap computation (2), tolerance check (1).

Question 3

(a) (6 marks) Any three of the tenets (2 marks each, cap 6, but part reserves confirmation point):

• The averages discount everything.
• The market has three trends (primary, secondary, minor).
• Primary trends have three phases (accumulation, public participation, distribution).
• Volume must confirm the trend.
• A trend is assumed in effect until it gives definite reversal signals.
• Averages must confirm each other: a new trend signal (e.g. new high in the Industrials) is only valid if a second index (e.g. Transports) confirms with a similar move; divergence casts doubt on the signal. (2 of the 6 for this explanation)

(b) (8 marks)

def volume_confirms(close, vol):
    if close[-1] <= close[0]:
        return False  # not an uptrend
    up_vols, down_vols = [], []
    for i in range(1, len(close)):
        if vol[i] is None:
            continue
        if close[i] > close[i-1]:
            up_vols.append(vol[i])
        elif close[i] < close[i-1]:
            down_vols.append(vol[i])
    if not up_vols:
        return False
    avg_up = sum(up_vols) / len(up_vols)
    avg_down = sum(down_vols) / len(down_vols) if down_vols else 0
    return avg_up > avg_down

Marks: uptrend guard (2), up/down classification vs previous close (3), averaging (2), comparison return (1).

(c) (4 marks)

• Day2 up (104>100): vol 500 → up. Day3 down (102<104): 300 → down. Day4 up (108>102): 700 → up. Day5 up (110>108): 600 → up. (2)
• avg up $=\frac{500+700+600}{3}=600$. avg down $=300$. (1)
• $600 > 300$ and final $110>100$ ⇒ uptrend is volume-confirmed (True). (1)

[
  {"claim":"Leg percentage gains are 18, 22.22..., 23.97... and increasing", "code":"g1=Rational(18,100)*100; g2=Rational(24,108)*100; g3=Rational(29,121)*100; result = bool(g1<g2<g3 and g1==18)"},
  {"claim":"Trendline through lows gives support 116 at t=18", "code":"m=Rational(8,9); b=100; p=m*18+b; result = (p==116)"},
  {"claim":"Log-scale gaps 40->80 and 80->160 are equal (both log10 2)", "code":"g1=log(80,10)-log(40,10); g2=log(160,10)-log(80,10); result = bool(simplify(g1-g2)==0)"},
  {"claim":"Volume confirmation: avg up 600 > avg down 300", "code":"up=(500+700+600)/S(3); down=S(300); result = bool(up==600 and up>down)"}
]