Asset Allocation & Rebalancing

Question 1

(a) Initial values: equity $= 600{,}000$, bond $= 400{,}000$. After returns: equity $= 600{,}000 \times 1.25 = 750{,}000$; bond $= 400{,}000$. Total $V_1 = 1{,}150{,}000$. $w_1 = \frac{750{,}000}{1{,}150{,}000} = \frac{75}{115} = \frac{15}{23} \approx 0.6522.$ (3 marks) Drift $= w_1 - 0.60 = \frac{15}{23} - \frac{3}{5} = \frac{75 - 69}{115} = \frac{6}{115} \approx 0.05217$. (2 marks) Since $0.05217 \ge 0.05 = \tau$, the threshold triggers. (1 mark)

(b) Proof. Post-return equity value $= V_0 w^\*(1+r_E)$; bond value $= V_0(1-w^\*)(1+r_B)$. Total $= V_0[w^\*(1+r_E)+(1-w^\*)(1+r_B)]$. Dividing equity by total ($V_0$ cancels): $w_1 = \frac{w^\*(1+r_E)}{w^\*(1+r_E)+(1-w^\*)(1+r_B)}. \checkmark$ (3 marks)

Trigger condition $w_1 \ge w^\*+\tau$. Let $a = w^\*(1+r_E)$, $b=(1-w^\*)(1+r_B)$. Then $\frac{a}{a+b} \ge w^\*+\tau \iff a \ge (w^\*+\tau)(a+b) \iff a[1-(w^\*+\tau)] \ge (w^\*+\tau) b.$ So $a \cdot (1-w^\*-\tau) \ge (w^\*+\tau)(1-w^\*)(1+r_B)$, giving $1+r_E \ge \frac{(w^\*+\tau)(1-w^\*)(1+r_B)}{w^\*(1-w^\*-\tau)}.$ With $r_B$ fixed, minimum outperformance: $\delta = r_E - r_B \ge (1+r_B)\left[\frac{(w^\*+\tau)(1-w^\*)}{w^\*(1-w^\*-\tau)} - 1\right].$ (5 marks) Check with $r_B=0$, $w^\*=0.6$, $\tau=0.05$: $\frac{0.65\cdot0.4}{0.6\cdot0.35}-1 = \frac{0.26}{0.21}-1 = 0.2381$. Any $r_E \ge 23.81\%$ triggers; $25\%$ does — consistent with (a).

(c) (6 marks: 4 correct logic, 1 zero-vector case, 1 complexity)

import numpy as np
def rebalance_threshold(weights, targets, tau):
    weights = np.asarray(weights, float)
    targets = np.asarray(targets, float)
    if np.any(np.abs(weights - targets) >= tau):
        return targets - weights   # buy/sell fractions to restore targets
    return np.zeros_like(weights)

Complexity: $O(n)$ — one pass to test bands, one pass to compute the trade.

Question 2

(a) In period $i$, units bought $= C/p_i$. Total units $U = \sum_i C/p_i = C\sum_i 1/p_i$. Total invested $= nC$. Average cost: $\bar c = \frac{nC}{C\sum 1/p_i} = \frac{n}{\sum 1/p_i} = \text{HM}(p). \checkmark$ (5 marks) By AM–HM inequality, for positive $p_i$: $\text{HM} \le \text{AM}$, i.e. $\bar c \le \bar p$, with equality iff all $p_i$ equal (proof: AM–HM follows from AM–GM or Cauchy–Schwarz $(\sum p_i)(\sum 1/p_i)\ge n^2$). (4 marks)

(b) Units: $100 \to 100$, $80 \to 125$, $125 \to 80$, $100 \to 100$. Total $U = 405$ units. (2) Total invested $= 40{,}000$. $\bar c = 40{,}000/405 = 98.765$. (2) Arithmetic mean price $= (100+80+125+100)/4 = 101.25$; indeed $\bar c < \bar p$. (1) Final value at price $110$: $405 \times 110 = 44{,}550$. (2)

(c) RCA outperforms lump sum when prices are volatile / dispersed and mean-reverting or dip early — higher price dispersion widens the AM–HM gap, lowering average cost. It underperforms in a steadily rising market, where early lump-sum capital captures more upside (opportunity cost of un-invested cash dominates). (3) Counter-example: monotone rising path $p=[80,90,100,110]$ — lump sum at $80$ beats RCA. (1)

Question 3

(a) Portfolio variance: $\sigma_p^2(c)=c^2\sigma_c^2+(1-c)^2\sigma_s^2+2c(1-c)\rho\sigma_c\sigma_s.$ (3) Minimise: $\frac{d}{dc}=0$ gives $c^\*=\frac{\sigma_s^2-\rho\sigma_c\sigma_s}{\sigma_c^2+\sigma_s^2-2\rho\sigma_c\sigma_s}.$ (3) Numerics: $\sigma_c=0.12,\sigma_s=0.25,\rho=0.3$. $\rho\sigma_c\sigma_s=0.3\cdot0.12\cdot0.25=0.009$. Numerator $=0.0625-0.009=0.0535$. Denominator $=0.0144+0.0625-0.018=0.0589$. $c^\*=0.0535/0.0589\approx0.9083.$ (3)

(b) Rebalance now (short-term): tax $=0.15\times200{,}000=₹30{,}000$. (2) Wait one period → long-term tax $=0.10\times200{,}000=₹20{,}000$, but incur tracking cost of holding the drift: $\kappa d^2 = 500{,}000\times0.04^2 = 500{,}000\times0.0016=₹800$. (3) Compare: rebalance now $=30{,}000$; wait $=20{,}000+800=20{,}800$. Wait — waiting saves $₹9{,}200$ (the long-term tax saving of ₹10,000 far exceeds the ₹800 drift cost). (2)

(c) (4 marks)

def tax_aware_sell(lots, target_value, price, st_rate, lt_rate):
    # lots: list of dicts {basis, qty, long_term(bool)}
    # compute tax per rupee sold for each lot, sell cheapest-tax first
    def tax_rate(lot): return lt_rate if lot['long_term'] else st_rate
    def unit_tax(lot):
        gain = max(price - lot['basis'], 0)
        return tax_rate(lot) * gain / price   # tax per rupee of proceeds
    lots_sorted = sorted(lots, key=unit_tax)   # greedy: lowest tax first
    proceeds, tax = 0.0, 0.0
    for lot in lots_sorted:
        cap = lot['qty'] * price
        take = min(cap, target_value - proceeds)
        units = take / price
        tax += tax_rate(lot) * max(price - lot['basis'],0) * units
        proceeds += take
        if proceeds >= target_value: break
    return proceeds, tax

This resembles a fractional knapsack / greedy selection problem (sort by tax-per-proceeds ratio, fill greedily), which is optimal because lots are divisible.

[
 {"claim":"Q1a equity weight after +25% equity, 0% bond from 0.6 target = 15/23",
  "code":"w1=Rational(600000*1.25,1150000); result=(w1==Rational(15,23))"},
 {"claim":"Q1a drift exceeds tau=0.05",
  "code":"drift=Rational(15,23)-Rational(3,5); result=(drift>=Rational(1,20))"},
 {"claim":"Q1b min outperformance ~0.2381 for wstar=0.6,tau=0.05,rB=0",
  "code":"val=(0.65*0.4)/(0.6*0.35)-1; result=abs(val-0.238095)<1e-4"},
 {"claim":"Q2b average cost = 40000/405 and less than 101.25",
  "code":"c=40000/405; result=(abs(c-98.7654)<1e-3 and c<101.25)"},
 {"claim":"Q2b final value at 110 = 44550",
  "code":"U=10000/100+10000/80+10000/125+10000/100; result=(abs(U*110-44550)<1e-6)"},
 {"claim":"Q3a min-variance core weight approx 0.9083",
  "code":"num=0.0625-0.009; den=0.0144+0.0625-0.018; result=abs(num/den-0.90832)<1e-3"},
 {"claim":"Q3b waiting (20800) cheaper than rebalancing now (30000)",
  "code":"now=0.15*200000; wait=0.10*200000+500000*0.04**2; result=(wait<now and abs(wait-20800)<1e-6)"}
]