Lec 20 Home Made Portfolio Insurance Lecture 20: “Home Made” Portfolio Insurance Risk Management: Trade a stock/bond portfolio as if it were a {+S, +P} hedged portfolio. Binomial Methodology Suppose we manage a well diversified stock portfolio and we expect the price to evolve as follows (1-yr Time horizon): Portfolio Price Tree Desired Portfolio B 140 D B 140D 120 ? 120 100 A 100 E ? A 100 E 80 ? 95.238 C 60 F C 100 F t=0 1 2 0 1 2 There is a chance the price will fall to 60 (in one year). Portfolio sponsor does not want to take that chance. We have two choices: a) Buy a put (over the counter) with K = 100. Or, b) Create our own {Stock+Put} combination. Lec 20 Home Made Portfolio Insurance dfdf

Lec 20 Home Made Portfolio Insurance “Home Made Portfolio Insurance”. Assume riskless rate 5%/6months at B set up a Portfolio {Δ shares, $B in bonds} to achieve the desired cash flows at D and E. Portfolio Prices Desired Portfolio t=1/2 T=1 t=1/2 T=1 140 D 140 D 120 ? 100 E 100 E Δ(140) + B(1.05) = 140 Δ(100) + B(1.05) = 100 Sol’n: Δ= 1, and B= $0 Therefore, at B we must be 100% in stock. No bonds, Desired Portfolio Value = 1(120) - 0 = $120 Lec 20 Home Made Portfolio Insurance dfdf

Lec 20 Home Made Portfolio Insurance at C set up a Portfolio {Δ shares, $B in bonds} to achieve the desired cash flows at D and E. Portfolio Prices Desired Portfolio t=1/2 T=1 t=1/2 T=1 100 E 100 E 80 ? C 60 F 100 F at C we want: Δ(100) + B(1.05) = 100 Δ( 60 ) + B(1.05) = 100 Sol’n: Δ = 0, B = 100/1.05 = $95.238 (no stock, all bonds) Therefore, at C we must be 100% in Bonds, No stock. Desired Portfolio Value = $95.238 Lec 20 Home Made Portfolio Insurance dfdf

Lec 20 Home Made Portfolio Insurance at A Portfolio Prices Desired Portfolio t=1/2 T=1 t=1/2 T=1 120 B 120 B A 100 ? 80 C 95.238 C Δ(120) + B(1.05) = 120 Δ( 80 ) + B(1.05) = 95.238 Sol’n: Δ= 0.61905 shares, B = $43.537. Therefore, at A we must have 0.61905 shares @ 100/share = $61.905 (amount in stock) $43.537 in Bonds Total amount to invest = 61.905 + 43.537 = $105.442 Lec 20 Home Made Portfolio Insurance dfdf

Lec 20 Home Made Portfolio Insurance In sum ▸ Invest $100 in stock, then there is a chance of losing $40. ▸ Invest $105, then the portfolio value will not drop below 100. ▸ If the floor of $90, then you only need to invest $104.082 Portfolio Price Tree Desired Portfolio B 140 D B 140D 120 120 100 A 100 E 105.44 100 E 80 95.238 C 60 F C 100 F t=0 1 2 0 1 2 Lec 20 Home Made Portfolio Insurance dfdf

Lec 20 Home Made Portfolio Insurance “Home Made Portfolio Insurance”, Black-Scholes Methodology At any point in time t, the Put price is: Pt = -St N(-d1 ) + K e-rT N(-d2 ) d1 = [ln(St /K)+(r+σ2/2)T ]/(σ√T) , and d2 = [ ln(St /K)+(r-σ2/2)T ]/(σ√T) a {+S, +P} combination will be: St + Pt = St [1 - N(-d1 )] + K e-rT N(-d2 ) = St Delta +Bonds ~=~S_t`~~~Delta_t ~+~~~~~ Bond_t Lec 20 Home Made Portfolio Insurance dfdf

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