Forward pricing Assets with no cash flows Assets with know discrete cash flows Assets with continuous cash flows (index) Finance 7523 Spring 1999 Assistant Professor Steven C. Mann Neeley School , TCU Forward price of a stock S(0) = $25. Stock pays no dividends is(6 month) = 7.12% ( T = 1/2 ) if you borrow $25 today you repay $25[1+ 0.0712(1/2)] = $25.89 f = six month forward price of stock Consider strategy: now T=6 months later borrow $25 25.00 -$25.89 buy stock - 25.00 S(T) sell 6-month forward 0 - [ S(T) - f ] total 0 f - 25.89 = f - S(0)[ 1 + is T] Arbitrage-free forward pricing: f (0,T) = S(0)(1 + isT) = $25.89 Cash and Carry forward pricing Forward contract with delivery date T; spot asset with no cash flows “Cash and carry” strategy: now at date T buy asset at cost S(0) - S(0) S(T) borrow asset cost + S(0) -S(0)(1+isT) sell forward at f(0,T) 0 -[ S(T) -f(0,T)] Total 0 f(0,T) - S(0)(1+isT) Forward price is “future value” f(0,T) = S(0)( 1 + isT) of spot price f(0,T) [ 1/(1+isT)] = S(0) S(0) = f(0,T)B(0,T) Spot price is “present value” of the forward price S(t) = f(t,T) B(t,T) Forward valuation (post-initiation and “off-market”) Value = V[forward price, time] at initiation, value is zero: V[ f(0,T), 0 ] ð 0. At maturity: V[ f(0,T), T ] = S(T) - f(0,T) at some time t (post initiation): V[ f(0,T), t ] = ? 1) Valuation by offset: At time t: long f(0,T). Sell f(t,T) value at T: S(T) - f(0,T) - [ S(T) - f(t,T) ] total value at T: f(t,T) - f(0,T) value at t: B(t,T)[ f(t,T) - f(0,T) ] 2) Valuation by algebra: V[f(0,T),t] = PV( V[f(0,T),T]) = PV[ S(T) - f(0,T)] note S(t) = PV[S(T)]; and S(t) = f(t,T)B(t,T) (prior page) so V(f(0,T),t] = f(t,T)B(t,T) - f(0,T)B(t,T) =B(t,T) [ f(t,T) - f(0,T)] = PV(price difference) Example - problem 2.3: forward pricing and valuation Non-dividend paying asset; S(0) = $65 contract maturity is 90 days simple interest rate is 4.50% ; daycount is actual/365. a) find forward price f(0,90/365) = S(0)(1+is(90/365)) = $65(1 + 0.045(90/365) ) = $65.72 value of contract is zero. b) You are asked to value a 90-day forward on this asset with delivery price = $60. This is an “off-market’ forward: value is nonzero. Value of long forward with off-market delivery price: value = PV( difference in forward prices) = B(0,T)[ market forward price - contract forward price] = B(0,T) [ 65.72 - 60.00] = (1 + 0.045(90/365)) -1 [$ 5.72] =(0.98903)($5.72) = $5.66 Example - problem 2.4: forward pricing and valuation Prob 2.4: S(0) = $45. Non-dividend paying asset contract maturity is 100 days simple interest rate is 4.75% ; daycount is actual/365. a) find current forward price f(0,100/365) f(0,100/365) = $45.00(1+ 0.0475(100/365)) = $45.59 b) You are long 100 day forward to buy asset at $50.25. If you sell a 100 day forward at current price, what is payoff at T? At maturity: long: S(T) - $50.25 short -[ S(T) - $45.59] net: $45.59 - $50.25 = - $4.66. c) what is the present value of your net position? PV = B(t,T)[ f(t,T) - f(0,T)] = [1/(1+0.0475(100/365)](-4.66) = (0.9872)(-4.66) = -$4.60 Assets with known cash flows Example: Zero-coupon yield curve 12 month T-note (simple interest) par = $1000. month T is(T) B(0,T) Coupon=10% semi-annual 6 1/2 7.18% 0.9653 $50 $1050 9 9/12 7.66% 0.9456 12 1 7.90% 0.9267 0 ! 1 Spot bond price Bc(0,12) = $50.00 B(0,6) + $1050.00 B(0,12) = $50.00(.9653) + $1050.00(.9268) = $48.27 + $ 973.12 = $1021.39 Forward contract does not receive coupon at T=6 months Forward pricing: assets with known cash flows Strategy 1: cost now t1 =6 months at T=9 months a) buy bond 1021.39 + 50.00 Bc(9,12) b) borrow PV(coupon) - 50.00 B(0,6) - 50.00 total 1021.39 - 48.27 0 Bc(9,12) [S(0) - d(t1)B(0,t1)] Strategy 2: cost now t1= 6 months at T=9 months a) enter long forward 0 0 Bc(9,12) - f (0,9) b) lend PV( f (0,9)) f (0,9) B(0,9) 0 f (0,9) (buy bill ) total f (0,9) B(0,9) 0 Bc(9,12) Each strategy has same payoff: must have same cost to avoid arbitrage f (0,9) B(0,9) = $1021.39 -28.27 = $973.12 f (0,9) = $ 973.12/0.9457 = $1029.03 in general: f (0,T)B(0,T) = S(0) - d(t1)B(0,t1) General forward pricing for assets with known cash flows Strategy 1: cost now at t1 at T a) buy asset S(0) + d(t1) S(T) b) borrow PV(d(t1)) - d(t1) B(0,t1) - d(t1) total S(0) - d(t1)B(0,t1) 0 S(T) Strategy 2: cost now at t1 at T a) enter long forward 0 0 S(T) - f (0,T) b) lend PV( f (0,T)) f (0,T) B(0,T) 0 f (0,T) (buy bill ) total f (0,T) B(0,T) 0 S(T) Each strategy has same payoff: must have same cost to avoid arbitrage in general: f (0,T)B(0,T) = S(0) - d(t1)B(0,t1) for N known flows: N f (0,T)B(0,T) = S(0) - S d(ti)B(0,ti) i=1 Example: forward pricing - asset pays dividends Problem #2.7 S(0) = 63 * = $63.375 Bill prices: stock pays dividends: $1.50 in 1 month 1 month: 0.9967 $2.00 in 7 months 7 month : 0.9741 12 month: 0.9512 Find price of one-year forward contract written on stock. Use: f (0,T) B(0,T) = S(0) - PV(dividends) f (0,12) B(0,12) = S(0) - d(1)B(0,1) - d(7)B(0,7) f (0,12) (0.9512) = $63.375 - $1.50(0.9967) - $2.00(0.9741) f (0,12) = $59.93/(0.9512) f (0,12) = $63.01 Assets with continuous payouts (index, currency) for N known flows: N f (0,T)B(0,T) = S(0) - S d(t )B(0,t ) i=1 i i = S(0) - PV(cash payout to time T) if asset pays continuous yield then PV(dividends to time T) = S(0)[ 1 - exp(-dyT)] (J&T ch 2 appendix - note typo ) so that f (0,T)B(0,T) = S(0) - [ S(0) [ 1 - exp(-dyT)]] f (0,T)B(0,T) = S(0) exp(-dyT) write B(0,T) as continuous discount factor: B(0,T) = exp(-rT) then f (0,T) = S(0) exp ( (r-dy)T) Example: Index forward pricing Problem #2.9: S&P500 Index = 495.00 div yield = 2.50% continuous (365 day year) a) Given 95-day discount rate =5.75% (360 day year), find f (0,95 days) B(0,95) = 1 - 0.0575(95/360) = 0.984826 exp(-dyT) = exp(-0.025(95/365)) = 0.993514 f (0,95)B(0,95) = S(0)exp(-dyT) f (0,95) = 495.00 (0.993514) / (0.98426) = 499.37. b) One day later index is at 493. 94-day discount is 5.75%. What is the value of the contract in part (a)? B(0,94) = 1 - 0.0575(94/360) = 0.984986 exp(-dyT) = exp(-0.025(94/365)) = 0.993582 f (0,94) = 493.00(0.993582)/(0.984986) = 497.20 value of prior contract = B(0,94) ( 497.20 - 499.37) = 0.984986 (-2.17) = -2.04 Commodity Forwards: Storage cost Storage: Define: G = cost of storing asset for (0,T) (per unit); paid time 0. “Cost of carry” strategy: cost now value at date T buy asset at cost S(0), pay storage S(0) + G S(T) borrow asset cost and storage cost -[S(0) + G] -[S(0)+G](1+isT) sell forward at f(0,T) 0 -[ S(T) -f(0,T)] Total 0 f(0,T) - [S(0) +G](1+isT) f (0,T)B(0,T) = S(0) + G Example: 180-day Gold forward (100 troy oz.) Spot Gold S(0) = $368 / oz. cost of storage for 180 days = 2.25 / oz., paid at time 0. 180 days simple interest rate = 3.875% annualized (actual/actual). f (0,180) = (368 + 2.25)(1 + 0.03875(180/365)) = 370.25(1.01911) = $377.33 / oz. If storage cost G is defined to be paid at T, then f(0,T)B(0,T) = S(0) + G B(0,T) so f (0,180) = 368(1+0.03875(180/365)) + 2.25 = $377.28 / oz. Commodity Forwards: Convenience yield Convenience yield: Define: Y(0,T) = present value (time 0) of benefits provided by holding asset.