ACTS 4308 FORMULA SUMMARY Section 1: Calculus review and effective rates of interest and discount 1 Some useful finite and infinite series: (i) sum of the first n positive integers: (ii) finite geometric series: 1 + 2 + + n = n(n + 1) 2 1 + r + r 2 + + r k = 1 rk+1 1 r ; r + r2 + + r k = r rk+1 1 r (iii) infinite geometric series: if r < 1 then 1 + r + r 2 + = 1 1 r ; r + r2 + = r 1 r (iv) increasing geometric series: 1 + 2r + 3r 2 1 + = (1 r) 2 (v) exponential and natural log series: e x = 1 + x + x2 2! + x3 3! + for x < 1 ln(1 + x) = x x2 2 + x3 3 2 Simple interest i: An amount P deposited for the time t accumulates to P (1 + it) 3 Compound interest i: An amount P deposited for the time t accumulates to P (1 + i) t 4 Cash flow principle: when using compound interest, for a series of deposits and withdrawals that occur at various points in time, the balance in the account at any given time point is the accumulated values of all deposits minus the accumulated values of all withdrawals to that time point 5 Present value: the present value of 1 due in one year is the amount required now to accumulate to 1 in one year v = 1 1 + i 6 Present value of 1 due in t years for simple interest 1 1 + it 7 Present value of 1 due in t years for compound interest 1 (1 + i) t = vt 8 d = annual effective rate of discount = A(1) A(0) A(1) 9 d = i 1+i 10 v = 1 d is the present value of 1 due in 1 year 11 i = d 1 d ; id = i d; 1 d = 1 i + 1 12 Compound discount: (1 + i) n = v n = (1 d) n 13 Simple discount: present value of 1 due in t years is 1 dt
2 Section 2: Nominal Rates of Interest and Discount 1 i (m) = nominal annual rate of interest compounded (or convertible) m times per year = m [ (1 + i) 1/m 1 ] 2 The effective rate of interest for each 1 i(m) -year is ( ) m m m 3 1 + i(m) = 1 + i m 4 d (m) = nominal annual rate of discount compounded (or convertible) m times per year = m [ 1 (1 d) 1/m] 5 The effective rate of discount for each 1 d(m) -year is ( ) m m m 6 1 d(m) = 1 d m ( ) p ( = (1 d) 1 = 1 + i = 7 1 d(p) p 1 + i(m) m ) m
3 1 i ( ) = d ( ) = δ t = A (t) A(t) = d dt [ln A(t)] ( 2 For a constant force of interest: e δ = 1 + i = Section 3: Force of Interest, Inflation 1 + i(m) m ) m ( = 1 d(p) 3 For a non-constant force of interest: δ t = 1 da(t) ln A(n) = A(t) dt 4 Simple interest: δ t = A (t) A(t) = i 1 + it 5 Real interest rate - interest rate adjusted for inflation: i REAL = i r 1+r 6 The equation of value for cash flow X = n X k : n p n X kv t k = Xv t 7 Exact solution ( n ) ( X n ) ln k X vt k X ln k X vt k t = = ln v δ 8 Method of equated time: uses the approximate solution t = 0 n X kt k X ) p δ t dt
4 Section 4: Annuity-immediate and annuity-due 1 The present value of an annuity-immediate with n level payments of 1 is a n i = 1 vn i 2 The accumulated value at time n of an annuity-immediate with n level payments of 1 is s n i = (1 + i)n 1 i 3 The present value of an annuity-due with n level payments of 1 is ä n i = 1 vn d 4 The accumulated value at time n of an annuity-due with n level payments of 1 is s n i = (1 + i)n 1 d 5 s n i = (1 + i) n a n i ; s n i = (1 + i) n ä n i 6 s n+1 i = 1 + (1 + i)s n i ; s n+1 i = 1 + (1 + i) s n i 7 A loan of 1 can be repaid by level payments throughout the n year period of the loan, or by paying interest every year, and making payments that accumulate to the loan amount at the end of the n year period of the loan: 1 a = 1 1 n i s + i; n i ä = 1 n i s n i + d 8 ä n = (1 + i)a n ; s n = (1 + i)s n 9 ä n = 1 + a n 1 ; s n = s n+1 1 10 The present value for a perpetuity-immediate is 11 The present value for a perpetuity-due is a i = 1 i ä i = 1 d
5 Section 5: Annuity valuation at any time point 1 Suppose this is the balance on an account at time n and there are no more deposits into the account, but the account continues to earn interest at an annual effective compound rate of i after time n Then, m periods after the final deposit into the account, the balance of the account is 2 The present value of a deferred annuity is s n i (1 + i) m = s n+m i s m i v m a n i = a n+m i a m i 3 Suppose that deposits of 1 are made at the end of each period into an account earning interest at effective rate i until the end of the nth period Then, just after the nth deposit the effective interest rate changes to j and deposits are made at the end of each period for m more periods (a) The accumulated value of this annuity at time n + m is (b) The present value of this annuity is s n i (1 + j) m + s m j a n i + v n i a m j = a n i + (1 + i) n a m j
6 Section 6: Annuities with differing interest and payment period 1 General Rule When payment frequency and interest compounding are different, use the payment frequency as your basis for calculation and adjust compounding to your payment frequency by finding the effective interest rate per payment period For example, if payments are monthly, but compounding is semi-annual with annual nominal rate of i (2), the period is one month and the effective interest rate per period is j = i(12) 12 To find j, use the equivalence principle between the nominal rates: ( ) 12 ( ) 2 ( ) 1 1 + i(12) = 1 + i(2) j = i(12) 12 2 12 = 1 + i(2) 6 1 2 2 Continuous Annuities Basic continuous annuity: payment of 1 per year is paid uniformly and continuously for an n-year period The present value is n ā n = v t dt = 1 vn 0 δ The accumulated value at time n is s n = n 0 (1 + i)n t dt = (1+i)n 1 δ ā n = lim ; s n = lim m a(m) n m s(m) n General continuous annuity: continuous rate of payment of c t at time t for an n-year period The present value is n 0 c tv t dt = n 0 c te δt dt The accumulated value at time n is n 0 c t(1 + i) n t dt = n 0 c te δ(n t) dt
7 Section 7: Annuities whose payments follow a geometric progression 1 Consider an annuity-immediate with payments which follow a geometric progression; ie, payments of K, K(1 + r), K(1 + r) 2,, K(1 + r) n 1 are made at times 1, 2, 3,, n 2 If i r, then the present value is P V = K ( 1 1+r 1+i i r 3 If i r, then the accumulated value at time n is [ (1 + i) n (1 + r) n ] F V = K i r 4 If i = r, then the present value is P V = nkv and the accumulated value at time t = n is ) n F V = nk(1 + i) n 1 5 Suppose a particular stock pays dividends at the end of each year forever (perpetuity with a geometric progression) 6 Dividend payments of K, K(1 + r), K(1 + r) 2, are made at times 1, 2, 3, where r is the growth rate for the dividends 7 If i > r, then the theoretical price of the stock (the present value of the dividend payments) is S = K i r
8 Section 8: Annuities whose payments follow an arithmetic progression 1 Increasing annuity-immediate: payments of 1, 2, 3,, n are made at the end of the 1st, 2nd, 3rd,, nth period, respectively 2 The present value of an increasing annuity-immediate is än nvn (Ia) n = i 3 The accumulated value of an increasing annuity-immediate at time n is denoted by (Is) n = s n n i 4 Increasing annuity-due: payments of 1, 2, 3,, n are made at the beginning of the 1st, 2nd, 3rd,, nth period, respectively; ie, payments are made at times 0, 1,, n 1 än nvn 5 (Iä) n = d 6 (I s) n = s n n d 7 Increasing perpetuities: (Ia) = 1 i + 1 i 2 and (Iä) = 1 d 2, if i > 0 8 Decreasing annuity-immediate: payments of n, n 1, n 2,, 1 are made at the end of the 1st, 2nd, 3rd,, nth period, respectively 9 The present value of a decreasing annuity-immediate is (Da) n = n a n i 10 The accumulated value of a decreasing annuity-immediate at time n is (Ds) n = n(1 + i)n s n i 11 Decreasing annuity-due: payments of n, n 1, n 2,, 1 are made at the beginning of the 1st, 2nd, 3rd,, nth period, respectively; ie, payments are made at times 0, 1,, n 1 12 (Dä) n = n a n d 13 (D s) n = n(1 + i)n s n d 14 Reinvestment rate for a single payment: Suppose interest is reinvested at a rate j on an investment of 1 made at time 0 Then the accumulated value of the principal and interest at time n is 1 + is n j 15 Reinvestment rate for a series of level payments: Suppose interest is reinvested at a rate j on payments of 1 at the end of each period for n periods Then the accumulated value of the principal and interest at time n is n + i (Is) n 1 j
9 1 Section 9: Amortization of a loan L = Ka n i = K = L a n i 2 OB t = outstanding balance after the t th payment (valued at time t) = Ka n t i (prospective formula) = L(1 + i) t Ks t i (retrospective formula) 3 The balance goes up between payments because the loan is accruing interest according to the formula OB t+s = (1 + i) s OB t, 0 < s < 1 4 I t = interest paid in the t th payment = i OB t 1 = K(1 v n t+1 ) 5 P R t = principal paid in the t th payment = K I t = Kv n t+1 6 So the outstanding balance after the t th payment OB t = OB t 1 P R t = (1 + i)ob t 1 K 7 The principle portion of consecutive payments increases by the factor (1 + i): P R t = (1 + i)p R t 1 8 The interest portion of consecutive payments decreases as follows: I t = I t 1 i(k I t 1 ) 9 For non-level payments: L = K 1 v + K 2 v 2 + + K n v n 10 I t = i OB t 1 P R t = K t I t OB t = OB t 1 P R t = (1 + i)ob t 1 K t
10 Section 10: The sinking fund method of loan repayment 1 i = effective interest rate per payment period paid by the borrower to the lender 2 j = effective interest rate earned by the borrower on the sinking fund 3 D = periodic sinking fund deposit = L s n j 4 Li = interest payment to the lender 5 K = periodic outlay by the borrower = Li + D
11 Section 11: Bond valuation 1 Frank formula: P = F ra n j + K where K = Cv n j is the present value of the redemption value 2 Premium-Discount formula: P = C + (F r Cj)a n j 3 There are different ways of computing bond prices at fractional points in the coupon periods BV t = F ra n t j + Cv n t j Purchase Price Accrued Coupon BV t+u MV t+u Compound Compound BV t(1 + j) u BV t(1 + j) u F r (1+j)u 1 j Compound Simple BV t(1 + j) u BV t(1 + j) u F ru Simple Compound BV t(1 + ju) BV t(1 + ju) F r (1+j)u 1 j Simple Simple BV t(1 + ju) BV t(1 + ju) F ru
12 Section 12: Bond amortization, callable bonds 1 BV t = Book value (or Amortized value): value at time t of the remaining payments of a bond after the t th coupon payment has been made 2 BV 0 = P, BV n = C 3 K t = t th coupon payment 4 BV t+1 = (1 + j)bv t K t 5 I t = Interest portion of the t th coupon payment 6 I t = j BV t 1 7 P R t = Amount of amortization of the premium with the t th coupon payment 8 P R t = BV t 1 BV t = K t I t 9 If BV t > BV t 1 (equivalently P R t < 0), then the bond is written up If BV t < BV t 1 (equivalently P R t > 0), then the bond is written down 10 BV t = F ra n t j + Cv n t j 11 BV t+1 = (1 + j)bv t F r BV t+k = (1 + j) k BV t F rs k j = C + (F r Cj)a n t j for k = 1,, n t 12 P R t = (F r Cj)v n t+1 13 P R t+1 = (1 + j)p R t P R t+k = (1 + j) k P R t for k = 1,, n t 14 If a bond is bought at a premium (P > C or equivalently, F r > Cj), then the bond is written up with each coupon payment 15 If a bond is bought at a discount (P < C or equivalently, F r < Cj), then the bond is written down with each coupon payment 16 Basic rules regarding callable bonds: If the bond is bought at premium (P > C), then for a given yield j, calculating the price based on the earliest redemption date is the maximum price that guarantees the yield will be at least j If the bond bought at discount (P < C), then for a given yield j, calculating the price based on the latest redemption date is the maximum price that guarantees the yield will be at least j
Section 13: Measures of the rate of return on a fund I 1 Dollar-weighted rate of return: i D =, where B = A + C + I n A + C k (1 t k ) 2 Time-weighted rate of return = i T = (1 + j 1 )(1 + j 2 ) (1 + j n+1 ) 1, where B k = account balance just before investment C k and 1 + j k = effective rate of return between two consecutive B k contributions C k 1 and C k = (C 0 = 0, B n+1 = B) B k 1 + C k 1 3 Fund accumulation methods: Portfolio method: All investors are pooled together in the same overall portfolio, and every investor gets the same return Investment year method: Segregate the money of all individuals who started in a given year and give them all the return on that segregated fund that is unique to them 13
14 Section 14: The term structure of interest rates, forward rates of interest and duration 1 Let s n be the spot rate for n years to maturity for n = 1, 2, 2 Let f j be the forward rate from time j until time j + 1 for j = 1, 2, 3 The spot rates can be determined from the forward rates using (1 + s n ) n = (1 + f 0 )(1 + f 1 ) (1 + f n 1 ) 4 The forward rates can be determined from the spot rates using 1 + f n = (1 + s n+1) n+1 (1 + s n ) n, f 0 = s 1 5 Duration: measure of sensitivity to changes in the interest rate 6 At time 0, suppose a series of future cashflows will occur at times t 1,, t n of amounts A 1,, A n, respectively 7 The present value as a function of the force of interest δ is n P = e δt k A k 8 (Macaulay) Duration = D = dp/dδ P = d[ln P ] dδ 9 dp n n dδ = t k e δt k A k = t k v t k A k 10 D = n t kv t ka k n vt k Ak = n t k w k where w k = vt ka k P 11 Modified duration (volatility) = D M = dp/di P = d[ln P ] di 12 dp di = dp dδ dδ di = 1 dp dp = v since δ = ln (1 + i) 1 + i dδ dδ 13 D M = D 1 + i = vd 14 Duration of a portfolio: Suppose there are m cashflow streams with respective present values P 1, P 2,, P m Then the duration (either Macaulay or modified) of the combined set of cashflows is D = P 1 P D 1 + P 2 P D 2 + + P m P D m where D k is the duration of the kth cashflow stream and P = P 1 + P 2 + + P m 15 The Macaulay duration of an annuity immediate with n payments is n D = kvk n vk = (Ia) n a n 16 The Macaulay duration of a coupon bond with face value F and coupon rate r for n periods and with redemption value C is D = F r n kvk + ncv n F r n vk + Cv n = F r(ia) n + nk F ra n + K 17 Cash flow matching: the present value of assets P B and the present value of liabilities P A are matched at interest rate i 0 18 The liabilities are immunized by the assets if the present value of assets is greater than or equal to the present value of liabilities for any small change in interest rates
15 Section 15: Introduction to financial derivatives, forward and futures contracts 1 If an asset pays no dividends, then the prepaid forward price is F0,T P = S 0 2 If an asset pays dividends with amounts D 1,, D n at times t 1,, t n, then the prepaid forward price is n F0,T P = S 0 D i e rt i 3 If an asset pays dividends as a percentage of the stock price at a continuous (lease) rate δ, then the prepaid forward price is F P 0,T = S 0e δt 4 Cost of carry = r δ 5 F 0,T = F P 0,T ert 6 Implied fair price: the implied value of S 0 when it is unknown based on an equation relating S 0 to F 0,T 7 Implied repo rate: implied value of r based on the price of a stock and a forward 8 Annualized forward premium = 1 T ln ( F0,T S 0 ) i=1
16 Section 16: Introduction to options 1 Assets (a) S 0 = asset price at time 0; it is assumed the purchaser of the stock borrows S 0 and must repay the loan at the end of period T (b) Payoff(Purchased Asset)= S T (c) Profit(Purchased Asset)= S T S 0 e rt (d) Payoff(Short Asset)= S T (e) Profit(Short Asset)= S T + S 0 e rt 2 Calls (a) C 0 = premium for call option at time 0; it is assumed the purchaser of the call borrows C 0 and must repay the loan when the option expires (b) Payoff(Purchased Call)= max {0, S T K} (c) Profit(Purchased Call)= max {0, S T K} C 0 e rt (d) Payoff(Written Call)= max {0, S T K} (e) Profit(Written Call)= max {0, S T K} + C 0 e rt 3 Puts (a) P 0 = premium for put option at time 0; it is assumed the purchaser of the put borrows P 0 and must repay the loan when the option expires (b) Payoff(Purchased Put)= max {0, K S T } (c) Profit(Purchased Put)= max {0, K S T } P 0 e rt (d) Payoff(Written Put)= max {0, K S T } (e) Profit(Written Put)= max {0, K S T } + P 0 e rt
17 Section 17: Option Strategies (Part I) 1 Floor: buying an asset with a purchased put option 2 Cap: short selling an asset with a purchased call option 3 Covered put (short floor): short selling an asset with a written put option 4 Covered call (short cap): buying an asset with a written call option 5 Synthetic Forward: combination of puts and calls that acts like a forward; purchasing a call with strike price K and expiration date T and selling a put with K and T guarantees a purchase of the asset for a price of K at time T 6 Short synthetic forward: purchasing a put option with strike price K and expiration date T and writing a call option with strike price K and expiration date T 7 No Arbitrage Principle: If two different investments generate the same payoff, they must have the same cost 8 C(K, T ) = price of a call with an expiration date T and strike price K 9 P (K, T ) = price of a put with an expiration date T and strike price K 10 Put-Call parity: C 0 P 0 = C(K, T ) P (K, T ) = (F 0,T K)v T 11 If the asset pays no dividends, then the no-arbitrage forward price is F 0,T = S 0 e rt S 0 = F 0,T v T 12 C(K, T ) + Kv T = P K, T ) + S 0
18 Section 18: Option Strategies (Part II) 1 Spread: position consisting of only calls or only puts, in which some options are purchased and some written 2 Bull: investor betting on increase in market value of an asset 3 Bull spread: purchased call with lower strike price K 1 and written call with higher strike price K 2 4 A bull spread can also be created by a purchased put with strike price K 1 and a written put with strike price K 2 5 Bear: an investor betting on a decrease in market value of an asset 6 Bear spread: written call with lower strike price K 1 and purchased call with higher strike price K 2 7 Ratio spread: purchasing m calls at one strike price K 1 and writing n calls at another strike price K 2 8 Box spread: using options to create a synthetic long forward at one price and a synthetic short forward at a different price 9 Purchased collar: purchased put option with lower strike price K 1 and written call option with higher strike price K 2 10 Collar width: difference between the call and put strikes 11 Written collar: writing a put option with lower strike price K 1 and purchasing a call option with higher strike price K 2 12 Zero-cost collar: choosing the strike prices so that the cost is 0 13 Purchased straddle: purchased call and put with the same strike price K 14 Written straddle: written call and put with the same strike price K 15 Purchased strangle: purchased put with lower strike price K 1 and purchased call with a higher strike price K 2 16 Written strangle: written put with lower strike price K 1 and written call with a higher strike price K 2 17 Butterfly spread: a written straddle with a purchased strangle 18 Asymmetric butterfly spread: purchasing λ calls with strike price K 1, purchasing 1 λ calls with strike price K 3, and writing 1 call with strike price K 2 where λ = K 3 K 2 K 3 K 1 and K 1 < K 2 < K 3
19 Section 19: Swaps 1 s n = n-year spot rate as defined in Section 14 2 r 0 (n, n + 1) = n-year forward rate as defined in Section 14: 1 + r 0 (n, n + 1) = (1 + s n+1) n+1 (1 + s n ) n 3 P (t 1, t 2 ) = price of a zero-coupon bond that is purchased at time t 1 and pays 1 at time t 2 1 P (0, n 1) 4 P (0, n) = = (1 + s n ) n 1 + r 0 (n 1, n) 5 R = Swap rate is the rate for which the present value of level interest payments at the end of each period equals the present value of interest payments at the forward rates at the end of each period n n 1 P (0, n) 6 R P (0, k) = r 0 (k 1, k)p (0, k) R = n P (0, k) 7 Deferred swap: a swap that starts at some time in the future but the swap rate is agreed upon today 8 For a deferred swap with the first loan payment in m periods on an n-period loan, the swap rate is R = P (0, m 1) P (0, m 1 + n) n P (0, m 1 + k)