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1 principle interest simple interest compound interest geometric growth 7-10 seven-ten rule /i i effective interest rate nominal rate continuous rate e r (r=nominal rate) exponential growth present value discounting discount factor d=1/(1+r) (r=rate) ideal bank (constant ideal bank) future value (cf. present value) PV= n k=0 x(t k)e rt k equivalent streams (x 0, x 1,, x n ) (y 0, y 1,, y n ) internal rate of return (IRR) P V = n k=0 x k/(1 + r) k = 0 r net present value 1

2 present worth of benefits present worth of costs cost of capital inflation rate constant dollars (real dollars) actual dollars (nominal dollars) real interest rate Definition: 1 + r 0 = (1 + r)/(1 + f) (r 0 : r: f: ) 3 financial instruments security fixed-income securities default fixed-income securities demand deposit tune deposit account certificate of deposit (CD) 1 money market

3 commercial paper banker s acceptance A B B 3 B A Eurodollar deposit Eurodollar CD T U.S. Treasury bill T 9,500 T U.S. Treasury note 1000 T 6 (coupon payment) Treasury notes (or T-Notes) mature in one to ten years. They have a coupon payment every six months, and are commonly issued with maturities dates between 1 to 10 years, with denominations of 1,000. In the basic transaction, one buys a 1,000 T-Note for say, 950, collects interest over 10 years of say, 3 per year, which comes to 30 yearly, and at the end of the 10 years cashes it in for So, 950 over the course of 10 years becomes 1300.(Wikipedia) T U.S. Treasury bond 10 T T (callable) Treasury bonds are issued by the government of the United States in order to pay for government projects. The money paid out for a Treasury bond is essentially a loan to the government. As with any loan, repayment of principal is accompanied by a fixed interest rate. These bonds are guaranteed by the full faith and credit of the U.S. government, meaning that they are extremely low risk (since the government can simply print money to pay back the loan). Additionally, interest earned on Treasury bonds is exempt from state and local taxes. Federal taxes, however, are still due on the earned interest. The government sells Treasury bonds by auction in the primary market, but they can also be purchased through a broker in the secondary market. A broker will charge a fee for such a transaction, but the government charges no fee to participate in auctions. Treasury bonds are marketable securities, meaning that they can be traded after the initial purchase. Additionally, they are highly liquid because there is an active secondary market for them. Prices on the secondary market and at auction are determined by interest rates. Treasury bonds issued today are not callable, so they will continue to accrue interest until the maturity date. One possible downside to Treasury bonds is that if interest rates increase during the term of the bond, the money invested will be earning less interest than it could earn elsewhere. Accordingly, the resale value of the bond will decrease as well. Rising inflation can also eat into the interest earned on Treasury bonds. Because there is almost no risk of default by the government, the return on Treasury bonds is relatively low, and a high inflation rate can erase most of the gains by reducing 3

4 the value of the principal and interest payments. Investors who wish to participate in auctions and purchase Treasury securities directly from the Federal Reserve Bank can open a Treasury Direct Account. There are no fees associated with the account unless it contains over 100,000, at which point a very small maintenance fee is incurred.( U.S. Treasury strips 10 0 (zero-coupon bond) Separate Trading of Registered Interest and Principal Securities (or STRIPS) are T-Notes, T-Bonds and TIPS whose interest and principal portions of the security have been separated, or stripped ; these may then be sold separately (in units of 1000 face value) in the secondary market. The name derives from the days before computerization, when paper bonds were physically traded; traders would literally tear the interest coupons off of paper securities for separate resale. The government does not directly issue STRIPS; they are formed by investment banks or brokerage firms, but the government does register STRIPS in its book-entry system. They cannot be bought through TreasuryDirect, but only through a broker. STRIPS are used by the Treasury and split into individual principal and interest payments, which get resold in the form of zero-coupon bonds. Because they then pay no interest, there is not any interest to re-invest, and so there is no reinvestment risk with STRIPS.(Wikipedia) municipal bond (general obligation bond) (revenue bond) corporate bond indenture callable bond sinking fund debt subordination (mortgage) (mortgage) 4

5 baloon payment adjustable-rate mortgage (mortgage-backed securities) (annuity) :annuitant (annual worth) bond (face value, par value) bid price ask price (AI) Def: AI= ( ) ( ) Moody s Standard Poor s (investment grade) (junk bond) yield to maturity :YTM P = F/[1 + ( /m)] n + n C/m k=1 P: F: [1+( /m)] k C: m: n: 10 6 (par bond) 5

6 current yield(cy) Def:CY=( )/()*100 yield to call(ytc) (long bond) (short bond) (duration) Definition: D = P V (t 0)t 0 + P V (t 1 )t P V (t n )t n ) P V D t 0 D t n Macaulay Duration Definition n k=1 D = (k/m)c k/(1 + (λ/m)) k n k=1 c k k (1+(λ/m)) YTM c y m n D = 1 + y my 1 + y + n(c y) mc((1 + y) n 1) + my 0 1 P P λ = D M (D M = D/(1 + (λ/m))) λ D M 1 P D M P λ 6

7 D M P P i D i (i = 1,,, m) m D m m D = w i D i (w i = P i / P i, D i = n k=0 t k P V (i) k /P (i) ) immunization (rebalance) T reinvestment risk (CD) convexity - (convexity) C = 1 P P λ P D M P λ + P C ( λ) ( ) yield curve inversed yield curve spot rate s t (t=0) t s 7

8 A s (1 + s ) A a) t (1 + s t ) t s t b) m m s t (1 + s t /m) mt c) s t e s tt (ii m ) spot rate curve discount factor (d t ) a) d k = 1 (1 + s k ) k b) m d k = 1 (1 + s k /m) mk c) d t = e s tt s 1 T P C F P = C + C + F 1 + s 1 (1 + s ) s 1 s s 3, s 4, A P A = 98.7 B 8 10 P B = A 0.8 B 1 0 8

9 P = P B 0.8P A = s 10 (1 + s 10 ) 10 P = 0 s 10 = 11. forward rate DEFINITION: t 1 < t t 1 t f t1,t t t 1 (market forward rate) (implied forward rate) a) (1 + s j ) j = (1 + s i ) i (1 + f i,j ) j i f i,j = ( ) 1/(j i) (1 + s j ) j 1 + s i ) i 1 b) m (1 + s j /m) j = (1 + s i /m) i (1 + f i,j /m) j i f i,j = m ( (1 + s j /m) j (1 + s i /m) i ) 1/(j i) m c) e st t = e st 1 t1 e ft 1,t (t t1) f t1,t = s t t s t1 t 1 t t 1 expectation hypothesis (liquid) expectation dynamics s 1, s,, s n s 1, s,, s n 1 foward rate f 1,j s j 1 s j 1 = f 1,j = ( ) 1/(j 1) (1 + s j ) j s 1 9

10 (expectation dynamics) (discount factor) d j,k k j forward rate ( ) k j 1 d j,k = 1 + f j,k k i k j j i i < j < k d j,k = d i,j d j,k 1 k r k r k = f k,k+1 s k k 1 (1 + s k ) k = (1 + r i ) k k 1 (1 + j j,k ) k j = (1 + r i ) r 0, r 1,, r n 1 r 1, r,, r n 1 n (1 + s n ) n i=0 i=j (running present value) P V (k) = x k + d k,k+1 P V (k + 1) PV(k) k x k k d k,k+1 = 1/(1+f k,k+1 ) k (reset) 4.1 (x t0, x t1,, x tn ) s t,t 0 t t n n P V = x ti e s t i t i i=0 10

11 (Fisher-Weil duration) D F W = 1 P V n t i x ti e s t i t i i=0 dp (λ) dλ P (λ) = n x ti e (st i +λ)ti i=0 n (λ = 0) = t i x ti e st i ti (relative price sensitivity) i=0 1 dp (0) P (0) dλ = D F W m k s k (x 0, x 1,, x n ) P (λ) = dp (0) dλ n k=0 x k (1 + s k + λ m ) k n = ( k m )x k(1 + s k m k=0 (k+1) P (0) D Q = 1 n dp (0) P (0) dλ = k=1 (k/m)x k(1 + s k /m) (k+1) n k=0 x k(1 + s k /m) k D Q (quasi-modified duration) 3 5 optimization capital budgeting 11

12 porfolio problem 0-1 zero-one variable 0-1 zero-one programming problem benefit-cost raito ) m i n i 1 C x ij 0-1 i j x ij 1 0 b ij benefit c ij cost m n i b ij x ij j=1 m n i c ij x ij C j=1 n i j=1 x ij 1 (i = 1,,, m) (optimal portfolio) (portfolio optimization) (cash matching problem) asset total return =/ rate of return =(-)/ short sales 1

13 σi σ i j σ ij σ = E [ (r r) ] [( n n ) ] = E w i r i w i r i = n w i w j σ ij i,j n 0 n feasible region minimum variance set minimum variance point :MVP risk averse nonsatiation efficient frontier minimize 1 w i w j σ ij i,j n n subject to w i r i = r w i = 1 r σ ( ) ( ) risk-free asset 1 F( ) F 1 13

14 7 market portfolio w i ( ) equilibrium r σ ( ) r = r f + r M r f σ σ M price of risk K = ( r M r f )/σ M (CAPM) M i r i r i r f = β i ( r M r f ) β i = σ im σ M β i CAPM σ im expected excess return r i r f r M r f i r i = r f + β i (r M r f ) + ε i CAPM ε i CAPM E(ε i ) = 0 Cov(ε i, r M ) = 0 σi = βi σm + var(ε i ) 1 (systematic risk) 0 (nonsystematic risk) (idiosyncratic risk) (specific risk) index fund 14

15 Jensen s index ˆ r r f = J + β(ˆ r M r f ) J Sharpe index ˆ r r f = Sσ S CAPM CAPM P = Q a + r f + β( r M r f ) r = Q P P P 1 + P = [ r f = r f 1 = 1 + r f [ 1 = 1 + r f P = 1 [ 1 + r f Q 1 Cov(Q 1, r M )( r M r f σ M Q Cov(Q, r ] M)( r M r f ) σm ] + 1 [ 1 + r f [ ( Q 1 + Q ) (Cov(Q 1, r M ) + Cov(Q, r M ))( r M r f ) σ M [ ( Q 1 + Q ) Cov(Q 1 + Q, r M )( r M r f ) σ M Q 1+ Cov(Q 1+, r M )( r M r f ) σ M Q Cov(Q ], r M )( r M r f ] ] = P 1+ ] σ M (NPV) NP V = P + 1 [ 1 + r f Q Cov(Q, r ] M)( r M r f ) σm harmony theorem NPV 9 risk neutral U(x) = x 15

16 V (x) = au(x) + b V (x) U(x) x y 1/ ( (U(x)+U(y))/) x/ + y/ U(x) + U(y) U( x + y ) ( ) ( ) U(x) = x ( ) = ( ) ( ) a(x) = U (x) U (x) (Arrow-Pratt absolute rsik aversion coefficient) U (x) U(x) = logx a(x) = U (x) U (x) = 1 x (x) (a(x) ) (a(x) ) C U(C) = E [ U(x) ] C 10 0 U(x) = x 0.04x C E(x) ( ) - U(x) = ax 1 bx E [ U(y) ] = E (ay 1 ) by = ae(y) 1 be(y ) = ae(y) 1 b[ E(y) ] 1 bv ar(y) 16

17 E(y) V ar(y) E [ U(y) ] - - U(x) µ Taylor U(x) = k=0 U (k) (µ) (x µ) k k! U(µ) + U (µ)(x µ) + U (µ) (x µ) E [ U(x) ] E [ U(µ) + U (µ)(x µ) + U (µ) (x µ) ] = U(µ) + U (µ)e(x µ) + U (µ) E(x µ) = U(µ) + U (µ) σ U(x) (U (x) < 0) E(U(x)) σ A () A (type A arbitrage) αd 1 + βd αp 1 + βp ( )(linear pricing)) B ( ) ( ) A B (P305) [ ( n )] Maximize E U θ i d i Subjecto to n θ i P i = W [ ( n L = E U )] ( n θ i d i λ ) θ i P i W 17

18 θ i E [ U (x d i ] = λpi R d i = R, P i = 1 λ = E [ U (x ] R λ E [ U (x )d i ] RE [ U (x ) ] d ( ) P P = E ( d R ) R d R R (R=1+r ) S (elementary state security)e s =< 0, 0,, 0, 1, 0,, 0 >, s = 1,,, S ψ s s 1 d =< d 1, d,, d s > d = S d ie i S P = d i ψ i ψ s, s = 1,,, S d =< d 1, d,, d S > P = S d i ψ i ψ 0 = S ψ i q s = ψ s /ψ 0 (q s ) P = ψ 0 S q s d s = ψ 0 Ê(d) Ê(d) q s ψ 0 ψ 0 = S ψ i ψ 0 1, 1,, 1 1 R 1/R P = 1 R Ê(d) (risk-neutral pricing) q s ( 1 ) q s 18

19 mature primary market: The market for new securities issues. In the primary market the security is purchased directly from the issuer. This differs from the secondary market. secondary market A market in which an investor purchases a security from another investor rather than the issuer, subsequent to the original issuance in the primary market. also called. InvestorWords.com issuer A company or municipality offering (or having already offered) securities for sale to investors. Examples include corporations, investment trusts. call callable bond zero-coupon bond 19

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, The Big Change of Life Insurance Companies in Japan Hisayoshi TAKEDA Although the most important role of the life insurance system is to secure economic life of the insureds and their