Interest Functions and Bond Pricing: Annuities, Loans, and Bonds - Prof. John Dinius, Study notes of Mathematics

Download Interest Functions and Bond Pricing: Annuities, Loans, and Bonds - Prof. John Dinius and more Study notes Mathematics in PDF only on Docsity! Interest Functions 1 (1 ) v i = + 1 1 i d v iv i = − = = + 1 d d i d v = = − i d id− = ln(1 )iδ = + Accumulation function: ( ) (1 )n na n i eδ= + = - for constant force of interest 0 ( ) ( ) n t dt a n e δ∫ = - for variable force of interest Definition of “force of interest”: ( ) [ ( )] / ( ) ( ) ( ) a t d a t dt t a t a t δ ′ = = Nominal interest rates: 1 ( ) [(1 ) 1]m mi m i= ⋅ + − ( ) 1 1 m m i i m   + = +    1 ( ) (1 )m md m v= ⋅ − ( ) 1 1 m m d d v m   − = − =    ( ) 1 1 m m d i m −   − = +    Level Annuities 1 n n v a i − = (1 ) 1 (1 ) n n n n i s a i i + − = ⋅ + = 1 (1 ) n n n v a i a d − = = + ⋅

1 1(1 ) 1 (1 ) (1 1) (1 ) (1 ) (1 ) n n n n n n n n i i s i s i a i a d i + ++ − + − + = = = + ⋅ = + ⋅ = + ⋅



( ) ( ) ( ) / 1 m n m mn i mn i m v a a i − = = (similarly for ( )m n a

, ( )m n s , ( )m n s

: use i (m) or d (m) in denominator) 1 n n v a δ − = (similarly for ( )m n s : use δ in denominator) Perpetuities: 1 a i∞ = 1 a d∞ =

1 a δ ∞ = ( ) ( ) 1m m a i∞ = ( ) ( ) 1m m a d∞ =

Bonds Assuming the bond’s maturity value = face value (F): Pr ice n n F v F r a= ⋅ + ⋅ ⋅ = PV of maturity value + PV of coupons ( ) n F F r i a= + ⋅ − = face value plus premium (or minus discount, if r < i) ( ) r K F K i = + − , where nK F v= ⋅ Amount of premium or discount amortized with kth coupon payment 1( ) n kF r i v − += ⋅ − ⋅ Price between payment dates [t = (no. days since last coupon)/(no. days in coupon period); i = yield rate (effective interest rate) per coupon period]: Total price (including accrued interest) = Price on prior coupon date (1 )ti⋅ + 0 (1 ) t P i= ⋅ + (Note that total price accumulates with compound interest at the yield rate.) Accrued interest = t F r⋅ ⋅ =t times amount of coupon (Actually, this is the accrued portion of the next coupon payment, but it is referred to as “accrued interest.” Note that coupons accrue at simple interest at the coupon rate.) Market Price 1 = (Total price) – (Accrued interest) 0 (1 ) ( ) t P i t Fr= ⋅ + − ⋅ If the market price exceeds the bond’s par value, because the bond’s coupon rate exceeds the market interest rate (the yield rate), the bond will be called at the earliest possible call date (unless market interest rates rise). Exception: If the call premium exceeds the bond’s current premium (calculated based on the bond’s maturity date and the current market interest rate), it will not be called. If the market price is less than the bond’s par value, because its coupon rate is less than the market interest rate, it will not be called before maturity (unless market interest rates fall). 1 “Price” (or “market price”) means the price of the bond excluding accrued interest. The buyer must pay the seller the total price (market price plus accrued coupon), but the bond’s price is quoted without the accrued interest. (Buyer beware!) Spot rates and Forward rates The n-year spot rate is the accumulation rate (expressed as an annual rate) for an n-year investment (an n-year zero-coupon bond). The symbol for this rate is sn in Module 6, and is 1/P(0,n) in Module 14 of the ACTEX manual. The n-year accumulation factor based on the spot rate is ( ) (1 )n n a n s= + . The “(n - t)-year-forward, t-year rate” is the rate that will apply during the t-year period from (n-t) to n. It can be calculated by comparing the accumulation factors for (n - t) years and for n years: , (1 ) (1 ) (1 ) n t n n t n n t n t s i s − − − + + = + , or ,(1 ) (1 ) (1 ) n n t t n n t n t ns s i − − − + = + ⋅ + The latter formula states that the accumulation factor for n years can be regarded as representing accumulation at sn for n years, or accumulation at sn-t for (n - t) years, followed by accumulation at in-t,n for t years. Forward rates are commonly quoted as “n-year forward rates,” which means the “n-year-forward, one-year rate,” which is the one-year rate that currently applies for the period between n and n+1: 1 1 ,1 (1 ) (1 ) (1 ) n n n n n s i s + + + + = + . Duration, Convexity, and Immunization Macaulay duration: t t t t t v CF D v CF ⋅ ⋅ = ⋅ ∑ ∑ Modified Duration: / 1 dP di D DM P i = − = + Duration of a level-payment loan: ( ) n n Ia D a = Duration of n-year zero-coupon bond: D n= Duration of coupon bond (with face F, coupon Fr, and redemption value C): ( ) ( ) BondPrice n n n n n n Fr Ia n C v Fr Ia n C v D Fr a C v ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ = = ⋅ + ⋅ Approximations of a security’s change in price due to a change, i∆ , in interest rate: 1 st approximation: ( )P DM P i i∆ = − ⋅ ⋅ ∆ , where P(i) = price at yield rate i 2 nd approximation: 2( )( ) ( ) ( ) 2 P i i P DM P i i Convexity ∆ ∆ = − ⋅ ⋅∆ + ⋅ Criteria for immunization 2 A portfolio is “immunized” if the assets and liabilities are equal in value and in duration, but the assets have greater convexity. PV(assets) = PV(liabilities) at i0: 0 0 t t t i t iA v L v⋅ = ⋅∑ ∑ Duration (assets) = Duration (liabilities): 0 0 t t t i t it A v t L v⋅ ⋅ = ⋅ ⋅∑ ∑ Assets have greater Convexity: 0 0 2 2t t t i t it A v t L v⋅ ⋅ > ⋅ ⋅∑ ∑ 2 Immunization describes the situation where, if the value of a company’s assets equals the value of its liabilities, then after a small change in the interest rate (either up or down), the value of the company’s assets will exceed the value of its liabilities. This will be the case if the assets and liabilities have the same duration, but the assets have a greater convexity.