Chapter 1 Formulas. Mathematical Object. i (m), i(m) d (m), d(m) 1 + i(m)

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1 F2 EXAM FORMULA REVIEW Chapter 1 Formulas Future value compound int. F V = P V (1 + i) n = P V v n Eff. rate of int. over [t, t + 1] Nominal, periodic and effective interest rates i t+1 := a(t+1) a(t) a(t) ( ) m i (m), i(m) m, and 1 + i(m) m 1 Present value of n term payments of 1 ν n := ( ) 1 n 1+i Discount rate d := iv = 1 v = i 1+i Nominal, periodic and effective discount rates ( d (m), d(m) m, and 1 1 d(m) m ) m Nominal interest-discount rate equivalence ( 1 + i(m) m ) m = ( 1 d(p) p ) p 1

2 2 F2 EXAM FORMULA REVIEW Chapter 1 (cont.) and Chapter 2 Formulas Force of interest, δ (i.e. cont. growth rate), for compound interest δ = ln(1 + i) so v = e nδ Derivative formula for the force of interest, δ(t) Accumulation, a(t) derived from a continuous force of interest, δ(t) δ = a (t) a(t) [ ] t a(t) = Exp 0 δ rdt (1) Exact day counting (2) Ordinary day counting (3) Banker s Rule day counting (1) actual/actual: count exact no. of days per month / 365 days in each year (2) 30/360: count 30 days per month / 360 days per year (3) actual/360: count exact no. of days per month / 360 days in each year

3 F2 EXAM FORMULA REVIEW 3 Chapter 2 Formulas Chapter 2 Geometric Series n=0 r n = 1 1 r Geometric Progression (i.e. Finite Geometric Series) n k=0 r k = 1 rn+1 1 r Annuity Immediate Present Value (PV) a n = 1 vn i Annuity Immediate Future Value (FV) s n = (1+i)n 1 i = (1 + i) n a n Annuity Due PV ä n = 1 vn d = (1 + i)a n Annuity Due FV s n = (1+i)n 1 d = (1 + i)s n

4 4 F2 EXAM FORMULA REVIEW Chapter 2 Formulas (cont.) Perpetuity Immediate PV a = 1 i Perpetuity Due PV ä = 1 d Continuous Annuity PV a n = 1 vn δ Continuous Annuity FV s n = (1+i)n 1 δ Annuity Increasing Payments PV (Ia) n = än nv n i Annuity Increasing Payments FV (Is) n = s n n i = (1 + i) n (Ia) n Annuity Due Increasing Payments PV (Iä) n = än nv n d = (1 + i)(ia) n Annuity Due Increasing Payments FV (I s) n = s n n d = (1 + i) n (Iä) n

5 F2 EXAM FORMULA REVIEW 5 Chapter 2 Formulas (cont.) P, P + Q,... Payments PV P a n + Q ( an nv n i ) P, P +Q,... Payments Perpetuity P i + Q i 2 Temporal Addition Rule a n+k = a n + v n a k Chapter 3: Ammortization Prospective Method = Retrospective Method (Verbally) PV of Future PMTs = Acc. Value of Loan Acc. Value of Past PMTs Prospective Method = Retrospective Method (Level PMT Formula) a n t = a n (1 + i) t s t P MT = 1; Totals of PMTs, interest, Princ. and Loan Bal. n, n a n, a n, 0, repsectively P MT = 1; Outstanding balance in period k a n k P MT = 1; Interest paid, period k 1 v n k+1 P MT = 1; Principal paid in period k v n k+1

6 6 F2 EXAM FORMULA REVIEW Chapter 3: Ammortization Formulas (cont.) Interest paid in P mt k+1 (General Payments) Int k+1 = i(bal k ) (Gen- Principal paid in P mt k+1 eral Payments) P Rin k+1 = P mt k+1 i(bal k ) Balance after P mt k+1 Payments) (General Bal k+1 = (1 + i)bal k P mt k+1 Level P MT amount for loan, L, lasting n periods P MT = L a n Level Payment Int k P MT (1 v n k+1 ) Level Payment P Rin k P MT v n k+1 Level Payment Temporal Addition Rule P Rin n+k = (1 + i) k P Rin n

7 F2 EXAM FORMULA REVIEW 7 Chapter 3 Formulas (cont.) Level Payment Balance (Prospective Method) Bal k+1 = a n k Level Payment Balance (Retrospective Method) Bal k+1 = P V (1 + i) k s k Sinking Fund Deposit L s nj Total Sinking Fund PMT L s nj + Li Sinking Fund Balance Sinking Fund Principle Paid SF Bal k = SF Ds k j = Ls k j s nj SF P Rin k = SF D(1 + j) k 1 Chapter 4 Price of a Par F Bond with Coupon Rate r, Interest i, and Redemption Value C P = (F r)a ni + Cv n i Bond Amortization (i.e. principal paid) in Period k F (r i)v n k+1 Bond Amortization in the Period k + m in Terms of the Period k Amoritzation (1 + i) m F (r i)v n k+1

8 8 F2 EXAM FORMULA REVIEW Chapter 4 Formulas (cont.) Price of a bond if F = C P = (F r)a ni + F v n i (Premium-Discount Formula) Price of a bond if F = C P = F F (r i)a ni (Makehelm Formula) Price of a bond if F = C P = K + r i (F K) where K = F vn i Chapter 5 Internal Rate of Return (IRR) or Yield Soluition of n C k v k = 0 k=0 Effective rate over [t k 1, t k ] 1 + j k = B k B k 1 +C k 1 Time-weighted rate (1 + j) = n (1 + j k ) k=1 Dollar-weighted rate i = B A C A+ C t (1 t) where A and B are the initial and final balances, resp., and C t costs

9 F2 EXAM FORMULA REVIEW 9 Chapter 6 Formulas n-year Forward Rate i n 1,n = (1+s n) n (1+s n 1 ) n 1 and i 0,1 = s 1 n-year Forward Rate Accumulation Formula (1+i 0,1 )(1+i 1,2 ) (1+i n 1,n ) = (1+s n ) n Chapter 7 Investment Price P = n v k CF k k=1 Macaulay Weights w k = vk CF k P Macaulay Duration D = n kw k k=1 Modified Duration MD = ( dp di ) P = D 1+i Duration of Level Payment Investment D = (Ia) n a n Macaulay Duration of a Face F Coupon Bond with Coupon Rate r, Interest i, and Value C D = F r(ia) n +ncv n F r(a n )+Cv n

10 10 F2 EXAM FORMULA REVIEW Chapter 7 Formulas (cont.) Convexity P (i) P (i) First Order Change in Price Approximation p (DM)P (i) i Second Order Change in Price Approximation p (DM)P (i) i + (Conv.) P (i)( i)2 2

Chapter 04 - More General Annuities

Chapter 04 - More General Annuities 4-1 Section 4.3 - Annuities Payable Less Frequently Than Interest Conversion Payment 0 1 0 1.. k.. 2k... n Time k = interest conversion periods before each payment n