An Introduction to Mathematics 金融学数学基础知识

Answer
FV 15,000 1.5 ln(1.5) 0.405 0.135 e0.135 1.144 r = C0 (1+r)T = 10,000(1+r) 3 = (1+r) 3 = ln(1+r) 3 = 3ln(1+r) = ln (1+r) = 1+r =1+r ≈ 0.144 or 14.4%
THE VALUE OF MONEY: FUTURE VALUE
• Let us assume that one invests an amount of money at time zero (C0) in an account for T years and that the going annual interest rate is r. Then, the value of this amount after the completion of T years shall equal: FV = C0 (1+r)T where FV = Future Value • However, the above assumes annual compounding of the interest, yet the latter could be compounded: Semiannually: FV = C0 [1+(r/2)]2T Quarterly: FV = C0 [1+(r/4)]4T Monthly: FV = C0 [1+(r/12)]12T Daily: FV = C0 [1+(r/360)]360T Continuously: FV = C0 erT
Introduction to Mathematics & Statistics
Dr. Bill Kallinterakis Durham Business School
SETS (1)
• • • • • • • A set is a well-specified collection of elements These elements can be finite or infinite Finite set: A = {1,2,3,4,5,6,7,8,9,10} Infinite set: B = {x | x>0} Membership to a set expressed as: a ∈ A Conversely: a ∉ A Empty set: A = { } ,also expressed as A = ∅
THE VALUE OF MONEY: PRESENT VALUE
• Let us assume that an investor wishes to retain an amount C after T years of investment with the going annual interest rate equal to r. Then the amount of money he would have to invest now to obtain the amount C in the future is calculated as:
THE VALUE OF MONEY: FUTURE VALUE (continued)
• In general, compounding an investment m times a year over many years provides wealth of FV = C0 [1+(r/m)]mT
SETS (2)
• Let A = {1,3,5,7}, B = {2,3,4,6,8} and C = {1,5}. Then it holds that: A ⊃C or C ⊂ A, or else that “C is a subset of A”
• Let A = {1,2,3} and B = {4,5,6}. These two sets bear no common elements and are known as Disjoint • Let A = {1,2,3,4,5} and B = {1,2,3,4,5}. Then these sets are called Equal
Present va lue P Present va lue P Present va lue P V4 n [1 ( r / 4)]4 n V12 n [1 ( r / 12)]12 n V360n [1 ( r / 360)]360n
• Continuously:
THE VALUE OF MONEY : PRESENT VALUE (continued)
• In case the expected cash flows are unequal in size, the present value is calculated as:
Vi V3 Vn V1 V2 PV ... i 2 3 n i 1 (1 r ) (1 r ) (1 r ) (1 r ) (1 r )
Answer
Vn Present value P n (1 r )
X X = 100,000/(1+0.05) 3 ≈ £86,384
Exercise
You have ordered a tugboat and the manufacturer has informed you that he can prepare it immediately at the cost of £ 40,000. However, she has also advised you that you could receive the tugboat after 3 years for the price of £ 50,000. What is the interest rate at which you would be indifferent between buying the tugboat now and in 3 years’ time?
SETS OF NUMBERS
• Natural (“counting”) numbers
– ℕ = {1, 2, 3, 4,.....}
• Integers
– ℤ. = {0, 1, -1, 2, -2, .....}
• Rational numbers: ℤ : ℤ = ℚ 1 1 1 – ℚ = 1 , , 2 , 2 , , 2 2 3 3 ,...... • Irrational numbers: cannot be expressed as ℤ / ℤ • Example: 2 • Real numbers: R • ℕ ⊂ ℤ⊂ ℚ ⊂R
Vn Present va lue P rn e
Example
• Your parents have set up an endowment fund that will allow you to receive £ 100,000 once you reach the age of 30. Assume at the moment that you are 27 and that the going interest rate is 5 percent. What is the present value of this future cash flow?
Vn Present value P n (1 r )
THE VALUE OF MONEY: PRESENT VALUE
• However, the above assumes annual compounding of the interest, yet the latter could be compounded: V2 n • Semiannually: Present va lue P [1 ( r / 2)]2 n • Quarterly: • Monthly: • Daily:
i n
THE VALUE OF MONEY: SPECIAL CASES
• Perpetuity: constant stream of equal-sized cash-flows with no end
V Present value of a perpetuity r
• Thus, if a regular perpetual cash flow of £ 100 is to be received and the discount rate is 8%, then the present value of this series is £ 1250 (calculated as £ 100/.08). • Example: UK government consols (bonds)
Answer
Vn Present value P (1 r ) n
40,000 = 50,000/(1+r)3 0.8 = 1/(1+r)3 1.25 = (1+r)3 ln(1.25) = ln(1+r)3 0.223 = 3 ln(1+r) 0.074 = ln (1+r) e0.074 = 1+r 1.077 = 1 + r r ≈ 0.077 or 7.7%
THE VALUE OF MONEY: SPECIAL CASES
• Growing perpetuity
V Present va lue of a growing perpetuity rg • where g is the rate of growth per period expressed as a percentage