108156.pdf

30 Mathematics for Finance Example 2.4 Consider a loan of $1,000 to be paid back in 5 equal instalments due at yearly intervals. ...

Risk-Free Assets 31 perpetuity can be obtained from (2.7) in the limit asn→∞: nlim→∞PA(r, n)×C= C 1+r+ C (1 +r)^2 + C (1 +r)^3 ...

32 Mathematics for Finance 2.1.4 Continuous Compounding Formula (2.5) for the future value at timetof a principalPattracting int ...

Risk-Free Assets 33 Exercise 2.18 How long will it take to earn $1 in interest if $1, 000 ,000 is deposited at 10% compounded ...

34 Mathematics for Finance Exercise 2.21 Find the present value of $1, 000 ,000 to be received after 20 years as- suming continu ...

Risk-Free Assets 35 2.1.5 How to Compare Compounding Methods As we have already noticed, frequent compounding will produce a h ...

36 Mathematics for Finance Definition 2.1 We say that two compounding methods areequivalent if the corresponding growth factors ...

Risk-Free Assets 37 In particular, in the case of periodic compounding with frequencymand raterthe effective rateresatisfies ( ...

38 Mathematics for Finance admits only time instantstbeing whole multiples of the compounding period 1 m. An argument similar to ...

Risk-Free Assets 39 2.2 Money Market The money market consists of risk-free (more precisely, default-free) securi- ties. An ex ...

40 Mathematics for Finance Typically, a bond can be sold at any time prior to maturity at the market price. This price at timeti ...

Risk-Free Assets 41 For an average bank customer the information that a one-year $100 bond can be purchased for $92.59 may not ...

42 Mathematics for Finance dollars. Observe that the total wealth at time 1 is V(1) +C=V(0)er. Six months later the bond will be ...

Risk-Free Assets 43 = rF 1+r + rF (1 +r)^2 + F (1 +r)^2 = rF 1+r + F(1 +r) (1 +r)^2 =F. Conversely, note that C 1+r + C (1 +r) ...

44 Mathematics for Finance at timet. As a result, the investment will reach A(t)= A(0) B(0,T) B(t, T)=A(0)ert at timet≤T. Exerci ...

Risk-Free Assets 45 An alternative way to prolong an investment in the money market for as long as required is to reinvest the ...

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3. Risky Assets................................................ The future prices of any asset are unpredictable to a certain ex ...

48 Mathematics for Finance The current stock priceS(0) known to all investors is simply a positive number, but it can be thought ...

Risky Assets 49 Figure 3.1 Tree of price movements in Example 3.2 Exercise 3.1 Sketch a tree representing the scenarios and pr ...