Notation: “Lx.y” refers to [Lecture x, Slide y] 1. Use l’Hopital’s rule to show that
Solution. First note that
ynt ln 1+n
When n → ∞ we get the indeterminate form 0/0, so l’Hopital’s rule yields that
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2. Prove that
ynt = lim exp ln 1+
– Tutorial week 3 –
MATH3090/7039: Financial mathematics Tutorial week 3
y =ntln 1+n
= tln(1 + y/n). 1/n
ynt lim ln 1 +
d ln(1+y/n) = t lim dn
n→∞ d (1/n) dn
n→∞ n =exp lim ln1+ynt
n→∞ n = exp{yt} = eyt.
1+a+a2 +…+aT−1 = (1−aT). 1−a
Solution. Direct calculation yields that (1 − a)(1 + a + a2 + . . . + aT −1) = 1 − aT . 3. Derive the discrete compounding annuity formulas:
(1+y)T −1 1−(1+y)−T FV=Cy andPV=Cy.
Derive the continuous compounding annuity formulas:
eyT −1 1−e−yT F V = C ey − 1 and P V = C ey − 1 .
Also derive the formulas for incorporating n (the cashflow frequency).
MATH 3090/7039 – 1 –
– Tutorial week 3 –
Solution. For the future value formula, draw a timeline to see that
FV =C(1+y)T−1 +C(t+y)T−2 +…+C(1+y)2 +C(1+y)+C
=C(1+y)T−1 +(t+y)T−2 +…+(1+y)2 +(1+y)+1
= C 1 − (1 + y)T 1−(1+y)
=C(1+y)T −1. y
The future value formula for continuous compounding is derived analogously. For the present value formula, draw a timeline to see that
PV=C+C+…+C+C 1+y (1+y)2 (1+y)T−1 (1+y)T
C111 = 1+y 1+ 1+y +…+ (1+y)T−2 + (1+y)T−1
1−1 = C (1+y)T
1+y1−1 1+y
1−1 =C (1+y)T .
The present value formula for continuous compounding is derived analogously.
Incorporating more frequent compounding involves replacing C by Cn , replacing T by nT , and replacing y
by ny . In particular
(We are assuming that C is always stated as an annual cashflow.)
CeyT −1 C1−e−yT FV=n eny −1 and PV=n eny −1 .
4. We have been assuming for an annuity cashflow that the payments begin at the end of the 1st period. Suppose the payments begin at the start of the first period, so the last payment is made at the start of the last period. What is the present and future value formulas for an annuity in this case? (Assume annual cashflows so n = 1.)
Solution. For the future value formula, draw a timeline to see that
FV =C(1+y)T +C(1+y)T−1 +…+C(1+y)2 +C(1+y)
=(1+y)C(1+y)T−1 +C(t+y)T−2 +…+C(1+y)+C =(1+y)C(1+y)T −1.
The formula for continuous compounding is calculated analogously.
MATH 3090/7039 – 2 – -Piggott
– Tutorial week 3 –
For the present formula, draw a timeline to see that
PV=C+C+…+C+C 1+y (1+y)T−2 (1+y)T−1
CCCC =(1+y) 1+y+(1+y)2 +…+(1+y)T−1 +(1+y)T
1−1 =(1+y)C (1+y)T .
The formula for continuous compounding is calculated analogously.
5. A bond is characterised by a promise to pay the principal F at maturity and an annuity cashflow of coupon payments C.
Derive the bond pricing formula with continuous compounding when payments are made semiannually.
C1−e−yT −yT P=y+Fe.
6. The Australian government issues three types of fixed interest securities (see Australian Office of Financial Management, where pricing formulas are also provided):
• Treasury bonds: Treasury Bonds are medium to long-term debt securities that carry an annual rate of interest fixed over the life of the security, payable six monthly.
• Treasury indexed bonds: Treasury Indexed Bonds are medium to long-term securities for which the capital value of the security is adjusted for movements in the Consumer Price Index (CPI). Interest is paid quarterly, at a fixed rate, on the adjusted capital value. At maturity, investors receive the adjusted capital value of the security – the value adjusted for movement in the CPI over the life of the bond.
• Treasury notes: Treasury Notes are a short-term debt security issued to assist with the Australian Government’s within-year financing task. Terms are generally less than six months.
Derive the pricing formula for a Treasury indexed bond.
Solution. Keep grappling with this one if you can’t get it.
7. A fixed interest loan is a classical example of an annuity. The loan amount is viewed as the present value of the loan payments, themselves an annuity cashflow stream. Suppose you want to buy a $500, 000 house. You have $100, 000 for a deposit, so you want to borrow $400, 000 from the bank. Suppose the interest rate is 6.5% fixed for a 30-year loan, with weekly loan payments. What is the size of the loan payments? How much money do you end up paying the bank overall? Draw up a loan schedule in Excel which works out the principal paid off and the interest paid for each loan payment. How much of the loan payments are principal and how much are interest early in the loan compared to late in the loan?
Solution. Loan schedule provided on Blackboard.
(We are assuming the coupon C is always expressed in annual terms.)
MATH 3090/7039 – 3 – -Piggott
– Tutorial week 3 –
8. Write a Matlab program that implement formulas on L1.30 for the case where all cashflows are the same and is equal to C, T is a positive integer, and all cashflows are one year apart. For the case of non-constant yield, assume that y(t) = 0.045 + 0.002t2.
Solution. Matlab code provided on Blackboard.
MATH 3090/7039 – 4 – -Piggott
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