Notation: “Lx.y” refers to [Lecture x, Slide y]
Interest rate securities
1. Consider the bond pricing formula without semiannual compounding:
1−(1+y)−T F
Copyright By PowCoder代写 加微信 powcoder
B = C y + (1 + y)T or B = C
1−e−yT ey − 1
– Tutorial week 4 –
MATH3090/7039: Financial mathematics Tutorial week 4
Recall that
Using a face value of F = $100, 000, a coupon rate of 5%, and T = 25 years to maturity, calculate prices for
C = coupon rate × F.
YTMs of y = 4%, y = 5%, and y = 6%. What is the relationship between the coupon rate and the YTM?
(Bonds are usually issued at par, which means that P = F . When P < F they are said to be trading at a discount, and at a premium when P > F .)
What is the relationship between the price of a bond and its YTM y? (Hence, volatility in interest rates creates risk in the form of fluctuating bond prices. We will see later in the course how to manage this risk.)
Solution. Calculating the bond prices is straightforward. By substituting the variable values for F, C, y, T into the discrete equation above we obtain the following results: $115,622.10; $100,000; $87,216.64. If y > coupon rate then P < F , and if y < coupon rate then P > F . Inverse relationship between P and y. Look up the terms duration and convexity as measures of interest rate risk.
2. Show that the value of a bond issued at par is equal to its face value. You can assume annual cashflows.
Solution. Let c be the coupon rate. By assumption, c = y so we find that
1−(1+y)−T P = cF y
1−(1+y)−T = yF y
+ (1 + y)T
+ (1 + y)T
=F−F(1+y)−T + = F.
3. Take the YTM example on L2.47 with the initial guess y0 = 0.07 and calculate by hand y1 and y2 as given by the Newton scheme on L2.46. You may be asked to perform similar tasks on the exams.
Solution. Straightforward, refer to L2.46: substitute the variable values for Ct, y, P and t into the summation equations for f(y) and f′(y). Use the results for f(y) and f′(y) with y0 to compute y1.
Investment analysis
MATH 3090/7039 – 1 – -Piggott
– Tutorial week 4 –
4. Use the idea of a geometric sequence to prove the constant-growth dividend discount model:
∞ D0(1 + g)t D0(1 + g)
T D0(1 + g)t T 1 + g t (1+k)t =D0 1+k ,
(1 + k)t = k − g . k = D0(1 + g) + g.
Also quickly show that
Solution. Calculate that
use the formulas for a sum of a geometric sequence, take the limit as T → ∞, and rearrange. The second
part is clear.
5. You are considering buying a share whose current dividend is $1 and is expected to grow at a constant rate of 3%. What would you expect the share to be worth in 5 years time if your required rate of return k = 12%?
P5 = [D0(1+g)5](1+g) =(1+g)5D0(1+g) =(1+g)5P0 =$13.27. k−g k−g
6. You are considering the following two investment projects which both require an initial outlay of 80 and whose cashflows over the next few years are:
Project A pays 93 in yr1, 1 in yr2, and 5 in yr3. Project B pays 10 in yr1, 10 in yr2, and 90 in yr3.
Calculate the internal rate of return on each project. Use Excel or Matlab to plot a graph of the net present value of each project as a function of k the required rate of return for k between k = 0% and k = 30%. At what k are we indifferent between the projects?
Solution. See spreadsheet on Blackboard.
7. Suppose a coupon-paying bond has T years to maturity and pays coupons of C and a principal of F . If yields y, the probability of default p, and the recovery rate α are all constant over time, derive the pricing formulas for a bond under both discrete and continuous compounding assuming annual coupon payments.
Solution. The expected coupon is the same each year at E[C ] = pαC + (1 − p)C , and the expected principal is E[F ] = pαF + (1 − p)F . Hence, we can just use the standard valuation formula:
1−(1+y)−T E[F]
B = E[C] y + (1 + y)T .
(What about continuous time compounding?)
8. Youareanalysinganinvestmentprojectwhichisexpectedtocost$5,000,000andhavenetprofitsof$1,500,000 for the next 5 years (and zero afterwards). You have arranged funding from the following sources:
MATH 3090/7039 – 2 – -Piggott
– Tutorial week 4 –
• $1,000,000 from a 180-day bank-accepted bill with a face value of $1,059,178.08 quoted in simple interest.
• $500,000 from a 90-day promissory note with a face value of $517,877.41 quoted at a discount.
• $500,000 from a 5-year zero coupon bond with a face value of $1,000,000.
• $1,000,000 from a 5-year coupon-paying bond which pays annual coupons at a coupon rate of 6% and has a face value of $1,244,204.
• $1,000,000fromanordinaryshareissuewhereadividendamountof$154,500willbepaidinoneyears time and the market is expecting it to grow at 3% annually.
• $1,000,000 from a preference share issue whose annual dividend will be $120,000. Should you invest in the project?
Solution. Please be sure to use our convention that there are 365 days in a year. Please read the lecture notes to see where I have mentioned this. Bank-accepted bill is quoted in simple interest:
r=F −11 =1,059,178.08−1 1 =12%. P T 1,000,000 180
Promissory note is quoted via the discount rate convention:
d=1−P1 =1− 500,000 1 =14%.
F T 517,877.41 90 365
Please also see the lecture notes for the difference between these discount security market conventions. The zero is:
F T1 1,000,00015
P − 1 = 500, 000 − 1 = 14.87%.
The coupon-paying bond requires use of Newton’s method; the yield is 11.36%. Important: You may need tocalculatethisbyhandinthemid-semestertest. Thesharesare:
k=D0(1+g)+g=D1 +g= 154,500 +0.03=18.45%. P P 1,000,000
The preference shares are:
k= D = 120,000 =12%. P 1, 000, 000
WACC = 0.2×0.12+0.1×0.14+0.1×0.1487+0.2×0.1136+0.2×0.1845+0.2×0.12 = 13.65%. We calculate that
NPV=−IO+ C1 + C2 + C3 + C4 + C5 1+k (1+k)2 (1+k)3 (1+k)4 (1+k)5
= −5, 000, 000 + 1, 500, 000 + 1, 500, 000 + 1, 500, 000 + 1, 500, 000 + 1, 500, 000
so invest.
MATH 3090/7039
= $193, 237.45 > 0
1.1365 1.13652 1.13653 1.13654
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com