Tutorial 9

Question 1

1) The 2, 4 and 5 year spot rates are 5%, 4.25% and 3.85% pa respectively. The 2 year forward
rate from time 4 is 3.5% pa. Calculate the following:
i) y,6
ii) f,5
iii) f2,2

Answer 1

So y,2 =5%, y,4 =4.25%, y,5 =3.85%, f4,2 = 3.5%

i) y,6

(1+f4,2)^2 = (1+y,6)^6/(1+y,4)^4 
y,6 = [1.035^2x1.0425^4]^1/6   -1 = 4%

ii) f5,1

(1+f,5) = (1+y,6)^6/(1+y,5)^5 = 1.04^6/1.0385^5 -1 = 4.75%

iii) f2,2

(1+f2,2)^2 = (1+y,4)^4/(1+y,2)^2  
f2,2 = [1.0425^4/1.05^2]^1/2   -1 = 3.5%

Question 2

2) Consider a nine-year fixed-interest bond that pays coupons annually in arrears at a rate of 7% pa and is redeemable at par.
i) Calculate the present value and the discounted mean term (DMT) of this bond, assume i = 5% pa.
ii) Using the DMT, calculate the change in the price of the bond if i is now equal to 6% pa.

Answer 2

i) DMT = (0.07x100x(I𝑎)9¬ + 100x9v^9)/(0.07x100x𝑎9¬ + 100v^9)
Top line = 0.07x100x33.2347 + 100x9/(1.05)^9 =812.79
Bottom line = 0.07x100x7.1078 + 100/(1.05)^9 =114.22
DMT = 812.79/114.22 =7.116 years

ii) Price of bond at 5% = £114.22
Revised price = 114.22x(1.05/1.06)^7.116 = £106.77
Check 0.07x100x6.8017 + 100/(1.06)^9 =£106.80
Change in price = 106.77-114.22 = £-7.45

Question 3

3) An asset provides the following cash flows:
£55,000 at t =5
£65,000 at t = 9
Calculate the present value, volatility, discounted mean term and convexity at i =4% pa.

Answer 3

PV = 55000(1.04)^-5 + 65000(1.04)^-9 = £90,874.13 
Vol = (55000x5(1.04)^-6 + 65000x9(1.04)^-10 )/PV =£612,541.53/90874.13 = 6.74 
DMT = (55000x5 (1.04)^-5 + 65000x9(1.04)^-9 )/PV= £637043.19/90874.13 = 7.01 yrs 
Test = Vol x1.04 = 7.01 yrs 
Convexity = (55000x5x6(1.04)^-7 + 65000x9x10(1.04)^-11)/PV= 5053912.84/90874.13 = 55.61

Question 4

4) A pension fund must make the following payments at i = 6% pa
£100,000 at t = 4
£100,000 at t = 8

Show that immunisation can be achieved by purchasing the following portfolio

A nominal amount Y of a 3 year zero coupon bond
A nominal amount Z of a 9 year zero coupon bond.

Answer 4

Condition 1
V,A(i,0) = V,L(i,0)
PV liabilities @ 6% pa = 100000v^4 + 100000v^8 = £141,950.60
PV assets @ 6% pa = Yv^3 + Zv^9 = £141,950.60 ………(Eqn1)

Condition 2
v,A(i,0) = v,L(i,0) or V’,A(i,0) = V’,L(i,0)
Volatility liabilities @ 6% pa = (100000x4v^5 + 100000x8v^9)/V,L(i,0)= £772422.04/ V,L(i,0)
Volatility assets @ 6% pa = (Yx3v^4 + Zx9v^10)/ V,A(i,0) = 772422.04/ V,A(i,0)
Yx3v^4 + Zx9v^10 = 772,422.04 …………………(Eqn2)

Yv^3 + Zv^9 = £141,950.60 ………(Eqn1)
Yx3v4 + Zx9v10 = 772,422.04 …………………(Eqn2)
Solve for Z
3v(141950.60 - Zv^9) + Zx9v^10 = 772422.04

Z = £110,637.10
Solve for Y
Yv^3 + 110637.1v^9 = £141950.60
Y = £91,070.65

Condition 3
c,A(i,0) > c,L(i,0) or V’’,A(i,0) = V’’,L(i,0)
Convexity liabilities @ 6% pa
(100000x4x5v^6 + 100000x8x9xv^10)/V,L(i,0) = £5,430,363.48/ V,L(i,0)
Convexity assets @ 6% pa
(91070.65x3x4v^5 + 110637.1x9x10v^11)/ V,A(i,0) = 6,062,041.42/ V,A(i,0)

All three conditions hold for this portfolio at i = 6% pa