2 a 11,460.09
b 42.62
c 8.33\%
4 26215.92
28575.35 - 12000 = 16575.35
19693.18 - 4000 = 15693.18
20323.12
---------Simple Interest----------
5 a 1012.50
b 201 days 207.71
6 a 304.17 days ---> 305 days
6th January
b 0.27
7 18.62%
8 a Note pays 10526.03
Bank pays 10503.58
b 7.99\% simple interest.
---------Present Value------------
9 21370.10
10 9837.74
--------Equations of Value-------
11 10500.00 = X + 2984.86
10000.00 = ((1.05)^-1 )X + 2842.73
X=7515.14
12 0 days 62 days 115 days
a i 10000 = x/(1+0.05 62/365) + 3937.96
= 0.9916 X + 3937.96
X = 6113.52
ii 10084.93 = X + 3971.16
X=6133.76
iii 10157.53 = 1.00726 X + 4000
X=6113.15
13 30000 = 12892.18 + 10101.36 + 20000 (1.05)^-n
n=21.50
Needs to wait 3.5 years
--------Nominal Rates of Interest---------
14 a 3135.76
b 3150.66
15 i 10.47%
ii 10.5%
iii 10.29%
16 a Note pays 13763.95
Bank pays 12057.88 (53 months)
b 5.36%
c 1.376395(1.0025)^(-12y)=1.055^(8-y)
.8968575052=.9766975894 ^y
4.62 years before maturity. So within the first 3.38
years. (about 3 years 4 months)
-------Rates of Discount----------------
17 1.0825 ---> 0.9238
10.16%
18 5.66%
--------Force of Interest---------------
19 5.13%
20 $1,008.37
------Inflation and the Real Rate of Interest---
21 a 20000(1.06)^t=17000(1.08)^t
t=8.69 years
b 4.39 years
------ANNUITIES-------------------------------
------Level Payment Annuities---------------
22 529 payments
$1,299,093.28
23 $9413.07
24 a $171,060.05
b $354.64
25 1st January 2003 (25) 2629.09
1st June 2004 (17) 2728.11 + 1746.10 = 4528.21
1st January 2005 (7) 4634.93 + 1060.56 = 5695.48
1st January 2007 (24) 6356.15 + 3796.28 = 10,152.43
1st January 2008 (12) 10995.08 + 1867.49 = 12862.57
1st April 2009 (15) 13520.92 + 2303.27 = 15824.19
1st January 2010 (9) 16305.29 + 912.09 = 17217.38
1st August 2012 (31) 19586.06 + 3301.77 = 22887.84
26 1190874.6121200000
3.58 years + 1 = 4.58 years
-------Level payment Annuities - Some Generalisations--------
27 32,049.20
28 monthly rate = 0.1663897%
109 payments
$172.69
29 (a)
0.8164846%
240 payments
$1903.33
Total accumulated value of payments=1407998.58
Total accumulated value of debt=1407997.74
overpayment=0.83
Final payment 1902.50
(b) (i) 80=(1-1.0081648^-n)/0.0081648
n=130.2
132=11 years.
(ii) 2481.15002 ---> 2481.16
Accumulated balance: 585054.4931785744
debt: 585052.1439800000
overpayment: 2.35
$2478.81
(c) (i) Monthly investment 596.67
$245,251.46
+0.83
$245,252.29
(ii) Monthly investment 18.84
3306.53 after 11 years.
+2.35
grows to 5184.15
plus 9 years of 2500
$250,821.63
$256,005.78
30 (a) 20=(1-1.025^-n)/0.025
-log 0.5/log 1.025=28.07
(b) Accumulated value of 28 payments =19929.90
Accumulated value of loan =19964.95
Balance owing = 35.05
Final payment 535.05
(c) 35.93
31 See sheet. 10.52%
32 6.14%
325,926.381 ---> 325,926.39
-------Annuities with non-constant payments---------
33 1st year 1st December balance = 2444.49
Annual effective rate 4.0741542919785%
value on 1st December in 25 years= 157,997.14
Value on 1st January in 25 years= 158,523.80
34 40000/year increasing by 5% =1156336.30
35000/year not increasing = 705490.85
Difference=$450,845.45
35 This year's dividends = 2.09 (at end of year).
Annual rate 12.55%
$29.09
---------Reinvestment Rates and Yield Rates-----------
36 (a) 3909.97
(b) semiannual coupons of $200 compound to 10324.54
Annual yield 9.53%
---------Depreciation--------------------
37 see sheet
38 (a) Cost at purchase time of maintainance:
8028.94
+43615.22
=51644.17
Costs of 91644.17 every 20 years in perpetuity
$133165.94
(b) Cost at purchase time of maintainance:
6798.57
+ 28777.28
= 35575.85
Costs of 75575.85 every 15 years in perpetuity
$129,691.64
(c) Perpetual maintainance costs:
33333.33
Purchase costs
92787.88
Total cost
$126,121.22
-----------Amortisation---------------
39 (a) 682.31
(b) See sheet
40 monthly rate=0.005750039
monthly payments 1751.04
(a) Value of payments made after 5 years: 125038.24
Value of loan after 5 years: 352649.69
Outstanding balance: $227,611.45
(b) New monthly rate=0.007363123
New monthly payment 2023.90
(c) (i) 429.25 months = 36 years
New payments 1749.47
(ii) Final payment at 7% had 240 payments to come, so
(1/1.00575)^241 of it was principal = $439.74.
This is 1/3.9784 of the new payment, which will be the
amount of principal repaid, when no of remaining payments
n has 1.00736^n<3.9784
188.2, so when 188 payments remaining, i.e. after 245
months or 20 years, 5 months.
41 Monthly rate 0.004938622
loan value after 7 years 605035.89
Accumulated value of 7 years payments 295678.65
Outstanding balance after 7 years = 309357.24
3 times monthly interest = 4583.40
Refinanced balance = 313940.63
See sheet j_12=5.67%
42 monthly payments 456.33
value of all regular payments 17950.24
value of loan 17950.21
reduction in final payment 0.04
(a) value of loan 15302.26
value of payments 1839.06
Outstanding balance 13463.20
(b) value of regular payments 13463.23
value of 0.04 reduction 0.03
Outstanding balance 13463.20
43 monthly rate 0.004938622
(a) 2136.57
(b) value of loan 318270
value of repayments 26364.85
outstanding balance 291923.15
penalty 4325.09
Refinanced balance 296248.25
value after 4 more years 375278.42
value of 4 years payments of 3000 = 162051.32
Balance after 5 years = 213227.09
No change:
value of original loan after 5 years 403174.91
value of first 5 years payments 148786.73
outstanding balance 254388.18
leftover payments of 863.43 a month at 3% interest
for 4 years = 43975.53
If he pays this at the 5-year point, balance will be
210412.65
44 monthly payments 2280.83
Interest rate 6.12% (see sheet)
45 (a) amortisation 10142.67
Sinking fund 7500
+ 2764.06
= 10264.06
(b) amortisation 3036.85
+ 10142.67
13179.52
value of sinking fund 258443.10
58443.10
after 9 more years, this grows to
125255.67
Need to accumulate 874744.33
New payments 5419.93
+7500
12919.93
46 (a) K=82387.35
183199.62
(b) see sheet
j_2=4.48%
47 10700
-2616.67
-4116.67
3966.67
48 (a) 1,183.58
(b) (69400t+35000)(1.035-0.07t) =
??+69379t-4858t^2
maximised when t=0.1400 = 1.68 months, or 51 days
49 (a) K=1781.39
P=5459.80
(b) K=1597.80
r=0.027
2103.14
50 see sheet j_2=5.82%
51 see sheet
52 (a) Price on 1st January 2011:
K=187.84
534.68
62 days
interest period has 181 days
flat price=538.78
(b) Accrued interest 4.28
Quoted price 534.49
53 (a) K=552.07
1111.98
1104.14
(b) Bond K=661.86
1190.21
strip 1323.71
54 see sheet
(a) 85.73
(b) 20.98
(c) 19.88%
(a)
Profitability index 225.954/140.219=1.61
100 (1+j)^10 + 10(1+j)^9+50=368.055
55 see sheet
56 see sheet
(a) j_12=17.75%
(b) -2.82%
57 (a)
1 -16200 2000 -14200
2 -15336 4000 -11336
3 -12242.88 4000 -8242.88
5 -9614.50 10000 385.50
25 572.84
(b)
1 -21600 2000 -19600
4 -24690.36 11500 -13190.36
5 -14245.58 14000 -245.58
25 -1144.65 1000 -144.65
(c)
5 139.92
25 207.92 1000 1207.92
-----------Spot Rates of Interest---------
58 See sheet
59 See sheet
60 See sheet
----------Forward Interest Rates----------
61 (a) 7.26%
(b) 7.51%
62 15.14%
-----------Applications and Illustrations--------
63 forward rate = 8.01%
1,134,225
1,219,291.875
1,225,043
income
5,751.125
64 54000
50000
4000
65
66 Forward rates:
5%
6.00%
5.80%
7.21%
6%
Interest payments:
30000
36014.29
34801.71
43254.68
36000
Present value of Outstanding balance after 5 years
448354.90
Present value of all interest payments
151645.10
=X(1.05^-1+1.055^-2+1.056^-3+1.06^-4+1.06^-5)
X=35770.57
67 See sheet.
5.97%
---------Duration of a Set of Cashflows and Bond Duration---------
68 K=86.17
D=(1+i)/i-(1+i+n(r-i))/(r((1+i)^n-1)+i)
(a) Price 95.39
Duration 6.24 periods = 3.12 years
(b) Price 109.22
Duration 5.31 = 2.65
(c) Price 155.33
Duration 4.73=2.36 years
69 (a) Present value 7835.26
1473.35
=9308.61
d/di = 37310.7698318309 + 27720.5246096556
=65031.2944414865
d/di / P = 6.986
yield 14.5 %
Face value 27523.47
(b) Present value 7651.34
+1559.02
=9210.37
d/di 36262.2916514895+29415.5402723886 = 65677.8319238781
Modified Duration=7.13
-----------Asset-Liability Matching and Immunisation----------
70 K=623.17
P=937.19
Duration=13.277=6.639 years.
So payment must be 1387.63
71 Present value of liability = 9330.15
d/di = 67855.6189395970
Present value of A_15 = 4976.0787222371
A_0=4354.07
72 (a) Duration of bond = 36.05 coupons=18.025 years
So the single payment must be after 18.025 years.
The present value of the bond is given by Makeham's formula
K=409295.97
P=1393802.69
Payment=2383977.33
Second Derivative of payment 1 after $n$ years is given by
n(n+1)(1+i)^{-(n+2)}P
This is (n+1)/(1+i) times the modified duration or
(n+1)/(1+i)^2 times the Macauley duration. For the single
payment, This means the second derivative puts more weight on
the later payments. For the bond, this means that the second
derivative is larger, so the bond cannot be immunised by a
single payment.
73 (a) d/di = 1.1785825430 +
2.2675736961+3.2801984470+4.2219491788+5.0969396247 +243.0935452476
=259.1387887372
present value = 97.5949670036
Modified duration = 2.6552
(b) Macauley duration = 5.64 periods = 2.82 years.
Modified duration 2.6614