lecture 101 macroeconomic analysis financial market and expectations
TRANSCRIPT
Lecture 10 1
Macroeconomic Analysis
Financial Market and Expectations
Lecture 10 2
Contents• Financial system and its importance in the Economy• Determination of Instalment payments• Growth of funds and their present values• Fisher Equation: Real and nominal interest rates • Bond market: Value of bonds and yield to maturity• Arbitrage condition• Stock market: value of stocks by the interest rate,
risk premium and growth of dividends• Investment decision problem• Impact of changes in the interest rate in the price of
assets
Lecture 10 3
Y= F(K,L)
SC T
Funds
K FA
EquityTreasury
Bonds
DepositBanks
Pension FundsProfit
Link Between Financial System and the Economy
Lecture 10 4
Savers
Households, Corporations and Government
Intermediaries
Banks, Insurance Companies, Building Societies, Trusts, Stock and Bonk Markets
Intermediaries
Banks, Insurance Companies, Building Societies, Trusts, Stock and Bonk Markets
InvestorsSmall, Medium and LargePrivate, Public, Domestic and Foreign
InvestorsSmall, Medium and LargePrivate, Public, Domestic and Foreign
Major Players in a Financial Market
CENTRAL
BANK
CENTRAL
BANK
GOVERNMENT&FSA
GOVERNMENT&FSA
CATS
http://www.dmo.gov.uk/http://www.fsa.co.uk
http://www.londonstockexchange.com/default.asp
Lecture 10 5
Financing an Investment
Project
Self FinanceBequests
Bonds:Debt Finance
Banks, BuildingSociety, Insurance
Equity FinanceStock Market
(LSE)
NoRisk Risk
HighRisk
MaturityInstalment
MethodRepayment
Method
Financing of an Investment Project
Demand for output
Need for Capital
Lecture 10 6
Total debt D
Regular and equal instalments T
Db
Declining balance at time t tbDBt
Effective interest rate for each instalment: 1 tT
i
where t = 0..T-1
Interest payment for declining balance: tBtT
i
1
Total payment tBtT
ib
1
Average instalment TBtTi
tbt
t
1
Calculations of Instalments on a Loan: Lower Interest Rate Reduces the Interest Payment
Lecture 10 7
D =20000 i = 8.9 APR 67.166612
20000b
Instalment payment in the seventh month
25.179067.16667200006
089.067.1666
6 6
BT
ib
Instalment payment in the ninth month
92.177767.16669200004
089.067.1666
8 8
BT
ib
What should be the instalment payment for the 5th month?
Numerical Example for the Instalment Calculations
Lecture 10 8
Balance effective r interest payment18333.33 0.0074 135.97 1802.6416666.67 0.0081 134.85 1801.5215000.00 0.0089 133.50 1800.1713333.33 0.0099 131.85 1798.5211666.67 0.0111 129.79 1796.4610000.00 0.0127 127.14 1793.818333.33 0.0148 123.61 1790.286666.67 0.0178 118.67 1785.335000.00 0.0223 111.25 1777.923333.33 0.0297 98.89 1765.561666.67 0.0445 74.17 1740.83
0.00 0.0890 0.00 1666.67
Total Payment 1319.69 21319.69Average Payment 109.97 1776.64
Calculation of Instalment Payment on 20000 at 8.9 APR at 12 Instalments (from the spreadsheet)
67.166612
20000b
D =20000
i = 8.9 APR
Lecture 10 9
Total debt D
Regular and equal instalments TN
Db
where N= no. years T=periods
Declining balance at time t nTbtbDB tn , Where n = 1 to N-1, t = 1 to T
Effective interest rate for each instalment: nTtTN
iN
1
where t = 0..T-1
Interest payment for declining balance: tnBnTtTN
iN,1
Total payment each period tnBnTtTN
iNb ,1
Average instalment TNBnTtTN
inTbtb
nttn
,
,1
Calculations of Instalments on a Loan for Multiple Years
Lecture 10 10
D =20000 i = 8.9 APR 33.333512
20000
b
Interest payment in the 12 month of the first year
306.145
33.33312033.33312200001201125.125*089.0
,1.
tnBnTtTNNi
b
Instalment payment in the 12 month of the first year
370.478
33.33312033.33312200001201125.125*089.0
33.333,1.
tnBnTtTNNi
b
What would be the interest and instalment payment in the 3rd month of the 4th year? 474.92
Numerical Example for the Instalment Calculations
Lecture 10 11
Calculation of Instalment Payment on 20000 at 8.9 APR for 5Years (from the spreadsheet)
33.333512
20000
b
D =20000
i = 8.9 APR
N=5 20000 0.089T=12 20000 0.445
Months RemainingN t Balance effect r interest Payment60 1 1 19666.67 0.007417 145.8611 479.194459 1 2 19333.33 0.007542 145.8192 479.152558 1 3 19000 0.007672 145.7759 479.109257 1 4 18666.67 0.007807 145.731 479.0643
48 1 12 16000 0.009082 145.3061 478.6395
37 2 12 12000 0.012027 144.3243 477.657736 3 1 11666.67 0.012361 144.213 477.546335 3 2 11333.33 0.012714 144.0952 477.428634 3 3 11000 0.013088 143.9706 477.303933 3 4 10666.67 0.013485 143.8384 477.1717
5 5 8 1333.333 0.089 118.6667 4524 5 9 1000 0.11125 111.25 444.58333 5 10 666.6667 0.148333 98.88889 432.22222 5 11 333.3333 0.2225 74.16667 407.51 5 12 0 0.445 0 333.3333
Lecture 10 12
Investment Questions• How much will amount grow if you invest £100 today at 10
percent interest rate in 100 years time?• What is the value of £100 that you will receive after 100 years
if market interest rate is 10 percent?• What is the yield to maturity of a bond with face value £1000
which sells in the market for 100 and matures in 10 years time?• What is the yield to maturity if both face value and market
prices are equal?• What is the value of a Share that has a face value of 1000 and
promised to pay dividend growing at 3 percent and has a risk factor about 2 percent?
Lecture 10 13
More Investment Questions• How do you finance a investment project?
– Self finance– Bond Finance– Stock Market
• What determines value of a bond?
• What is the value of a stock that now pays £1000 as dividend and promises this to grow by 3 percent every year if the market interest rate is 5 percent and risk premium is 8 percent?
• How to make a decision to invest or not to invest?
Lecture 10 14
Years 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.11 101.0 102.0 103.0 104.0 105.0 106.0 107.0 108.0 109.0 110.02 102.0 104.0 106.1 108.2 110.3 112.4 114.5 116.6 118.8 121.03 103.0 106.1 109.3 112.5 115.8 119.1 122.5 126.0 129.5 133.14 104.1 108.2 112.6 117.0 121.6 126.2 131.1 136.0 141.2 146.45 105.1 110.4 115.9 121.7 127.6 133.8 140.3 146.9 153.9 161.16 106.2 112.6 119.4 126.5 134.0 141.9 150.1 158.7 167.7 177.27 107.2 114.9 123.0 131.6 140.7 150.4 160.6 171.4 182.8 194.98 108.3 117.2 126.7 136.9 147.7 159.4 171.8 185.1 199.3 214.49 109.4 119.5 130.5 142.3 155.1 168.9 183.8 199.9 217.2 235.8
10 110.5 121.9 134.4 148.0 162.9 179.1 196.7 215.9 236.7 259.411 111.6 124.3 138.4 153.9 171.0 189.8 210.5 233.2 258.0 285.312 112.7 126.8 142.6 160.1 179.6 201.2 225.2 251.8 281.3 313.813 113.8 129.4 146.9 166.5 188.6 213.3 241.0 272.0 306.6 345.214 114.9 131.9 151.3 173.2 198.0 226.1 257.9 293.7 334.2 379.715 116.1 134.6 155.8 180.1 207.9 239.7 275.9 317.2 364.2 417.716 117.3 137.3 160.5 187.3 218.3 254.0 295.2 342.6 397.0 459.594 254.8 643.3 1609.5 3991.5 9812.8 23919.5 57819.6 138622.3 329678.6 777879.695 257.4 656.2 1657.8 4151.1 10303.5 25354.6 61867.0 149712.1 359349.7 855667.696 259.9 669.3 1707.6 4317.2 10818.6 26875.9 66197.7 161689.0 391691.2 941234.497 262.5 682.7 1758.8 4489.9 11359.6 28488.5 70831.5 174624.1 426943.4 1035357.898 265.2 696.3 1811.5 4669.5 11927.6 30197.8 75789.7 188594.1 465368.3 1138893.699 267.8 710.3 1865.9 4856.2 12523.9 32009.6 81095.0 203681.6 507251.4 1252782.9
100 270.5 724.5 1921.9 5050.5 13150.1 33930.2 86771.6 219976.1 552904.1 1378061.2
How does 100 invested today grows at different interest in 100 years
Lecture 10 15
Investors Care about the Real not the Nominal Interest Rate: Approximation Rule (Fisher Equation)
Example: if i=4% and e = 2 %
%2%96.102.0104.01
111
rir
Approximation using the Fisher equations ir ; %2%2%4 r
This approximation does not hold for large r , i and
%30%54%84 i where as
%1919.154.0184.01
111
rir
ir approximation should be used only for small interest rates.
Lecture 10 16
Expectation and the financial market: Fisher equation
Gross nominal return next year on P amount invested today =
tP tPi
1 .
Gross real return depends on nominal interest rate as well as the expected prices in the next period
tP etPtPir1
11
tP eieir
111
where et
etPtP
11
1; and
tPtP
etP
et
1 .
Lecture 10 17
Market Price of a consoleWhat is the market price (value) of a console that pays 100 each year forever from the beginning of the next year at the interest rate, r?
nrrrr
PV
1100...........
31
1002
1
1001
1
100
r
nr
rPV
111
11
11
11
100 as n n 01
1
1
nr
;
r
r
rPV 1
11
100 = r100=
1.0100=1000.
Thus market value of this console should equal 1000.
Lecture 10 18
Value of Bonds and MaturityValue of £100 console by maturity and the interest rate
Years 10% 2% 5%9 575 816.22 710.78
19 836 1567.85 1208.5329 937 2184.44 1514.1159 996 3445.61 1887.57
Observation:
1, Longer the maturity period higher is the price of the console.
2. Lower the interest rate higher the market price of console that promises to£100.
Why? Mainly because future streams of revenue are discounted at a lowerrate.
Lecture 10 19
Yield Curve for Government Bonds: Long Term and Short Term Interest Rates in the UK
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
0.5
2.0
3.5
5.0
6.5
8.0
9.5
11.0
12.5
14.0
15.5
17.0
18.5
20.0
21.5
23.0
24.5
Maturity
Inte
rest
rat
es
24 Dec 0203 Jan 0104 Jan 00
Higher Long Run Interest Rates are Better for Investment
Lecture 10 20
Discount rates d 0.990099 0.980392 0.970874 0.961538 0.952381 0.943396 0.934579 0.925926 0.917431 0.909091Interest rate r 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1Years
1 99.0 98.0 97.1 96.2 95.2 94.3 93.5 92.6 91.7 90.912 98.0 96.1 94.3 92.5 90.7 89.0 87.3 85.7 84.2 82.643 97.1 94.2 91.5 88.9 86.4 84.0 81.6 79.4 77.2 75.134 96.1 92.4 88.8 85.5 82.3 79.2 76.3 73.5 70.8 68.305 95.1 90.6 86.3 82.2 78.4 74.7 71.3 68.1 65.0 62.096 94.2 88.8 83.7 79.0 74.6 70.5 66.6 63.0 59.6 56.457 93.3 87.1 81.3 76.0 71.1 66.5 62.3 58.3 54.7 51.328 92.3 85.3 78.9 73.1 67.7 62.7 58.2 54.0 50.2 46.659 91.4 83.7 76.6 70.3 64.5 59.2 54.4 50.0 46.0 42.41
10 90.5 82.0 74.4 67.6 61.4 55.8 50.8 46.3 42.2 38.5511 89.6 80.4 72.2 65.0 58.5 52.7 47.5 42.9 38.8 35.0512 88.7 78.8 70.1 62.5 55.7 49.7 44.4 39.7 35.6 31.86
100 37.0 13.8 5.2 2.0 0.8 0.3 0.1 0.0 0.0 0.0
Present value of console 6302.9 4309.8 3159.9 2450.5 1984.8 1661.8 1426.9 1249.4 1110.9 999.9
What is the Present Value of a Console that pays £100 every year for 100 years at various
discount factors (interest rates)
Lecture 10 21
Discount rates d 0.990099 0.980392 0.970874 0.961538 0.952381 0.943396 0.934579 0.925926 0.917431 0.909091Interest rate r 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1Years 0 0 0 0 0 0 0 0 0 0
1 99.0 98.0 97.1 96.2 95.2 94.3 93.5 92.6 91.7 90.92 197.0 194.2 191.3 188.6 185.9 183.3 180.8 178.3 175.9 173.63 294.1 288.4 282.9 277.5 272.3 267.3 262.4 257.7 253.1 248.74 390.2 380.8 371.7 363.0 354.6 346.5 338.7 331.2 324.0 317.05 485.3 471.3 458.0 445.2 432.9 421.2 410.0 399.3 389.0 379.16 579.5 560.1 541.7 524.2 507.6 491.7 476.7 462.3 448.6 435.57 672.8 647.2 623.0 600.2 578.6 558.2 538.9 520.6 503.3 486.88 765.2 732.5 702.0 673.3 646.3 621.0 597.1 574.7 553.5 533.5
95 6114.3 4238.0 3132.3 2439.8 1980.6 1660.1 1426.3 1249.2 1110.8 999.996 6152.8 4252.9 3138.1 2442.1 1981.5 1660.5 1426.4 1249.2 1110.8 999.997 6190.9 4267.6 3143.8 2444.3 1982.4 1660.8 1426.6 1249.3 1110.9 999.998 6228.6 4282.0 3149.3 2446.5 1983.2 1661.1 1426.7 1249.3 1110.9 999.999 6265.9 4296.0 3154.7 2448.5 1984.0 1661.5 1426.8 1249.4 1110.9 999.9
100 6302.9 4309.8 3159.9 2450.5 1984.8 1661.8 1426.9 1249.4 1110.9 999.9
What is the Present Value of a Console that pays £100 every year and matures at
a given year at Various interest rates
Lecture 10 22
Bond Market and Bond PricesBond Ratings: AAA ABB ACC ABC CCC Market price of bond differs by risks and interest rate Price of bond maturing at the end of the period 1.
ti
valueFaceBP t
,11
,1
Price of bond that matures at the end of period two.
eti
ti
valueFaceBt
P
1,11
,11,2
Arbitrage condition
tP
et
P
ti
2
1,1,1
1 where
eti
et
P,1
11
1,1
Price of two period bond using this arbitrage condition becomes
etiti
valueFace
ti
etPB
tP
1,11,11,111,1
2
Lecture 10 23
Yield to maturity depends upon expected bond prices and interest rates
Suppose the price of two period bond is
22
1
1002
21
2
ti
ti
valueFaceBt
P
Yield to maturity of this two period bond is:
902
;054.1901002
1
Bt
P
ti
Thus its return is 5.4%.
Arbitrage condition for two periods
eti
ti
ti
1,11
11
22
1
eti
ti
ti
2121
2
Higher long-run rate and lower short run interest rate is good for investment.
Yield curve for n period bonds
entie
tie
ti
tinnti ...
3211
Rising short run interest rates gives an upward sloping and falling short run interest give downward sloping yield curves
Lecture 10 24
Face Market 1 2 3 4 5 6 7Value Price 1000 100 900.00% 216.23% 115.44% 77.83% 58.49% 46.78% 38.95%1000 200 400.00% 123.61% 71.00% 49.53% 37.97% 30.77% 25.85%1000 300 233.33% 82.57% 49.38% 35.12% 27.23% 22.22% 18.77%1000 400 150.00% 58.11% 35.72% 25.74% 20.11% 16.50% 13.99%1000 500 100.00% 41.42% 25.99% 18.92% 14.87% 12.25% 10.41%1000 600 66.67% 29.10% 18.56% 13.62% 10.76% 8.89% 7.57%1000 700 42.86% 19.52% 12.62% 9.33% 7.39% 6.12% 5.23%1000 800 25.00% 11.80% 7.72% 5.74% 4.56% 3.79% 3.24%1000 900 11.11% 5.41% 3.57% 2.67% 2.13% 1.77% 1.52%1000 1000 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%1000 1100 -9.09% -4.65% -3.13% -2.35% -1.89% -1.58% -1.35%1000 1200 -16.67% -8.71% -5.90% -4.46% -3.58% -2.99% -2.57%
Yield to maturity (Return on a bond that matures at year t)
Lecture 10 25
2000/01 2001/02 2000/01 2001/02Government securities: new issuesKQGA 25789.8 43433.4 Pensioners Guaranteed Income BondKJDW 687.2 603.5National savings securities: Treasurer’s account KWNF 12.5 15.2National savings certificatesKQGB 3086.2 2580.7 Individual Savings AccountZAFC 265.9 397.8Capital bonds KQGC 29.0 40.9 Fixed Rate Savings BondsZAFD 284.7 192.7Income bonds KQGD 760.5 625.6 Guaranteed Equity BondsECPU .. 27.2Deposit bonds KQGE - - Certificate of tax depositKQGL 76.5 77.6British savings bonds KQGF - - Nationalised industries’, etcPremium savings bondsKQGG 3296.0 3859.6 temporary deposits KQGM 56106.6 62150.0Save As You Earn KQGH 0.3 - British Gas corporation depositsKQGN - -Yearly plan KQGI - - Sterling Treasury bills (net receipt)KQGO - -National savings stamps and gift ECU Treasury bills (net receipt)KQGP - -tokens KQGJ - - ECU Treasury notes (net receipt)KDZZ - -National Savings Bank InvestmentsKQGK 955.3 864.9 Ways and means (net receipt)KQGQ 12126.0 12095.3Children’s Bonus BondsKGVO 53.4 45.0 Other debt : payable in sterling :First Option Bonds KIAR - - Interest free notes KQGR 972.7 1427.2
Other debt : payable in external currenciesKHCY - -
Total receipts KHCZ 104502.6 128436.6
Government Borrowing (Million £)
Source: Bank of England
Lecture 10 26
Stock Market and Stock Prices S t o c k p r i c e s d e p e n d s u p o n t h e e x p e c t e d s t r e a m o f d i v i d e n d
p a y m e n t s a n d t h e e x p e c t e d f u t u r e i n t e r e s t r a t e s . M a r k e t p r i c e o f a s t o c k i s c l o s e l y r e l a t e d t o t h e m a r k e t
p r i c e o f b o n d s b y a n a r b i t r a g e c o n d i t i o n b e t w e e n t h er e t u r n s o n b o n d s a n d s t o c k s .
A r b i t r a g e c o n d i t i o nb e t w e e n s t o c k a n d b o n dp r i c e s
tSP
etS
Pet
D
ti
,
1,1,1
1
R e a r r a n g i n g t h e s e t e r m s t h ec u r r e n t s t o c k p r i c e s a r eg i v e n b y
ti
etS
P
ti
et
D
tSP
,11
1,
,11
1,
etn
iet
it
i
entS
P
etn
iet
it
i
entD
et
it
i
et
D
ti
et
D
tSP
,1..
,11
,11
,
,1..
,11
,11
...
,11
,11
2
,11
1,
Lecture 10 27
Total number of UK and International companies at end December
0
500
1,000
1,500
2,000
2,500
3,000
3,500
1998 1999 2000 2001 2002
Nu
mb
er O
f C
om
pan
ies
Listed UK Listed International AIM
AIM
Lecture 10 28
Total equity turnover value as at end September
2,482
2,872
4,1704,383
3,588
0
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
4,500
5,000
1998 1999 2000 2001 2002
£b
n
UK International
Lecture 10 29
UK indices - daily index movements January to December 2002
60
65
70
75
80
85
90
95
100
105
110
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Re
ba
se
d t
o 1
00
FTSE 100 FTSE 250 FTSE SmallCap
FTSE Fledgling FTSE AIM FTSE All-Share
Lecture 10 30
FTSE100 Index
0.00
1000.00
2000.00
3000.00
4000.00
5000.00
6000.00
7000.00
8000.00
Jan-
85
Jan-
86
Jan-
87
Jan-
88
Jan-
89
Jan-
90
Jan-
91
Jan-
92
Jan-
93
Jan-
94
Jan-
95
Jan-
96
Jan-
97
Jan-
98
Jan-
99
Jan-
00
Jan-
01
Jan-
02
Inde
x
FTSE100
Lecture 10 31
Value of a Stock: An example
M a r k e t v a l u e o f a s h a r e w i t h f i x e d r e t u r n a n d a
c e r t a i n g r o w t h r a t e a n d r i s k
xgrPShare
PV 1
r = i n t e r e s t r a t e , g = g r o w t h r a t e o f d i v i d e n d , x = r i s k
p r e m i u m .
I f a s h a r e ( s t o c k ) h a s a f a c e v a l u e o f £ 1 0 0 0 a n d r
= 5 % a n d g = 3 % h a s a r i s k x = 0 ;
000,5002.0
100003.005.0
11000
SharePV
i f r = 8 % 000,2005.0
100003.008.0
11000
SharePV
Lecture 10 32
T h i s s h a r e h a s a f a c e v a l u e o f £ 1 0 0 0 a n d r = 5 % a n d
g = 3 % h a s a r i s k x = 8 % ;
100001.0
100008.003.005.0
110001
xgrPShare
PV
w h e n r = 8 %
31.769213.0
100008.003.008.0
110001
xgrPShare
PV
Value of a Stock: An example
Lecture 10 33
Company
Market Capitalisation
As At 31st December
2002
% Increase since 31st December
2001
Reckitt Benckiser
8,499 34%
Gallaher Group
4,016 32%
Liberty International Plc
1,765 32%
SAB Miller 4,750 21%
National Grid Transco
14,112 19%
Associated British Foods
4,649 17%
Imperial Tobacco Group
5,497 16%
Scottish & Newcastle
3,944 16%
Rexam 1,846 15%
Kingfisher 5,813 12%
Largest increases in equity market valuation - FTSE 100
Lecture 10 34
1. Lower the market interest rate, higher is the value of stock. Because
future earnings are discounted at lower rate.
2. Higher the growth rate of dividend higher the value of stock. As
dividend grows earning from the share rises
3. Higher the risk premium lower is the value of the share. A decrease in
the risk premium will increase the market value of a stock.
Observations From the above Analysis of Stock Markets
Lecture 10 35
Comparative UK indices' Performance - January to December
2002
-24.5%
-27.3%-29.4%
-25.0%
-18.4%
-32.8%-35%
-30%
-25%
-20%
-15%
-10%
-5%
0%
% C
ha
ng
e
Lecture 10 36
Five Largest UK sector indices' movements - January to December
2002
20.8% 20.5%
1.8%
-3.6%-0.6%
-5%
0%
5%
10%
15%
20%
25%
PersonalCare &
HouseholdProducts
Water HouseholdGoods &Textiles
% C
ha
ng
e
Lecture 10 37
Suppose a manufacturer is considering buying a machine that costs
£100,000. The machine will depreciate by 8% per year. It will generate real
profits equal to £18000 next year, £18,000(1-8%) two years from now and
£18,000(1-8%)2 three years from now and so on. How can you determine
whether the manufacturer should buy the machine if the real interest rate is
(a) 5% (b) 10% or (c) 15%?
A Question on Investment Decision
Lecture 10 38
Investment Decision AnalysisB r e a k s e v e n p o in t : RCr
T h e v a lu e o f th is in v e s tm e n t p r o je c t : V = r18000
C o s t o f th e P r o je c t : 1 0 0 ,0 0 0
r 0 .0 5 0 .1 0 .1 5
P V 1 3 8 4 6 1 .5 1 0 0 0 0 0 7 8 2 6 0 .8 7
P r o je c t b r e a k s e v e n a t 1 0 % in te r e s t r a te a n d m a k e sp o s i t iv e r e a l p r o f i t a t 5 % in te r e s t r a te
I n v e s tm e n t s h o u ld n o t b e r e c o m m e n d e d w h e n th ein te r e s t r a te is 1 5 % b e c a u s e th e m a n u f a c tu r e r w i l llo o s e a lm o s t 2 2 k .
Lecture 10 39
Exercises• Difference between real and nominal and short
term and long term interest rates• Instalment payments• |Arbitrage conditions and Port-folio allocations in
the financial markets• Value of a console• Market value of bonds and stocks• Life time income• Yield to maturity