Lecture 101 Macroeconomic Analysis Financial Market and Expectations.

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