iii)Monthly
FV= PV(1+( i/f))^n x f
= 5000(1 + (0.12 / 12))^2 x 12
=5000(1 +0.01)^24
=5000(1.01)^24
= 5000(1.2697)
= Rs.6348.5
iii)Future Value of Annuity -Even Cash flow
An annuity is a series of equal sums of receipts or payments occurring at equal time intervals over a specified number of years.In other words, constant periodic sums are called annuities.The future value of even cash flow, if it is deposited at a particular rate of interest for a specified period of time is calculated in similar lines as that of single cash flow
FVA = R(1 +i)^n-1 + R(1 +i)^n-2 + R(1 +i)^n-3 +.......
where FVA = future value of annuity,R = even cash flow
i= interest rate , n=number of years
1.Calculate future value of annuity of Rs.8000 deposited at the end of each year at 6% for a period of 5 years.
R = 8000 i = 0.06 n= 5 years
FVA = R(1 +i)^n-1 + R(1 +i)^n-2 + R(1 +i)^n-3 + R(1 +i)^n-4 + R(1 +i)^n-5
= 8000(1 +0.06)^5-1 + 8000(1 +0.06)^5-2 + 8000(1 +0.06)^5-3 + 8000(1 +0.06)^5-4 + 8000(1 +0.06)^5-5
= 8000(1.06)^4 + 8000(1.06)^3 + 8000(1.06)^2 + 8000(1.06)^1 +
8000(1.06)^0
= 8000(1.2625) + 8000(1.1910) +8000(1.1206) +8000(1.06) +8000(1)
=10100 + 9528 + 8988.8 + 8 + 80 + 8000
Ans =Rs.45096.8
2.Mr. SMV deposits Rs.12000 at the end of every year for 5 years and the deposit earns compound interest at 18% per annum. Determine how much will he have at the end of 5 years.
FVA = R(1 +i)^n-1 + R(1 +i)^n-2 + R(1 +i)^n-3 + R(1 +i)^n-4 + R(1 +i)^n-5
= 12000(1 +0.12)^5-1 + 12000(1 +0.12)^5-2 + 12000(1 +0.12)^5-3 +
12000(1 +0.12)^5-4 + 12000(1 +0.12)^5-5
= 12000(1.12)^4 + 12000(1.12)^3 + 12000(1.12)^2 + 12000(1.12)^1 +
12000(1.12)^0
= 12000(1.5735) + 12000(1.4049) +12000(1.2544) +12000(1.12) +12000(1)
=18882 + 16858.8 + 15052.8 + 13440 + 12000
Ans =Rs.76233.6
iv)Future Value of Annuity -Uneven Cash flow
FVA(UCF) = R1 (1 +i)^n-1 + R2 (1 +i)^n-2 + R3 (1 +i)^n-3 + R4 (1 +i)^n-4
where FVA(UCF = future value of uneven cash flow,
R1, R2, R3,... = uneven cash flow, i= interest rate, n= number of years
1.Calculate future value of the following cash flow if it is compounded at 8% per annum.
At the end of each year Amount Deposited
1 1000
2 2000
3 3000
4 4000
Solution
FVA(UCF) = R1 (1 +i)^n-1 + R2 (1 +i)^n-2 + R3 (1 +i)^n-3 + R4 (1 +i)^n-4
=1000(1 +0.08)^4-1 +2000(1 +0.08)^4-2 +3000(1 +0.08)^4-3 +
4000(1 +0.08)^4-4
=1000(1.08)^3 +2000(1.08)^2 +3000(1.08)^1 +4000(1.08)^0
=1000(1.2597) +2000(1.1664)+3000(1.08) +4000(1)
=1259.7 + 2332.8 + 3240 + 4000
= 10832.5
2.Calculate future value at the end of 5 years of the following series of payment at 9% rate of interest.
Rs 2000 at the end of 1st year
Rs 4000 at the end of 2nd year
Rs 6000 at the end of 3rd year
Rs 8000 at the end of 4th year
Rs 10000 at the end of 5th year
Solution
FVA(UCF) = R1 (1 +i)^n-1 + R2 (1 +i)^n-2 + R3 (1 +i)^n-3 + R4 (1 +i)^n-4
+ R5 (1 +i)^n-5
=2000(1 +0.09)^5-1 +4000(1 +0.09)^5-2 +6000(1 +0.09)^5-3 +
8000(1 +0.09)^5-4 + 8000(1 +0.09)^5-5
=2000(1.09)^4 +4000(1.09)^3 +6000(1.09)^2 +8000(1.09)^1 +10000(1.09)^0
= 2000(1.4115) +4000(1.2950)+ 6000(1.1881)+8000(1.09)
+10000(1)
=2823 + 5180 +7128.6 +8720 +10000
FVA = Rs. 33851.6
Discounting
Discounting refers to finding the present value of cash inflow or outflow happening on a future date.It refers to finding today's value of future cash flows.
i)Annual discounting
PV = FV / (1+i)^n
1.Calculate PV of Rs.25000 received after 2 years at dividend rate at 8% per annum.
PV = FV / (1+i)^n
=25000/ (1 +0.08)^2
=25000/1.1664
=Rs. 21433.4785
2.Mr.A receives rs.30000 after 5 year from now. His time preference for money is 10% per annum. Calculate the PV.
PV = FV / (1+i)^n
=30000/ (1 +0.1)^5
=30000/1.6105
PV=Rs. 18625
ii)Multiple Discounting
MDPV= FV / [1 +(i/f)^ n x f]
iii)Present value of annuity-Even cash flow
If the investor expects a uniform cash flow over a period of time at a definite discount rate, such cash flows present value will always be less, compared to the future flows.
PVA = FV/(1+i)^1 +FV/(1+i)^2 +FV/(1+i)^3 +FV/(1+i)^4 +FV/(1+i)^5
where PVA is present value of annuity,
FV = future value or uniform series of payments,
i= rate of interest expressed in decimals
1.Find out the present value of annuity(PVA) receipt of Rs.8000 received for 5 years at the rate of 8% discount rate.
PVA = FV/(1+i)^1 +FV/(1+i)^2 +FV/(1+i)^3 +FV/(1+i)^4 +FV/(1+i)^5
=8000/(1+0.08)^1 +8000/(1+0.08)^2 +8000/(1+0.08)^3+8000/(1+0.08)^4 + 8000/(1+0.08)^5
=8000/(1.08)^1 +8000/(1.08)^2 +8000/(1.08)^3 +8000/(1.08)^4+8000/(1.08)^5
=8000/(1.08) +8000/(1.1664) +8000/(1.2597) +8000/(1.3604) +8000/(1.4693)
=7407.4+6858.71+6350.7+ 5880.6 + 5444 .76
=Rs.31942
2.What is the present value of Rs.9000 received at the end of 5 years if it is discounted at the rate of 10% per annum.
PVA = FV/(1+i)^1 +FV/(1+i)^2 +FV/(1+i)^3 +FV/(1+i)^4 +FV/(1+i)^5
=9000/(1+0.1)^1 +9000/(1+0.1)^2+9000/(1+0.1)^3+9000/(1+0.1)^4 + 9000/(1+0.1)^5
=9000/(1.1)^1 +9000/(1.1)^2 +9000/(1.1)^3 +9000/(1.1)^4 +9000/(1.1)^5
=9000/(1.1) +9000/(1.21) +9000/(1.331) +9000/(1.4641) +9000/(1.6105)
=8181.81+7438+6761.83+6147.52+ 5588.32
=Rs.34117.09
iv)Present value of annuity-Uneven cash flow
PVA(UEC) = FV1/(1+i)^1+FV2/(1+i)^2 +FV3/(1+i)^ +FV4/(1+i)^4+...FVn/(1+i)^n
1.Calculate the PV of the following series of payments received at the end of each year for a period of 5 years at a discount rate of 8% per annum.
Year Cash flow
1 8000
2 10000
3 12000
4 14000
5 16000
PVA(UEC) = FV1 /(1+i)^1 + FV2 /(1+i)^2 + FV3 /(1+i)^ +FV4 /(1+i)^4 +...FVn/(1+i)^n
=8000/(1+0.08)^1 +10000/(1+0.08)^2+12000/(1+0.08)^3+14000/(1+0.08)^4 + 16000/(1+0.08)^5
=8000/(1.08)^1+10000/(1.08)^2+12000/(1.08)^3+14000/(1.08)^4+16000/(1.08)^5 = 8000/1.08 +10000/(1.1664) +12000/(1.2597) +14000/(1.3604) +
16000/(1.4093)
=7404.4 +8573.38 +9526.07 + 10291.09 + 10999.5
= Rs 46,687.47
2.How much should Mr. A invest to get cash flows of Rs 16000, Rs 14000, Rs 12000 ,Rs 10000 and Rs 8000 at end of years respectively at discounted rate 8%.
PVA = 16000/(1.08)^1 + 14000/(1.08)^2 + 12000/(1.08)^3+ 10000/(1.08)^4+
8000/(1.08)^5
= 16000/(1.08)+ 14000/(1.1664) + 12000/(1.2597)+ 10000/(1.3604)+
8000/(1.4693)
=14814.81+12009.74+9526.07+7350.77+5444.7
=Rs.49139
Doubling Period
Doubling period means the period required for an investment to get doubled.For eg, the time required for Rs. 1000 to become Rs.2000 is known as doubling period.
Sometimes, investor should know how long it will take to double his money at a given rate of interest. In this case, a rule of thumb called the rule of 72, can be used.You can calculate the doubling by dividing 72 by the interest rate.
i)Rule 72 : Doubling period= 72 / Rate of interest
A more accurate method used for doubling your money is using the rule of 69.
ii)Rule 69 : Doubling period= 0.35 + 69 / Rate of interest
Note:
1.The amount of money is irrelevant
2.Under doubling period the rate of interest is not converted into decimal because only the doubling time period is calculated and does not involve any monetary calculation.
1.Calculate the doubling period for a sum of Rs.8000 at 6% per annum.
i)Rule 72 :
Doubling period= 72 / Rate of interest
=72 / 6 = 12 years
ii)Rule 69 :
Doubling period= 0.35 + 69 / Rate of interest
= 0.35 + 69 / 6
=0.35 + 11.5
= 11.85 years , Approx. 12 years
2.In how many years a sum of money will be doubled if it is invested at 8%.
i)Rule 72 :
Doubling period= 72 / Rate of interest
=72 / 8 = 9 years
ii)Rule 69 :
Doubling period= 0.35 + 69 / Rate of interest
= 0.35 + 69 / 8
=0.35 + 8.625
= 8.975 years , Approx. 9 years
3.Calculate the doubling period using rule 72 and rule 69 if an investor invest a sum of money at 12% per annum.
i)Rule 72 :
Doubling period= 72 / Rate of interest
=72 / 12 = 6 years
ii)Rule 69 :
Doubling period= 0.35 + 69 / Rate of interest
= 0.35 + 69 / 12
=0.35 + 5.75
= 6.1 years , Approx. 6 years
4.If the interest rate is 10%, what are the doubling periods of an investment at this rate ?
Solution
(a) As per rule of 72, the doubling period will be
Doubling period= 72 / Rate of interest
72/10 = 7.2 years, Approx. 7 years
(b) As per the rule of 69, the doubling period will be
Doubling period= 0.35 + 69 / Rate of interest
0.35 + 69/10 = 0.35 + 6.9 = 7.25 years, Approx. 7 years
Concept of valuation
Valuation refers to finding the value or worth of any asset or resource.Assets can be real assets and financial assets.
Real assets refer to those assets which can be put into some use.Examples of real assets are building, machinery,furniture,vehicles,etc.
Financial assets refer to instruments which represent investments made on which return are expected.Examples of financial assets are shares,debentures,mutual funds etc
Valuation of Bonds and Debentures
A bond or a debenture is a debt instrument.It is an acknowledgement of debt given by an investor.Being a financial asset a bond or a debenture is valued by taking the present value of future inflows from the bond or debenture.It is used to evaluate the securities either to buy or not to buy.
Debentures
i)Redeemable debentures-carry a specific data of redemption on the certificate.The company is legally bound to repay the principle amount to the debenture holder on that day.
ii)Irredeemable debentures-They do not carry any date of redemption.The principle amount will be returned to the debenture holder at the time of winding up of the company or when the company decides to repay the amount.
i)Redeemable debentures
PVD/B(Rede) = I1/(1+d)^1+ I2/(1+d)^2+.....In/(1+d)^n+M/(1+d)^n
where I = interest amount received ,d=debentures rate of interest
n= number of years, M= maturity value of debentures or bond
1.What is the present value of bond redeemable after 5 years yielding Rs.60 every year with the maturity value of Rs.110 capitalised at 8%.
n=5 years, I= 60 rupees, d= 8%=0.08 , M=110 rupees
PVD/B(Rede) = I1/(1+d)^1+ I2/(1+d)^2+.....In/(1+d)^n+M/(1+d)^n
= 60/(1+0.08)^1 +60/(1+0.08)^2 +60/(1+0.08)^3 +60/(1+0.08)^4 +60/(1+0.08)^5
+110/(1+0.08)^5
= 60/(1.08)^1 +60/(1.08)^2 +60/(1.08)^3 +60/(1.08)^4 +60/(1.08)^5
+110/(1.08)^5
=60/(1.08) +60/(1.1664) +60/(1.2597) +60/(1.3604) +60/(1.4693)
+110/(1.4693)
=55.515+51.44+47.63+44.104+40.83+74.86
=Rs. 314.44 , Approx Rs.314
2.A debenture is available in the market for Rs.1000 with Rs.80 as the interest for a year for a period of 4 years with maturity value of Rs.1120.The debenture capitalisation rate is 10%.Advice Mr.A whether to buy this debenture or not.
80 rupees x 4 years = 320
MV(Maturity value) - PV(Present Value) = 1120-1000= 120
There is a profit of 440. So Mr.A can be advised to buy this debenture.
ii)Irredeemable Debentures
PVD/B(Irr) = I / d
1.What is the value of an irredeemable debenture which has 260 as interest for infinite period of time at a discount rate of 9% per annum.
PVD/B(Irr) = I / d
= 60 / 0.09
= Rs.667
2.What is the value of an irredeemable bond yielding Rs.110 per annum as interest until dissolution of the company at 7% rate of return.
PVD/B(Irr) = I / d
= 110 / 0.07
= Rs.1571
Valuation of Preference shares
Preference shares are also called as preference stocks.These are the shares of the company with dividends having a fixed rate and that are paid out to the shareholders before equity dividend are distributed.
Preference shares are of two types
1.Redeemable
2.Irredeemable
Valuation of Redeemable Preference shares
PVP= D1/(1+d)^1+D2/(1+d)^2+D3/(1+d)^3+D4/(1+d)^4+M4/(1+d)^4
PVP= Present value of preference shares
D1, D2,...=Dividend
M= maturity value
d= discount rate
Problems
1.How much an investor has to pay to a preference share which has the dividend of Rs.70 per year for 4 years and the maturity value of same is Rs.1150.The capitalisation rate of this share is 8% per annum.
PVP= D1/(1+d)^1+D2/(1+d)^2+D3/(1+d)^3+D4/(1+d)^4+M4/(1+d)^4
=70/(1+0.08)^1+70/(1+0.08)^2+70/(1+0.08)^3+70/(1+0.08)^4+1150/(1+0.08)^4
=70/(1.08)^1+70/(1.08)^2+70/(1.08)^3+70/(1.08)^4+1150/(1.08)^4
=70/(1.08)+70/(1.1664)+70/(1.2597)+70/(1.3604)+1150/(1.3604)
=64.81+60.01+55.56+51.45+845.33
=Rs.1077
2.What is the PV of the preference shares which earns Rs.60 dividend for 5 years and carrying maturity value at the end of 5th year Rs. 1130 at capitalisation rate 10% per annum
PVP=D1/(1+d)^1+D2/(1+d)^2+D3/(1+d)^3+D4/(1+d)^4+D5/(1+d)^5
+M4/(1+d)^5
=60/(1+0.1)^1+60/((1+0.1^2+60/(1+0.1^3+60/(1+0.1)^4+60/(1+0.1)^5
+1130/(1+0.1^5
=60/(1.1)^1+60/(1.1)^2+60/(1.1)^3+60/(1.1))^4+ 60/(1.1)^5+1130/(1.1)^5
=60/(1.1)+60/(1.21)+60/(1331)+60/(1.4641)+60/(1.6105)+1130/(1.6105)
=54.54+49.59+45.07+40.98+37.25+701.64
=Rs.929.05 Approx.Rs.929
Valuation of Irredeemable Preference shares
PVIR/PS = D / d where D is the dividend amount, d= capitalisation rate
Problems
1.A company issues 8% irredeemable preference shares of Rs.100 each.The capitalisation rate is 6%.Calculate its present value
PVIR/PS = D / d = 8 / 0.06 = Rs.133 [D=100 x 8%= 8]
2.Compute the present value of 7% irredeemable preference share of Rs.100 each if it is capitalised at 11% rate.
PVIR/PS = D / d = 7 / 0.11 = Rs.64 [D=100 x 7%=7]
Valuation of Equity share
Equity share refer to that part of equity capital directly contributed by the investor with a view to enjoy the ownership of the company.The reward given by the investor to the company at the year is known as dividend.This percentage of dividend is known to the investor at the commencement of issue of preference shares.
The valuation of equity shares are determined by using two approaches
i.Dividend capitalization model
ii.Earning capitalization model
Dividend capitalization model
1.Single Period valuation model-under this method it is assumed that the investor is holding the equity shares for one year and enjoying the dividend of that year and later sells his shares.
PVES= D1 / (1+d)^1 + P1 /(1+d)^1
where PVES is the present value of equity share
D is the dividend paid in first year
P1 is the sale price of the equity share at the end of first year
d is the discount rate
1.Mr.A holds an equity share which has the feature of getting Rs.20 as the dividend for the first year.He expect to sell the same share for Rs.190 at the end of the year.What is the value today if the capitalization rate is 11%
PVES= D1 / (1+d)^1 + P1 /(1+d)^1
= 20 / (1+0.11)^1+ 190 /(1+0.11)^1
=20 / (1.11)^1+ 190 /(1.11)^1
=20 / (1.11) + 190 /(1.11)
=18.01 + 171.17
=Rs.189
2.1.Mr.A purchases an equity share which has the following feature .The dividend for the year is Rs.10.The sale price of equity at the end of first year is 140 rupees.The rate of discount is 12%.How much Mr.A should pay as of today to make investment.
PVES= D1 / (1+d)^1 + P1 /(1+d)^1
= 10 / (1+0.12)^1+ 140 /(1+0.12)^1
=10 / (1.12)^1+ 140 /(1.12)^1
=10 / (1.12) + 140 /(1.12)
=8.92 + 125
=Rs.134
2.Double period valuation
Under this method it is assumed that the shareholder holds the equity for a period of 2 years and sells it at the end of second year.There is a possibility that the dividend for both the years may or may not be the same.
Note:Multiple period valuation-This method is similar to double period valuation where the shareholder holds the equity for certain number of years and then sell it off.
PVES= D1 / (1+d)^1 + D2 / (1+d)^2 + P /(1+d)^2
1.Mr.A is investing in an equity share with the following features. It earns a dividend of 8 and 10 Rupees respectively for the first two years .The equity is capitalized at 9% per annum. At the end of the second year Mr.A can earn 160 rupees by selling this equity.Calculate the PV of the equity.
PVES= D1 / (1+d)^1 + D2 / (1+d)^2 + P /(1+d)^2
Plz First sem FA notes NEP syllabus
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