TIME VALUE OF MONEY

TIME VALUE OF MONEY

Learning Objectives

After studying this chapter, you will be able to understand

♦ The concept of time value of money;

♦ Techniques of Discounting and Compounding;

♦ Identify the equation for calculating the present value of an annuity and calculation of the present value of an annuity; and

♦ Identify the equation for calculating the future value of an annuity and calculation of the future value of an annuity.

Money NOW is worth more than money LATER!

THE CONCEPT OF TVM REFERS TO THE FACT THAT THE MONEY RECD TODAY IS DIFFERENT IN ITS WORTH FROM THE MONEY RECEIVABLE AT SOME OTHER TIME IN THE FUTURE.IN OTHER WORDS THE SAME PRINCIPLE CAN BE APPLIED TO STATE THAT MONEY RECEIVABLE IN FUTURE IS LESS VALUABLE THAN VALUE RECEIVABLE TODAY ITSELF.

Time Preference for Money

Time preference for money is an individual’s preference for possession of a given amount of money now, rather than the same amount at some future time. Three reasons may be attributed to the individual’s time preference for money:

n   risk

n   preference for consumption

n   investment opportunities

For example Rs 1000 now is not equal to Rs 1000 in 1 year’s time. Somebody who does not know finance will also prefer the money now to money anytime in the future. That is time preference for money

Required Rate of Return

The time preference for money is generally expressed by an interest rate. This rate will be positive even in the absence of any risk. It may be therefore called the risk-free rate.

An investor requires compensation for assuming risk, which is called risk premium.

The investor’s required rate of return is:

          Risk-free rate + Risk premium.

Why TIME?

Why is TIME such an important element in your decision?

TIME allows you the opportunity to postpone consumption and earn INTEREST.

The reason why there is time value of money is as follows:

Opportunity Cost: There are alternative productive uses of money. The cost of any decision includes the cost of the next best opportunity forgone. You can save and invest, get interest and spend.

Inflation: It erodes the value of money.

Risk: There are always financial and non-financial risks involved.

The trade-off between money now and money later depends on, among other things, the rate of interest you can earn by investing. It impacts business finance, consumer finance and government finance. Time value of money results from the concept of interest.

Interest rate is the cost of borrowing money as a yearly percentage. For investors, interest rate is the rate earned on an investment as a yearly percentage.

SIMPLE INTEREST

Interest paid (earned) on only the original amount, or principal, borrowed (lent).

SIMPLE INTEREST IS COMPUTED ON THE PRINCIPAL FOR THE ENTIRE PERIOD IT IS BORROWED.

simple interest   =     

S = P + Prt        

total amount A  in simple interest  = 

Example 1:

Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years.  What is the accumulated interest at the end of the 2nd year?

SI       = P (r)(t) = $1,000(.07)(2)         = $140

Compound Interest:

Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent).

Compound interest means interest calculated on the principal for the first year and then interest is not withdrawn – is reinvested. Interest is added to the principal and interest is calculated again

1. Compound interest  sum (S)   = 

the amount calculated on the basis of compound interest rate is higher than when calculated with the simple rate. The time interval between successive additions of interests is known as conversion (or payment) period. Typical conversion periods are given below:

Conversion Period	Description
1 day	Compounded daily
1 month	Compounded monthly
3 months	Compounded quarterly
6 months	Compounded semiannually
12 months	Compounded annually

EXAMPLE 2: Rs. 2,000 is invested at annual rate of interest of 10%. What is the amount after 2 years if the compounding is done:

(a) Annually? (b) Semi annually? (c) Monthly? (d) Daily?

Solution

(a) The annual compounding is given by: = 2,000 (1.1)² = 2,000 × 1.21 =

Rs. 2,420

(b) For Semiannual compounding, n = 2 × 2 = 4, I = 0.1/2 = 0.05

A = 2,000 ( 1 + 0.05)⁴ = 2,000 × 1.2155 = Rs. 2,431

(c) For monthly compounding, n = 12 × 2 = 24, i = 0.1/12 = 0.00833

A = 2,000 (1.00833)²⁴ = 2,000 × 1.22029 = Rs. 2440.58

(d) For daily compounding, n = 365 × 2 = 730, i = 0.1/(365) = 0.00027

A = 2,000 (1.00027)⁷³⁰ = 2,000 × 1.22135 = Rs. 2,442.70

COMPOUND INTEREST VERSUS SIMPLE INTEREST

The given figure shows graphically the differentiation between compound interest and simple interest. The top two ascending lines show the growth of Rs. 100 invested at simple and compound interest. The longer the funds are invested, the greater the advantage with compound interest. The bottom line shows that Rs. 38.55 must be invested now to obtain Rs. 100 after 10 periods. Conversely, the present value of Rs. 100 to be received after 10 years is Rs. 38.55.

                              Compound Interest versus Simple Interest

Example 3: Julie wants to know how large her deposit of $10,000 today will become at a compound annual interest rate of 10% for 5 years.

Calculation based on general formula:  FVn   = P(1+i)n                                            

FV5     = $10,000 (1+ 0.10)5 =  $16,105.10

Calculation based on Table:   FV5     = $10,000 (FVIF10%, 5)

= $10,000 (1.611)       = $16,110           [Due to Rounding]

Example 4: If you invest $1,000 today at an interest rate of 10 percent, how much will it grow to be after 5 year ?

FVn = P (1+i)n

FVn = 1,000(1.10)5

FVn = $1,610.51

Example 5: Julie Miller has $1,000 to invest for 2 Years at an annual interest rate of 12%.

Annual    FV           = 1,000(1+ [.12/1])(1)(2)            = 1,254.40

Semi - annual FV   = 1,000(1+ [.12/2])(2)(2)            = 1,262.48

Qrtly  FV                 = 1,000(1+ [.12/4])(4)(2)          = 1,266.77

Monthly FV            = 1,000(1+ [.12/12])(12)(2)        = 1,269.73

Daily  FV                = 1,000(1+[.12/365])(365)(2)     = 1,271.20

Effective Annual Interest Rate

The actual rate of interest earned (paid) after adjusting the nominal rate for factors such as the number of compounding periods per year.

(1 +  [ i / m ] )m – 1)

Example 6: Basket Wonders (BW) has a $1,000 CD at the bank.  The interest rate is 6% compounded quarterly for 1 year.  What is the Effective Annual Interest Rate (EAR)?

EAR = ( 1 + 6% / 4 )4 - 1         = 1.0614 - 1  = .0614 or 6.14%

Example 7: If the interest is 10% payable quarterly, find the effective rate of interest.

E = ( 1 + 10% / 4 )4 - 1         = = 0.1038 or 10.38%

TECHNIQUES OF DISCOUNTING

The present value of a sum of money to be received at a future date is determined by discounting the future value at the interest rate that the money could earn over the period. This process is known as Discounting. The figure below shows graphically how the present value interest factor varies in response to changes in interest rate and time. The present value interest factor declines as the interest rate rises and as the length of time increases.

Present Value

Present value of a future cash flow (inflow or outflow) is the amount of current cash that is of equivalent value to the decision-maker.

n  Discounting is the process of determining present value of a series of future cash flows.

n  The interest rate used for discounting cash flows is also called the discount rate.

Present Value of a Single Cash Flow

n  The following general formula can be employed to calculate the present value of a lump sum to be received after some future periods:

             PV =FV x PVF _(r,n)  or  S   [   1/ (1+i)ⁿ] or S(1 + i) ⁻ⁿ

n  The term in parentheses is the discount factor or present value factor (PVF), and it is always less than 1.0 for positive i, indicating that a future amount has a smaller present value.

Example 8: Suppose that an investor wants to find out the present value of Rs 50,000 to be received after 15 years. Her interest rate is 9 per cent. First, we will find out the present value factor, which is 0.275. Multiplying 0.275 by Rs 50,000, we obtain Rs 13,750 as the present value:     

PV =FV x PVF _(r,n)  =  FV x PVF _{(.09,15)  =} 50,000 x .275 = 13,750.

Example 9 : Mr. X has made real estate investment for Rs. 12,000 which he expects will have a maturity value equivalent to interest at 12% compounded monthly for 5 years. If most savings institutions currently pay 8% compounded quarterly on a 5 year term, what is the least amount for which Mr. X should sell his property? Given that (1 + i)ⁿ = 1.81669670 for i = 1% and n = 60 and that (1 + i) ⁻ⁿ = 0.67297133 for i = 2% and n = 20.

Solution :

It is a two-part problem. First being determination of maturity value of the investment of Rs. 12,000 and then finding of present value of the obtained maturity value.

Maturity value of the investment may be found from A_n = P (1+i)ⁿ

Now, A_n = 12,000 (1+1%)⁶⁰ = 12,000 × 1.81669670  = 21,800.36040000 = Rs. 21,800.36

Thus, maturity value of the investment in real estate = Rs. 21,800.36

The present value, P of the amount A_n due at the end of n interest periods at the rate of i% interest per period is given by P = A_n (1 + i) ⁻ⁿ

Thus, P = 21,800.36 (1+ 2%)⁻²⁰

= 21,800.36 × 0.67297133 = Rs. 14,671.02

Mr. X should not sell the property for less than Rs. 14,671.02

Future Value of a Single Cash Flow

The general form of equation for calculating the future value of a lump sum after n periods may, therefore, be written as follows:

The term  (1 + i)ⁿ is the compound value factor (CVF) of a lump sum of Re 1, and it always has a value greater than 1 for positive i, indicating that CVF increases as i and n increase.

Example 10: If you deposited Rs 55,650 in a bank, which was paying a 15 per cent rate of interest on a ten-year time deposit, how much would the deposit grow at the end of ten years?

We will first find out the compound value factor at 15 per cent for 10 years which is 4.046. Multiplying 4.046 by Rs 55,650, we get Rs 2,25,135 as the compound value:

FV =  PV  x CVF _(r,n)   ⁼ PV  x CVF _(.15,10)   =    55,650 X 4.045555 = 2,25,135.28

ANNUITY

An annuity is a stream of regular periodic payment made or received for a specified period of time. A recurring deposit with the bank is typical example of an annuity.

An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

Ordinary Annuity:  Payments or receipts occur at the end of each period.

Annuity Due:  Payments or receipts occur at the beginning of each period.

Examples of Annuities

• Student Loan Payments
• Car Loan Payments
• Insurance Premiums
• Mortgage Payments
• Retirement Savings

If you make payments of $2,000 per year into a retirement fund, it is an annuity.

If you receive pension checks of $1,500 per month, it is an annuity.

If an investment provides you with a return of $20,000 per year, it is an annuity.

For the future value of an annuity:

FV = PMT  [(1+i)n - 1]/I]

FV = PMT x FVIFA _{( i, n)}

Example 11: If you save $50 per month at 12 percent per annum, how much will you have at the end of 20 years?

Note that since time periods are months, i = 12%/12 months = 1% per period, for 240 periods.

    FV = PMT[(1+i)n - 1]/i

    FV = 50[(1.01)240 - 1]/.01

    FV = $49,463

Example 12: If you want to save $500,000 for retirement after 30 years, and you earn 10 percent per annum, how much must you save each year?

FV = PMT[(1+i)n - 1]/i

500,000 = PMT[(1.1)30 - 1]/.1

PMT = $3,040 per year

Example 13: Find the amount of an annuity if payment of Rs. 500 is made annually for 7 years at interest rate of 14% compounded annually.

Solution

Here P = 500, n = 7, i = 0.14

A = Rs. 500 × A _{(7, 0.14)} = 500 × 10.7304915 = Rs. 5,365.25

Present Value of an Annuity

The computation of the present value of an annuity can be written in the following general form:

The term within parentheses is the present value factor of an annuity of Re 1, which we would call PVFA, and it is a sum of single-payment present value factors.

Example 14: An investor deposits a sum of Rs 1, 00,000 in a bank account on which interest is credited @10% per annum. How much amount can be withdrawn annually for a period of 15 years?

Answer:

in  this case the deposit of Rs 1,00,000 can be viewed as the present value of future annuity of 15 years@10%. the situation can be also presented as follows:

Rs 1,00,000 = Annuity x PVAF_{(10% ,15 years)}

                   = annuity  x 7.606

Annuity  = 1,00,000 / 7.606 =13,148.

This means the investor can withdraw Rs 13,148 for the next 15 years.

Future Value of an Annuity

Annuity is a fixed payment (or receipt) each year for a specified number of years. If you rent a flat and promise to make a series of payments over an agreed period, you have created an annuity.

FV =   A x CVAF _{(r, n)}

FVA ( future value of an annuity)  =  S =   

(The term within brackets is the compound value factor for an annuity of Re 1, which we shall refer as CVFA or CVAF)

Example 14: Suppose that a firm deposits Rs 5,000 at the end of each year for four years at 6 per cent rate of interest. How much would this annuity accumulate at the end of the fourth year? We first find CVFA which is  4.3746. If we multiply 4.375 by Rs 5,000, we obtain a compound value of Rs 21,875:

A x CVAF _{(r, n)     =} 5000  x  CVAF _(4,.06) = 5000 x 4.3746   =  21,873

Perpetuities

A perpetuity is simply an annuity that continues forever  perpetually).The formula for finding the present value of a perpetuity is:

PV = PMT/i

A variation to perpetuity problems is the case of growing perpetuities

If an annuity continues forever, and grows in amount each period at a rate g, then

PV = PMT1/(i - g)

Example 15: If you invest in a stock that will pay a dividend of $10 next year and grow at 5 percent per year, and you require a 14 percent rate of return, how much is the stock worth to you today?

   PV = PMT1/(i - g)

   PV = 10/(.14-.05)

   PV = $111.11 

Example 16: Ramesh wants to retire and receive Rs. 3,000 a month. He wants to pass this monthly payment to future generations after his death. He can earn an interest of 8% compounded annually. How much will he need to set aside to achieve his perpetuity goal?

Solution     C = Rs. 3,000

r = 0.08/12 or 0.00667

Substituting these values in the above formula, we get

PV = 

If he wanted the payments to start today, we must increase the size of the funds to handle the first payment. This is achieved by depositing Rs. 4,52,775 which provides the immediate payment of Rs. 3,000 and leaves Rs. 4,49,775 in the fund to provide the future Rs. 3,000 payments.

SINKING FUND

It is the fund created for a specified purpose by way of sequence of periodic payments over a time period at a specified interest rate.

Size of the sinking fund deposit is computed from A=P.A (n,i), where A is the amount to be saved, P, the periodic payment, n, the payment period.

Example 17: How much amount is required to be invested every year so as to accumulate Rs. 3,00,000 at the end of 10 years if the interest is compounded annually at 10%?

Solution

Here, A= 3,00,000 n = 10 i = 0.1

Since, A=P.A (n,i)

3,00,000=P.A(10, 0.1)

= P*15.9374248

Therefore, P = 3,00,000/15.9374 = Rs. 18,823.62

P = Rs. 18,823.62

Summary

1. FV  of a single amount = PV  x  CVF _(r,n)
2. PV of a single cash flow at the end of year n = FV  x PVF _(r,n)
3. FV of a series of equal payments  or Annuities :    A x CVAF _(r,n)
4. PV of a series of annual payments : FV x PVAF _(r,n)

Mathematical formulae’s

1. present value    (P) =      S (1+i)^-n   or      S   [   1/ (1+i)ⁿ]

Present value    (P) =         (if compounded continuously)

2. effective rate of  interest  =(r _eff ) = 
• Annuities are equal payments at equal intervals.
• Annuity Payments are made at the end of the periods.

3. FVA ( future value of an annuity)  =  S =   
4. PVA (present value of an annuity )  = p  =   
5. uniform payments for capital recovery – amortization   =       R =  
6. uniform payments for capital formation – sinking fund =  =       R =  

7. Amortization: A loan with a fixed rate of interest is said to be amortized if both principal and interest are paid by a sequence of equal payments made over equal period of time.

                                                                                      

                 Where R is the equal amt of annual  payment.

8. Amount of deferred annuity:  deferred annuity is an ordinary annuity in which the first payments are postponed until the expiry of ascertain no of years or payment intervals .if the first payment is being made at the end      of 7 years then it deferred for 6 years.

S =

9. Perpetuity: Perpetuity is an annuity whose payments being on a fixed date and continues forever.

        

10. Sinking funds: A sinking fund is a fund that’s accumulated for the purpose of paying off a financial obligation at some future designated date.

                         And   S =

Where R is the periodic payment required to accumulate a sum of S over n periods of time.

The “Rule-of-72”

Approx. Years to Double = 72 / i%

Exhibit 18 : How long does it take to double $5,000 at a compound rate of 12% per year (approx.) ?

Approx. Years to Double = 72 / i%

72 / 12% = 6 Years

[Actual Time is 6.12 Years]

Rule of 69

0.35+    69 / int. rate %

Following previous example,

0.35+69/12 = 0.35+5.75 =6.10

Example 19: If you invest $1,000 today at an interest rate of 12 percent, how much time it will take to double?

FVn = P0(1+i)n

2000 = 1,000 (1.12)n

interpolation method

(1.12)n = 2000 /1000= 2

6Yrs + 1Yr* (1.974 - 2.000/1.974-2.211)

= 6Yrs+ 1yr (-0.026/-0.237) =  6Yrs+0.11Yrs= 6.11 Years

Example 20 : How long will it take for $10,000 to grow to $20,000 at an interest rate of 15% per year?

Rule of 72 =72 / 15   =4.80 years

Rule of 69= .35+69/15=.35+4.6 = 4.95 years

 FVn = P0(1+i)n

 20,000 = 10,000 (1.15)n

 n = 4.96 years

Interpolation method:

(1.15)n   =20000/10000 = 2.000

= 4+1x[1.749-2.000/1.749-2.011]

= 4+1x[0.251/0.262]

= 4+1x[0.958] =4.958 =4.96

Example 21: Julie wants to know how large of a deposit to make so that the money will grow to $10,000 in 5 years at a discount rate of 10%.

Calculation based on gen formula: PV0  = FVn / (1+i)n 

 PV0   = $10,000 / (1+ 0.10)5          = $6,209.21

Calculation based on Table I: 

PV0    = $10,000 (PVIF10%, 5)  = $10,000 (.621) = $6210.00  

Example 22: Y bought a TV costing Rs. 13,000 by making a down payment of Rs. 3,000 and agreeing to make equal annual payment for 4 years. How much would be each payment if the interest on unpaid amount be 14% compounded annually ?

Solution

In the present case, present value of the unpaid amount was (13,000 – 3,000) = Rs. 10,000. The periodic payment, A may be found from

PV = A X PVAF _{r%, n} = A X PVAF_14%,4

10,000 =  A X 2.9103  

A = 10,000/2.9103 = 3,432.05

Example 23: A person is required to pay four equal annual payments of Rs. 5,000 each in his deposit account that pays 8% interest per year. Find out the future value of annuity at the end of 4 years.

SOLUTION

FVA =ANNUITY ( CVAF_R%,N) = ANNUITY ( CVAF_8%,4)  =Rs. 5,000 (4.507)  = Rs. 22,535

Example 24: Assume you will receive an inheritance of Rs100,000, six years from now.  How much could you borrow from a bank today and spend now, such that the inheritance money will be exactly enough to pay off the loan plus interest when it is received?  Assume the bank charges an interest rate of 12 percent?

ANSWER:

_{= 1,00,000/1.9738}

Example 25: A company offers to refund an amount of Rs 44,650 at the end of 5 years for a deposit of Rs 6,000 made annually. Find out the implicit rate of interest offered by the company.

Answer: in this case the refund amount is the future amount of annuity of Rs 6,000 after 5 years at a particular rate of interest. This can be presented as follows:

Rs 44,650 = 6000 x CVAF_(r%,5years)

Or

CVAF_(r%,5years)=44,650 /6000 = 7.442

Now in the CVAF table the value 7.442 corresponding to 5 years row is found in 20% column. So the implicit rate of interest is 20%.

Example 26: Assume that a deposit is made at zero years into an account that will earn 8% compounded annually. it is desired to withdraw Rs 5,000 three years from now and Rs 7,000 six years from now. What is the size of the year zero deposit that will produce these future payments?

Answer: let the deposit be the sum of the present values of the later withdrawals by using the present value table:

PV = FV x PVF _(r%,n)

= 5000 x PVF _(8%,3)+ 7000 x PVF _(8%,6)

= 5000 x .794+ 7000 x .630

= 3,970 + 4,410

= 8,380.

Amortizing a Loan Example

Example 27: Julie is borrowing $10,000 at a compound annual interest rate of 12%.  Amortize the loan if annual payments are made for 5 years.

                             Payment

    PV0            = R (PVIFA i%,n)

                              $10,000        = R (PVIFA 12%,5)

                              $10,000        = R (3.605)

                             R = $10,000 / 3.605 = $2,774

Example 28:  Assume that Rs 20,00,000 plant extension is to be financed through the following plan: Pay 15% of the cost as down payment and the balance is to be repaid in 8 equal annual installments beginning 4 years from now. What is the size of the required annual loan payments? Compound interest is 9% per annum.

Answer:

The firm borrows 17,00,000 which is (Rs 20,00,000-Rs 3,00,000).compound interest is 9% per annum.the first payment will be made after 3 years or alternatively first payment  will be made in the fourth year. By compounding we get the balance of loan in the fourth year as follows:

FV = PV (1+r)ⁿ

FV = 17,00,000 (1+.09)³ = 22,01,550

Now the FV becomes the present value of the 8 payment annuity discounted at 9%.so compute the equal  yearly payment by using the following equation.

PV = Annuity x PVAF_(9%,8)

22,01,550 = Annuity x 5.535

Or Annuity = 22,01,550 / 5.535 = 3, 97,750

So annual payment for the next year is going to be Rs 3, 97,750

Example 29:  A 10 year annuity of Rs 2000 per year is beginning at the end of current year. The payment of retirement annuity is to begin 16 years from now and continue to provide a 20 year payment annuity. Interest is 7% per annum , what will be the size of annuity to be received?

Answer: First find the compounded amount at the end of year 10.

FV =annuity x CVAF _(7%,10)

FV = 2,000 X 13.816 = 27,632

The amount of Rs 27,632 is available immediately after the last payment. This amount invested for next 5 years will become,(beginning of 16 year)

FV = PV x CVF _(7%,5)

FV =27,632 x CVF _(7%,10) =27,632 x1.403 = 38,768

Finally find the retirement annuity,

PV  = Annuity x PVAF _(7%,20)

38,768 = Annuity x 10.594

Annuity = 38,768/ 10.594 = 3,659

Example 30: (sinking fund) A machine costs Rs 98,000 and its effective life is estimated at 12 years . it's the effective life is estimated at 12 years. If the scrap value is Rs 3, 000, what should be cut out of the profit at the end of each year to accumulate at compound rate of 5% per annum so that a new machine can be purchased after 12 years ?

Answer: Effective cost of the machine = Rs 98,000 –Rs 3,000 = Rs 95,000.

FV =annuity x CVAF _(5%,12)

95,000 = Annuity  X 15.917

So annuity = 95,000/15.917 =Rs 5,968.

Example 31: A finance company offers to give Rs.20,000 after 14 years in return for Rs.5,000 deposited today. Using the rule of 69, figure out the approximate interest rate offered.

Solution:

In 14 years Rs.5,000 grows to Rs.20,000 or 4 times. This is 2² times the initial deposit. Hence doubling takes place in 14 / 2 = 7 years.

According to the Rule of 69, the doubling period is 0.35 + 69 / Interest rate

We therefore have 0.35 + 69 / Interest rate = 7

Interest rate = 69/(7-0.35) = 10.38 %

Example 32: Someone offers to give Rs.80,000 to you after 18 years in return for Rs.10,000 deposited today. Using the rule of 69, figure out the approximate interest rate offered.

Solution:

In 18 years Rs.10,000 grows to Rs.80,000 or 8 times. This is 23 times the initial deposit. Hence doubling takes place in 18 / 3 = 6 years.

According to the Rule of 69, the doubling period is

0.35 + 69 / Interest rate.

We therefore have 0.35 + 69 / Interest rate = 6

Interest rate = 69/(6-0.35) = 12.21 %

Example 33: You can save Rs.5,000 a year for 3 years, and Rs.7,000 a year for 7 years thereafter.

What will these savings cumulate to at the end of 10 years, if the rate of interest is 8 percent?

Solution:

Saving Rs.5000 a year for 3 years and Rs.6000 a year for 7 years thereafter is equivalent to saving Rs.5000 a year for 10 years and Rs.2000 a year for the years 4 through 10.

Hence the savings will cumulate to:

5000 x FVIFA (8%, 10 years) + 2000 x FVIFA (8%, 7 years)

= 5000 x 14.487 + 2000 x 8.923 = Rs.90281

Example 34: Krishna saves Rs.24,000 a year for 5 years, and Rs.30,000 a year for 15 years thereafter. If the rate of interest is 9 percent compounded annually, what will be the value of his savings at the end of 20 years?

Solution:

Saving Rs.24,000 a year for 5 years and Rs.30,000 a year for 15 years thereafter is equivalent to saving Rs.24,000 a year for 20 years and Rs.6,000 a year for the years 6 through 20.

Hence the savings will cumulate to:

24,000 x FVIFA (9%, 20 years) + 6,000 x FVIFA (9 %, 15 years)

= 24,000 x 51.160 + 6, 000 x 29.361 =Rs. 1,404,006

Example 35: You plan to go abroad for higher studies after working for the next five years and understand that an amount of Rs.2,000,000 will be needed for this purpose at that time. You have decided to accumulate this amount by investing a fixed amount at the end of each year in a safe scheme offering a rate of interest at 10 percent. What amount should you invest every year to achieve the target amount?

Solution:

Let A be the annual savings.

A x FVIFA (10%, 5years) = 2,000,000

A x 6.105 = 2,000,000

So, A = 2,000,000 / 6.105 = Rs. 327,600

Example 36: How much should Vijay save each year, if he wishes to purchase a flat expected to cost Rs.80 lacs after 8 years, if the investment option available to him offers a rate of interest at 9 percent? Assume that the investment is to be made in equal amounts at the end of each year.

Solution:

Let A be the annual savings.

A x FVIFA (9 %, 8 years) = 80,00,000

A x 11.028 = 80,00,000

So, A = 80,00,000 / 11.028 = Rs. 7,25,426

Example 37: A finance company advertises that it will pay a lump sum of Rs.100,000 at the end of 5 years to investors who deposit annually Rs.12,000. What interest rate is implicit in this offer?

Solution:

12,000 x FVIFA (r, 5 years) = 100,000

FVIFA (r, 5 years) = 100,000 / 12,000 = 8.333

From the tables we find that

FVIFA (24%, 5 years) = 8.048

FVIFA (28%, 5 years) = 8.700

Using linear interpolation in the interval, we get:

Example 38: Someone promises to give you Rs.5,000,000 after 6 years in exchange for Rs.2,000,000 today. What interest rate is implicit in this offer?

Solution:

2,000,000 x FVIF (r, 6 years) = 5,000,000

FVIF (r, 6 years) = 5,000,000 / 2,000,000 = 2.5

From the tables we find that

FVIF (16%, 6 years) = 2.436

FVIF (17%, 6 years) = 2.565

Using linear interpolation in the interval, we get:

Example 39: At the time of his retirement, Rahul is given a choice between two alternatives:

(a) an annual pension of Rs120,000 as long as he lives, and

(b) a lump sum amount of Rs.1,000,000.

If Rahul expects to live for 20 years and the interest rate is expected to be 10 percent throughout , which option appears more attractive.

Solution:

The present value of an annual pension of Rs.120,000 for 20 years when r = 10% is:

120,000 x PVIFA (10%, 20 years)

= 120,000 x 8.514 = Rs.1,021,680

The alternative is to receive a lumpsum of Rs 1,000,000

Rahul will be better off with the annual pension amount of Rs.120,000.

Example 40: What is the present value of an income stream which provides Rs.30,000 at the end of year one, Rs.50,000 at the end of year three , and Rs.100,000 during each of the years 4 through 10, if the discount rate is 9 percent ?

Solution:

The present value of the income stream is:

30,000 x PVIF (9%, 1 year) + 50,000 x PVIF (9%, 3 years)

+ 100,000 x PVIFA (9 %, 7 years) x PVIF(9%, 3 years)

= 30,000 x 0.917 + 50,000 x 0.772 + 100,000 x 5.033 x 0.0.772 = Rs.454,658.

Example 41: What is the present value of an income stream which provides Rs.25,000 at the end of year one, Rs.30,000 at the end of years two and three , and Rs.40,000 during each of the years 4 through 8 if the discount rate is 15 percent ?

Solution:

The present value of the income stream is:

25,000 x PVIF (15%, 1 year) + 30,000 x PVIF (15%, 2 years)

+ 30,000 x PVIF (15%, 3 years)

+ 40,000 x PVIFA (15 %, 5 years) x PVIF (15%, 3 years)

= 25,000 x 0.870 + 30,000 x 0.756 + 30,000 x 0.658

+ 40,000 x 3.352 x 0.658 = Rs.152,395.

Example 42: What is the present value of an income stream which provides Rs.1,000 a year for the first three years and Rs.5,000 a year forever thereafter, if the discount rate is 12 percent?

Solution:

The present value of the income stream is:

1,000 x PVIFA (12%, 3 years) + (5,000/ 0.12) x PVIF (12%, 3 years)

= 1,000 x 2.402 + (5000/0.12) x 0.712

= Rs.32,069

Example 43: What is the present value of an income stream which provides Rs.20,000 a year for the first 10 years and Rs.30,000 a year forever thereafter, if the discount rate is 14 percent ?

Solution:

The present value of the income stream is:

20,000 x PVIFA (14%, 10 years) + (30,000/ 0.14) x PVIF (14%, 10 years)

= 20,000 x 5.216 + (30,000/0.14) x 0.270

= Rs.162,177

Example 44: Mr. Ganapathi will retire from service in five years .How much should he deposit now to earn an annual income of Rs.240,000 forever beginning from the end of 6 years from now ? The deposit earns 12 percent per year.

Solution:

To earn an annual income of Rs.240,000 forever , beginning from the end of 6 years from now, if the deposit earns 12% per year a sum of

Rs.240,000 / 0.12 = Rs.2,000,000

is required at the end of 5 years. The amount that must be deposited to get this

sum is:

Rs.2,000,000 PVIF (12%, 5 years) = Rs.2,000,000 x 0.567 = Rs. 1,134,000

Example 45: Suppose someone offers you the following financial contract. If you deposit Rs.100,000 with him he promises to pay Rs.50,000 annually for 3 years. What interest rate would you earn on this deposit?

Solution:

Rs.100,000 =- Rs.50,000 x PVIFA (r, 3 years)

PVIFA (r,3 years) = 2.00

From the tables we find that:

PVIFA (20 %, 3 years) = 2.106

PVIFA (24 %, 3 years) = 1.981

Using linear interpolation we get:

Example 46: If you invest Rs.600,000 with a company they offer to pay you Rs.100,000 annually for 10 years. What interest rate would you earn on this investment?

Solution:

Rs.600,000 =- Rs.100,000 x PVIFA (r, 10 years)

PVIFA (r,10 years) = 6.00

From the tables we find that:

PVIFA (10 %, 10 years) = 6.145

PVIFA (11 %, 10 years) = 5.889

Using linear interpolation we get:

Example 47: Suppose you deposit Rs.200,000 with an investment company which pays 12 percent interest with compounding done once in every two months, how much will

this deposit grow to in 10 years?

Solution:

FV10 = Rs.200,000 [1 + (0.12 / 6)]^10x6

= Rs.200,000 (1.02)⁶⁰

= Rs.200,000 x 3.281

= Rs.656,200

Example 48: A bank pays interest at 5 percent on US dollar deposits, compounded once in every six months. What will be the maturity value of a deposit of US dollars 15,000 for three years?

Solution:

Maturity value = USD 15 ,000 [1 + (0.05 / 2)]^3x2

= 15,000 (1.025)⁶

= 15,000 x 1.1597

= 17,395.50

Example 49: You have a choice between Rs.200,000 now and Rs.600,000 after 8 years. Which would you choose? What does your preference indicate if market rate of interest is 14% ?

Solution:

The interest rate implicit in the offer of Rs.600,000 after 8 years in lieu of

Rs.200,000 now is:

Rs.200,000 x FVIF (r,8 years) = Rs.600,000

FVIF (r,8 years) = Rs.600,000/Rs.200,000  = 3.000

From the tables we find that

FVIF (15%, 8years) = 3.059

This means that the implied interest rate is nearly 15%.

I would choose Rs.600,000 after 8 years from now because I find a return of 15% quite attractive.

Example 50: Ravikiran deposits Rs.500,000 in a bank now. The interest rate is 9 percent and compounding is done quarterly. What will the deposit grow to after 5 years? If the inflation rate is 3 percent per year, what will be the value of the deposit after 5 years in terms of the current rupee?

SOLUTION

FV5 = Rs.500,000 [1 + (0.09 / 4)]²⁰

 = Rs.500,000 (1.0225)²⁰

= Rs.500,000 x 2.653

= Rs.780,255

If the inflation rate is 3 % per year, the value of Rs.780,255 5 years from now, in terms of the current rupees is:

Rs.780,255 x PVIF (3%, 5 years)

= Rs.780,255 x 0. 863 = Rs.673,360

Example 51: A person requires Rs.100,000 at the beginning of each year from 2015 to 2019. Towards this, how much should he deposit ( in equal amounts) at the end of each year from 2007 to 2011, if the interest rate is 10 percent.

Solution:

The discounted value of Rs.100,000 receivable at the beginning of each year from 2015 to 2019, evaluated as at the beginning of 2014 (or end of 2013) is:

Rs.100,000 x PVIFA (10%, 5 years)

= Rs.100,000 x 3.791= Rs.379,100

The discounted value of Rs.379,100 evaluated at the end of 2011 is

Rs.379,100 x PVIF (10 %, 2 years)

= Rs.379,100 x 0.826= Rs.313,137

If A is the amount deposited at the end of each year from 2007 to 2011 then

A x FVIFA (10%, 5 years) = Rs.313,137

A x 6.105 = Rs.313,137

A = Rs.313,137/ 6.105 = Rs.51,292

Example 52: You require Rs.250 ,000 at the beginning of each year from 2010 to 2012. How much should you deposit( in equal amounts) at the beginning of each year in 2007 and 2008 ? The interest rate is 8 percent.

Solution:

The discounted value of Rs.250,000 receivable at the beginning of each year from 2010 to 2012, evaluated as at the beginning of 2009 (or end of 2008) is:

Rs.250,000 x PVIFA (8 %, 3 years)

= Rs.250,000 x 2.577= Rs.644,250

To have Rs. 644,250 at the end of 2008, let A be the amount that needs to be deposited at the beginning of 2007 and 2008.We then have

Ax (1+0.08) x FVIFA ( 8%, 2years) = 644,250

A x 1.08 x 2.080 = 644,250 or A = 286,792

Example 53: What is the present value of Rs.120,000 receivable annually for 20 years if the first receipt occurs after 8 years and the discount rate is 12 percent.

Solution:

The discounted value of the annuity of Rs.120,000 receivable for 20 years,

evaluated as at the end of 7th year is:

Rs.120,000 x PVIFA (12%, 20 years) = Rs.120,000 x 7.469 = Rs.896,290

The present value of Rs. 896,290 is:

Rs. 896,290 x PVIF (12%, 7 years)

= Rs. 896,290 x 0.452

= Rs.405,119

Example 54: What is the present value of Rs.89,760 receivable annually for 10 years if the first receipt occurs after 5 years and the discount rate is 9 percent.

Solution:

The discounted value of the annuity of Rs.89,760 receivable for 10 years, evaluated as at the end of 4th year is:

Rs. 89,760 x PVIFA (9%, 10 years) = Rs. 89,760 x 6.418 = Rs.576,080

The present value of Rs. 576,080is:

Rs. 576,080x PVIF (9%, 4 years)

= Rs. 576,080x 0.708

= Rs.407,865

Example 54: Metro Corporation has to retire Rs.20 million of debentures each at the end of 6, 7,and 8 years from now. How much should the firm deposit in a sinking fund account annually for 5 years, in order to meet the debenture retirement need? The net interest rate earned is 10 percent.

Solution:

The discounted value of the debentures to be redeemed between 6 to 8 years evaluated at the end of the 5th year is:

Rs.20 million x PVIFA (10%, 3 years) = Rs.20 million x 2.487

= Rs.49.74million

If A is the annual deposit to be made in the sinking fund for the years 1 to 5,

then

A x FVIFA (10%, 5 years) = Rs.49.74 million

A x 6.105 = Rs.49.74 million

A = Rs.8,147,420

Example 55: Ankit Limited has to retire Rs.30 million of debentures each at the end of 7, 8, 9 and 10 years from now. How much should the firm deposit in a sinking fund account annually for 5 years, in order to meet the debenture retirement need? The net interest rate earned is 12 percent.

Solution:

The discounted value of the debentures to be redeemed between 7 to 10 years evaluated at the end of the 6th year is:

Rs.30 million x PVIFA (12%, 4 years) = Rs.30 million x 3.037

= Rs.91.11 million

If A is the annual deposit to be made in the sinking fund for the years 1 to 6,

then

A x FVIFA (12%, 6 years) = Rs.91.11 million

A x 8.115 = Rs. 91.11 million

A = Rs.11,227,357

Example 56: Mr.Mehta receives a provident fund amount or Rs.800,000. He deposits it in a bank which pays 9 percent interest. If he plans to withdraw Rs.100,000 at the end of each year, how long can he do so ?

Solution:

Let `n’ be the number of years for which a sum of Rs.100,000 can be withdrawn annually.

Rs.100,000 x PVIFA (9%, n) = Rs.800,000

PVIFA (9%, n) = Rs.800,000 / Rs.100,000 = 8 .000

From the tables we find that

PVIFA (9%, 14 years) = 7.786

PVIFA (9%, 15 years) = 8.060

Using a linear interpolation we get

Example 57: Mr. Naresh wants to invest an amount of Rs. 400,000, in a finance company at an interest rate of 12 percent, with instructions to the company that the amount with interest be repaid to his son in equal instalments of Rs.100,000, for his education expenses . How long will his son get the amount ?

Solution:

Let `n’ be the number of years for which a sum of Rs.100,000 can be withdrawn annually.

Rs.100,000 x PVIFA (12%, n) = Rs.400,000

PVIFA (12 %, n) = Rs.400,000 / Rs.100,000 = 4

From the tables we find that

PVIFA (12%, 5 years) = 3.605

PVIFA (12%, 6 years) = 4.111

Using a linear interpolation we get

Example 58: As a winner of a competition, you can choose one of the following prizes:

a. Rs. 800,000 now

b. Rs. 2,000,000 at the end of 8 years

c. Rs. 100,000 a year forever

d. Rs. 130,000 per year for 12 years

e. Rs. 32,000 next year and rising thereafter by 8 percent per year forever.

If the interest rate is 12 percent, which prize has the highest present value?

Solution:

(a) PV = Rs.800,000

(b) PV = 2,000,000PVIF12%,8yrs = 2,000,000 x 0.0.404 = Rs.808,000

(c ) PV = 100,000/r = 100,000/0.12 = Rs. 833,333

(d) PV = 130,000 PVIFA12%,12yrs = 130,000 x 6.194 = Rs.805,220

(e) PV = C/(r-g) = 32,000/(0.12-0.08) = Rs.800,000

Option c has the highest present value viz. Rs.833,333

Example 59: You want to borrow Rs.3,000,000 to buy a flat. You approach a housing company which charges 10 percent interest. You can pay Rs.400,000 per year toward loan amortisation. What should be the maturity period of the loan?

Solution:

Let n be the maturity period of the loan. The value of n can be obtained from

the equation.

400,000 x PVIFA(10%, n) = 3,000,000

PVIFA (10%, n) = 7.5

From the tables we find that

PVIFA (10%,14 years) = 7.367

PVIFA (10 %, 15 years) = 7.606

Using linear interpolation we get

Example 60: You want to borrow Rs.5,000,000 to buy a flat. You approach a housing company which charges 11 percent interest. You can pay Rs.600,000 per year toward loan amortisation. What should be the maturity period of the loan?

Solution:

Let n be the maturity period of the loan. The value of n can be obtained from

the equation.

600,000 x PVIFA(11%, n) = 5,000,000

PVIFA (11%, n) = 8.333

From the tables we find that

PVIFA (11%,20 years) = 7.963

PVIFA (11 %, 25 years) = 8.422

Using linear interpolation we get,

CASE STUDY 1

As an investment advisor, you have been approached by a client called Vikas for your advice on investment plan. He is currently 40 years old and has Rs.600,000 in the bank. He plans to work for 20 years more and retire at the age of 60. His present salary is Rs.500,000 per year. He expects his salary to increase at the rate of 12 percent per year until his retirement.

Vikas has decided to invest his bank balance and future savings in a balanced mutual fund scheme that he believes will provide a return of 9 percent per year. You agree with his assessment.

Vikas seeks your help in answering several questions given below. In answering

these questions, ignore the tax factor.

(i) Once he retires at the age of 60, he would like to withdraw Rs.800,000 per year for his consumption needs from his investments for the following 15 years (He expects to live upto the age of 75 years). Each annual withdrawal will be made at the beginning of the year. How much should be the value of his investments when Vikas turns 60, to meet this retirement need?

(ii) How much should Vikas save each year for the next 20 years to be able to withdraw Rs.800,000 per year from the beginning of the 21st year ? Assume that the savings will occur at the end of each year.

(iii) Suppose Vikas wants to donate Rs.500,000 per year in the last 5 years of his life to a charitable cause. Each donation would be made at the beginning of the year. Further, he wants to bequeath Rs.1,000,000 to his son at the end of his life. How much should he have in his investment account when he reaches the age of 60 to meet this need for donation and bequeathing?

(iv) Vikas is curious to find out the present value of his lifetime salary income. For the sake of simplicity, assume that his current salary of Rs.500,000 will be paid exactly one year from now, and his salary is paid annually. What is the present value of his life time salary income, if the discount rate applicable to the same is 7 percent?

Remember that Vikas expects his salary to increase at the rate of 12 percent per

year until retirement.

ANSWER

(i) This is an annuity due

Value of annuity due = Value of ordinary annuity (1 + r)

The value of investments when vikas turns 60 must be:

800,000 x PVIFA (9%, 15 years) x 1.09

= 800,000 x 8.060 x 1.09 = Rs.7,028,320

(ii) He must have Rs.7,092,800 at the end of the 20th year.

His current capital of Rs.600,000 will grow to:

Rs.600,000 x FVIF (9%, 20yrs) = 600,000 x 5.604

= Rs.3,362,400

So, what he saves in the next 15 years must cumulate to:

7,028,320 – 3,362,400 = Rs.3,665,920

A x FVIFA (9%, 20 yrs) = Rs.3,665,920

A x 51.160 = 3,665,920

A = 3,665,920/51.160 = Rs.71,656

(iii)

To meet his donation objective, Vikas will need an amount equal to:

500,000 x PVIFA (9%, 5years) when he turns 69.

This means he will need

500,000 x PVIFA (9%, 5yrs) x PVIF (9%, 9yrs) when he turns 60.

This works out to:

500,000 x 3.890 x 0.460 = Rs.894,700

To meet his bequeathing objective he will need

1,000,000 x PVIF (15%, 9yrs) when he turns 60

This works out to:

1,000,000 x 0.275 = Rs.275,000

So, his need for donation and bequeathing is: 894,700 + 275,000

= Rs.1,169,700

(iv)

CASE STUDY 2

As an investment advisor, you have been approached by a client called Ravi for  advice on his investment plan. He is 35 years and has Rs.200, 000 in the bank. He plans to work for 25 years more and retire at the age of 60. His present salary is 500,000 per year. He expects his salary to increase at the rate of 12 percent per year until his retirement.

Ravi has decided to invest his bank balance and future savings in a balanced mutual fund scheme that he believes will provide a return of 9 percent per year.

You concur with his assessment.

Ravi seeks your help in answering several questions given below. In answering these questions, ignore the tax factor.

(i) Once he retires at the age of 60, he would like to withdraw Rs. 900,000 per year for his consumption needs for the following 20 years (His life expectancy is

80years).Each annual withdrawal will be made at the beginning of the year.

How much should be the value of his investments when he turns 60, to meet his

retirement need?

(ii) How much should Ravi save each year for the next 25 years to be able to withdraw Rs.900, 000 per year from the beginning of the 26th year for a period

of 20 years?

Assume that the savings will occur at the end of each year. Remember that he already has some bank balance.

(iii) Suppose Ravi wants to donate Rs.600, 000 per year in the last 4 years of his life to a charitable cause. Each donation would be made at the beginning of the year.

Further he wants to bequeath Rs. 2,000,000 to his daughter at the end of his life.

How much should he have in his investment account when he reaches the age of

60 to meet this need for donation and bequeathing?

(iv) Ravi wants to find out the present value of his lifetime salary income. For the sake of simplicity, assume that his current salary of Rs 500,000 will be paid

exactly one year from now, and his salary is paid annually. What is the present value of his lifetime salary income, if the discount rate applicable to the same is

8 percent? Remember that Ravi expects his salary to increase at the rate of 12 percent per year until retirement.

ANSWER

(i)

900,000 x PVIFA ( 9 %, 20 ) x 1.09

900,000 x 9.128 x 1.09

= Rs. 8,954,568

(ii)

Ravi needs Rs. 8,954,568 when he reaches the age of 60.

His bank balance of Rs. 200,000 will grow to : 200,000 ( 1.09 )25

= 200,000 ( 8.623 ) = Rs. 1,724,600

This means that his periodic savings must grow to :

Rs. 8,954,568 - Rs. 1,724,600 = Rs. 7,229,968

His annual savings must be:

(iii)

Amount required for the charitable cause:

600,000 x PVIFA ( 9% , 4yrs ) x PVIF ( 9%, 15yrs )

= 600,000 x 3.240 x 0.275

Rs. 534,600

Amount required for bequeathing

2,000,000 x PVIF ( 9%, 20yrs )

= 2,000,000 x 0.178 = Rs.356,000

(iv)