Time Value of Money

The Time Value of Money

Would you prefer to

have Rs1 million now or

Rs1 million 10 years

from now?

Of course, we would all Of course, we would all prefer the money now!prefer the money now!

This illustrates that there This illustrates that there is an inherent monetary is an inherent monetary

value attached to timevalue attached to time..

“Money has time value” means that the valueOf money changes over a period of time.

The Value of rupee, today is different from what it will be, say, after one year.

Why should money have time value?

1. In an inflationary period, a rupee today has a higher Purchasing power than a rupee in the future.

2. Since future is characterized by uncertainty, individuals Prefer current consumption to future consumption

3. Money can be employed productively to earn real return.

Factors Contributing to the time value of money:

1. Individuals generally prefer current consumption to future consumption.

2. An investor can profitably employ a rupee received today, to give him a higher value to be received tomorrow

3. In Inflationary economy, the money received today has more purchasing power than the money received in future.

4. “ A bird in hand is worth two in bush” ie uncertainty connected to future

Nominal or Market interest rate: = Real rate of interest + Expected rate of Inflation

OR

Real interest rate = Nominal - inflation

Eg:

If somebody lends $ 1000 for a year at 10%, and receives $ 1100 back at the End of the year, this represents a 10% increase in purchasing power if prices of the average goods and services that they buys are unchanged from what they were at the beginning of the year.

If price of Food, clothing, housing purchases have increased 20% over this period, than he has suffered a real loss of about 10% in his purchasing power.

The Inflation rate will not be known in advance

Borrower hope to repay in cheaper money in the future, while lender hope to collect on more expensive money.

When an inflation and currency risk are underestimated by lenders, then they will suffer a net reduction in buying power.

An interest rate that has been adjusted to remove the effects of inflation to reflect the real cost of funds to the borrower, and the real yield to the lender. The real interest rate of an investment is calculated as the amount by which the nominal interest rate is higher than the inflation rate.

Real Interest Rate = Nominal Interest Rate - Inflation (Expected or Actual)

The real interest rate is the growth rate of purchasing power derived from an investment. By adjusting the nominal interest rate to compensate for inflation, you are keeping the purchasing power of a given level of capital constant over time.

For example, if you are earning 4% interest per year on the savings in your bank account, and inflation is currently 3% per year, then the real interest rate you are receiving is 1% (4% - 3% = 1%). The real value of your savings will only increase by 1% per year, when purchasing power is taken into consideration.

Nominal Interest rate is the periodic interest rate times the number of periods perYear.

Uses of Time Value of Money

• Time Value of Money, or TVM, is a concept that is used in all aspects of finance including:– Bond valuation– Stock valuation– Accept/reject decisions for project management– Financial analysis of firms– And many others!

Objectives

• Understand what gives money its time value.

• Explain the methods of calculating present and future values.

• Highlight the use of present value technique (discounting) in financial decisions.

• Introduce the concept of internal rate of return.

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:–  risk–  preference for consumption–  investment opportunities

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.

Required rate of return is also called Opportunity Cost of Capital

Time Value Adjustment

• Two most common methods of adjusting cash flows for time value of money: – Compounding—the process of calculating future

values of cash flows and – Discounting—the process of calculating present

values of cash flows.

• Principal: The amount borrowed or invested

• Interest rate: A percentage of the outstanding principle.

• Time: the number of years or fractional portion of a year that principal is outstanding.

Variables in Interest Variables in Interest ComputationsComputations

Basic Time DiagramBasic Time Diagram

Future Value/ Compounding value concept

• Compounding is the process of finding the future values of cash flows by applying the concept of compound interest.

• Compound interest is the interest that is received on the original amount (principal) as well as on any interest earned but not withdrawn during earlier periods.

• Simple interest is the interest that is calculated only on the original amount (principal), and thus, no compounding of interest takes place.

Q1. Rs 1,000 invested at 10% is compounded annually for 3 years.

Calculate the compounded value after three years

Solution:

Amount at the end of 1st year will be: 1100( 1000*110/100) P(1+i)

Amount at the end of 2nd year will be: 1210(1100*110/100) P(1+i)2

Amount at the end of 3rd Year will be 1331(1210*110/100)

A = P (1+i)n

A = Amount at the end of period ‘n’P = Principal at the beginning of the periodI = Interest rateN= Number of years

A = P(1+i)n

A= 1000(1.10) + 1000(1.10)(1.10) + 1000(1.10)(1.10)(1.10)

A = 1000 (1+ .10)3

A = 1331

Compound value table can be used (Compound value of Re 1 at 10% p.a. at the end of 3 years as 1331)

1000* 1.331 = Rs 1331

Future Value/ Compound value

• 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.

nn iPF )1(

= CVFn n,iF P

Example

• 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 225,159.90 as the compound value:

10, 0.12FV 55,650 × CVF 55,650 4.046 Rs 225,159.90

Future value of series of cash flows

An investor investing in installments may wish to know the value of his savings after n years

Q. Mr. Manoj invests Rs 500, Rs 1,000, Rs 1,500, Rs 2,000 and Rs 2500 at the end of each year. Calculate the compounded value at the end of 5 years, compounded annually, when the interestcharged is 5% p.a.

FVIF = Future value interest factorFV = Future value

Year Amount Deposited

No. of years compounded (at

the end of the year)

FVIF from table

FV

1 500 4 1.216 608.00

2 1000 3 1.158 1158.00

3 1500 2 1.103 1654.50

4 2000 1 1.050 2100.00

5 2500 0 1.000 2500.00

Compound interest formula for each seperately:

CV = 500 (1+.005)4 = Rs 608

= 1000 (1+ .005)3 = Rs 1158

0 1 2 3 4 5Rs 500 Rs1000 Rs 2000 Rs 2500

Rs 2100.00 Rs 1654.50 Rs 1158.00 Rs 608.00

Rs 8020.50

Double Your Money!!!Double Your Money!!!

Quick! How long does it take to double $5,000 at a compound rate of 12% per year

(approx.)?

We will use the ““Rule-of-72Rule-of-72””..

The “Rule-of-72”The “Rule-of-72”

Quick! How long does it take to double $5,000 at a compound rate of 12% per year

(approx.)?

Approx. Years to Double = 7272 / i%

7272 / 12% = 6 Years6 Years[Actual Time is 6.12 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.

• The term within brackets is the compound value factor for an annuity of Re 1, which we shall refer as CVFA.

(1 ) 1n

n

iF A

i

= CVFAn n, iF A

Example

• Suppose that a firm deposits Rs 2,000 at the end of each year for 45 years at 5 per cent rate of interest. How much would this annuity accumulate at the end of the fifth year? We first find CVFA which is 5.527. If we multiply 5.527 by Rs 2,000, we obtain a compound value of Rs 11,054:

F5 = 2000 (CVFA 5, 0.005 ) = 2000 x 5.527 = Rs 11,054

Year Amount Deposited

No. of years compounded (at

the end of the year)

FVIF from table

FV

1 2000 4 1.216 2,432

2 2000 3 1.158 2,316

3 2000 2 1.103 2,206

4 2000 1 1.050 2,100

5 2000 0 1.000 2,000

11,054

How Much you should save annually

Q. You want to buy a house after 5 years when it is expected cost Rs 2 million. How much should you save annually if your savings earn a compound return of 12%?

FVIFA (n= 5, r =12%) = 6.353

(1+ .12)2 – 1/0.12

Annual Savings should be: 20,000,00/6.353 = Rs 314,812

Annual Deposit in a Sinking Fund

Q Futura limited has an obligation to redeem Rs 500 million bonds 6 years hence. How much should the company deposit annually in a sinking fund account wherein it earns 14% interest to cumulate Rs 500 million in 6 years time?

The future value interest factor for a 6 year annuity, given an interest rate of 14% is 8.536

FVIFA = (1+ .14)6 – 1/.14 = 8.536

The annual sinking fund deposit should be = Rs 500 Million/8.536 = Rs 58.575 Million

Sinking Fund Factor:

A fund which is created out of fixed payments each period to accumulate to a future sum after a specified period.

Co. generally create sinking fund to retire bonds (debentures) on maturity.

It is equal to the reciprocal of compound value factor for an annuity

F = A * CVFA n,i

A = Fn * 1/ CVFA n,i

A = F * SFF

Sinking Fund Annuity =

Future value/ Compound value factor of an annuity of Re 1

A = F ( i /(1+i)n – 1)

Suppose that we want to accumulate Rs 21875 at the end of four years from now. How much should we deposit each year at an interest rate of 6% so that it grows to Rs 21,875 at the end of fourth year.

A = 21875 * (1/4.375)

A = 21875 * .2286

Rs 5000

The sinking fund factor is useful in determining the annual amount to be put in a fund to repay bonds or debentures at the end at a specified period.

Compounding Semi Annually

Q You deposit Rs 10000 in bank which offers 10% interest per annum compundedSemi annually .

Solution:

Amount in the beginning Rs 10,000Interest @ 10% p.a. for first six months Rs 500 ( 10000* .1/2)

Rs 10,500

Interest for second six months Rs 525(10500 * .1/2) Rs 11,025

If compounding is done annually, than amount at the end of the year is10000 (1+ .1) = Rs 11000

Rs 25 difference is due interest for first six months earn interest

FV n = PV(1+ k/m) m*n

Where,

FVn = Future value after “n” yearsPV = Cash flow todayK = Nominal interest rate per annumm = number of times compounding is done in an yearN = number of years for which compounding is done

Q Under the cash certificate scheme of ICICI bank, deposits can be made for periodsFrom 6 months to 10 years. Every quarter, interest will be added on to principle. The Rate of interest applied is 9 % p.a. for period from 12 to 23 months and 10 % p.a.For periods from 24 to 120 months

An amount of 1000 invested for 2 years will grow to: FV 2 = 1000( 1 + 0.10/4)

= 1000 (1.025)8

= 1000 * 1.2184 = Rs 1218

Interest Rate is usually specified on an annual basis in a loan agreement or security and is known as the Nominal Interest rate

If compounding is done more than once a year, the actual Annualised rate of interest would be higher than the nominal interest rate and it is called effective interest rate.

Q. Compound value of Rs 1000 interest rate being 12% per annum if compounded Annually, semi annually, quarterly and monthly for 2 years.

1. Annual Compounding = 1000 * (1.12)2 = 1000 * 1.254 = Rs 1254

2. Half yearly compounding : 1000 * (1 + .12/2)2*2 = 1000 * (1.06)4 = Rs 1262

3. Quarterly compounding : 1000 * (1 + .12/4)2*4 = 1000 * (1.03)8 = Rs 1267

4. Monthly compounding : 1000* (1 + .12/12)2*12 = 1000* (1.01)24 = Rs 1270

Present Value Discounting value concept

• 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.

• Discounting is the process of determining present

value of a series of future cash flows.

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

Present Value of a Single Cash Flow

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

• 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.

(1 )(1 )

nnnn

FP F i

i

,PVFn n iPV F

Example

• 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:

15, 0.09PV = 50,000 PVF = 50,000 0.275 = Rs 13,750

Present value of series of cash flows

Q Given the time value of money as 10%. You are required to find out the presentValue of future cash inflows that will be received over the next four years:

Year Cash Flow1 1,0002 2,0003 3,0004 4,000

Year Cash Flow PVIF at 10% PV

1 1,000 .909 909

2 2,000 .826 1,652

3 3,000 .751 2,253

4 4,000 .683 2,732

7546

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.

1 1

1n

P Ai i i

= × PVAFn, iP A

Q. Mr. Bhat wishes to determine the PV of the annuity consisting of cash flows of Rs 4000 per annum for 6 years. The rate of interest he can earn from this Investment is 10%

Rs 40,00 * PVIPARs 4,000 * 4.355 = Rs17,420

Year Cash Flow PVIF at 10% PV

1 4000 .91 3636

2 4000 .826 3304

3 4000 .751 3004

4 4000 .683 2732

5 4000 .621 2484

6 4000 .564 2256

PV at annuity 17416

Capital recovery or Loan Amortization:

If you make an invest today for a given period of time at a specified rate of interestYou may like to know annual income.

Capital Recovery is the annuity of an investment for a specified time at a given Rate of interest.

The reciprocal of Present value annuity factor (PVAF) is called Capital recoveryFactor.

Amount = P [ i (1+i)n/ (1+ i)n -1 ]

PV = A (PVAF)

A = P ( 1/PVFA n, i)

A = PV * CRF n, i

Q. Suppose you have borrowed a 3 year loan of Rs 10,000 at 9%From your employer to buy a motorcycle. IF your employer requires three equal end of the year repayment, then annual installment will be

P = A (PVAF 3,.09)10000 = A (2.531)A = 10000/ 2.531A = 3951

By paying Rs 3951 each year for three years Completely pay off loan with 9% interest.

End of year

Payment Interest PrincipalRepayment

Outstanding Balance

1 3951 900 3051 6959

2 3951 625 3326 3623

3 3951 326 3625 0

Loan Amortizations Schedule

Present value of a perpetuity

Perpetuity is annuity that occurs Indefinitely. For eg irredeemable preference shares.

P = A/i (1+i)n is equal to zero

An investor expect a perpetual sum of Rs 500 annually from His investmentP = 500/.10

Present value of Growing Annuity

Assume that to finance your post graduate studies in an evening college, you undertake a part time job for 5 years. Your employer fixes an annual salary of Rs 1000 with the provision that you will get annual increment at the rate of 10%.

End of year Amount of salary (Rs)

1 1,000 = 1000*1.100 1000

2 1000 *1.10 = 1000*1.101 1100

3 1100 *1.10 = 1000*1.102 1210

4 1210 *1.10 = 1000*1.103 1331

5 1331 *1.10 = 1000*1.104 1464

Year Amount of salary PVF@12% PV of salary

1 1000 .893 893

2 1100 .797 877

3 1210 .712 862

4 1331 .636 847

5 1464 .567 830

6105 4306

P = A/(1+i) + A(1+g)1/(1+i)2 + A(1+g)2/(1+i)3 + …………….+ A(1+g)n-1/(1+i)n

P = A[1/(1+I ) + (1+g)1/(1+i)2 + (1+g)2/(1+i)3 + …………….+ (1+g)n-1/(1+i)n]

P = A/i – g [1 – (1+g/1+i)n ]

P = (1000/.12-.10) [1- (1.10/1.12)5]

P = 50000 * (1 -.9138) = Rs4309

Q A company paid dividend of Rs 60 last year. The Dividend stream commencingOne year is expected to grow at10% annum for 15 years and then ends. If the Discount is 21%, what is the present value of the expected series?

P = A/I – g [1 – (1+g/1+i)n ]

P = 66/(.21 - .10) [ 1 – (1.10/ 1.21)15]

P = Rs 456.36

Present Value of anUneven Periodic Sum

• Investments made by of a firm do not frequently yield constant periodic cash flows (annuity). In most instances the firm receives a stream of uneven cash flows. Thus the present value factors for an annuity cannot be used. The procedure is to calculate the present value of each cash flow and aggregate all present values.

Value of an Annuity Due

• Annuity due is a series of fixed receipts or payments starting at the beginning of each period for a specified number of periods.

• Future Value of an Annuity Due

• Present Value of an Annuity Due

, = CVFA × (1 )n n iF A i

= × PVFA × (1 + )n, iP A i

Future Value of an Annuity Due

Concept of Compound value and present value are based on concept of cash flow at the end of the period.

In Practice cash flow can be at the beginning of the period.

Q . Suppose you deposit Re 1 in a saving account at the beginning of each year for 4 years to earn 6% interest rate. How much will be the compound value at the end 4 years?

F = 1*1.064 + 1*1.063 + 1*1.062 + 1*1.061 = Rs 4.63

Future value of Annuity due = Future value of annuity * (1+i) = A * CVFA n,i * (1+i)

= 1* 4.375 * 1.06 = 4.637

Present Value of an Annuity Due

Annuity due would be higher than the present value of an Annuity.

Present Value of an Annuity Due

Q . Suppose you deposit Re 1 in a saving account at the beginning of each year for 4 years to earn 6% interest rate. How much will be the present value at the annuity if each payment is made at the beginning of the year?

P = 1/(1.10)0 + 1/(1.10)1 + 1/ (1.10)2 + 1/(1.10)

P = 3.487

P = 1 *3.170 *1.10 Rs 3.487

= × PVFA × (1 + )n, iP A i

Net Present Value

• Net present value (NPV) of a financial decision is the difference between the present value of cash inflows and the present value of cash outflows.

01

NPV = (1 + )

nt

tt

CC

k

Present Value and Rate of Return

• A bond that pays some specified amount in future (without periodic interest) in exchange for the current price today is called a zero-interest bond or zero-coupon bond. In such situations, you would be interested to know what rate of interest the advertiser is offering. You can use the concept of present value to find out the rate of return or yield of these offers.

• The rate of return of an investment is called internal rate of return since it depends exclusively on the cash flows of the investment.

Summary of Compounding and discounting Formulae

1. Compound Value of a lump sum:

2. Compound Value of an Annuity:

3. Present Value of a lump sum :

4. Present value of an Annuity :

(1 ) 1n

n

iF A

i

(1 )(1 )

nnnn

FP F i

i

1 1

1n

P Ai i i

nn iPF )1( = CVFn n,iF P

= CVFAn n, iF A

,PVFn n iPV F

= × PVAFn, iP A

5. Capital recovery: Amount = P [ I (1+i)n/ (1+ i)n -1 ]

6. Present Value of Perpetuity: P = A/i

7. Present value of a constantly growing perpetuity: P = A/ i - g

8. Compounded value of an annuity due:

9. Present value of Annuity due:

, = CVFA × (1 )n n iF A i

, = CVFA × (1 )n n iF A i

Q1. Assume as annual rate of interest of 15%. The sum of Rs 100 received immediately is equivalent to what quantity received in ten equal annual payments, the first payment to be received one year from now. What could be the annual amount if the first payment were received immediately? (at the beginning of the year)

Capital recovery factor of annuity and annuity due

A. Rate of interest 15%B. Sum received now (Rs) 100C. Period (years) 10D. Present value factor (annuity) at 15% 5.0188E. Capital recovery factor (annuity) at 15% : [1/D]: 1/5.0188 0.1993F. Annual instalment (end of period) [B x E] 19.93

G. Present value factor (annuity due) at 15%: : 5.0188 x 1.15 5.7716H. Capital recovery factor (annuity due) at 15% [1/G] 0.1733I. Annual instalment (beginning of period) [B x H] 17.33

93.19Rs0188.5/100A

A0188.5100

15.1

1A100

10

1tt

Q Assume that you are given a choice between incurring an immediate outlay of Rs 10,000 and having to pay Rs 2310 a year for 5 years (first payment due one Year from now); the discount rate is 11%.

What would be your choice?

Will your answer change if Rs 2310 is paid at the beginning of each year for 5 years?

A. Discount rate 11%

B. Outlay now 10,000

C. Period of installments (years) 5

D. Present value factor (annuity) at 11% 3.6959

E. Capital recovery factor (annuity) at 11% [1/D] 0.27057

F. Annual installment (end of period) [B x E or B/D] 2705.70

G. Present value factor (annuity due) at 11% 4.1024

H. Capital recovery factor (annuity due) at 11% [1/G] 0.24376

I. Annual installment (beginning of period) [B x H] 2437.57

Q. Exactly 20 years from now Mr. Ahmed will start receiving a pension of Rs 10,000 a year. The payment will continue for twenty years . How much is pension worth now, assuming money is worth 15% per year.

A. Discount rate 15%

B. Annual pension 10,000

C. Periods of pension 20

D. Present value factor, 20 years, 15% 6.25933

E. Present value of pension at the end of 20 years 62593.31

F. Present value factor, end of 20 years 0.06110

G. Present value of pension now [E x F] 3824.47

Q1. Mr. Srinavas is going to retire after 6 months. He has a choice between (a) an annual pension of Rs 8000 as long as he lives, and (b) a lump sum amount of Rs 50,000. If he expects to live for 20 years and the interest rate is 10%, which option would you suggest him to go for?

Present value annuity

Present value of annual pension

= Rs 8,000 * PVFA (10,20)

= Rs 8000 * 8.514

= Rs 68,112

The second choice is Rs 50,000 lump sum now

Hence he should opt for the first choice.

Q. Jai Chand is planning for his retirement. He is 45 years old today, and would like to have Rs 3,00,000 when he attains the age of 60. He intends to deposit a constant amount of money at 12% at each year in the PPF in the state bank of India to achieve his objective. How much money should Jai chand invest at the end of each year for the next 15 years to obtain Rs 3,00,000 at the end of that period?

Sinking Fund

8,047.27 Rs=.2830,0000/37=A

30,000=A 28.37

000,3012.0

112.1 15

A

A. Needed future sum after 15 years 300,000

B. Period (years) 15

C. Interest rate 12%

D. Future value factor of an annuity, 15 years, 12% 37.2797

E. Annuity value [A/D]: 8047.27

Q. Which Alternative you will Choose:

1. An annuity of Rs 5000 at the end of each year for 30 years.

2. An annuity of Rs 6600 at the end of each year for 20 years.

3. Rs 50000, in cash right now.

4. In each case time value of money is 10%

Present Value annuity

A. Time value of money 10%

B. 30-year annuity 5,000

C. PVAF, 10%, 30 year 9.4269

D. Present value of 30-year annuity 47,134.57

E. 20-year annuity 6,600

F. PVAF, 10%, 20 year 8.5136

G. Present value of 20-year annuity 56,189.52

H. Cash right now 50,000.00

You should choose 20-year annuity of Rs 6,600 as it has highest PV.