Financial Management I 6. Time Value Value of Money
Dr. Suresh
[email protected] Phone: 40434399, 25783850
Course Con onte tent nt
- Sy Syll llab abus us
Sr
Title
ICMR Ch.
PC Ch.
IMP Ch.
1
Introduction to Financial Management
1*
1
1
2
Overview of Financial Markets
2*
2
-
3
Sources of Long-Term Finance
10*
17
20, 21
4
Raising Long-term Finance
-
18*
20, 21, 23
5
Introduction to Risk and Return
4*
8, 9
4, 5
6
T Tiime Value of Money
3*
6
2
7
Val alua uati tio on of Se Secu curi riti ties es
8
Cost
9
Basics of Capital Expenditure Decisions
of Capital
10 Anal alys ysis is of Pr Proj ojec ectt Cash Flows O ptimal Capital 11 Risk Analysis and Optimal Expenditure Decision
*Book preference
2 / 57
Course Con onte tent nt
- Sy Syll llab abus us
Sr
Title
ICMR Ch.
PC Ch.
IMP Ch.
1
Introduction to Financial Management
1*
1
1
2
Overview of Financial Markets
2*
2
-
3
Sources of Long-Term Finance
10*
17
20, 21
4
Raising Long-term Finance
-
18*
20, 21, 23
5
Introduction to Risk and Return
4*
8, 9
4, 5
6
T Tiime Value of Money
3*
6
2
7
Val alua uati tio on of Se Secu curi riti ties es
8
Cost
9
Basics of Capital Expenditure Decisions
of Capital
10 Anal alys ysis is of Pr Proj ojec ectt Cash Flows O ptimal Capital 11 Risk Analysis and Optimal Expenditure Decision
*Book preference
2 / 57
Time Value of Money Reference Books 1. Fi Fina nanc ncia iall Man Manag agem emen ent, t, ICMR Book, Chapter 3 2. Fi Fina nanc ncial ial Man Manage agem men ent, t, Pras Prasan anna na Chandra, 7th Edition, Chapter
6
3. Fi Fina nanc ncial ial Man Manage agem men ent, t, I. I. M. Pande Pandey, y, 9th Edition, Chapter
2
3 / 57
Syllabu Syll abuss ± Time Val Value ue of Mone Money y 1. Introduction 2. Types of Cash Flows 3. Futu ture re Va Valu luee of a Si Singl glee Cash Flow 4. Mu Mult ltip iple le Fl Flow owss an and d An Annu nuit ity y 5. Pre ressen entt Va Valu luee of of a Sin ingl glee Cash Flow 6. Mu Mult ltip iple le Fl Flow owss an and d An Annu nuit ity y 7. Growing Annuity 8. Pe Perp rpet etui uity ty and and Gro Growi wing ng Pe Perp rpet etui uity ty
4 / 57
1. Introduction Companies
take up new projects, where they invest money
for the benefits expected over a period of time in future. 4 Cr Investment
1 Cr
1 Cr
1 Cr
1 Cr
1 Cr
How to determine whether the project is financially viable or not? Answer to this question will be to sum up the benefits accruing over the future period and compare the total value of benefits with the initial investment. If the total of benefits exceed the initial investment, then the project is considered to be financially viable.
5 / 57
1. Introduction Concept
of Time Value of Money
One rupee at present is worth more than one rupee next year. Because one rupee at present can be invested to earn an income. Value of money at present is more than value of money in future time. Hence we can say money has a value based on time. This concept is called as µtime value of money¶ or an µinterest¶.
4 Cr Investment
1 Cr
1 Cr
1 Cr
1 Cr
Present values total 3.26 Cr
0.86Cr
0.74Cr
0.64Cr
0.55Cr
1 Cr 0.47Cr 6 / 57
1. Introduction How to determine the time value of money and use it into cash flows of a project? We are going to study these things in this chapter. Nominal or market interest rate depends on real interest rate, inflation and uncertainty about future. Nominal or market interest rate = Real interest rate + expected inflation rate + risk premium for uncertainty Due to inflation, a rupee today has higher purchasing power than rupee in future. Future is characterized by uncertainty, which require some risk premium.
7 / 57
Compounding 0
1
-1000
2 250
3 500
4 750
750 + FV(750) + FV(500) + FV(250) compare with FV(1000)
Under the method of compounding, we find the future values (FV) of all the cash flows at the end of the time horizon, at a particular rate of interest. 8 / 57
Discounting 0
1 -1000
2 250
3 500
4 750
750
compare with the sums of PV(250) + PV(500) + PV(750) + PV(750)
Under the method of discounting, we calculate the time value of money at present. So we will be comparing the initial outflow with the sum of the present values (PV) of the future inflows at a given rate of interest. 9 / 57
2. Types of Cash Flows Types of Cash Flows: Cash flows from Operating Activities, Investing Activities and Financing Activities. Cash
flow is the difference between amount of cash
flowing in and out a company. 10 / 57
3. Future Value of a Single Cash Flow FVn
!
PV (1 k)
n
Where FVn = future value in n years PV = present value or initial value of cash flow k = annual rate of interest n = duration or the life of investment (1+k)n = future value of unit investment, say Rs. 1 = FVIF(k, n) = future value interest factor Use Table 1 to get FVIF value for k & n Multiply PV and FVIF to get the future value 11 / 57
3. Future Value of a Single Cash Flow Example If the bank offers a compounded rate of interest of 11% per annum on the deposit. An amount of Rs. 10,000 deposited today, will become how much after 3 years? Solution Formula
n !
(1 k) n
FVn = PV x FVIF(k,n) = PV x FVIF(11,3) = 10,000 x 1.368 = Rs. 13 680
12 / 57
3. Future Value of a Single Cash Flow Doubling Period Frequent question asked by the investors is µHow many years required to double an investment with a given rate of interest?¶ Solution Above question can be answered by a rule known as µRule of 72.´ It is an approximate method of calculating. As per the rule, the period within which the amount will be doubled is obtained by dividing 72 by the rate of interest. 13 / 57
3. Future Value of a Single Cash Flow For example, if the given rate of interest is 10%, then doubling period is 72 / 10 = 7.2 years. However, more accurate way of calculating doubling period is the µRule of 69¶, according to which the doubling period is
0.35
69 Interest Rate
= 0.35 + 69/10 = 0.35 + 6.9 = 7.25 years
14 / 57
3. Future Value of a Single Cash Flow By calculator Doubling Period Calculation Formula
FV n
!
FVn
PV
PV (1 k) n !
(1
k)
n
By taking log on both sides log (FVn/PV) = n log(1+k)
n = log(FVn/PV) log(1 k)
=
log 2 log 1.1
!
7.27 years 15 / 57
3. Future Value of a Single Cash Flow Growth Rate Compounded
Annual Growth Rate (CAGR) can be
calculated for a series of incomes by using Future Value Interest Factor (FVIF) Table, (Table 1) Example: Years
1
2
3
4
5
6
Profits in lakh
95
105
140
160
165
170
Calculate
the compound rate of growth. 16 / 57
3. Future Value of a Single Cash Flow Calculate
the compound rate of growth.
Years
1
2
3
4
5
6
Profits in lakh
95
105
140
160
165
170
Solution: Ratio of profits for last year to first year = 170 / 95 = 1.79 Refer FVIF(k,n-1) table (Table 1) Look for the value close to 1.79 for 5 years The value close to 1.79 is 1.762 and corresponding interest rate is 12 %. Therefore compound rate of growth (CAGR) is 12%.
17 / 57
3. Future Value of a Single Cash Flow
By calculator FV = PV (1+k)n
170 = 95 (1+k)5 1+k = (170/95)1/5 =
5
1.79
= 1.1235
k = 0.1235 or 12.35%
18 / 57
3. Future Value of a Single Cash Flow Increased Frequency of Compounding For example, half yearly compounding, quarterly compounding etc. Example You have deposited Rs. 10, 000 in a bank, which offers 10% interest p.a. compounded semi-annually. Calculate the amount at end of the year. Formula n !
(1
k
)
m
n
m
Where, FVn = future value after n years
19 / 57
3. Future Value of a Single Cash Flow Example An amount of Rs. 1000 is invested in a bank for 2 years. Rate of interest is 12 % compounded quarterly. Calculate
the amount at the end.
Solution FV n
!
PV (1
k
)
n
m
m
where m = 4, frequency of compounded in a year.
FVn = 1000 (1+0.12/4)8 = 1000 (1.03)8 = 1000 x 1.267 = Rs. 1267
20 / 57
3. Future Value of a Single Cash Flow Effective vs. Nominal Rate of Interest Rs. 100 with 10 % interest amounts to Rs. 110 at the end of year. However semi-annually 100(1.05)2 = 100 x 1.1025 = 110.25
The principal amount grows 10.25% p.a.
This 10.25% is called effective rate of interest. Hence the accumulation under semi-annual compounding exceeds the accumulation under annual compounding. 21 / 57
3. Future Value of a Single Cash Flow Effective vs. Nominal Rate of Interest Formula for effective rate of interest r
!
(1
k
)
m
1
m
Where r = effective rate of interest k = nominal rate of interest m = frequency of compounding per year
22 / 57
3. Future Value of a Single Cash Flow Example Calculate
the effective rate of interest, for the 12%
nominal rate of interest quarterly compounded. Solution Effective rate of interest r
!
(1
k
)
m
1
m
!
(1
0.12 4
4 ) 1
= 1.034 -1
«Refer FVIA(3,4) table
= 1.126 -1 = 0.126 or 12.6%
23 / 57
4. Future Value of Multiple Flows and Annuity 0 1000
1 2000
2
3 3000
Accumulation FV(3000) + FV(2000) + FV(1000)
Suppose we invest Rs. 1000 now i.e. at the beginning of year 1, Rs. 2000 at the beginning of year 2 and Rs. 3000 at the beginning of year 3. How much will these cash flows accumulate at the end of year 3 at the rate of 12% p. a. 24 / 57
4. Future Value of Multiple Flows and Annuity 0
1
1000
2000
2
3 3000
Accumulation FV(3000) + FV(2000) + FV(1000)
Future value at the end of 3 years, at 12% interest p.a. = FV(Rs. 1000) + FV (Rs. 2000) + FV(Rs. 3000) =1000 x FVIF(12,3) +2000 x FVIF(12,2)+3000 x FVIF(12,1)
= 1000 x 1.405 + 2000 x 1.254 + 3000 x 1.12 = Rs. 7273
25 / 57
4. Future Value of Multiple Flows and Annuity By calculator Future value at the end of 3 years = 1000 x 1.123 + 2000 x 1.122 + 3000 x 1.12 = Rs. 7273.73
26 / 57
4. Future Value of Multiple Flows and Annuity Annuity Annuity is a stream or series of periodic flows of equal amounts. For example life insurance premium. Regular Annuity or Deferred Annuity: When the equal amounts of cash flows occur at the end of each period, over the specified time horizon. Annuity Due: When cash flows occur at the beginning of each period, over the specified time horizon. 0
1
2
3
4
5
6
Future value of regular annuity FVA 6 ! A(1 k)
6 1
A(1 k)
6 -2
A(1 k)
6 -3
... A(1 k)
6-6
27 / 57
4. Future Value of Multiple Flows and Annuity Future value of regular annuity FVA
n !
A(1
Which reduces to
k)
n 1
A(1
k)
n -2
« (1 k) n 1» FVAn ! A ¬ ¼ k ½
A(1
k)
n -3
...
A
Where, A = amount deposited /invested at the end of every year for n years k = rate of interest, expressed in decimals n = time horizon FVAn = Accumulation at the end of n years. The expression «¬ (1 k) 1 »¼ is called the Future Value k ½ Interest Factor for Annuity (FVIFA) and it accumulates Rs. 1 invested or paid at the end of every year for a period of n years at the rate of interest k. FVIFA values are iven in Table 2. 28 / 57
n
4. Future Value of Multiple Flows and Annuity Example Calculate
the µAnnuity regular¶ for yearly annuity of Rs.
1000 invested with a 12 % interest for a period of 10 years. Solution n !
« (1 k) n 1» ¬ ¼ k ½
FVAn = A x FVIFA(k,n) = 1000 x FVIF(12,10) = 1000 x 17.549 = Rs. 17549
29 / 57
4. Future Value of Multiple Flows and Annuity By calculator « (1 k) n 1» FVAn ! A ¬ ¼ k ½
« (1 0.12)10 1» FVAn ! 1000 ¬ ¼ 0.12 ½
= 1000 x 17.54874 = Rs. 17548.74
30 / 57
4. Future Value of Multiple Flows and Annuity Example Under a recurring deposit scheme of the bank, a fixed amount is deposited every month. The rate of interest is 9% compounded quarterly. Calculate the maturity value for a monthly installment of Rs. 500 for 12 months. Solution Amount of deposit = Rs. 500 per month Rate of interest
= 9% compounded quarterly
Effective rate of interest
r
!
(1
k
)
m
1
m
!
(1
0.09 4
) 4 1 ! 0.0931 or 9.31% 31 / 57
4. Future Value of Multiple Flows and Annuity Rate of interest per month
!
(1 r )1/12 1
!
(1 0 .0931 )1/12 1
!
0.0074 or 0.74
Maturity value of an annuity « (1 k) n 1» n ! ¬ ¼ k ½ « (1 0.0074) 121» ! 500 ¬ ¼ 0.0074 ½ !
500 x 12.50 !
Rs.
6250 32 / 57
4. Future Value of Multiple Flows and Annuity
If the payments are made at the beginning of every year, then the value of such an µannuity due¶ is calculated by modifying the formula of µannuity regular¶ as follows
FVAn(due) = A (1+k) FVIFA(k,n)
33 / 57
4. Future Value of Multiple Flows and Annuity Example A person aged 20 is insured for a policy of Rs. 10,000. The term of policy is 25 years and the annual premium is Rs. 41.65. Calculate the rate of return the person gets. Solution Premium
= Rs. 41.65 per annum
Term of policy = 25 years Value at the maturity = P (1+K) FVIFA(k,n), since the premium is paid at the beginning of the year.
10,000 = 41.65 (1+k) FVIFA (k,25)
(1+k) FVIFA(k,25) = 240.1
34 / 57
4. Future Value of Multiple Flows and Annuity From Table 2, we get (1+0.14) FVIFA(14,25) = 1.14 x 181.871 = 207.33 and (1+0.15) FVIFA(15,25) = 1.15 x 212.793 = 244.71 By interpolation k = 14% + (15% - 14%) x
240.1 207.33 244.71 207.33
= 14 + 1 x 32.77 /37.38 = 14 + 0.87 % = 14.87 % 35 / 57
5. Present Value of a Single Cash Flow Discounting: Using this approach we can find present value of a future cash flow or a stream of future cash flows. Present value approach is commonly followed for evaluating financial viability of projects. FVn = PV(1+k)n = PV x FVIF(k,n)
PV !
FVn FVIF(k, n)
or
PV !
FVn (1 k) n
The inverse of FVIF(k,n) is defined as PVIF(k,n)
The Present Value Interest Factor for k and n PV
= FVn x PVIF(k,n)
36 / 57
5. Present Value of a Single Cash Flow To determine the present value of a future sum, we have to just multiply future value by PVIF factor for the values of k and n. PVIF values are provided in Table 3. Example: Calculate the issue price of a zero coupon bond with a face value of Rs. 1000, redeemable after 5 years with 12% interest p.a. Solution:
PV
= FV x PVIF(k,n) = 1000 x PVIF(12,5) = 1000 x 0 567
= Rs 567
37 / 57
5. Present Value of a Single Cash Flow By calculator PV ! !
!
FVn (1 k) n 1000 (1 0.12) 5 R s. 567.43
38 / 57
5. Present Value of a Single Cash Flow Example: A bank certificate having value of Rs. 100 after a year is to be issued with a 12% interest p.a. quarterly compounded. Calculate the issue price of a certificate. Solution Effective rate of interest
r
!
(1
k
)
m
1
m
!
(1
0.12 4
)4 1
!
0.1255
or 12.55
Issue price of the certificate is PV ! FVn n !
(1 k) 100
(1 0.1255)
!
Rs. 88.85 39 / 57
5. Present Value of a Single Cash Flow Example: A bank wishes to issue a cash certificate of Rs. 1,00,000 to be received after 10 years, at the rate of 10% interest p.a. compounded quarterly. Calculate the issue price of this certificate. Solution Effective rate of interest
r
!
(1
k
)
m
1
m
!
(1
0.10 4
)4 1
!
0.1038
or 10.38
Issue price of the certificate is PV ! FVn n !
(1 k) 100000
(1 0.1038)
10
!
Rs. 37247.41 40 / 57
6. Present Value of Multiple Flows and Annuity 0 Accumulation
1 1000
2 2000
3 3000
PV(1000) + PV(2000) + PV(3000)
Present value at the end of 3 years, at 12% interest p.a. = PV(Rs. 1000) + PV (Rs. 2000) + PV(Rs. 3000) =1000 x PVIF(12,3) +2000 x PVIF(12,2)+3000 x PVIF(12,1)
= 1000 x 0.893 + 2000 x 0.797 + 3000 x 0.712 = Rs. 4623 41 / 57
6. Present Value of Multiple Flows and Annuity
Project is said to be financially viable if the present value of the cash flows exceeds the present value of the cash outflows.
42 / 57
6. Present Value of an Annuity When an annuity is receivable at the end of every year for a period of n years at the interest rate of k, then the present value is equal to n !
(1 k)
This reduces to
(1 k)
n !
2
(1 k)
3
...
(1 k) n
« (1 k) n 1» ¬ n ¼ k(1 k) ½
« (1 k) n 1» The expression ¬ k(1 k) n ¼ is called the PVIFA, Present ½
Value Interest Factor for Annuity and it represents the present value of a regular annuity of Rs. 1 for a given value of k and n.
43 / 57
6. Present Value of an Annuity
PVIFA values are given in Table 4 for various values of k and n.
These values are used with following conditions
The cash flows are equal
The cash flows occur at the end of every year
44 / 57
6. Present Value of an Annuity Example: Calculate the investment to receive a yearly regular income of Rs. 10,000 for 10 years at the rate of 12 % p.a. Solution:
PVAn = 10,000 x PVIFA(12,10) = 10,000 x 5.65 = Rs. 5650
« (1 k) n 1» By calculator PVAn ! A ¬ n ¼ k(1 k) ½ « (1 0.12)10 1 » ! 10,000 ¬ 10 ¼ 0.12(1 0 . 12 ) ½ « 2.10585 » ! 10,000 ¬ 0.3727 ¼ ! R s. 5650.23 ½
45 / 57
6. Present Value of an Annuity Example: Calculate the initial deposit amount to receive a monthly payment of Rs. 1000 for 12 month. The rate of interest is 12% p.a. quarterly compounded. Solution: Effective rate of interest
r
!
k
(1
)
m
1
m
!
(1
0.12 4
) 4 1 ! 0.1255 or 12.55
Effective rate of interest per month = (1.1255)1/12-1 = 0.0099 n !
« (1 k) n 1» « (1 0.0099) 12 1 » ! 1000 ¬ ¬ n ¼ 12 ¼ k(1 k) 0.0099(1 0.0099) ½ ½ « 0.1255 » ! 1000 ¬ 0.01114¼ ! 1000 x 11.26336 ! Rs.11,263.36 ½ 46 / 57
6. Present Value of an Annuity Example: If an initial deposit of Rs. 4610 is done in a bank for an annuity period of 60 months. Rate of interest is 11% p. a. quarterly compounded. Calculate the value of monthly annuity income a person can receive. Solution: Effective rate of interest
r
!
(1
k
)
m
1
m
!
(1
0.11 4
) 4 1 ! 0.1146 or 11.46%
Effective rate of interest per month = (1.1146)1/12-1. = 0.00908 Monthly annuity can be calculated as
4610
!
« (1 0.00908) 60 1 ¬ - 0.00908(1 0.00908)
» 60 ¼ ½
n !
« (1 k) n 1» ¬ n ¼ k(1 k) ½
A = Rs. 99.8833 47 / 57
Capital
Recovery Factor
Rearranging the equation for present value of an annuity, we get
« (1 k) n 1» PVAn ! A ¬ n ¼ k(1 k) ½ « (1 k) 1» A ! PVAn ¬ n ¼ k(1 k) ½ n
1
« k(1 k) n » ! PVAn ¬ ¼ n (1 k) 1 ½ « k(1 k) n » The term ¬ ¼ is known as a capital recovery n - (1 k) 1½
factor.
48 / 57
Capital
Recovery Factor
Example: A loan of Rs. 10,000 is repaid in 5 equal installments. The loan carries a interest rate of 14% p. a. Calculate
the amount of each installment.
Solution PVAn = A x PVIFA(k,n)
10,000 = A x 3433
A = 10,000 / 3,433 = Rs. 29,129 49 / 57
Capital
Recovery Factor
By Calculator n !
!
« (1 k) n 1» ¬ n ¼ k(1 k) ½ « k(1 k) n » n ¬ ¼ n (1 k) 1 ½
« 0.11(1 0.11) 5 » ! 10,000 ¬ ¼ ! R s. 29,129 5 - (1 0.11) 1 ½
This method is very useful for deciding an equal amount of installments for a loan repayment scheme.
50 / 57
7. Present Value of Growing Annuity 0
1 A(1+g)
2 A(1+g)2
3
n
A(1+g)3
A(1+g)n
A cash flow that grows at a constant rate for a specified period of time is a growing annuity. Formula for the present value of growing annuity PV of Growing Annuity !
« (1 g) n ¬ 1 - (1 k) n A(1 g) ¬ ¬ k - g ¬ -
» ¼ ¼ ¼ ¼ ½
where g is a growth rate and k is a discount rate. Above formula is used when g
k. However it does not work when g=k. In this case the present value is equal to nA. 51 / 57
7. Present Value of Growing Annuity Example: Suppose you have a right to harvest a teak plantation for the next 20 years over which you expect to get 1,00,000 cubic feet of teak per year. The current price of teak per cubic feet is Rs. 500 and it is expected to increase at a rate of 8% p.a. The discount rate is 15%. Calculate the present value of the teak that you can n « » (1 g) harvest. 1 ¬ (1 k) n ¼ Solution: PV of Growing Annuity ! (1 g) ¬ k - g ¼ ¬ ¬ -
PV of
« (1 0.08) 20 ¬1 - (1 0.15) 20 teak ! Rs. 500 x 1,00,000 x (1 0.08) ¬ ¬ 0.15 - 0.08 ¬ -
¼ ¼ ½
» ¼ ¼ ! Rs.55,17,3 6,683 ¼ ¼ ½ 52 / 57
8. Present Value of Perpetuity and Growing Perpetuity An annuity of an infinite duration is known as perpetuity. The present value of such perpetuity is expressed as P = A x PVIFA(k,) Where PVIFA(k,) is a present value interest factor for a perpetuity. Value of PVIFA(k,) is ! § 1 ! 1 g
t !1
Pg
!
(1 k) t
k
A r
Present value interest factor of a perpetuity is one divided by interest rate (expressed in decimal form). Present value of a perpetuity is equal to the constant annuity divided by the interest rate. 53 / 57
8. Present Value of Perpetuity and Growing Perpetuity Example: How much I should invest in a bank to offer a scholarship of Rs. 10,000 per annum perpetually (forever) to a bright student with 10% p.a. Solution Pg
!
!
A r
10,000 0.10
!
Rs.1,00,00
0
54 / 57
8. Present Value of Perpetuity and Growing Perpetuity Example: How much I should invest in a bank to offer a scholarship of Rs. 1000 per month perpetually (forever) to a bright student with 10% p.a. Solution: Effective rate of interest per month = (1.10)1/12-1 = 0.007974 or 0.7974% Pg
!
!
A r
1000 0.007974
!
Rs.1,25,40 8 55 / 57
8. Present Value of a Growing Perpetuity 0
1
2
3
A(1+g)2
A(1+g)
A(1+g)3
A perpetuity growing at a constant rate is called growing perpetuity. We assume that the increase will continue indefinitely. For example, rent is a growing perpetuity. PV of Growing Perpetuity 2 n -1 !
(1 k)
This reduces to PV of growing Perpetuity
(1 g)
(1 k)
!
2
(1 g)
(1 k)
3
...
(1 g)
(1 k)
n
...
A k - g
Where g is a growth rate and k is a discount rate.
56 / 57
