# You plan to make 5 deposits of 100 each by ARY9dd0f

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```									TIM E VALUE OF M ONEY
TUTORIAL QUESTION AN D
SOLUTION
Question 1            Find the following present and future values:
a.     An initial £ 500 compounded for 1 year at 6 percent.
Present and future   b.     An initial £ 500 compounded for 2 years at 6 percent.
values
c.     The present value of £ 500 due in 1 year at a discount rate of 6 percent.
d.     The present value of £ 500 due in 2 years at a discount rate of 6 percent.
S O L U T IO N
a.     Given,
Present value (PV) = £. 500
Interest rate (i) = 6%

0         6%                          1

- 500                                FV = ?

FVn         = PV(1 + i)n
FV1         = PV (1 + i)1
= £ 500 (1 + 0.06)1 = £. 530
b.     Present value (PV) = £. 500
Interest rate (i) = 6%
0         6%            1              2

- 500                                FV = ?
FVn       = PV(1 +   i)n
FV2       = PV (1 + i)2
= £. 500 (1 + 0.06)2 = £. 561.80
= £. 471.70
d.     Future value (FV) = £. 500
Interest rate (i) = 6%
No. of periods (n) = 2
Present value (PV) = ?

0         6%            1              2

PV = ?                                FV = 500

FVn        FV2
PV        = (1 + i)n = (1 + i)2
= £ 445
Question 2           Suppose John deposits £ 10,000 in a bank account that pays 10 percent interest annually.
Future value    How much money will be in his account after 5 years?

SOLUTION
Here, Present value (P) = £ 10,000,
Interest rate (k) = 10%
Number of years (n) = 5 years,
Future value (FV5) = ?
0          1     2        3                   5
10%

£10,000                                       FV = ?
We have,
FV5 = PV × (1 + k)n = £ 10,000 × (1.10)5
= £ 10,000 × 1.6105
= £ 16,105.10
John will have £ 16,105.10 at the end of year 5 in his account.

Question 3           What is the present value of a security that promises to pay you £ 5,000 in 20 years?
Present value    Assume that you can earn 7 percent if you were to invest in other securities of equal risk?
SOLUTION
Here, Future value (FV) = £ 5,000
Number of years (n) = 20 years
Interest rate (k) = 7%
Present value (PV) = ?
0           1     2        3                   20
7%

PV = ?                                        £5,000
We have,
FV20
PV = (1 + k)n = 5000/ (1.07)20
=5000/3.8697 = £ 1,292.09

Question 4           If you deposit money today into an account that pays 6.5 percent interest, how long will it
Time for a lump   take for you to double your money?
sum to double
SOLUTION
Here, Interest rate (i) = 6.5%
Number of period (n) = ?
Present value (PV) = £ 1000 (assume)
Future value (FV) = £ 2000
0           1     2        3                   n=?
6.5%

PV = £1000                                      FV = £ 2,000
We have,
FV
Present value (PV) = (1 + i)n
or,   £ 1000 = 2000 / (1+0.065)n
or, (1 + 0.065)n = 2000/1000
or, (1.065)n = 2                      .... (i)
Trying at n = 11
We get,
If n = 11, the left hand side in above equation (i) is approximately equal to 2. Hence
the required no. of years to double the sum of money is 11 years.

Question 5             Your broker offers to sell a note for £ 13250 that will pay £ 2345.05 per year for 10 years. If
Effective rate of   you buy the note, what rate of interest will you be earning? Calculate to the closest
interest    percentage.
S O L U T IO N
Here,
Present value of annuity (PVA) = £. 13,250
Periodic equal payment (PMT) = £. 2345.05
No. of periods (n) = 10 years
Interest rate (i) = ?
Time Line
0        1          2         3         4        5        6          7          8        9        10

PVA =     2345.05    2345.05    2345.05   2345.05 2345.05   2345.05    2345.05   2345.05   2345.05 2345.05
13250

We have,
PVA = PMT × PVIFA i × n yrs.
or,   £. 13,250 = £. 2345.05 × PVIFAi% 10 yrs
or,   PVIFAi%, 10 yrs = 5.6502
From the PVIFA table, the value of 5.6502 in 10 years lies at 12%.
     The required interest rate is 12%.

Question 6             Your parents are planning to retire in 18 years. They currently have £ 250,000, and they
would like to have £ 1,000,000 when they retire. What annual rate of interest would they
Effective rate of
interest    have to earn on their £ 250,000 in order to reach their goal, assuming they save no more
money?
SOLUTION
Here, Future value (FV) = £ 1,000,000
Present value (PV) = £ 250,000
Time period (n) = 18 years
Interest rate (i) = ?
0          1         2         3                       18
i=?

£ 250,000                                                    £ 1,000,000
We have,
FV = PV (1 + i)n
or, £ 1,000,000 = £ 250,000 (1 + i)18
or, (1 + i)18 = 1,000,000/ 250,000
or, (1 + i)18 = 4
or, 1 + i = (4)1/18
or, i = 1.08 - 1 = 0.08 or 8%
     The required rate of interest to reach the goal is 8%.
Question 7            What is the future value of a 5-year ordinary annuity that promises to pay you £ 300 each
Future value of an   year? The rate of interest is 7 percent.
annuity
SOLUTION
Here, Future value of annuity (FVA) = ?
Payment (PMT) = £ 300
Number of period (n) = 5 years
Interest rate (i) = 7%

0            1         2    3                5
7%

£300           £300   £300           £300
We have,                                                  FVA = ?
(1 + i) - 1
FVA = PMT 
n

 i 
(1 + 0.07)5 - 1
= £ 300 
 0.07 
= £ 300 × 5.7507 = £ 1,725.21

Question 8            What is the future value of a 5-year annuity due that promises to pay out £ 300 each year?
Assume that all payments are reinvested at 7% a year, until year 5.
Future value of an
annuity due

SOLUTI ON
Here, Future value of annuity due (FVAdue) = ?
Payment (PMT) = £ 300
Number of period (n) = 5 years
Interest rate (i) = 7%

0            1         2    3                5
7%

£300      £300           £300   £300           FVA (due) = ?
We have,
(1 + i)n - 1
    i       (1 + i)
(1 + 0.07)5 - 1
= £ 300 
 0.07  (1 + 0.07)
= £ 300 × 5.7507 × 1.07 = £ 1,845.97
Question 9            A company invests £ 4 million to clear a tract of land and to set out some young pine
trees. The trees will mature in 10 years, at which time the company plans to sell the forest
Expected rate of   at an expected price of £ 8 million. What is company's expected rate of return?
return

SOLUTION
Here, Future value (FV) = £ 8,000,000
Present value (PV) = £ 4,000,000
Time period (n) = 10 years
Expected rate of return (i) = ?
First set up time line as follows:
0            1      2       3                    10
i=?

We have,
£ 4 million                                              £ 8 million
FV = PV (1 + i)n
or, £ 8,000,000            = £ 4,000,000 (1 + i)10
or, (1 + i)10              =8,000,000/ 4,000,000
or, (1 + i)10              =2
or, 1 + i                  = (2)1/10
    i                     = 1.0718 - 1
= 0.0718 or 7.18%
Question 10            Rachel wants a refrigerator that costs £ 12000. She has arranged to borrow the total
purchase price of refrigerator from a finance company at a simple interest rate equal to 12
Solving for payment
percent. The loan requires quarterly payments for a period of three years. If the first
payment is due three months after purchasing the refrigerator, what will be the amount
of her quarterly payments on the loan?
SOLUTION
Given,
Present value of an annuity (PVA) = £. 12,000
Simple interest rate per annum (i) = 12%
Loan requires quarterly payment i.e. m = 4
Number of years (n) = 3 year
Periodic equal payment (PMT) = ?
Time line:
0 12% 1            2       3          4                 12

-12000   PMT     PMT        PMT   PMT                    PMT
We have,
1 -     1
n × m

PVA = PMT      (1 + i/m)      
         i         
        m          
1 -       1
3 × 4

or,   12,000 = PMT  (1 + 0.12/4) 
         0.12          
          4            
or,   12,000 = PMT × 9.9540
     PMT = 12,000/ 9.9540
= £ 1,205.55

Question 11            You need to accumulate £ 10,000. To do so, you plan to make deposits of £ 1,250 per year,
with the first payment being made a year from today, in a bank account which pays 12
Reaching a financial   percent annual interest compounded annually. Your last deposit will be less than £ 1,250
goal   if less is needed to round out to £ 10,000. How many years will it take you to reach your £
10,000 goal, and how large will the last deposit be?
SOLUTION
Here,
Annual payment (PMT) = £ 1,250
Future value of annuity (FVAn) = £ 10,000
Interest rate (i) = 12%
Time to maturity (n) = ?
Last deposit = ?
First, we determine the 1
0        number of periods of the financialngoal. This is calculated using
2       3                       =?
12%
future value of annuity formula as follows:
We have,
FVAn                    PMT 1,250
1,250 =1,250 × FVIFAi, n               Last deposit = ?
£ 10,000              = £ 1,250 × PViFA12, n         FVA = £ 10,000
10‚000
FVIFA12, n            = 1‚250 = 8
Looking FVIFA table the value 8 at 12 percent interest rate lies approximately in 6 years.
Therefore the number of years to reach the financial goal is 6 years. Now we calculate the
future value of £ 1,250 for 5 years at 12%, it is £ 7,941.06
FV = £ 1,250 × FVIFA12, 5 = £ 1,250 × 6.3528 = £ 7,941
Compounding this value after 6 years and before the last payment is made, it is £ 7,941
(1.12) = £ 8,893.92. Thus, we will have to make a payment of £ 10,000 - £ 8,893.92 = £
1,106.08 at year 6, therefore it will take 6 years, and £ 1,106.08 must be paid in the last
installment.

Question 12          Your Company is planning to borrow £ 1,000,000 on a 5-year, 15 percent, annual
Loan amortization   payments, fully amortized term loan. What fraction of the payment made at the end of
the second year will represent repayment of principal?
SOLUTION
Here, Loan amount (PVA) = £ 1,000,000
Number of years (n) = 5 years
Interest rate (i) = 15%
First we determine the annual installment or payment (PMT)
We have,
PVA
PMT = PVIFA = 1,000,000/PVIFA15,5 =1,000,000/ 3.3522 = £ 293,311.55
i‚ n

Preparation of Amortization Schedule,
Amortization schedule
Year            Payments                Payment of Principal
Interest                      Ending Balance
1          £ 298,311.5566      £ 150,000     £ 148,311.5566    £ 851,688.4434
2            298,311.5566   127,753.2665       170,558.2901      681,130.1533
Principal payment in 2nd year
% principal in 2nd year =         Payments            = 57.17%
That is 57.17% of the payment in second year represents the principal.

Question 13          You are branch manager of town centre Natwest Bank, Manchester. A borrower
Loan amortization
approaches you for a term loan of £ 500,000. You agreed to give loan to be fully
amortized in a period of 5 year at 10 percent, annual payment. What will be the size of
each installment? What fraction of the payment made at the end of second year represents
repayment of interest?
SOLUTION
Here, Loan amount (PVA) = £ 500,000
Number of years (n) = 5 years
Interest rate (i) = 10%
First we determine the annual installment or payment (PMT)
We have,
PVA
PMT = PVIFA            = £ 500,000/ PVIFA10,5 = 500,000/3.7908
i‚ n

= £ 131898.28

Preparation of Amortization Schedule,
Amortization schedule
Beginning                                           Repayment of       Ending
Year                               PMT          Interest
balance                                              principal        balance
1               £500,000        £131,898.28       £50,000            £81898.28   £418,101.72
2              418,101.72        131,898.28      41,810.17           90,088.11    328,013.61
Interest payment in 2nd year
% interest in 2nd year =    Payments           = 31.7%
That is 31.7% of the payment in second year represents the interest.

Question 14         a.        It is now January 1, 2007. You plan to make 5 deposits of £ 100 each, on every 6
months, with the first payment being made today. If the bank pays a nominal
Non annual
compounding              interest rate of 12 percent, but uses semiannual compounding, how much will be
in your account after 10 years?
b.        Ten years from today you must make a payment of £ 1,432.02. To prepare for this
payment, you will make 5 equal deposits, beginning today and for the next 4
quarters, in a bank that pays a nominal interest rate of 12 percent, quarterly
compounding. How large must each of the 5 payments be?
SOLUTION
a.        Here,    Number of deposits (n) = 5 deposits;
Semiannual (every 6 months), payment = £ 100;
Nominal interest rate (i) = 12%,
Present value of annuity (PVA) = ?

0                      1               2             10
6%

£100        £100        £100    £100   £100          FV = ?

We have,
(1 + i)n - 1                   (1 + 0.06)5 - 1
FVA = PMT ×               (1 + i) = £ 100 
      i                        0.06  (1 + 0.06)
= £ 100 × 5.6371 × 1.06 = £ 597.5326
Now remaining period is 15 periods (20 periods - 5 periods), so we calculate the
future value of this £ 597.5326 for remaining periods.
We have,
FV = PV (1 + i)n = £ 597.5326 (1 + 0.06)15 = £ 1,432.02
b.    Here,
Future value at the end of 10 years = £ 1,432.02;
n = 35 periods because quarterly compounding (in 10 years there are 40 quarters);
Quarterly interest rate = 3%,
PMT = ?, PV = ?

0                                     1             10
3%

PMT = ? PMT = ? PMT = ? PMT = ? PMT = ?             FV = £1,432.02
We have,
FV
PV = (1 + i)n = 1,432.02/(1+0.03)35 = £ 508.91
Now we calculate the payment (PMP)
Here, n = 5 periods, i = 3%, PV = ?; FV = £ 508.91, FVA = £ 508.91
PMT = ?
We have,
(1 + i)n - 1
FVA = PMT 
 i  (1 + i)
(1 + 0.03)5 - 1
or, £ 508.91         = PMT 
 0.03  (1 + 0.03)
or, £ 508.91         = PMT × 5.3091 × 1.03
    PMT             = £ 508.91/ 5.4684 = £ 93.06

Question 15           The prize in last week's Lottery was estimated to be worth £ 35 million. If you were lucky
enough to win, then it will pay you £ 1.75 million per year over the next 20 years. Assume
Value of an annuity   that the first installment is received immediately.
a.      If interest rates are 8 percent, what is the present value of the prize?
b.      If interest rates are 8 percent, what is the future value after 20 years?
year?
SOLUTION
Here, Payment (MPT) = £ 1.75 million
Number of periods (n) = 20 years,
a.    Present value of annuity (PVA) = ? interest rate (i) = 8%
1 - 1 n                      1 -      1      
PVA = PMT ×
   (1 + i)                       (1 + 0.08)20
     i      (1 + i) = £ 1.75       0.08       (1 + 0.08)
= £ 1.75 × 9.8181 × 1.08
= £ 18.56 million
b.     Future value of annuity (FVA) = ?, Interest rate (i) = 8%
(1 + i)n - 1
FVA = PMT 
      i       (1 + i)
(1 + 0.08)20 - 1
= £ 1.75 
        0.08        (1 + 0.08)
= £ 1.75 × 45.7620 × 1.08
= £ 86.49 million
c.    PVA and FVA assuming payments received at the end of year.
Present value of annuity (PVA) = ?, Interest rate (i) = 8%
We have,
1 - 1 n
PVA = PMT ×
 (1 + i) 
     i    
1 -      1      
= £ 1.75
    (1 + 0.08)20
      0.08      
= £ 1.75 × 9.8181
= £ 17.18 million
Future value of annuity (FV) = ?, Interest rate (i) 8%
(1 + i)n - 1            (1 + 0.08)20 - 1
FVA    = PMT                  = £ 1.75 
      i                      0.08       
= £ 1.75 × 45.7620
= £ 80.08 million

Question 16           Ashley has £ 42,180.53 in brokerage account, and plans to contribute an additional £ 5,000
Solving for time   every year at an annual interest rate of 12 percent. If Ashley has to accumulate £ 250,000,
how many years will it take for him to reach his goal?
SOLUTION
Here, Present value (PV) = £ 42,180.53
Payment (PMT) = £ 5,000
Annual return (i) = 12%
Future value (FV) = £ 250,000
Number of years to reach goal (n) = ?

0         1         2      3                          n
12%

42,180.53     5000      5000      5000                   2,50,000

We have,
(1 + 0.12)n - 1
£ 42,180.53 (1 + 0.12)n + £ 5,000 
       0.12       = £ 250,000
or, 5,061.66 (1.12)n + £ 5,000 (1.12)n - £ 5,000 = £ 30,000
or, 10,061.66 (1.12)n = £ 35,000
or, (1.12)n = £ 35,000/£ 10,061.66
or, (1.12)n = 3.4786            ... (i)
In above eqn. (i) if we go for trying several values of 'n', the left hand side is exactly
equal to right hand side at n = 11.
      The required no. of years to reach the goal is 11 years.

Question 17           Your client is 40 years old and wants to begin saving for retirement. You advise the client
to put £ 5,000 a year into the stock market. You estimate that the market's return will be,
Future value of an    on average, 12 percent a year. Assume the investment will be made at the end of the year.
annuity
a.     If the client follows your advice, how much money will she have by age 65?
b.     How much will she have by age 70?
SOLUTION
Here, Your client is 40 years old, Payment (PMT) = £ 5,000, Interest rate (i) = 12%
Investment will be made at the end of the year
a.    Future value of annuity (FVA) at the age of 65?
Number of periods (n) = 65 - 40 = 25 years
(1 + i)n - 1             (1 + 0.12)25 - 1
FVA = PMT                      = £ 5,000 
      i                        0.12       
= £ 5,000 × 133.3338 = £ 666,669
b.    Future value of annuity (FVA) at the age of 70?
Number of periods (n) = 70 - 40 = 30 years
(1 + i)n - 1               (1 + 0.12)25 - 1
FVA = PMT                      = £ , 5,000 
      i                         0.12       
= £ 5,000 × 241.3327
= £ 1,206,66

Question 18            Jason has inherited £ 25,000 and wishes to purchase an annuity that will provide him with
a steady income over the next 12 years. He has heard that the local savings and loan
Solving for payment    association is currently paying 6 percent compound interest on an annual basis. If he were
to deposit his funds, what year-end equal pound amount (to the nearest pound) would he
be able to withdraw annually such that he would have a zero balance after his last
withdrawal 12 years from now?
SOLUTION
Here,
Present value of annuity (PVA) = £ 25,000
Number of years (n) = 12 years
Interest rate (i) = 6%
Equal annual withdraw (PMT) = ?
1 - 1 n
PVA = PMT ×
 (1 + i) 
    i      
1 -      1   
or,    £ 25,000 = PMT
 (1 + 0.06)12
      0.06   
or,    £ 25,000 = PMT × 8.3838
      PMT = £ 2,981.9414

Question 19            You need to have £ 50,000 at the end of 10 years. To accumulate this sum, you have
decided to save a certain amount at the end of each of the next 10 years and deposit it in
Solving for payment    the bank. The bank pays 8 percent interest compounded annually for long term deposits.
How much will you have to save each year (to the nearest Pound)?
SOLUTION
(1 + i)n - 1
FVA              = PMT 
 i          
(1 + 0.08)10 - 1
or, £ 50,000     = PMT 
        0.08       
or, £ 50,000     = PTM × 14.4866
 PMT            = 50,000/14.4866
= £ 3,451.46

Question 20            Louise wishes to borrow £ 10,000 for three years. A group of individuals agrees to lend
Annual interest rate   her this amount if she contracts to pay them £ 16,000 at the end of the three years. What is
the implicit compound annual interest rate you receive (to the nearest whole percent)?
SOLUTION
Here, Present value (PV) = £ 10,000
Number of year (n) = 3 years
Future value (FV) = £ 16,000
End payment, interest rate (i) = ?
We have,
FV = PV (1 + i)n
or, £ 16,000 = £ 10,000 (1 + i)3
or, 1.6 = (1 + i)3
or,   (1.6)1/3 - 1 = 1
or,   i = 0.1695 or 16.95%

Question 21           Calculate the present value of the following cash flow stream. Assume that the stated rate
Uneven cash flow   of interest is 14 percent per annum discounted semiannually.
stream           Cash flow                       1600        1500              850
1000

SOLUTION
End of year                                                3
1
If stated annual rate is014 percent, discounted semiannually, first we calculate the effective
2
annual rate as follows:
Effective interest rate (EAR) = (1 + 0.14/2)2 - 1 = 14.49%
Now present value of given cash flow stream discounted at 14.49 percent effective annual
rate is calculated as follows:
Year     Cash flow       14.49% PVIF           PV
0             £ 1,000     1.0000          £ 1,000.00
1               1,600     0.8734            1,397.44
2               1,500     0.7629            1,144.35
3                 850     0.6663              566.36
Total present value                £ 4,108.15

Question 22           National Lottery has offered you the choice of the following alternative payments.
Alternative 1: £ 10,000 one year from now
Uneven cash flow
stream     Alternative 2: £ 20,000 five years from now.
a.     Which should you choose if the discount rate is 0 percent? 20 percent?
b.     What rate makes the options equally attractive?

SOLUTION
a.     Calculation of present value if discount rate is 0 percent
Alternative 1:
FV    10000
PV = (1+i)n = (1+0)1 = £ 10,000

Alternative 2:
FV      20000
PV = (1+i)n = (1+0)5 = £ 20,000

If discount rate is 0 percent, Alternative 2 is preferable because of higher present
value.
Calculation of present value if discount rate is 20 percent
Alternative 1:
FV      10000
PV = (1+i)n = (1+0.20)1 = £ 8,333.33

Alternative 2:
FV        20000
PV = (1+i)n = (1+0.20)5 = £ 8,037.55

If discount rate is 20 percent, Alternative 1 is preferable because of higher present
value.
b.    Calculation of rate of interest (i) at which both the options are equally likely
PV of Alternative 1 = PV of Alternative 2
10000 20000
(1+i)1 = (1+i)5
(1+i)4 = 2
i = (2)1/4 – 1 = 0.1892 or 18.92%
That is, if the discount rate is 18.92 percent both the alternative would produce equal
present value so that both are equally likely.

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