TIME VALUE OF MONEY Answer the following questions:
a. Assuming a rate of 10% annually, find the FV of $1,000 after 5 year.
B. What is the investment's FV at rates of 0%, 5%, and 20% after 0, 1, 2, 3, 4, and 5 years?
c. Find the PV of $1,000 due in 5 years if the discount rate is 10%.
d. What is the rate of return on a security that costs $1,000 and returns $2,000 after years?
e. Suppose California's population is 36.5 million people and its population is expected to grow by 2% annually. How long will it take for the population to double?
f. Find the PV of an ordinary annuity that pays $1,000 each of the next S years if the interest rate is 15%. What is the annuity's FV?
g. How will the PV and FV of the annuity in Part f change if it is an annuity due? h. What will the IN and the PV be for $1,000 due in 5 years if the interest rate is 10%, semiannual compounding?
i. What will the annual payments be for an ordinary annuity for 10 years with a PV of $1,000 if the interest rate is 8%? What will the payments be if this is an annuity due?
j. Find the PV and the FV of an investment that pays 8% annually and makes the following end-of-year payment:
0. 1 2 3
$100 $200 $400
k. Five banks offer nominal rates of 6% on deposits; but A pays interest annually, 13 pays semiannually, C pays quarterly, D pays monthly, and E pays daily.
1. What effective annual rate does each bank pay? If you deposit $5,000 in each bank today, how much will you have in each bank at the end of 1 year? 2 years?
2. If all of the banks are insured by the government (the FDIC) and thus are equally risky, will they be equally able to attract funds? If not (and the TVM is the only consideration), what nominal rate will cause all of the banks to provide the same effective annual rate as Bank A?
3. Suppose you don't have the $5,000 but need it at the end of 1 year. You plan to make a series of deposits—annually for A, semiannually for B, quarterly for C, monthly for D, and daily for F-with payments beginning today. How large must the payments be to each bank?
4. Even if the five banks provided the same effective annual rate, would a rational investor be indifferent between the banks? Explain.
I. Suppose you borrow $15,000. The loan's annual interest rate is 8%, and it requires four equal end-of-year payments. Set up an amortization schedule that shows the annual payments, interest payments, principal repayments, and beginning and ending loan.
Answers
Answer:
The correct answers for each part are as follows:
a) FV = $1,500
b) Year 0:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,000
@ 20% p.a. = $1,000
Year 1:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,050
@ 20% p.a. = $1,200
Year 2:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,100
@ 20% p.a. = $1,400
Year 3:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,150
@ 20% p.a. = $1,600
Year 4:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,200
@ 20% p.a. = $1,800
Year 5:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,250
@ 20% p.a. = $2,000
c) PV = $620.92
d) Rate of return on security = 14.87%
e) No. of years that it will take to double the population = 35
f) PV of the annuity = $3,352.15
FV of the annuity = $6.742.38
g) PV of the annuity due = $3854.80
FV of the annuity due = $7,753.74
h) IN or Semiannual Interest Rate = 5% (Solution is attached in the form of an image)
PV = $613.91
i) Annual payment for ordinary annuity = $149.03 (Solution is attached in the form of an image)
Annual payment for annuity due = $137.99
j) FV at the end of 3 years of above payment = $732.64 (Solution is attached in the form of an image)
k) 1. i) EAR of Bank A = 6% (Solution is attached in the form of an image)
Payments after Year 1 = $4,716.98
ii) EAR of Bank B = 6.09%
Payments after Year 1 = $2,462.50
iii) EAR of Bank C = 6.14%
Payments after Year 1 = $1,221.60
iv) EAR of Bank D = 6.17%
Payments after Year 1 = $405.02
v) EAR of Bank E = 6.18%
Payments after Year 1 = $13.28
2. No, they will not be able to do so as they all have different effective annual rates.
A = 6%
B = 5.91%
C = 5.86%
D = 5.84%
E = 5.82%
3. Payment By:
A = $4,716.98
B = $2,462.50
C= $1,221.60
D= $405.02
E= $13.28
4. Even if all five banks provided the same effective annual rate, a rational investor NOT be indifferent between the banks because they all compound the value of the deposited money differently.
l. Year------Balance ----------Payment---------- Interest-------- Principal------- Balance
1)____$15,000.00____ $ 4,528.81____ $1,200.00 _____$3,328.81____ $ 11,671.19
2) ___$11,671.19____ $ 4,528.81___ $ 933.70______ $3,595.12_____ $ 8,076.07
3) ___$ 8,076.07 ______$ 4,528.81___ $ 646.09_____ $3,882.73______ $ 4,193.34
4) __$ 4,193.34_______ $ 4,528.81____ $ 335.47_____ $4,193.34_____ $ 0.00
(Solution is attached in the form of an image)
Explanation:
a) Given:
Rate (i) = 10% p.a.
Investment (PV) = $1,000
Time (n) = 5 years
To find:
The FV of $1,000 after 5 years.
Solution:
FV => PV x (1 + i/n) = 1000 x [1 + (0.10 x 5)]
= 1000 x 1.50
= $1,500
Therefore, FV of $1000 after 5 years @ 10% p.a.= $1,500
b) Given:
Rate of interest= 0% p.a., 5% p.a., 20% p.a.,
Years= 0,1,2,3,4,5
PV= $1000
To find:
The FV for all years at all given rates of interest.
Solution:
We shall use the same formula that we used in part (a) of this question, i.e., FV = PV x (1 + i/n)
Therefore, for Year 0, by substituting the value of PV, ‘i’ and ‘n’ in the given formula by taking i = 0% first, we get:
FV => 1000 x (1 + 0) = $1000
Because we obtain zero upon multiplying zero with any number, irrespective of what year it is, @ 0% p.a., the FV will remain $1000.
Similarly, for Year 0, irrespective of what the rate of interest is, FV = $1000
Therefore, for Year 0:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,000
@ 20% p.a. = $1,000
For Year 1:
@ 0% p.a. = $1,000
We then take the rate of interest as 5%p.a. By substituting the value of PV, i = 5% and n = 1, in the given formula, we get:
FV => 1000 x [1 + (0.05 x 1)] = $1,050
Similarly, we substitute i = 20% in the given equation, we get:
FV => 1000 x [1 + (0.20 x 1)] = $1,200
Therefore, for Year 1:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,050
@ 20% p.a. = $1,200
We follow this exact same pattern for years 2, 3, 4 and 5 until we obtain the following answers:
Year 2:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,100
@ 20% p.a. = $1,400
Year 3:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,150
@ 20% p.a. = $1,600
Year 4:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,200
@ 20% p.a. = $1,800
Year 5:
@ 0% p.a. = $1,000
@ 5% p.a. = $1,250
@ 20% p.a. = $2,000
c) Given:
FV = $1,000
Time (i) = 5 years
Discount rate (n) = 10%.
To find:
The PV of $1,000 due in 5 years @ 10%.
Solution:
In order to calculate the PV of $1,000 due in 5 years @ 10%, we take use the following formula:
PV = FV / (1 + i)^n
Therefore, by substituting the values for i and n in the given formula, we get:
PV => 1000/ (1 + 0.10)^5 = 1000/1.61051
Therefore, PV = $620.92
d) Given:
PV= $1,000
FV= $2,000
Time = 5 years
To find:
Rate of Return of Security
Solution:
In order to find the Rate of Return of Security, we must go through the following steps:
R => [(FV / PV)^1/n ] - 1 = [(2000/1000)^⅕ ] - 1
R = 2^0.2 - 1
R = 1.1487 - 1
Therefore, R = 0.1487
In order to represent it in percent form, we multiply it by 100.
Therefore, R = 14.87%
e) Given:
California’s Population (PV) = 36.5 million people
r = 2% p.a.
To find:
Time (n) it will take for the population to double
Solution:
We first calculate FV = 2 x PV
Therefore, FV = 36.5 x 2
We then use the following formula to calculate n:
FV = PV (1 + r)^n
=> 73 = 36.5 x (1 + 0.02)^n
=> 73/36.5 = (1.02)^n
=> 2 = (1.02)^n
Taking log,
=> n = 35 years
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