time value money questions answers

Economics and Financial Studies for Managers (EM601)Summer 2012

Assignment # 3

Time Value of Money

Question # 1

You sold a car and accepted a note with the following cash flow stream as your payment. What was the effective price you received for the car assuming an interest rate of 8%?

Years: 0 1 2 3 4| | | | |

CFs: $0 $1,500 $2,000 $2,2000 $2,500

Ans 1.

FV = PV (1 + I)n

PV=FV/(1 + I)n

PV1=1500/1.08

PV1=$1388.88---------------------1

PV2= 2000/(1.08) 2

PV2=1714.67$ ………………2

PV3= 22000/(1.08 )3

PV3=$ 17464.309-----------3

PV4= 2500/(1.08) 4

PV4= $1837.5-----------------------4

Now adding all four values of eq 1,2,3 & 4

EM601 Assignment # 3

P(total)= $ 22405.4

Question # 2

To complete your last year in business school and then go through law school, you will need $14,000 per year for 4 years, starting next year (that is, you will need to withdraw the first $14,000 one year from today). Your rich uncle offers to put you through school, and he will deposit in a bank paying 8.5% interest, compounded annually, a sum of money that is sufficient to provide the 4 payments of $14,000 each. His deposit will be made today.

a) How large must the deposit be?b) How much will be in the account immediately after the sthird withdrawal?

Ans2

PV 14000 14000 14000 14000

PVA = PMT [(1 - (1 / (1 + i)n)) / i]…………1

where

PVA=?

i=8.5%

PMT=$ 14000

n=4

Putting values in eq 1

PV=14000 (1 / (1 + .085)4)) / 0.085]

PVA= $ 45858.35……………ans to part a

Now for part b amortization table is required

years Beginning amount installment intrset principle amount ending principle


0 45858.35 ---- ----- ------ 45858.35

1 45858.35 14000 3898 10102 35756.35

2 35756.35 14000 3039.28 10960.7 24795.789

3 24795.789 14000 2107.642 11892.35 12903.43

4 12903.43 14000 1096.57 12903.43 0

Ans for part b is 12903.43$

Question # 3

The prize in last week’s Florida lottery was estimated to be worth $10 million. If you were lucky enough to win, the state will pay you $2 million per year over the next 5 years. Assume that the first installment is received immediately. If the interest rates are 7 percent, what is the present value of the prize?

Ans #3

PVA = PMT [(1 - (1 / (1 + i)n)) / i]…………1

PMT= 2 million

I=7%

N= 5 years

PVA=?

Putting values in eq 1

PVA = 2 [(1 - (1 / (1 + 0.07)5)) / 0.07]

PVA= 8.774 million


Question # 4

An investment pays you $800 at the end of each of the next 3 years. The investment will then pay you $900 at the end of Year 4, $950 at the end of Year 5, and $990 at the end of Year 6. If the interest rate earned on the investment is 6 percent, what is its present value? What is its future value?

Ans #4

Future value

FV = PV (1 + i)n

FV1 = 800 (1 + .06)5 = $1070.58

FV2 = 800 (1 + .06)4 = $1009.98

FV3 = 800 (1 + .06)3 = $952.81

FV4 = 900 (1 + 0.06)2 =$1011.24

FV 5= 950 (1 +0.06 )1= $1007

FV6 = $990

FV (Total)= $ 6041.61

Present value

FV = PV (1 + I)n

PV = FV / (1 + I)n


PV1 = FV / (1 + I)n

PV1 = 800 / (1 + 0.06)1

PV1 = $754

PV2 = 800 / (1 + 0.06)2

PV2 = $711

PV3 = 800 / (1 + 0.06)3

PV3 = $671.69

PV4 = 900 / (1 + 0.06)4

PV4 = $ 712.88

PV5 = 950 / (1 + 0.06)5

PV5 = $ 709.84

PV6 = 990 / (1 + 0.06)6

PV6 = $ 679.91

Question # 5

Your company is planning to borrow $8,000,000 on a 5-year, 11% annual payment fully amortized term loan. Prepare an amortization table to show the full payments for five years along with breakdown of each payment into interest and reduction in principal amount.

Ans #5


PVA = PMT [(1 - (1 / (1 + i)n)) / i]…….1

PVA= $ 8000000

I= 11%

N=5

So

PMT from eq 1

PMT= $ 2164502.165

Years Begning amount

Installment INTREST Principal amount

Ending amount

0 8000000 ------ ------ ------ 8000000

1 8000000 2164502.17 880000 1284502.17 6715497.83

2 6715497.83 2164502.17 738704.76 1425797.41 5289700.42

3 5289700.42 2164502.17 581867.046 1582635.124 3707065.29

4 3707065.296 2164502.17 407777.18 1756724.987 1950340.309

5 1950340.309 2164502.17 214161.86 1950340.309 ----nil----

Question # 6

After graduation, you plan to work for a company for 15 years and then start your own business. You expect to save and deposit $5,000 a year for the first 8 years (t = 1 through t = 8) and $8,000 annually for the following 7 years (t = 9 through t = 15). The first deposit will be made a year from today. In addition, your grandfather just gave you a $20,000 graduation gift which you will deposit immediately (t = 0). If the account earns 7% compounded annually, how much will you have when you start your business 15 years from now?

Ans #6


FV = PV (1 + I)n

as $ 20000 graduation gift at t=0 so future value at t= 15 will be

FV = 20000 (1 + 0.07)15

FV1 = $ 55180.630…………….1

Deposit pf $ 5000 for 8 years

so PMT= $ 5000

n=8

i=7%

FV(t=1 to t=8) = PMT [((1 + i)n - 1) / i]

FV(t=1 to t=8) = 5000 [((1 + 0.07)8 - 1) / 0.07]

FV(t=1 to t=8) = $ 51299As the amount will be in account for next 7 more years i.e t= 9 to 15 so’Future value of $ 5000 annuity at 15 years will be’

FV = 51299 (1 + 0.07)7

FV (at t=15) = $ 82374.98--------2

Now for PMT= $ 8000 for 7 years future value will be

FVA(t=9 to t=15) = PMT [((1 + i)n - 1) / i]

FVA(t=9 to t=15) = 8000 [((1 + 0.07)7 - 1) / 0.07]

FVA(t=9 to t=15) = $ 69232.168 ……….3

Now adding three eqs 1,2 and 3


FV total at 15th year will be=

55180.63 + 82374.98+69232.168= $ 206787.7 ............

Question # 7

Your sister turned 32 today, and she is planning to save $12,000 per year for retirement, with the first deposit to be made one year from today. She will invest in a mutual fund that's expected to provide a return of 6.5% per year. She plans to retire when she turns 65, and she expects to live for 20 years after retirement, to age 85. Under these assumptions, how much can she spend each year after she retires? Her first withdrawal will be made at the end of her first retirement year.

Ans # 7

FVA(t=32 to t=65) = PMT [((1 + i)n - 1) / i]

N= 33

I= 0.065

PMT= $ 12000

FVA(t=32 to t=65) = 12000 [((1 + 0.065)33 - 1) / 0.065]

FVA(at t=65) = $ 1290428.5

Now at t= 65th year my FVA will be my PVA

PVA= $ 1290428.5

PMT= ?

N= 20

I= 7%

PVA = PMT [(1 - (1 / (1 + i)n)) / i]

1290428.5=PMT(11.0185)


PMT=$ 117114.63

She can spend $ 117114.63

Question # 8

You just deposited $6,500 in a bank account that pays a 9% interest rate, compounded quarterly. If you also add another $8,000 to the account one year (4 quarters) from now and another $9,500 to the account two years (8 quarters) from now, how much will be in the account three years (12 quarters) from now?

I= 9% annually or for quarterly=0.09/4=0.0225n=3 years or 3x4= 12

FV = PV (1 + I)n

FV(for $ 6500at t=3 years) = 6500 (1 + .0225)12

FV(for $ 6500at t=3 years) = $8489.32

FV( for $ 8000 at 2 year)= 8000 (1 + .0225)8

FV( for $ 8000 at 2 year)= $ 9558.64

FV( for $9500 at 1 year)= 9500 (1 + .0225)4

FV( for $ 8000 at I year)= $ 10384.29

FV (Total at 3 years or at 12th quater)= $8489.32+$ 9558.64+$ 10384.29= $ 28432.25


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