Time Value of money (ch5) Flashcards
Suppose someone invests $1,000 today for a 5-year term and receives 10% annual simple interest on the investment. How much money would the investor have after 5 years?
1,000 x .1 = $100 per year
Suppose someone invests $1,000 today for a 5-year term and receives 10% annual simple interest on the investment. How much money would the investor have after 50 years?
1,000 x .1 = $100 per year
1,000 + (50 x 100)
= 6,000
can also use the calculator
Suppose someone invests $1,000 today for a 5-year term and receives 10% annual compound interest. How much would the investor have after 5 years? Use a financial calculator
o PMT -1,000 PV 10 I/Y 5 N CPT FV = 1,610.51
Suppose someone invests $1,000 today for a 5-year term and receives 10% annual compound interest. How much would the investor have after 50 years? Use a financial calculator
0 PMT
-1,000 PV
10 I/Y
50 N
CPT FV = 117,390.85
An investor estimates that she needs $1 million to live comfortably when she retires in 40 years. How much does she have to invest today, assuming a 10% interest rate on the investment? Use a financial calculator
0 PMT - 1,000,000 FV 10 I/Y 40 N CPT PV = 22,094.93
Suppose we modify the lottery example used earlier. The “prize” is now a $20,000 investment that has a payoff of $31,000 in 5 years. We have the present and future values and the period, so we can solve for the interest rate. This is an important interest rate, called the internal rate of return (IRR), because it is the rate of return that is internal to the value in the problem. Many problems in finance are IRR problems for which we need to compare the rates of return earned on different investments.
o PMT 1,000 FV -20,000 PV 5 N CPT I/Y = 9.161%
In 2011, the average NHL player earned $2.4 million per year, while the average career was approximately 6 years. Assuming a 41% tax rate, this would generate $1,416,000 in annual “disposable” (Ie. after-tax) income. Assume a player earning this average decides to invest 10% of his disposable income (ie. $141,600) at the end of each year for the next 6 years (ie. the average career span) and expects to earn 8% per year.
a. How much will he have accumulated after 6 years?
141,600 PMT 6 N 0 PV (no deposit today) 8 I/Y CPT FV = -1,038,768
In 2011, the average NHL player earned $2.4 million per year, while the average career was approximately 6 years. Assuming a 41% tax rate, this would generate $1,416,000 in annual “disposable” (Ie. after-tax) income. Assume a player earning this average decides to invest 10% of his disposable income (ie. $141,600) at the end of each year for the next 6 years (ie. the average career span) and expects to earn 8% per year.
b. How much would he need to deposit today to have the same results?
141,600 PMT 6 N 0 FV 8 I/Y CPT PV = -654,600
Sometimes annuities are structured so that the cash flows are paid at the beginning of a period, rather than at the end. For example, leasing arrangements usually set up like this, with the lessee making an immediate payment on takin possession of the equipment, such as a car. Such an annuity is called an annuity due.
We will repeat example5-8 except we assume that the payments are made at the beginning rather than the end of each year.
a. How much will the investor have after 6 years?
2nd BGN 2nd set
141,600 PMT 6 N 0 PV 8 I/Y CPT FV
Sometimes annuities are structured so that the cash flows are paid at the beginning of a period, rather than at the end. For example, leasing arrangements usually set up like this, with the lessee making an immediate payment on takin possession of the equipment, such as a car. Such an annuity is called an annuity due.
We will repeat example5-8 except we assume that the payments are made at the beginning rather than the end of each year.
b. How much would the investor have the deposit today to have the same results?
BGN 141,600 PMT 6 N 0 FV 8 I/Y CPT PV = -706,968
a. An annuity pays $3,000 per year at year end and earns an annual return of 12% per year for 30 years.
3,000 PMT
30 n
FV 0
I/Y 12
CPT PV = -24,165.55
What is the PV of a $3,000 per year annuity that goes on forever – that is, in perpetuity – if k= 12%?
PV0 = PMT/K
3,000 / .12 = 25,000
a. Suppose someone invests $1,000 today or one year at a quoted annual rate of 16% compounded annually. What is the FV at the end of the year?
FV 1,000 (1.26) exponent of 1 = 1,160
this means that each dollar grows to $1.16 by the end of the period, so we can say that the “effective” annual interest rate is 16%
What if someone invests $1,000 at a quoted rate of 16% compounded quarterly?
16 /4 = 4% per quarter
FV 1,000 (1.04) exponent 4 = 1,170 (rounded)
therefore the effective annual interest rate is 17% because each dollar grows by 1.17
What are the effective annual rates for the following quoted rates?
a. 12%, compounded annually
2nd Iconv 2nd clrwork
Nom 12 enter down arrow down arrow
C/Y = 1 (annually) enter down arrow down arrow
CPT = 12%
What are the effective annual rates for the following quoted rates?
12%, compounded Semi annually
2nd Iconv 2nd clrwork
Nom 12 enter down arrow down arrow
C/Y = 2 enter down arrow down arrow
CPT = 12.36%
What are the effective annual rates for the following quoted rates?
12% compounded quarterly
2nd Iconv 2nd clrwork
Nom 12 enter down arrow down arrow
C/Y = 4 enter down arrow down arrow
CPT = 12.55%
What are the effective annual rates for the following quoted rates?
12% compounded monthly
2nd Iconv 2nd clrwork
Nom 12 enter down arrow down arrow
C/Y = 12 enter down arrow down arrow
CPT = 12.68%
What are the effective annual rates for the following quoted rates?
12% compounded daily
2nd Iconv 2nd clrwork
Nom 12 enter down arrow down arrow
C/Y = 365 enter down arrow down arrow
CPT = 12.75%
What are the effective annual rates for the following quoted rates?
12% compounded continuously
12.75%
.12 2nd ex -1
Determine the required year-end payments for a 3 year $5,000 loan with a 10% annual interest rate. Complete an amortization schedule.
0 FV 5,000 PV 3 N 10 I/Y CPT PMT = 2010.57
Determine the principal outstanding on the loan in example 5-14 after one year, without referring to the amortization schedule found in the solution.
PMT = 2010.57
2010.57 PMT
FV 0
10 I/Y
2 N (2 years left)
CPT PV = 3489.42
Determine the monthly and the amortization schedule for the first three months of a $200,000 mortgage loan with an amortization period of 25 years, based on a quoted rate of 6% and a 10 year term.
PV -200,000
N = 25 x 12 = 300
I/Y (1 + 0.06/2) exponent 2/12 - 1
FV 0
CPT PMT = - 1279.61
Determine the monthly and the amortization schedule for the first three months of a $200,000 mortgage loan with an amortization period of 25 years, based on a quoted rate of 6% and a 10 year term.
Determine the principal outstanding on the mortgage in Example 5-16 at the end of the 10 year term. pMT = 1279.61
PMT 1279.61
FV 0
I/Y 0.4938622 (see last example on calculation)
(1 + .06 / 2) exponent 2/12-1
N = 180 (25 - 10 =15 x 12)
CPT PV = -152 355.78
Multiple annuities 5-18What is the PV of $1,000 received at year end for the next 4 years, followed by $2,000 per year end for years 5 and 7, assuming a 10% rate of interest, compounded annually?
add a
multiple annuities 5-18b. Suppose an investor needed $15,000 at the end of 7 years and can only invest $1,000 per year for years 1 to 4 (as above). How much would the investor need to deposit in each of years 5 to 7 to achieve this objective, assuming a 10% interest rate as above?
add b
An Investor plans to retire 35 years from today and have sufficient savings to guarantee $48,000 each year for 20 years. Assume retirement withdrawals will be made at the beginning of each of the 20 years. The investor estimates that at the time of retirement, he can sell his business for $200,000. The expectation is that interest rates will be relative stable at 8% a year for the next 35 years. Thereafter, the interest rate is expected to decline to 6% forever. The investor wants to make equal annual deposits at the end of each of the next 35 years. How much should be deposited each year in order to meet the stated objective?
Step 1: set calculator to BGN mode
step 2: FV = 0 PMT = 48,000 I/Y = 6 N = 20 CPT PV = - 583,589.59
Step 3: take calculator out of BGN mode
step 4: PV = 0 FV = 383,589.59 I/Y = 8 N = 35
CPT PMT = 2226.07
5a-1 valuing a growing perpetuity
You are attempting to determine the present value of cash flows to be generated from a rental property you are considering purchasing. You have estimated that the after-tax cash flows form tis property will grow at 4% per year indefinitely due to rental increases. The cash flow this past year was $100,000, and the appropriate discount rate is 15%. Find the present value of these cash flows.
PMT = 100,000 g = 4% K = 15%
PV = PMT0 (1 = g) / K-G
PV =( 100,000 x 1.04) /(.15 - .04)
PV = 945,454.55
5A-2 Valuing a growing (shrinking) Annuity
A mining company is attempting to determine the present value of cash flows to be generated form a new mining operation. They have estimated that the after-tax cash flows from this mine will shrink at a rate of 10% per year as the reserves are depleted, and that after 10 years, the mine will be abandoned. Next year’s cash flow is estimated to be $200,000, and the appropriate discount rate is 20%. Find the present value of these cash flows.
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