Finance Chapters 3&4
Suppose we were considering three, ten, and fifty years from the original deposit date at the annual 5% interest rate.
3 years: $100 x (1 + 0.05)^3 = $115.76 10 years: $100 x (1 + 0.05)^10 = $162.89 50 years: $100 x (1 + 0.05)^50 = $1,146.74
Ben and Donna determine that upon retirement they will need to withdraw $50,000 annually at the end of each year for the next thirty years. They now that they cam earn 4% each year on their investment. What is the present value of this annuity? In other words, how much will Ben and Donna need in their retirement account (at the beginning of their retirement) to generate this future cash flow?
1 - [1/1(1 + 0.04)^30]/0.04 = 1 - 0.308319 / 0.04 = 17.29203 17.29203 x $50,000 =$864,601.25 TVM Keys: N = 30 I/Y = 4 PV = ? PMT = -50,000 FV = 0 PV = $864,601.67
Define a growth rate and a discount rate. What is the difference between them?
A growth rate implies going forward in time, a discount rate implies going backwards in time.
Time Value of Money (TVM)
A key financial principle stating that a dollar today is worth more than a dollar tomorrow
Time Line
A linear representation of the timing of cash flows over a period of time
Interest-Only Loan
A loan in which the interest is paid regularly and the principal and final interest payment are repaid at the end of the loan
Amortized loan
A loan in which the principal and interest are paid each period
Discount loan
A loan wherein all interest and principal are repaid at maturity
Annuity due
A series of equal and regular payments in which the payments occur at the beginning of each period
Ordinary Annuity
A series of equal and regular payments in which the payments occur at the end of each period
Annuity
A series of equal cash flows at regular intervals across time FV = PMT x ((1 +r)^n - 1)/r PMT = same annual cash flow, payments r = interest rate five payments n = number of payments of the annuity
Which of the following will result in a future value of greater than $100?
All the future values are greater than $100
What does the amortization schedule tell you about a loan repayment?
An amortization schedule tells you the amount of each payment that is applied against the interest expense, the amount applied against the principal and the principal balance after the payment at each scheduled payment.
Growth Rate
The annual percentage increase (of dividends, investment values, and so forth)
What if you are willing to wait two years for your lump-sum payment?
$100 x 0.05 + $5 x 0.05 = $5 + $0.25 = $5.25 AT THE END OF 2 YEARS: $100 + $5 + $5.25 = $110.25
Determine the new lower principal at the end of year two. It is the principal at the beginning of the year minus the principal reduction amount
$21,592.12 - $3,680.51 = $17,911.61
Say you are shopping for a Canadian console that pays $30 a year in interest forever. What price will you if you want a 5% yield on this investment?
$30 x 1/0.05 = $600
What is the difference between a series of payments and an annuity? What are the two specific characteristics of a series of payments that make it an annuity?
An annuity is a series of payments of equal size of equal intervals. Uniform payments and equal time intervals such as months, quarters or years, are the two characteristics that make a series of payments an annuity. So, a series of payments can be an annuity but not all series of payments are annuities. If the series of payments is of different values or at different intervals, it is not an annuity.
Perpetuity
An infinite series of regular and equal payments
What is the difference between an ordinary annuity and an annuity due?
An ordinary annuity has the payments at the end of the period and an annuity due has the payment due at the start of the period.
Which of the following choices will result in a greater future value at age sixty-five? Choice 1 is to invest $3,000 per year from ages twenty through twenty-six (a total of seven investments) into an account and then leave it untouched until you are sixty-five years old, which is another forty years. Choice 2 is to begin at age twenty-seven and make $3,000 deposits into an investment account every year until you are sixty-five years old ( a total of thirty-nine investments). Each account earns an average 10% per year.
Choice 1: FV = CF x (1 + r)^n - 1 / r FV = $3,000 x (1.10)^7 - 1 / .10 FV = $28,461.51 FV = $28,461.51(1.10)^40 FV = $1,288,146.89 Choice 2: FV = $3,000 x (1.10)^39 - 1 / .10 FV = $1,204,343.33 Choice 1 is better than choice 2 because it has a future value of $1,288,146.89, which is greater than the choice 2 future value of $1,204,343.33
What does the term compounding mean?
Compounding means that interest is earned on prior interest available in the account.
What effect does decreasing the interest rate have on the present value of an annuity? Does a decrease from 7% to 5% have the same dollar effect as a decrease from 5% to 3%?
Decreasing the interest rate (discount rate) increases the present value of an annuity. The impact is different as the discount rates get smaller.
To determine the present value of a future amount, one should ___ the future cash flows.
Discount
Determine the new lower principal at the end of year one:
End-of-year remaining principal = $25,000 - $3,407.88 = $21,592.12
Time Value of Money
Equation 1: FV = PV(1 + r)^n Answers: "How much money will I have in m account at a specific point in the future given a specific interest rate?" Equation 2: PV = FV x [1/(1 + r)^n] Answers: "What is the current value of an amount of cash that I will receive at a specific time in the future given a known discount rate?" Equation 3: r = (FV/PV)^1/n - 1 Answers: "At what rate is my money growing over time?" and "What is the discount rate on my future cash?" Equation 4: n = ln(FV/PV)/ln(1 + r) Answers: "How long will it take my $500 savings bond to turn into its face value of $1,000 if the government is now paying 3.5% on its savings bonds?"
Original deposit (PV) = $15,000 Interest rate (r) = 5% Years (n) = 18
Equation Method: FV = PV(1+r)^n FV = $15,000(1 + 0.05)^18 FV = $36,099.29 TVM Keys: N = 18 I/Y = 5.0 PV = 15,000 PMT = 0 CPT FV =
Previous Equation using FVIFA
FV = $1,000 x ((1 + 0.06)^5 - 1)/0.06 FV = $5,637.10
What would the account look like after ten years of payments? Twenty years of payments? Fifty years of payments?
FV = $1,000 x ((1.06)^10 - 1)/0.06 FV = $13,180.79 FV = $1,000 x ((1.06)^20 - 1)/0.06 FV = $36,785.60 FV = $1,000 x ((1.06^50 - 1)/0.06 FV = $290,335.90
What if the interest rate over the eighteen years in the preceding example was 3%, or 9%, 12%?
FV = $1,500 x ((1.03)^18-1)/0.03 FV = $35,121.65 FV = $1,500 x ((1.09)^18 -1)/0.09 FV = $61,952.01 FV = $1,500 x ((1.12)^18 - 1)/0.12 FV = $83,624.57
Kitty and Red put $1,500 into a college fund every year for their son, Eric, on his birthday, with the first deposit one year from his birth (at his very first birthday). The college fund has a guaranteed annual growth or interest rate of 7%. At his eighteenth birthday, they will pay the last $1,500 into the fund. How much will be in the college fund for Eric immediately following his last payment?
FV = $1,500 x ((1.07)^18 - 1)/0.07 FV = $50,998.55 TVM Keys: N = 18 I/Y = 7 PV = 0 PMT = -1,500 FV = 50,998.55
John and Jane Smith are in the market for a vacation split. They find a small but pleasant condo in Malibu listed at $400,000. They decide that now is not the right time to buy and that they will wait six years. The condos in Malibu appreciate each year at 3.5% and the Smiths will want to know what a similar condo will sell for in six years. Can you help them?
FV = $400,000(1.035)^6 FV = $491,702.13
Future Value (FV)
The cash value of an asset in the future that is equivalent in value to a specific (lower) amount today Future value = deposit x (1 + r) x (1 + r) FV = PV x (1 + r)^1 FV = Future Value PV = Present Value r = interest rate n = number of time periods
In 1867, Secretary of State William H. Seward purchased Alaska fro Russian for the sum of $7,200,000, or about two cents per acre. At the time, the deal was dubbed Seward's Folly, but from our vantage point today, did Seward get a bargain after all? What would it cost today (assume it is 2015) if the land were in exactly the same condition as it was 148 years ago and the prevailing interest rate over this time were 4%?
FV = $7,200,000 (1 + 0.04)^148 FV = $2,389,278,156 TVM Keys: N = 148 I/Y = 4.0 PV = 7,200,000 PMT = 0 FV = ?
With the same $100 paid four times, but now at the beginning of the period, we have each payment with an extra year of interest.
FV = PMT x (1 +r)^n - 1 / r x (1 + r) or FV annuity due = FV ordinary annuity x (1 + r) $100 x (1.08)^4 - 1 / 0.08 x (1.08) FV = $100 x 4.5061 x 1.08 FV = $486.66
DISCOUNT LOAN We'll use the time value of money tools you've learned to figure out the repayment schedule for $25,000 borrowed today ( principal of the loan) fo a period of six years with an annual interest rate of 8%. We'll examine this loan using each of the three common repayment periods If you agreed to pay back the principal and interest at the end of the six years, the total repayment at the end is simply the future value of the $25,000 over six years at the 8% loan rate.
FV = PV (1 + r)^n FV6 = $25,000 (1.08)^6 FV6 = $39,671.86
Suppose you plan to put away some money each year to build up a nest egg to use as a down payment on a house. You start off by putting away $2,000 today, and over the next three years, you are able to put away $3,000 at the end of the first year, $4,000 at the end of the second year, and $5,000 at the end of the third year. How much will you have saved by the end of the third year if your investment rate is 5% per year?
FV of cash flow at T0 = $2,000 x (1.05)^3 = $2,315.25 FV of cash flow at T1 = $3,000 x (1.05)^2 = $3,307.50 FV of cash flow at T2 = $4,000 x (1.05)^1 = $4,2000 FV of cash flow at T3 = $5,000 x (1.05)^0 = $5,000 TOTAL = $14,822.75
Say you decide to put away $1,000 at the end of every year for the next five years. If you can earn 6% on the account, what is the value of the account at the end of the five years? Notice that, unlike the previous problem, you do not put any money away today.
FV of payment 1 = $1,000 + (1.06)^4 = $1,262.48 FV of payment 2 = $1,000 x (1.06)^3 = $1191.02 FV of payment 3 = $1,000 x (1.06)^2 = $1,123.60 FV of payment 4 = $1,000 x (1.06)^1 = $1,060 FV of payment 5 = $1,000 x (1.06)^0 = $1,000 TOTAL = $5,637.10
If you won the lottery and had he choice of a lump-sum payoff or an annuity payoff, what factors would you consider besides the implied interest rate (indifference interest rate) in selecting the payoff style?
Factors such as your current wealth or debts could influence your decision. If you have a large amount of debt and want to be debt free you might elect the lump-sum option. If you are terrible at budgeting money and know you would probably squander the money you might elect a slower payment method, i.e. the annuity. If you wanted to be philanthropic you might want to take the lump sum to give more money away to important causes. These are just a few of the potential non- financial impacts that could influence your decision on the pay out style.
Discounting
The compound reduction in value from future values to present values
If you increase the number of payments on an amortized loan, does the payment increase or decrease?
Increasing the number of payments (all else held constant) decreases the size of each payment. Reverse logic provides the rationale for the answer. If the payment was increased with the same interest payment, the loan would be paid off sooner with the higher payments. Therefore increasing the time to pay off the loan means the payments have been lowered. As the amount applied to the interest is the same but the amount applied to the principal is lower so it takes more payments to eliminate the principal.
Determine the interest expense for one year when the principal is $25,000 and interest rate is 8%
Interest expense first year = $25,000 x 0.08 = $2,000
A home improvement firm has quoted a price of $9,800 to fix up John's backyard. Five years ago John put $7,500 into a home improvement account that has earned an average of 5.25% per year. Does John have enough money in his account to pay for the backyard fix up?
No. John has only $9,687 in his home improvement account.
Denise has her heart set on being a millionaire. She decides that at the end of every year she will put away $5,000 into her "I want to be a. millionaire account" at her local bank. She expects to earn 6% annually on her account. How many years must Denise faithfully put away her money to succeed at becoming a millionaire?
PMT = $5,000 r = 6% FV = $1,000,000 FV = PMT x (1 + r)^n - 1 / r n = ln(FV x r/PMT +)/ ln (1 + r) n = ln(1,000,000 x 0.06/5,000) / ln (1.06) n = 44.0192
AMORTIZED LOAN Known values: N = 6 I/Y = 8 PV = 25,000 PMT = ? FV = 0
PMT = $5,407.88
You have an annuity of equal annual end-of-the-year cash flows of $500 that begin two years from today and last for a total of ten cash flows. Using a discount rate of 4%, what are those cash flows worth in today's dollars?
PMT x (1-(1/(1+r)^n)))/r $500 x (1 -(1/(1+0.04)^10)))/0.04 = $4,055.45 PV = $4,055.45/1.04 PV = $3,899.47
Say you want to buy a new laptop next year and the one you have in mine should be selling for $1,000 a year from now. How much do you need to put away today at 5% interest to have $1,000 a year from now?
PV = $1,000 x 1/(1+0.05)^1 PV = $952.38
You have purchased a savings bond that will pay $10,000 to your newborn child in fifteen years. If the bank discounts this bond at a rate of 3.875% per year, what is today's price (the present value) for this bond?
PV = $10,000 x 1/(1.03875)^15 PV = $5,654
4 equal payments of $100
PV = $100 x (1 - (1/(1.08)^4)/0.08 x (1 + 0.08) PV = $357.71
Your retirement goal is $2,000,000. The bank is offering you a certificate of deposit that is good for forty years at 6.0%. What initial deposit do you need to make today to reach your $2,000,000 goal at the end of forty years?
PV = $2,000,000 x 1/(1+0.06)^40 PV = $194,444.38 TVM Keys: N = 40 I/Y = 6.0 PV = ? PMT = 0 FV = 2,000,000
Donna wants to buy a savings bond for her newborn niece. The face value of the savings bond is $500, the amount the niece would receive in twenty years (future value). The government is currently paying 4% per year on savings bonds. How much will it cost Donna today to buy this savings bond?
PV = $500 x 1/(1 + 0.04)^20 PV = $228.19 TVM Keys: N = 20 I/Y = 4.0 PV = ? PMT = 0 FV = 500
Which form of the TVM equation best answers this question: What is the current value of an amount of cash that I will receive at a specific time in the future?
PV = FV/(1 + r)^n
Take the case of receiving four equal payments of $250 over the next four years (at the end of each year) with a discount rate of 8%. Using the separate lump-sum approach, the present value is
PV = FV1 x 1/(1 + r)^1 PV = $250 x 1/(1.08)^1 = $231.48 FV2 = $250 x 1/(1.08)^2 = $214.33 FV3 = $250 x 1/(1.08)^3 = $198.46 FV4 = $250 x 1/(1.08)^4 = $183.76 TOTAL = $828.03
Present value of an annuity due
PV = PMT x (1 - (1/(1+r)^n)/ r x (1 + r) or PV annuity due = PV ordinary annuity x (1 + r)
Equation
PV = PMT x 1 - [1/(1 + r)^n]/r
Annuity stream equation
PV = PMT x 1 - [1/(1+r)^infinity)/ r PV = PMT / r
Consols
Stocks that pay interest forever, have no date of maturity, and make no promise to repay the principal. They are priced as perpetual bonds
Your company just sold a product with the following plan: $50,000 today, $25,000 next year, and $10,000 the following year. If your firm places the payments into an account earning 10% per year, how much money will be in the account after collecting the last payment?
T0 = $50,000 x (1.10)^2 = $60,500 T1 = $25,000 x (1.10)^1 = $27,500 T2 = $10,000 x (1.10)^0 = $10,000 TOTAL = $98,000
What is the Rule of 72?
The Rule of 72 is a method that estimates how long it takes an investment to double in value, or for given a specific time period, what growth rate will double the value of an investment
Interest
The amount the lender charges for borrowing money
Compounding
The earning of interest on interest into the future
What happens to a future value as you increase the interest (growth) rate?
The future value gets larger as you increase the interest rate
What happens to the future value as the time to the future value increases?
The future value increases as you increase the time to the future
What effect does increasing the interest rate have on the future value of an annuity? Does a change from 4% to 6% have the same dollar effect as a change from 6% to 8%?
The greater the interest rate the greater the future value of an annuity everything else held constant. Changing the interest rate from 4% to 6% will increase the annuity but with a smaller dollar increase when compared to the 6% to 8% change.
Which of the following is not an example of annuity cash flows?
The grocery bill that changes every week
Future Value Interest Factor (FVIF)
The growth rate raised to the power of a number of periods (1 + r)^n, where r is the interest rate and n is the number of periods
Compound Interest
The interest earned in subsequent periods on the interest earned in prior periods
Amortization Schedule
The listing of the periodic interest expense, the reduction in principal each period, and the ending balance for each period
Future Value Interest Factor of an Annuity (FVIFA):
The mathematical factor, [(1 + r)^n - 1]/r, used to multiply the annuity to calculate the future value of the annuity stream FVIFA: = ((1 + r)^n - 1)/r
Present Value Interest Factor of an Annuity (PVIFA)
The mathematical factor, [1-1/(1 + r)^n]/r, used to multiply the annuity to calculate the present value of the annuity stream
Lump-Sum Payment
The one-time payment of money at a future date
Principal
The original loan amount borrowed
Explain the meaning of this statement: The current principal balance of a loan repaid as an amortized loan is the present value of the future payment stream.
The outstanding balance or remaining unpaid principal after the application of a scheduled payment reflects the current amount needed to pay off the loan. The remaining scheduled payment stream is another way to pay off the loan. Because both the principal and the remaining scheduled payments are sufficient to pay off the loan the current principal is therefore the present value of the remaining payments.
If you increase the interest rate on an amortized loan, does the payment increase or decrease?
The payment increases with a rise in interest rates all else held constant. The reason is that more of the payment is applied to the interest and so to reduce the principal at the same pace as before a higher payment is needed.
What happens to the present value as the time to the future value increases?
The present value decreases as you increase the time between the future value date and the present value date
What happens to a present value as you increase the discount rate?
The present value gets smaller as you increase the discount rate.
When a lottery prize is $10,000,000, but will pay out as a series of $250,000 payments over forty years, is it really a $10,000,000 lottery prize?
The present value of the lottery is not worth $10,000,000. The total payments over time are $10,000,000 but this is not a value of the lottery because these payments are at different points in time, and next year's $250,000 is not the same as this year's $250,000.
Discount Rate
The rate used to determine the present value of future cash flows
Present Value Interest Factor (PVIF)
The reciprocal of the FVIF 1/(1+r)^n
Present Value (PV)
The value today of a cash flow in the future PV = FV x 1/(1+r)^n
What are the four basic parts (variables) of the time value of money equation?
Years, interest rate, present value, future value
Is the present value always less than the future value?
Yes, as long as interest rates are positive - and interest rates are always positive - the present value of a sum of money will always be less than its future value.
Determine the amount available for reducing the principal in the second year by subtracting the second year's interest expense from the annual payment
amount available for reducing the principal = $5,407.88 - $1,727.37 = $3,680.51
INTEREST-ONLY LOAN Another acceptable repayment schedule is to pay the annual interest each year and then repay the principal at the maturity date with the last year's interest. Each year the interest payment is the 8% loan rate times the principal:
annual interest payment: $25,000 x 0.08 = $2,000 At the end of the sixth year, the final payment is $27,000, reflecting the $25,000 principal originally borrowed and the final $2,000 interest payment. Thus, the total outflow over the six years is six $2,000 interest payments, or $12,000 of interest, and the $25,000 principal repayment for a total of $37,000
Assume you have the option to convert from an ordinary annuity to an annuity due for which the current payment is $100 per year. Let's use the same setup with an interest rate of 8%. To switch to an annuity due payment, you simply divide the current ordinary annuity payment by (1 + 0.8)
annuity due payment = $100/(1.08) = $92.60
You can also adjust a payment from an ordinary annuity to one from an annuity due by dividing the ordinary annuity payment by (1 + r)
annuity due payment = ordinary annuity payment/ (1 + r)
Determine the amount available for reducing the principal after you have subtracted the interest expense from the payment:
available for principal reduction first year = $5,407.88 - $2,000 = $3,407.88
Determine the interest expense for the second year by multiplying the outstanding or remaining principal by the 8% interest rate:
interest expense second year = $21,592.12 x 0.08 = $1,727.37
How much money will you have one year from today if you put $100 in a savings account that promises to pay you 5% over the coming year?
interest rate = $100 x 0.05 = $5 Future value = $100 x 1.05 = $105 $100 + $5 = $105
Your goal in life is to be a millionaire. Today your financial portfolio is worth $3,733.24. Having studied this chapter carefully and being a shrewd investor, you determine that you can earn 15% every year on your portfolio. You do not plan to invest any additional money in this portfolio, nor will you withdraw any funds from it before it grows to $1 million. Given your 15% interest rate, how long will you have to wait to become a millionaire if this investment represents all your wealth?
n = ln($1,000,000/$3,733.24)/ln(1.15) n = 40 years TVM Keys: N = ? I/Y = 15 PV = 3,733.24 FV = 1,000,000
How long will it take my $500 savings bond to turn into its face value of $1,000 if the government is now paying 3.5% on its savings bonds?
n = ln($1,000/$500)/ln(1 + 0.035) n = ln(2)/ln(1.035) n = 20.15 years TVM Keys: n = ? I/Y = 3.5 PV = -500 PMT = 0 FV = 1,000
For much of the twentieth century, new car prices rose at an annual rate of 5.73%. Given a beginning new car price of $600, how long did it take the average new car price to rise up to $16,950? Round to the nearest year?
n = ln(FV/PV)/ln(1 + r) n = ln(16,950/600)/ln(1.0573) n = 59.96 or 60 years
If you deposit $250 in the bank today and in five years will get back $400, what is your growth rate?
r = ($400/$250)^1/5 - 1 r = 0.0986 or 9.86%
John, a college student, needs to borrow $5,000 today for his tuition bill. He agrees to pay back the loan in a lump-sum payment five years from now, after he is out of college. The bank states that the payment will need to be $7,012.76. If John borrows the $5,000 from the bank, what interest rate is he paying on his loan?
r = ($7,012.76/$5,000)^1/5 - 1 r = (1.402552)^0.2 -1 r = 0.070 or 7% TVM Keys: N = 5 I/Y = ? PV = $5,000 PMT = 0 FV = $7,012.76
The Millville School District had 3,071 students enrolled five years ago. Today the district enrollment is 2,418 students. What has been the annual rate of change of student enrollment in the Millville School District over this time period?
r = (2,418/3,071)^1/5 - 1 r = -0.0467 or -4.67%
You are the planning commissioner for Boomtown, a growing city in the Southwest. The city council has estimated that the city's population will increase very rapidly over the next twenty years, reaching an estimated 250,000. Today the population is 94,222. What is the projected growth rate of this city?
r = (250,000/94,222)^1/20 - 1 r = 0.05 or 5% TVM Keys: N = 20 years I/Y = ? PV = 94,222 PMT = 0 FV = 250,000
The average U.S. wage in 1990 was $28,960, far higher than the average wage in 1930 of $1,970. What was the average annual increase in wages over this sixty year period?
r = (28,960/1,970)^1/60 - 1 r = 0.0458 or 4.58%
The dividends per share paid by Going Going Gone (GGG) doubled for a starting value of $1.50 in 2000 to a value of $3.00 in 2006 (a six-year period). What was the approximate average annual rate of growth of GGG's dividends per share? Use the Rule of 72 to determine your answer.
r = 72/# of years r = 72/6 r = 12% GGG's dividends grew at an annual rate of approximately 12% per year
A manufacturer of LCD television sets has seen sales increase from 125,000 units per year to 500,000 units per year in eight years. What has been firm's average annual rate of increase in the number of television sets sold? Use the rule of 72 to determine your answer.
r = 72/4 r = 18%
In the second year, the beginning outstanding principal is now $21,592.12, and it is the lower amount that is earning interest for the lender. This ending balance is also the present value of the remaining payments. So for the second year we have the following steps:
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