Chapter 4 Introduction to Valuation: The Time Value of Money
What does it mean to compound interest? How does compound interest differ from simple interest?
Compound interest is measured based on principal amount and also on the interest rate earned in the previous periods at the interest rate. Compound interest 1000*(1+.05)^5=1276 1276-1000= $276 The amounts increase rapidly when using compound interest. Simple interest is based on the principal amount, while compound interest is based on the principal amount and the interest that it accumulates on it every period. The amount remains uniform when using simple interest.
What do we mean by discounted cash flow, or DCF, valuation?
Discounting cash flow valuation method is used to estimate the attractiveness of an investment opportunity. Analyzes the future free cash flow projecting and discounts them to arrive at a present value. PV= $1*[1*/(1+r)^t]= $1/(1+r)^t
The process of discounting a future amount back to the present is the opposite of doing what?
Discounting the future amount is the process of figuring out what that future value is in present day money.
In general, what is the future value of $1 invested at r per period for t periods?
FV= $1 (1+r)^t periods
You are considering a one-year investment. If you put up $1,250, you will get back $1,350. What rate is this investment paying?
First, in this single-period case, the answer is fairly obvious. You are getting a total of $100 in addition to your $1,250. The implicit rate on this investment is thus $100/$1,250 = .08, or 8 percent. More formally, from the basic present value equation, the present value (the amount you must put up today) is $1,250. The future value (what the present value grows to) is $1,350. The time involved is one period, so we have: $1,250 = $1,350/(1 + r) t 1 + r = $1,350/$1,250 = 1.08 r = .08, or 8% In this simple case, of course, there was no need to go through this calculation, but, as we describe later, it gets a little harder when there is more than one period.
You have been offered an investment that promises to double your money every 10 years. What is the approximate rate of return on the investment?
From the Rule of 72, the rate of return is given approximately by 72/r% = 10, so the rate is approximately 72/10 = .072, or 7.2%. Verify that the exact answer is 7.177 percent.
Future Value Formula
$1 * (1+r)^2
investing for a single period
$100 deposit into a savings account 10% interest rate per year annually ?? how much will you have in one year 100*.10=110
Investing for more than one period
$100 deposit into a savings account 10% interest rate per year annually ?? how much will you have in two year 100*.10=110 leave in back -> same interest 110*.10= 11 110+11=121 $ 100 original $ 10 interest 1st year $ 10 interest 2nd year $ 1 interest on interest
discounted cash flow (DCF) valuation
(a) Calculating the present value of a future cash flow to determine its value today. (b) The process of valuing an investment by discounting its future cash flows.
compound interest
Interest earned on both the initial principal and the interest reinvested from prior periods.
interest on interest
Interest earned on the reinvestment of previous interest payments
simple interest
Interest earned only on the original principal amount invested
In general, what is the present value of $1 to be received in t periods, assuming a discount rate of r per period?
Present Value = PV= $1*[1*/(1+r)^t]= $1/(1+r)^t t= Number of periods r= Discount Rate
Simple Interest Formula
SI= PRT P = principal R = % rate per annum T = Time period
You would like to retire in 50 years as a millionaire. If you have $10,000 today, what rate of return do you need to earn to achieve your goal?
The future value is $1,000,000. The present value is $10,000, and there are 50 years until retirement. We need to calculate the unknown discount rate in the following: $10,000 = $1,000,000/(1 + r) 50 (1 + r) 50 = 100 The future value factor is thus 100. You can verify that the implicit rate is about 9.65 percent.
What do we mean by the future value of an investment?
The future value is the value of an investment at a future date on a growth rate. Example $1000 is invested today at a rate of 5% for five years. $1000(1+.05)^5 = 1,276.
Compounding
The process of accumulating interest on an investment over time to earn more interest
discount rate
The rate used to calculate the present value or future cash value
Your company proposes to buy an asset for $335. This investment is very safe. You will sell the asset in three years for $400. You know you could invest the $335 elsewhere at 10 percent with very little risk. What do you think of the proposed investment?
This is not a good investment. Why not? Because you can invest the $335 elsewhere at 10 percent. If you do, after three years it will grow to: $335 × (1 + r) t = $335 × 1.13 = $335 × 1.331 = $445.89 Because the proposed investment only pays out $400, it is not as good as other alternatives we have. Another way of saying the same thing is to notice that the present value of $400 in three years at 10 percent is: $400 × [1/(1 + r) t] = $400/1.13 = $400/1.331 = $300.53 This tells us that we only have to invest about $300 to get $400 in three years, not $335.
4.3 Concept Questions
What is the basic present value equation? What is the Rule of 72?
Future Value (FV)
the amount an investment is worth after one or more periods. Also Compound Value
Present Value (PV)
the current value of future cash flows discounted at the appropriate discount rate
Compound Interest Formula
A = P(1 + r/n)^(n x t), P= principal amount invested r = Annual interest rate, n = Number of times compounded per year t = Number of years A= Amount of aster time
discount
A calculation of present value of some future amount
You would like to buy a new automobile. You have $50,000, but the car costs $68,500. If you can earn 9 percent, how much do you have to invest today to buy the car in two years? Do you have enough?
Assume the price will stay the same. What we need to know is the present value of $68,500 to be paid in two years, assuming a 9 percent rate. Based on our discussion, this is: PV = $68,500/1.092 = $68,500/1.1881 = $57,655.08 You're still about $7,655 short, even if you're willing to wait two years.
You've been saving up to buy the Godot Company. The total cost will be $10 million. You currently have about $2.3 million. If you can earn 5 percent on your money, how long will you have to wait?
At 16 percent, how long must you wait? At 5 percent, you'll have to wait a long time. From the basic present value equation: $2.3 = $10/1.05t 1.05t = 4.35 t = 30.12 years At 16 percent, things are a little better. Verify for yourself that it will take about 10 years.
4.1 Suppose you locate a two-year investment that pays 14 percent per year. If you invest $325, how much will you have at the end of the two years? How much of this is simple interest? How much is compound interest?
At the end of the first year, you will have $325 × 1.14 = $370.50. If you reinvest this entire amount, and thereby compound the interest, you will have $370.50 × 1.14 = $422.37 at the end of the second year. The total interest you earn is thus $422.37 − 325 = $97.37. Your $325 original principal earns $325 × .14 = $45.50 in interest each year, for a two-year total of $91 in simple interest. The remaining $97.37 − 91 = $6.37 results from compounding. You can check this by noting that the interest earned in the first year is $45.50. The interest on interest earned in the second year thus amounts to $45.50 × .14 = $6.37, as we calculated.
4.2 You've located an investment that pays 12 percent. That rate sounds good to you, so you invest $400. How much will you have in three years? How much will you have in seven years? At the end of seven years, how much interest have you earned? How much of that interest results from compounding?
Based on our discussion, we can calculate the future value factor for 12 percent and three years as: (1 + r) t = 1.123 = 1.4049 Your $400 thus grows to: $400 × 1.4049 = $561.97 After seven years, you will have: $400 × 1.127 = $400 × 2.2107 = $884.27 Thus, you will more than double your money over seven years. Because you invested $400, the interest in the $884.27 future value is $884.27 − 400 = $484.27. At 12 percent, your $400 investment earns $400 × .12 = $48 in simple interest every year. Over seven years, the simple interest thus totals 7 × $48 = $336. The other $484.27 − 336 = $148.27 is from compounding.
You estimate that you will need about $80,000 to send your child to college in eight years. You have about $35,000 now. If you can earn 20 percent per year, will you make it? At what rate will you just reach your goal?
If you can earn 20 percent, the future value of your $35,000 in eight years will be: FV = $35,000 × 1.208 = $35,000 × 4.2998 = $150,493.59 The minimum rate is the unknown r in the following: FV = $35,000 × (1 + r) 8 = $80,000 (1 + r) 8 = $80,000/35,000 = 2.2857 Therefore, the future value factor is 2.2857. Looking at the row in Table A.1 that corresponds to eight periods, we see that our future value factor is roughly halfway between the ones shown for 10 percent (2.1436) and 12 percent (2.4760), so you will reach your goal if you earn approximately 11 percent. To get the exact answer, we could use a financial calculator or we could solve for r: (1 + r) 8 = $80,000/35,000 = 2.2857 1 + r = 2.2857(1/8) = 2.2857.125 = 1.1089 r = 10.89%
Symbols
PV = Present value, what future cash flows are worth today FVt = Future value, what cash flows are worth in the future r = Interest rate, rate of return, or discount rate per period—typically, but not always, one year t = Number of periods—typically, but not always, the number of years C = Cash amount II. Future value of C invested at r percent per period for t periods FVt = C × (1 + r) t The term (1 + r) t is called the future value factor. III. Present value of C to be received in t periods at r percent per period PV = C/(1 + r) t The term 1/(1 + r) t is called the present value factor. IV. The basic present value equation giving the relationship between present and future value is: PV = FVt /(1 + r) t
What do we mean by the present value of an investment?
Present Value of an investment that will both be worth a specific amount at a future date.
What is the basic present value equation?
Present value = Future value/(1+r)n Future value = the amount that will be received in future R = rate on interest N = number on years/period
4.3 To further illustrate the effect of compounding for long horizons, consider the case of Peter Minuit and the Indians. In 1626, Minuit bought all of Manhattan Island for about $24 in goods and trinkets. This sounds cheap, but the Indians may have gotten the better end of the deal. To see why, suppose the Indians had sold the goods and invested the $24 at 10 percent. How much would it be worth today?
Roughly 392 years have passed since the transaction. At 10 percent, $24 will grow by quite a bit over that time. How much? The future value factor is approximately: (1 + r) t = 1.1392 ≃ 16,824,000,000,000,000 That is, 17 followed by 15 zeroes. The future value is thus on the order of $24 × 16.824 quadrillion, or about $404 quadrillion (give or take a few hundreds of trillions). Well, $404 quadrillion is a lot of money. How much? If you had it, you could buy the United States. All of it. Cash. With money left over to buy Canada, Mexico, and the rest of the world, for that matter.
Suppose you need $800 to buy textbooks next year. You can earn 7 percent on your money. How much do you have to put up today?
We need to know the PV of $800 in one year at 7 percent. Proceeding as earlier: Present value × 1.07 = $800 We can now solve for the present value: Present value = $800 × (1/1.07) = $747.66 Thus, $747.66 is the present value. Again, this means that investing this amount for one year at 7 percent will result in a future value of $800.
4.1 Concept Questions
What do we mean by the future value of an investment? What does it mean to compound interest? How does compound interest differ from simple interest? In general, what is the future value of $1 invested at r per period for t periods?
4.2 Concept Questions
What do we mean by the present value of an investment? The process of discounting a future amount back to the present is the opposite of doing what? What do we mean by discounted cash flow, or DCF, valuation? In general, what is the present value of $1 to be received in t periods, assuming a discount rate of r per period?
Recently, some businesses have been saying things like "Come try our product. If you do, we'll give you $100 just for coming by!" If you read the fine print, what you find out is that they will give you a savings certificate that will pay you $100 in 25 years or so. If the going interest rate on such certificates is 10 percent per year, how much are they really giving you today?
What you're actually getting is the present value of $100 to be paid in 25 years. If the discount rate is 10 percent per year, then the discount factor is: 1/1.125 = 1/10.8347 = .0923 This tells you that a dollar in 25 years is worth a little more than nine cents today, assuming a 10 percent discount rate. Given this, the promotion is actually paying you about .0923 × $100 = $9.23. Maybe this is enough to draw customers, but it's not $100.
What is the Rule of 72?
a very close estimate for seeing how long it takes for an investment to double. You just divide 72 by the interest rate. If the interest rate is 8% you divide 72/8=9
