Introduction to Valuation: The Time Value of Money

Dividend

A payment made by firms to stockholders. A dividend represents part of the investor's return for buying the stock (the other part of the return is any capital gain made when the stock is sold)

Discount

Calculation of the present value of some future amount

Future value question

Deposit £5,000 today in an account paying 12% . how much will you have in 6 years with compound interest? PV x (1 + r)^t = 5,000 x (1 + 0.12)^6 = 5,000 x 1.974 = 9,869 how much will you have in 6 years with simple interest? PV x (1 + r x t) = 5,000 x (1 + 0.12 x 6) = 8,600 how much is the compound interest? 9,869 — 5,000 = 4,869 what is the interest on interest? 4,869 — (8,600 — 5,000) = 1,269

Spreadsheet formulae

FV = FV(rate, nper, pmt, pv) PV = PV(rate, nper, pmt, fv) Discount rate = RATE(nper, pmt, pv, fv) Number of periods = NPER(rate, pmt, pv, fv)

FV with simple interest

FV = PV x (1+r x t)

Finding R for a multiple-period investment

Future value divided by present value Square root the future value by the number of years Subtract the Ans by 1 Multiply answer by 100 and that is the discount rate i.e. (FV/PV)^(1/t) - 1

Present vs future value

Future value factor = (1+r)^t Present value factor = 1/(1+r)^t

Investing for more than one period

Going back to our $100 investment, what will you have after two years, assuming the interest rate doesn't change? If you leave the entire $110 in the bank, you will earn $110 x to = $11 in interest during the second year, so you will have a total of $110 11 = $121. This $121 is the future value of $100 in two years at 10 percent. Another way of looking at it is that one year from now you are effectively investing $110 at 10 percent for a year. This is a single-period problem, so you'll end up with $1.10 for every dollar invested, or $110 x 1.1 = $121 total. This $121 has four parts. The first part is the $100 original principal. The second part is the $10 in interest you earn in the first year, and the third part is another $10 you earn in the second year, for a total of $120. The last $1 you end up with (the fourth part) is interest you earn in the second year on the interest paid in the first year: $10 x .10 = $1. This process of leaving your money and any accumulated interest in an investment for more than one period, page 99 thereby reinvesting the interest, is called compounding. Compounding the interest means earning interest on interest, so we call the result compound interest. With simple interest, the interest is not reinvested, so interest is earned each period only on the original principal.

Dividend growth

Suppose an investor buys 1 share in BT PLC. The company pays a current dvidend of £1.10, which is expected to grow at 40% per year for the next five years. What will the dividend be in fiver years? £1.1 x (1 + 0.40)^5 = £5.916064 = £5.92 (2dp)

Interest on interest

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 x 1.14 = $370.50. If you reinvest this entire amount, and thereby compound the interest, you will have $370.50 x 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 x .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 x .14 = $6.37, as we calculated.

Present value for multiple periods

Suppose you need to have $1,000 in two years. If you can earn 7 percent, how much do you have to invest to make sure that you have the $1,000 when you need it? In other words, what is the present value of $1,000 in two years if the relevant rate is 7 percent? Based on your knowledge of future values, you know that the amount invested must grow to $1,000 over the two years. In other words, it must be the case that: $1,000 = PV x 1.07 x 1.07 = PV x 1.072 = PV x 1.1449 Given this, we can solve for the present value: Present value = $1,00011.1449 = $873.44 Therefore, $873.44 is the amount you must invest in order to achieve your goal.

Investing for a single period

Suppose you were to invest $100 in a savings account that pays 10 percent interest per year. How much would you have in one year? You would have $110. This $110 is equal to your original principal of $100 plus $10 in interest that you earn. We say that $110 is the future value of $100 invested for one year at 10 percent, and we mean that $100 today is worth $110 in one year, given that 10 percent is the interest rate. In general, if you invest for one period at an interest rate of r, your investment will grow to (1 + r) per dollar invested. In our example, r is 10 percent, so your investment grows to 1 + .10 = 1.10 dollars per dollar invested. You invested $100 in this case, so you ended up with $100 x 1.10 = $110.

Future value (FV)

The amount of money an investment will grow to over some period of time at some given interest rate

Rule of 72

The number of years it takes for a certain amount to double in value is equal to 72 divided by its annual rate of interest.

The Single-Period Case

We've seen that the future value of $1 invested for one year at 10 percent is $1.10. We now ask a slightly different question: How much do we have to invest today at 10 percent to get $1 in one year? In other words, we know the future value here is $1, but what is the present value (PV)? The answer isn't too hard to figure out. Whatever we invest today will be 1.1 times bigger at the end of the year. Because we need $1 at the end of the year: Or solving for the present value: Present value x 1.1 = $1 Present value = $111.1 = $.909 In this case, the present value is the answer to the following question: What amount, invested today, will grow to $1 in one year if the interest rate is 10 percent? Present value is thus the reverse of future value. Instead of compounding the money forward into the future, we discount it back to the present.

Finding R for a single-period investment

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)tl+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.

Discounted cash flow valuation

calculating the present value of a future cash flow to determine its value today the process of valuing an investment by discounting its future cash flows

How to find the number of periods

ln(fv/pv) / ln(1+r)

Discount rate

the rate used to calculate the present value of future cash flows