Introduction to valuation: The time value of money CH 5

Introduction One of the basic problems faced by the financial manager is how to determine the value today of cash flow expected in the future. Time value of money: is refer to the fact that a dollar in hand today is worth more than a dollar promised at some time in the future. The reason for this is that you could earn interest while you waited; so a dollar today would grow to more than a dollar later.

5.1 future value and compounding future value: refer to the amount of money an investment will grow to over some period of time (one or more period) at some given interest rate.

Investing for a single period Suppose you invest $100 in a saving account that pay 10% interest per year, how much will you have in one year? You will have $110, this $110 is equal to yours original principal plus $10 in interest that you earn 100+(100*10%)=$110. this is the future value of $100. In general if you invest for one period at an interest rate of , your investment will grow to (1+r) per dollar invested. in our example: r is 10%, so yours investment grows up (1+0.10)=1.1 dollar per dollar invested. You invest $100 so you ended up with $100*(1.1)=$110.

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*0.10= $11in interest during the second years. So you will have a total of $110+11=$121. This $121 is the future value of $100 in two years. You will end up with $1.10 for every dollar invested, or $110*1.1=$121

Investing for more than one period This process of leaving your money and any accumulated interest in an investment for more than one period, thereby reinvesting the interest is called compounding. Compounding the interest means earning interest on interest. So we called the result compound interest. With simple interest, the interest is not reinvested, so interest is earned each period only on the original principal.

Investing for more than one period Future value = $1 * (1+r)t The growth of your $100 for 5 years Or $100*(1+0.10)5= $161.05 year Beginning amount Simple interest Compound interest Total interest earned Ending amount 1 $100 $10 $ 0 $110 2 $1 $11 $121 3 $2.1 $12.1 $133.1 4 $3.31 $13.31 $146.41 5 4.64 $14.64 $161.05 total $50 $11.05 61.05

Future value, simple interest, and compound interest

Future value of $1 for different period and rates

example The TICO corporation currently pays a cash dividends of $5 per share. You believe the dividends will be increased by 4% each year indefinitely. How big will the dividends be in eight years? Future vale = $5 * 1.048 = = $5 * 1.3686= $6.84 The dividends will grow by $1.84 over the period.

Present value and discounted Suppose you need to have $10,000 in 10 years, and you can earn 6.5% on your money. How much do you have to invest today to reach your goal?

The single period Present value: is the current value of future cash flow discounted at the appropriate discount rate. Present value * (1+r) t = future value Present value = future value/ (1+r) t Instead of compounding the money forward into the future, we discount it back to the present. Discount: calculate the present value of some future amount

example Suppose you need $400 to buy textbook next year. You can earn 7% on your money. Hoe much do you have to put up today? Present value = $400 * (1/1.07) = $373.83

Present value for multiple period Suppose you need to have $1,000 in two year. If you can earn 7%, how much do you have to invest to make sure you have the $1,000 when you need it? In other words, what is the percentage value of $1,000 in two year if the relevant rate 7%? $1,000 = PV * 1.07 * 1.07 $1,000 = PV * 1.072 $1,000 = PV * 1.1449 PV = $1,000/ 1.1449 = $873.44

Calculating present value is quite similar to calculating future value. PV = $1 * [ 1/ (1+r) t] The quantity [1/(1+r) t], goes by several different names. Because it’s used to discount a future cash flow, it is often called a discount factor. This quantity also called present value interest factor or just present value factor. The rate used in the calculation is often called discount rate. Calculating the present value of a future cash flow determine its worth today is commonly called discounted cash flow (DCF) valuation.

Present value of $1 for different periods and rates

Present Value – Important Relationship For a given interest rate – the longer the time period, the lower the present value For a given time period – the higher the interest rate, the smaller the present value

5.3 more about present and future value There is a simple relationship between present and future cash flow.

Present versus future value What we called the present value factor is just the reciprocal of (that is, 1 dividends by) the future value factors Future value factor = (1+r) t Present value factor = 1/(1+r) t It is easy to calculate future value first. PV * (1+r) t = FV PV = FV / (1+r) t = FV * [1/(1+r) t] basic present value equation PV: present value FV: future value T: period of time.

example Your company proposes to buy an assets for $335. this investment is very safe. You would sell off the assets in three years for $400. you know you could invest the $335 elsewhere a1 10% with very little risk. What do you think of the proposed investment? if you invest this $335 in elsewhere at 10%, you will get FV = $335 * (1+0.10)3 FV= $445.89 $445 in elsewhere is higher than $400 in your company proposed. So your company proposed is not a good investment.

Determine the discount rate We need to determine what discount rate is implicit in an investment. We can do this by looking to the basic present value equation: PV = FV / (1+r) t FV = PV(1 + r)t r = (FV / PV)1/t – 1

You are considering a one-year 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? You are getting $100 in addition to your $1,250. this implicit rate on thus investment is $100/1,250 = 8% PV = FV / (1+r)1 1+r = $1,350 / 1,250 = 1.08 r = 0.08 r = 8%

We need to calculate r. there are three ways we could use: Use a financial calculators. Solve the equation for 1+r. Use a future value table.

Finding a number of period Suppose we are interested in purchase an assets that cost $50,000. we currently have $25.000. if you can earn 12% on this $25,000, how long until we have the $50,000? PV = FV / (1+r) t $25,000 = $50,000 / (1+0.12)t $50,000/$25,000= 1.12t=2 t= 2

You’ve been saving up to buy the Godot company 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% on your money, how long will you have to wait? At 16% how long you have to wait? $2.3 million = $10 million / (1+0.05)t 1.05t = 4.35 t= 30 years Or use this formula to find t= ln (fv/pv) / ln 1+r At 5% you will have to wait a long time. At 16% things are little better.

Review Questions Luis is going to receive $20,000 six years from now. Soo Lee is going to receive $20,000 nine years from now. Which one of the following statements is correct if both Luis and Soo Lee apply a 7 percent discount rate to these amounts?  A. The present values of Luis and Soo Lee's monies are equal. B. In future dollars, Soo Lee's money is worth more than Luis' money. C. In today's dollars, Luis' money is worth more than Soo Lee's. D. Twenty years from now, the value of Luis' money will be equal to the value of Soo Lee's money. E. Soo Lee's money is worth more than Luis' money given the 7 percent discount rate. Gerold invested $6,200 in an account that pays 5 percent simple interest. How much money will he have at the end of ten years?  A. $8,710 B. $9,000 C. $9,300 D. $9,678 E. $10,099

Review Questions You invested $1,650 in an account that pays 5 percent simple interest. How much more could you have earned over a 20-year period if the interest had compounded annually?  A. $849.22 B. $930.11 C. $982.19 D. $1,021.15 E. $1,077.94 Today, you earn a salary of $36,000. What will be your annual salary twelve years from now if you earn annual raises of 3.6 percent?  A. $55,032.54 B. $57,414.06 C. $58,235.24 D. $59,122.08 E. $59,360.45

Review Questions . One year ago, you invested $1,800. Today it is worth $1,924.62. What rate of interest did you earn?  A. 6.59 percent B. 6.67 percent C. 6.88 percent D. 6.92 percent E. 7.01 percent . Some time ago, Julie purchased eleven acres of land costing $36,900. Today, that land is valued at $214,800. How long has she owned this land if the price of the land has been increasing at 10.5 percent per year?  A. 13.33 years B. 16.98 years C. 17.64 years D. 19.29 years E. 21.08 years