Note 4. Valuing Level Cash Flows

Size: px

Start display at page:

Download "Note 4. Valuing Level Cash Flows"

Vivien Whitehead
5 years ago
Views:

1 Note 4. Valuing Level Cash Flows 1 Key Concepts The present/future value of multiple cash flows Valuing Level Cash Flows: Annuities Perpetuities 2 1

2 I. PV of Multiple Future Cash Flows Suppose that your investment produces returns of $100 in the first year, $400 in the second year, $600 in the third year, and $900 in the fourth year. What is the fair price of the investment? All the return payments are made at the end of each year. The appropriate interest rate is 12%. - Find the PV of each cash flows and add them $100/(1.12) + $400/(1.12) 2 + $600/(1.12) 3 + $900/(1.12) 4 = $1,

3 Future Value of Multiple Future Cash Flows If you put $2,000 today, $3,000 in a year, and $4,000 in two years in a savings account that pays 10% a year, how much will you get after three years from now? 5 Future Value of Multiple Future Cash Flows ,000 4,000 4,000(1.1) 2,000 3,000(1.1)(1.) 2,000(1.1)(1.1)(1.1) FV = 2,000(1.1) 3 + 3,000(1.1) 2 + 4,000(1.1) 1 = 10,

4 II. Valuing Level Cash Flows How do we find out the value of periodic future cash flows of an equal amount? Perpetuity Annuity Annuity Due 7 RALEIGH, N.C. A Triangle couple is $1 million richer after they found a diamond while playing a North Carolina lottery scratch-off. Gerald and Phyllis Callahan beat the one in 1.1 million odds on Thursday when their Diamond Dazzler proved to be a winning ticket, according to the North Carolina Education Lottery. The Callahans bought their instant lottery ticket for $20 from the Handy Mart on Wilson s Mills Road in Smithfield, the agency said in a written statement. The couple was offered a choice between a $1 million annuity that provides 20 payments of $50,000 a year or a lump sum of $600,

5 Annuity An annuity is an equally spaced stream of level cash flows to be paid for a fixed period of time. Year Final Payment 9 Future Value of an Annuity Suppose you are to receive $2,000 every year for the next five years. The appropriate interest rate is 10%. What is the value of the sum of these cash flows five years from now? 10 5

6 Future Value of an Annuity 11 Future Value of Annuity (Formula) [ Variable C is the periodic cash flow from the annuity] Formula: Future Value Annuity Factor: 12 6

7 Example: Future Value of an Annuity Mr. Jon Snow is planning to save money to purchase a sedan five years from now. The model he is interested in buying is expected to be sold around $50,000 in five years. In order to have enough money to buy the car, he will deposit $8,000 a year for the next five years in a bank savings account that yields 7% a year. One concern, though, is that the savings might not be enough for the car. If that happens, he will pay the remaining balance using a bank loan with the car as a collateral. How much will he need to borrow five years from now? 13 Present Value of an Annuity Suppose you are to receive $1,000 every year for the next five years. The appropriate interest rate is 6%. What is the present value of these payments? 14 7

8 Present Value of Level Cash Flows 15 Future Value of Annuity (Formula) Formula: Present Value Annuity Factor: 16 8

9 Annuity PV Example Q. Suppose you need $20,000 each year for the next three years to make your tuition payments. Assume you need the first $20,000 in exactly one year from now. Suppose you can place your money in a savings account yielding 8% compounded annually. How much do you need to have in the account today? (Note: Ignore taxes, and keep in mind that you don t want any funds to be left in the account after the third withdrawal, nor do you want to run short of money.) 17 Annuity Present Value - Solution PV = 18 9

10 Example: Saving For Retirement You are offered the opportunity to put some money away for retirement. You will receive five equal annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%? 19 Saving For Retirement Timeline Notice that the cash flows from years 1 39 are 0 (i.e., CF 1 to CF 39 = 0) The cash flows years are $25,000 (i.e., CF 40 to CF 44 = 25,000) K 25K 25K 25K 25K 20 10

11 Saving for retirement (solution) PV = 21 PV of Annuity: Question Your friend, who borrowed $20,000 from you at 10% interest four years ago and disappeared, suddenly came back to town yesterday. He gave you an apology and promised that he would work hard to pay off the debt in ten equal annual payments. Now, because you don t trust your friend any more, you think the appropriate rate for the payments should be higher this time at 20%. How much does your friend have to pay you each year? 22 11

12 Solution K C C C.. C C 10% 20% 23 Solution 24 12

13 Application: Annuity Due Suppose that you just won a lottery worth $18 million. The lottery agency will pay you the winnings in 20 annual installments of $900,000 each with the first payment made immediately (this means you will get nothing at the end of year 20). If the interest rate is 10%, what is the value of this lottery today?

Spectacular scratch-off ticket at a local convenience store. Considering that the average life expectancy of Korean males is now 77.5 years and if Mr. Kang, now 27, lives until he gets 77.

14 (Application) Big Bonanza for a Korean-American in his 20s.: $10,000 a week for the rest of his life! [Herald Economy] Tuesday, Sept. 5, 2006 Lottery pays weekly money prizes until death A Korean-American man named Kang, Dae-Sung from Queens, New York, won the jackpot prize of $10,000 a week for life with a $20 Win for Life Spectacular scratch-off ticket at a local convenience store. Considering that the average life expectancy of Korean males is now 77.5 years and if Mr. Kang, now 27, lives until he gets 77.5 years old, the young man will collect a colossal sum of money, $26 million (KRW billion), in total. The lottery ticket was purchased by Mr. Kang s father at a local convenience store for a birthday gift. The lottery prize was the largest ever in the history of the New York State lotteries. The chance of winning the first price is estimated to be one in 3,258, Is he really that rich? Nominal Value $10, weeks/year 50.5 years = $26,260,000 Real Value (PV; Annuity) Assume an interest rate of 7% per year PV = $10,000 [1-1/( /52) 52*50.5 ]/(0.07/52) = $7,211,

RALEIGH, N.C. A Triangle couple is $1 million richer after they found a diamond while playing a North Carolina lottery scratch-off. Gerald and Phyllis Callahan beat the one in 1.

15 RALEIGH, N.C. A Triangle couple is $1 million richer after they found a diamond while playing a North Carolina lottery scratch-off. Gerald and Phyllis Callahan beat the one in 1.1 million odds on Thursday when their Diamond Dazzler proved to be a winning ticket, according to the North Carolina Education Lottery. The Callahans bought their instant lottery ticket for $20 from the Handy Mart on Wilson s Mills Road in Smithfield, the agency said in a written statement. The couple was offered a choice between a $1 million annuity that provides 20 payments of $50,000 a year or a lump sum of $600, An interest of 2.7% a year assumed: PV $50, $764,932 What is the break-even rate that makes the values of the two options equal? r 600,000 $50,000 r r 5.45% 30 15

16 Question You just won a lottery with the unlimited prize money! According to the game rules, you can either choose a perpetuity of $1,000 a year forever or a cash prize of $10,000. If the appropriate discount rate is 10%, which will be your choice? 31 Perpetuity An endless stream of equally spaced, level cash flows No maturity Equal amount payment 32 16

17 Perpetuity Valuation 33 Perpetuity: Examples U.K. Consols (war bond, issued in 1752, 3% coupons) Doosan Infracore issued perpetual bonds of the size of $500 million in Singapore in October * Coupon Rate: 5 yr. T-Note yield % Industrial Bank of Korea (Sept. 2016) issued perpetual contingent convertible bonds (300 billion won or US$271 million) with a call option of advanced redemption after 5 and 10 years

18 HSBC issues 1 billion perpetual securities to meet EU capital rules Sept. 21, 2018 HSBC plans to issue 1 billion (US$1.33 billion) of perpetual debt, with an interest rate of per cent, in order to strengthen its capital base to meet European Union regulations. The company intends to use the net proceeds from the sale of the securities for general corporate purposes and to further strengthen the company s capital base pursuant to requirements under CRD IV, HSBC said in a filing to the Hong Kong stock exchange on Friday. CRD IV, introduced by the European Union in 2013, is a legislative package that is intended to strengthen regulation of the EU s banking sector and implement the Basel III agreement a set of global measures drawn up in response to the financial crisis. It contains enhanced requirements for investment banks and credit institutions about the quality and quantity of capital, new liquidity and leverage rules, and macroprudential standards

Simple Interest: Interest earned only on the original principal amount invested.

53 Future Value (FV): The amount an investment is worth after one or more periods. Simple Interest: Interest earned only on the original principal amount invested. Compound Interest: Interest earned on