Time Value of Money & BONDS

Interest is The cost associated with the use of money Is an important cost to debtor Is an Important revenue to the creditor Since “Time is Money” : Important to any business decision

Present Value: Amount that must be invested today at a given rate to produce a given future value

Present Value is based on 3 Variables: Future Amount $ amount to be received Number of periods = N Length of time until the amount is received Interest rate = i Market / yield / discount rate of interest

Present Value can be calculated Manually (using mathematical formulas) Factor Tables (included in textbook) Using Financial calculator Using Excel (PV function)

PV Cash Flow Principles: Amounts to be paid/received LUMP SUM : a single sum at a point in time Receipt of face amount of the note ORDINARY ANNUITY: Equal end of the period payments at equal intervals of time Periodic interest payments made at the end of the period ANNUITY DUE: Beginning of the period payments at equal intervals of time Periodic interest payments made at the beginning of the period

Time Value of $ Calculations Using Tables in KIMMEL Textbook APPENDIX C Present Value of a Lump Sum = Present Value of $1: TABLE 3 (page c8) PV of a Series of payments made at the end of the period = PV of an Ordinary Annuity of $1: Table 4 (page c10)

PV Example #1 Assume you win the lottery. Your choice: $30,000 per year for the next 15 years (total of $450,000) OR $1,000,000 in 15 years. Current interest rate is 9% Which is a better deal? Compare using Present Value…

Choice B is worth more in TODAY’S DOLLARS Choice A: PV of an Ordinary Annuity $30,000 * 8.06069 (n=15, I = 9%) $241,821 Choice B: PV of a Lump Sum $274,540 $1,000,000 * .27454 (n=15, I = 9%) Choice B is worth more in TODAY’S DOLLARS

PV Example #2 Assume your rich Aunt Edith leaves you some inheritance. You have the option of receiving: $24,000 over the next 12 years ($2,000/yr) OR You can have all her government bonds which mature in 10 years at a value of $30,000. Assume you invest the $ you accumulate in a 10% Money Market account. Which option is more attractive? (In other words, which is worth more in today’s dollars? ) Compare using Present Value…

Choice A is worth more in TODAY’S DOLLARS Choice A: PV of an Ordinary Annuity $2,000 * 6.81369 (n=12, i = 10%) $13,627 Choice B: PV of a Lump Sum $11,566 $30,000 * .38554 (n=10, I = 10%) Choice A is worth more in TODAY’S DOLLARS

Using Present Value to Value a Bond Present value is relevant to the study of bonds because the value of a bond is based on the present value of two components of future cash flow: A series of fixed interest payments. (ordinary annuity) A single payment at maturity. (lump sum payment) The amount of interest a bond pays is fixed over its life.

PV & BONDS BONDS are recorded at the PV of the Cash expected to be collected May involve TWO types of future cash flows: Payment of the face value of the note at maturity (a LUMP SUM payment) & Periodic Interest Payments (an ordinary annuity @ the stated interest rate)

This interest rate is called the Effective Interest Rate OR The value of the Bond at any point in time should be = to the PV of those cash flows computed at an interest Rate = the rate of return desired by the lender. This interest rate is called the Effective Interest Rate OR Discount Rate of Interest OR Yield Rate of Interest OR Market Rate of Interest

IF the Stated Rate of Interest = Market rate Bond will sell (be recorded) at FACE value Investors willing to pay MORE: Will earn MORE! IF the Stated Rate of Interest > Market rate Bond will sell (be recorded) at a Premium Investors willing to pay LESS: Will earn LESS! IF the Stated Rate of Interest < Market rate Bond will sell (be recorded) at a Discount

Selling Price of Bond Stated Rate(12%) = Market Rate (10%) Issued at a PREMIUM Face amount of the Bond: $10,000,000 Stated Interest Rate: 12% Semi-annual Payments: $10,000,000 * 12% / 2 = $600,000 10 year Bond Present Value at 5%(Market rate of 10% / 2=5%) : Lump sum : $10,000,000 * .37689 (n=20, i=5%) = $3,768,900 Interest Payments: $600,000 * 12.46221(n=20, i=5%) = $7,477,326 PV of Bond (Record @ Face Value ) $11,246,226 1/1/04 Cash 11,246,226 Bonds Payable 10,000,000 Premium on B/P 1,246,226

BOND PRICING STEP 1 – Compute the PV of the principal to be repaid use MARKET rate PRINCIPAL x PV of $1 (n=number of periods, i=market rate) STEP 2 – Compute the PV of interest to be paid (use BOTH rates) PRINCIPAL x STATED Interest = Interest Payment Interest Payment x PV of an ANNUITY (n=number of periods, i =market rate) STEP 3 – ADD PV of Principal plus PV of Interest Payments TOTAL = SELLING PRICE OF THE BONDS

Selling Price of Bond Stated Rate(12%) = Market Rate (14%) Issued at a DISCOUNT Face amount of the Bond: $10,000,000 Stated Interest Rate: 12% Semi-annual Payments: $10,000,000 * 12% / 2 = $600,000 10 year Bond Present Value at 7%(Market rate of 14% / 2=7%) : Lump sum : $10,000,000 * .258 (n=20, i=7%) = $2,580,000 Interest Payments: $600,000 * 10.594(n=20, i=7%) = $ 6,356,400 PV of Bond (Record @ Face Value ) $8,936,400 1/1/04 Cash 8,936,400 Discount on B/P 1,063,600 Bonds Payable 10,000,000

Selling Price of Bond Stated Rate(10%) = Market Rate (10%) Face amount of the Bond: $10,000,000 Stated Interest Rate: 10% Annual Interest Payments: $10,000,000 * 10% = $1,000,000 10 year Bond Present Value at 10%(Market rate) : Lump sum : $10,000,000 * .38554 (n=10, i=10%) = $3,855,400 Interest Payments: $1,000,000* 6.14457(n=10, i=10%) = $6,144,570 ROUND $9.999,970 PV of Bond (Record @ Face Value ) $10,000,000 1/1/04 Cash 10,000,000 Bonds Payable 10,000,000 12/31/04 Bond Interest expense 1,000,000 Cash 1,000,000

Selling Price of Bond Question 4A: No stated interest Market Rate (10%) Face amount of the Bond: $10,000,000 Present Value at Market rate of 10% : Lump sum : $10,000,000 *.38554 (n=10, i=10%) = $3,855,400 Investors would be willing to PAY: $3,855,400 for a $10,000,000 Bond with no interest payments