Bond Markets,Analysis and Strategies (2)
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n 1牋 ? / 1 r PV A r 8 1牋 ? / 1 0.09 PV $100 0.09
PV = $100 [5.534811] = $553.48
Review of Time Value (continued)
1 r n 1 Pn A r 1 0.0815 1 Pn $2,000,000 0.08
Pn = $2,000,000 [27.152125] = $54,304.250
Review of Time Value (continued)
Present Value
The present value is the future value process in reverse. We have:
1 Pn n 1 r
r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years
The present value of an annuity due (PV) is:
n 1牋 ? / 1 r PV A r
A = the amount of the annuity (in dollars) r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years
Learning Objectives
(continued)
After reading this chapter, you will understand
that the relationship between price and yield of an optionfree bond is convex the relationship between coupon rate, required yield, and price how the price of a bond changes as it approaches maturity the reasons why the price of a bond changes the complications of pricing bonds the pricing of floating-rate and inverse-floating-rate securities what accrued interest is and how bond prices are quoted
Review of Time Value (continued)
Present Value of an Ordinary Annuity
When the same amount of money is received (or paid) each period, it is referred to as an annuity. When the first payment is received one period from now, the annuity is called an ordinary annuity. When the first payment is immediate, the annuity is called an annuity due.
Review of Time Value
Future Value
The future value (Pn) of any sum of money invested today is: Pn = P0(1+r)n n = number of periods Pn = future value n periods from now (in dollars) P0 = original principal (in dollars) r = interest rate per period (in decimal form) (1+r)n represents the future value of $1 invested today for n periods at a compounding rate of r
Review of Time Value (continued)
Present Value of a Series of Future Values
To determine the present value of a series of future values, the present value of each future value must first be computed. Then these present values are added together to obtain the present value of the entire series of future values.
1 r n 1 Pn A r
Review of Time Value (continued)
Example of Future Value of an Ordinary Annuity Using Annual Interest:
If A = $2,000,000, r = 0.08, and n = 15, then Pn = ?
For a given future value at a specified time in the future, the higher the interest rate (or discount rate), the lower the present value. For a given interest rate, the further into the future that the future value will be received, then the lower its present value.
Review of Time Value (continued)
Future Value of an Ordinary Annuity
When the same amount of money is invested periodically, it is referred to as an annuity. When the first investment occurs one period from now, it is referred to as an ordinary annuity. The equation for the future value of an ordinary annuity (Pn) is:
Chapter 2 Pricing of Bonds
Learning Objectives
After reading this chapter, you will understand the time value of money how to calculate the price of a bond that to price a bond it is necessary to estimate the expected cash flows and determine the appropriate yield at which to discount the expected cash flows why the price of a bond changes in the direction opposite to the change in required yield
1 r n 1 Pn A r 1 0.04 30 1 Pn $1,000, 000 0.04
Pn = $1,000,000 [56.085] = $56,085,000
Review of Time Value (continued)
Present Value When Payments Occur More Than Once Per Year
If the future value to be received occurs more than once per year, then the present value formula is modified so that
Example of Future Value of an Ordinary Annuity Using Semiannual Interest: If A = $2,000,000/2 = $1,000,000, r = 0.08/2 = 0.04, and n = 15(2) = 30, then Pn = ?
Review of Time Value (continued)
Future Value
When interest is paid more than one time per year, both the interest rate and the number of periods used to compute the future value must be adjusted as follows: r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years The higher future value when interest is paid semiannually, as opposed to annually, reflects the greater opportunity for reinvesting the interest paid.
Review of Time Value (continued)
Example of Present Value of an Ordinary Annuity (PV) Using Annual Interest: If A = $100, r = 0.09, and n = 8, then PV = ?
i. the annual interest rate is divided by the frequency per year ii. the number of periods when the future value will be received is adjusted by multiplying the number of years by the frequency per year
A = the amount of the annuity (in dollars). r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years
PV = $100 [5.534811] = $553.48
Review of Time Value (continued)
1 r n 1 Pn A r 1 0.0815 1 Pn $2,000,000 0.08
Pn = $2,000,000 [27.152125] = $54,304.250
Review of Time Value (continued)
Present Value
The present value is the future value process in reverse. We have:
1 Pn n 1 r
r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years
The present value of an annuity due (PV) is:
n 1牋 ? / 1 r PV A r
A = the amount of the annuity (in dollars) r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years
Learning Objectives
(continued)
After reading this chapter, you will understand
that the relationship between price and yield of an optionfree bond is convex the relationship between coupon rate, required yield, and price how the price of a bond changes as it approaches maturity the reasons why the price of a bond changes the complications of pricing bonds the pricing of floating-rate and inverse-floating-rate securities what accrued interest is and how bond prices are quoted
Review of Time Value (continued)
Present Value of an Ordinary Annuity
When the same amount of money is received (or paid) each period, it is referred to as an annuity. When the first payment is received one period from now, the annuity is called an ordinary annuity. When the first payment is immediate, the annuity is called an annuity due.
Review of Time Value
Future Value
The future value (Pn) of any sum of money invested today is: Pn = P0(1+r)n n = number of periods Pn = future value n periods from now (in dollars) P0 = original principal (in dollars) r = interest rate per period (in decimal form) (1+r)n represents the future value of $1 invested today for n periods at a compounding rate of r
Review of Time Value (continued)
Present Value of a Series of Future Values
To determine the present value of a series of future values, the present value of each future value must first be computed. Then these present values are added together to obtain the present value of the entire series of future values.
1 r n 1 Pn A r
Review of Time Value (continued)
Example of Future Value of an Ordinary Annuity Using Annual Interest:
If A = $2,000,000, r = 0.08, and n = 15, then Pn = ?
For a given future value at a specified time in the future, the higher the interest rate (or discount rate), the lower the present value. For a given interest rate, the further into the future that the future value will be received, then the lower its present value.
Review of Time Value (continued)
Future Value of an Ordinary Annuity
When the same amount of money is invested periodically, it is referred to as an annuity. When the first investment occurs one period from now, it is referred to as an ordinary annuity. The equation for the future value of an ordinary annuity (Pn) is:
Chapter 2 Pricing of Bonds
Learning Objectives
After reading this chapter, you will understand the time value of money how to calculate the price of a bond that to price a bond it is necessary to estimate the expected cash flows and determine the appropriate yield at which to discount the expected cash flows why the price of a bond changes in the direction opposite to the change in required yield
1 r n 1 Pn A r 1 0.04 30 1 Pn $1,000, 000 0.04
Pn = $1,000,000 [56.085] = $56,085,000
Review of Time Value (continued)
Present Value When Payments Occur More Than Once Per Year
If the future value to be received occurs more than once per year, then the present value formula is modified so that
Example of Future Value of an Ordinary Annuity Using Semiannual Interest: If A = $2,000,000/2 = $1,000,000, r = 0.08/2 = 0.04, and n = 15(2) = 30, then Pn = ?
Review of Time Value (continued)
Future Value
When interest is paid more than one time per year, both the interest rate and the number of periods used to compute the future value must be adjusted as follows: r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years The higher future value when interest is paid semiannually, as opposed to annually, reflects the greater opportunity for reinvesting the interest paid.
Review of Time Value (continued)
Example of Present Value of an Ordinary Annuity (PV) Using Annual Interest: If A = $100, r = 0.09, and n = 8, then PV = ?
i. the annual interest rate is divided by the frequency per year ii. the number of periods when the future value will be received is adjusted by multiplying the number of years by the frequency per year
A = the amount of the annuity (in dollars). r = annual interest rate ÷ number of times interest paid per year n = number of times interest paid per year times number of years