Long Term Debt Calculations

Please watch this video on long term debt

http://bit.ly/company_expansion

Workings for the calculations in the video;

Loan:

PV = $32,000,000

n = 10 x 2 = 20

i = 0.07 ÷ 2 = 0.035 = (3.5%)

PMT = 32,000,000 ÷ [1 – (1+ 0.035)-20]/0.035 = 2,251,554.46

Bond:

FV = 30,000,000

PMT (Coupon) = 30,000,000 x .09 ÷ 2 = 1,350,000

n = 10 x 2 = 20

i = .07 ÷ 2 = 0.035 = (3.5%)

PV (Price) = 1,350,000 [1-(1+0.035)-20]/0.035 + 30,000,000 (1+0.035)-20 = 34,263,720.99

Information provided in pre-requisite subject – Financial Management Lectures 4 and 5:

Terminology:

To solve financial mathematics problems we need to understand the terms used

PV         = present value, or principal, or price of the bond

i         = interest rate, yield to maturity, discount rate of market interest rate

n         = number of periods

FV          = future value or Face Value

PMT         = periodic payment or coupon payment = face value x coupon rate

Normally you require three variables to find the fourth

Compound Interest:

Calculated on the principal and interest earned in prior periods

Referred to as compounding – compounding can be any designated length of time e.g. annual, semi-annual, monthly, daily etc.

Used in the valuation of long term financial instruments e.g. bonds – term is 12 months or more – usually used in bank deposits and loans.

FV = PV(1 + i)n 

PV = FV/(1 + i)n = FV(1 + i)-n

Where: i = interest per compound period

             n = number of compound periods

Example: What is the future value of $1,000 invested at 12% compounding annually in three years?

FV = 1000(1+0.12)3 = 1404.93

Example: Your uncle has a superannuation policy that promises to pay him $1,250,000 on his 60th birthday. He is currently 35 and the fund earns 12% p.a. compounded monthly, what is the present value of this policy?

PV = 1,250,000(1 + 0.12/12)-25 x 12 = 1,250,000(1 + 0.01)-300 = 63,168.11

Annuities:

Annuities are a special case of multiple cash flows.

An annuity is a number of equal cash flows occurring at equal time intervals

An ordinary annuity assumes all cash flows occur at the end of each period

FV = PMT [(1 + i)n – 1]/i

PV = PMT [1- (1+i)-n]/i

Where: PMT = the annuity payment

                  n = the number of payments

                  i = the per period interest rate

Example:  June has put aside $500 each quarter for the past five years into an account with an interest rate of 8% p.a. compounded quarterly. What amount would June be able to spend if she withdrew her money today?

FV = 500 [(1 + .08/4)4 x 5 – 1]/0.02 = 500 [(1 + 0.02) 20 – 1]/0.02 = 12,148.68

Example: What is the present value of a series of $150 payments received at the end of each quarter for 5 years if the interest rate is 6% p.a. compounded quarterly?

PV = 150 [1- (1+0.06/4)-4 x 5]/.006/4  = 150 [1- (1+0.015)-20]/0.015 = 2,575.30

A perpetuity is an annuity that continues forever so there is no future value.

PV = PMT/i

Example: What amount would have to be invested today to provide $5,000 p.a. forever if you can earn 5% p.a.?

PV = 5000/.05 = 100,000

Bond Valuation:

Calculated by discounting all the future cash flows you will receive from holding the bond back to the present value i.e. the price of the bond.

The future cash flows are the coupon payments you receive (usually six monthly) and the face value at maturity.

PV =  PMT [1- (1+i)-n]/i + FV(1 + i)-n

Where: i = yield per compound period

             n = number of payments or compound periods

       PMT = coupon payment

           FV = Face Value

Example: What is the price of a $100,000 ten year bond that was issued two years ago and has eight years to run before the bonds maturity date? Interest is paid half yearly, the last payment was made yesterday (so you will not receive that payment) and the coupon rate is 12% p.a. payable semi-annually. The current market yield is 14% p.a. compounded semi-annually.

PMT = 100,000 x 0.12 ÷ 2 = 6,000

n = 8 years x 2 = 16

i = 14% ÷ 2 = 7%

PV = 6000 [1- (1+0.07)-16]/0.07 + 100,000FV(1 + 0.07)-16 = 90,553.35

Practice Questions:

1. What is the present value of:

a. $1,000 payable in five years if the interest rate is 7% p.a.?

b. $5,000 payable in two years if the interest rate is 8% p.a. compounding monthly?

c. $2,500 payable in three and a half years if the interest rate is 6% p.a. payable quarterly?

d. $150,000 payable in nine years if the interest rate is 3% per month?

2. What is the future value of:

a. $1,000 invested today in five years if the interest rate is 7% p.a.?

b. $5,000 invested today in two years if the interest rate is 8% p.a. compounding daily?

c. $2,500 invested in three and a half years from now if the interest rate is 6% p.a. payable quarterly?

d. $150,000 payable in nine years if the interest rate is 3% per quarter?

3. What is the future value of:

a. An annuity with payments of $100 per month for 10 years and a yield of 12% p.a. compounding monthly?

b. An annuity with payments of $35 per quarter for 8 years and a yield of 6% p.a. compounding quarterly?

c. An annuity with payments of $1800 per week for 9 and a half years and a yield of 5% p.a. compounding weekly?

d. An annuity with payments of $400 per fortnight for six months and a yield of 6% p.a. compounding fortnightly?

4. What is the present value of:

a. An annuity with payments of $100 per month for 3 years and a yield of 15% p.a. compounding monthly?

b. An annuity with payments of $5 per day for 6 years and a yield of 6% p.a. compounding daily?

c. An annuity with payments of $100 per week for 7 years and a yield of 18% p.a. compounding weekly?

d. An annuity with payments of $2000 per six months  for one and a half years and a yield of 6% p.a. compounding six monthly?

5. What is the payment of:

a. An annuity paid monthly which has a future value of $500 in 4 years and has a yield of 8% p.a. compounding monthly?

b. An annuity paid quarterly which has a future value of $3000 in six years and has a yield of 9% p.a. compounding quarterly?

c. An annuity paid fortnightly which has a future value of $1,500 in three and a half years and has a yield of 10% p.a. compounding fortnightly?

d. An annuity paid annually which has a future value of $80 in two years and has an interest rate of 2% p.a.?

6. What is the payment of:

a. An annuity paid monthly which has a future value of $500 in 4 years and has a yield of 8% p.a. compounding monthly?

b. An annuity paid quarterly which has a future value of $3000 in six years and has a yield of 9% p.a. compounding quarterly?

c. An annuity paid fortnightly which has a future value of $1,500 in three and a half years and has a yield of 10% p.a. compounding fortnightly?

d. An annuity paid annually which has a future value of $80 in two years and has an interest rate of 2% p.a.?

7. What is the price of a bond if:

a. The face value is $100, the coupon rate is 7% p.a. paid semi –annually, has five years until maturity and has a yield of 6% p.a. compounding semi-annually?

b. The face value is $300,000, the coupon rate is 5% p.a. paid semi –annually, has six and a half years until maturity and has a yield of 6% p.a. compounding semi-annually?

c. The face value is $2,000, the coupon rate is 18% p.a. paid semi –annually, has  seven years until maturity and has a yield of 22% p.a. compounding semi-annually?

d. The face value is $100, the coupon rate is 3% p.a. paid annually, has five years until maturity and has a yield of 3% p.a.?

Answers to practice questions: