Business F723

Fixed Income Analysis

Week 2

Measuring yields and returns

2

Current Yield

• The simplest yield measure– Divide annual coupon payment by price

• Ignores capital gain/loss

• Time value of money also ignored– i.e. compounding is not considered

Why use it?

3

Yield to Maturity

• Similar to IRR for capital budgeting

• The discount rate that sets the present value of future cash flows equal to the price

• Cannot be used to calculate future value of an investment due to reinvestment risk

• Usually stated on a bond equivalent basis to allow comparison to coupon rates

4

Calculating YTM

• Easy for pure discount bonds– face = P x (1 + YTM/2)t; where t is in 1/2 years– YTM = 2 x [(face/Price)(1/t)-1]

• For Coupon paying bonds the calculation is more complex

n

ttYTM

CFP

1 1

5

How to Calculate YTM

• Typical methods include– Financial calculator – Trial and error– Spreadsheet tools; goal seek or solver– Spreadsheet function IRR

6

Goal SeekRate per 1/2 year = 4.97%

Period CF PV1 40 38.106182 40 36.302023 40 34.583274 40 32.945915 40 31.386066 40 29.900077 40 28.484448 40 27.135829 40 25.85106

10 1040 640.3053

Price = 925.0001YTM = 9.940%

7

SolverRate per 1/2 year = 4.97%

Period CF PV1 40 38.106182 40 36.302013 40 34.583274 40 32.945915 40 31.386066 40 29.900077 40 28.484438 40 27.135829 40 25.85106

10 1040 640.3052

Price = 925YTM = 9.940%

8

IRR Function

Period CFPrice 925.00-$

1 40.00$ 2 40.00$ 3 40.00$ 4 40.00$ 5 40.00$ 6 40.00$ 7 40.00$ 8 40.00$ 9 40.00$

10 1,040.00$ IRR= 4.97%

YTM= 9.94%

9

Approximate YTM

• YTM approximation formula– Average cash flow per period

divided by average value of investment

• Gives a reasonable starting point for trial and error

2FacePrice

nPriceFace

CouponYTM

10

Yield to Call

• Sometimes a YTM may not make any sense because the recent trades in that bond have been priced to reflect a probable call.

• In those cases, it helps to solve for what the discount rate is that sets the price equal to the present value of the cash flows assuming that the bond will be called.

• We call this the yield to call.

11

Yield to Call Example

• Consider the following bond;• $1,000 face value

• 16% coupon rate

• 5.5 years to maturity

• $1,192.31

• callable 5 years before maturity at 16% premium

• YTM = 11.22%

Period CFPrice -1192.31

1 80.00$ 2 80.00$ 3 80.00$ 4 80.00$ 5 80.00$ 6 80.00$ 7 80.00$ 8 80.00$ 9 80.00$

10 80.00$ 11 1,080.00$

IRR= 5.61%YTM= 11.22%

12

Yield to Call Example

• Consider the following bond;• $1,000 face value

• 16% coupon rate

• 5.5 years to maturity

• $1,192.31

• callable 5 years before maturity at 16% premium

• $1,000 + 160 + 80 = $1,192.31(1 + YTC/2)

• YTC = 8%

13

More Yield to Call

• Most callable issues have a Call Schedule which lays out multiple call prices depending on when the issue is called

• p. 146 shows Anheuser-Bucsh call schedule for a 30 year bond; 10% premium if called in the first year, decreasing by 0.5% per year, to par in 2008 or later

• Bond was non-refundable

14

Other Yields to Call

• To deal with call schedules, investors can calculate– Yield to first call or next call– Yield to first par call– Yield to refunding

• Often all possible call dates are considered for yield to call analysis

15

Yield to Put

• Only of significant value for a discount bond… since the company cannot force the buyer to exercise the put provision

• Similar to yield to call, except using the put schedule… usually par value or below as opposed to a premium for calls

16

Yield to Put Example

• Consider the following bond– $1,000 face value– 8 years to maturity– 5% coupon rate– Putable at par within

5 years to maturity– Current price $900– Find YTM and YTP

Period CF CFpPrice -900 -900

1 25.00$ 25.00$ 2 25.00$ 25.00$ 3 25.00$ 25.00$ 4 25.00$ 25.00$ 5 25.00$ 25.00$ 6 25.00$ 1,025.00$ 7 25.00$ 8 25.00$ 9 25.00$

10 25.00$ 11 25.00$ 12 25.00$ 13 25.00$ 14 25.00$ 15 25.00$ 16 1,025.00$

IRR= 3.315% 4.435%YTM/P= 6.631% 8.869%

17

Yield to Worst

• Calculate all possible Yield to ________

• The lowest yield is called the yield to worst

• Using optional yields can give a misleading picture… a premium bond that has just paid a coupon and is putable now for 95% of par you have a YTP of -5%, though nobody is likely to use the put option today

18

Cash Flow Yield

• Similar to YTM except for amortizing securities, e.g. mortgage or asset backed

• Instead of having an annuity and face value to find the PV, you have a stream of cash flows that may change over time

• Mortgage borrowers not only have an amortization schedule, but can prepay principal, causing prepayment risk

19

Cash Flow Yield Example

• An asset backed security has forecast cash flows of $100 in six months, decreasing by 5% per period

• The current price is $795

• Find the cash flow yield

Period CFPrice -795

1 100.00$ 2 95.00$ 3 90.25$ 4 85.74$ 5 81.45$ 6 77.38$ 7 73.51$ 8 69.83$ 9 66.34$

10 63.02$ 11 59.87$ 12 56.88$ 13 54.04$ 14 51.33$ 15 48.77$ 16 46.33$

IRR= 5.069%CFY= 10.138%

20

Yield on Portfolio

• Given a portfolio of bonds, each with a different yield to maturity, how do you find the YTM of the portfolio

• Averages and weighted averages don’t work

• Method requires summing all cash flows and then finding the IRR of the portfolioBond CR maturity par,000 Price YTM WAA 7.0% 5 10,000$ 9,209$ 9.0% 0.0145B 10.5% 7 20,000$ 20,000$ 10.5% 0.0367C 6.0% 3 30,000$ 28,050$ 8.5% 0.0416

9.28%

21

Portfolio Yield Example, p. 44Period A B C Portfolio

0 -9,209 -20,000 -28,050 -57,2591 350 1,050 900 2,3002 350 1,050 900 2,3003 350 1,050 900 2,3004 350 1,050 900 2,3005 350 1,050 900 2,3006 350 1,050 30,900 32,3007 350 1,050 1,4008 350 1,050 1,4009 350 1,050 1,400

10 10,350 1,050 11,40011 1,050 1,05012 1,050 1,05013 1,050 1,05014 21,050 21,050

IRR= 4.50% 5.25% 4.25% 4.77%YTM= 9.00% 10.50% 8.50% 9.54%

22

Floating Rate Securities

• Yield spread analysis can be used.

• Calculate cash flows assuming that the reference rate does not change

• Find price based on required spread over the reference rate

• Ignores that the reference rate changes over time and may have cap or floor provisions

23

Sources of Bond Returns

• Bond returns have 3 components– Coupon payments– Capital gain/loss on sale or maturity– Reinvestment income or interest on interest

• Final component can be very important if investing for a significant time period

24

FVcoupons 1,806.79$

nC 1,050.00$ Int. on int. 756.79$ Face 1,000.00$

FVall 2,806.79$

% Int on Int 26.96%

Interest on Interest Example

• Given a par bond;• $1,000 face value

• 7% coupon rate

• 15 years to maturity

– Assume reinvestment at 7%

– find the portion of future value that is interest on interest

25

YTM and Reinvestment Risk

• From the previous example almost 27% of future value was dependent on reinvestment of coupon payments

• Reinvestment risk increases with maturity

• Reinvestment risk increases with coupon rate, no reinvestment risk in coupon rate is set to zero (pure discount bond)

26

Cash Flow Yield and RR

• Reinvestment risk is greater for amortizing securities than for bonds

• Reinvestment is required for both interest and principal

• Prepayments also increase as interest rates decrease (refinancing) further increasing the reinvestment risk of amortizing securities

27

Finding Total Return

• By forecasting reinvestment rates and YTM at the time of sale, we can calculate a total expected return

• Find future value of reinvested coupons and sale value of bond

• Return is [FV/Price](1/t)-1

28

Total Return Example

• Using previous bond• $1,000 face value

• 7% coupon rate

• 15 years to maturity

– 5 year horizon– reinvestment at 8%– YTM; 7.5% in 5 years

• Find the total return

FVcoupons 420.21$

Price, t=10 $965.26

FVall 1,385.47$

rsemi-annual 3.314%

YTM 6.628%

29

Total Return Notes

• The total return analysis can be done with rates that change over time

• Accuracy of total return analysis is based on accuracy of forecasts used

• Total return is quite sensitive to planning horizon, so it is also known as horizon return

30

Yields Used

• Current yield– ignores capital gain/loss

• YTM, YTC, YTP, yield to worst, etc.– includes capital gain/loss– assumes reinvestment at calculated yield

• Total return or horizon return– includes capital gain/loss and reinvestment