Question

1. A U.S. government bond with 5 years to maturity pays a coupon of 3.5%. The yield to maturity is 2.8%. Use Excel or Python to produce a graph of the bond price as a function of different yields to maturity.

2. Carla earns $100,000 per year now, and pays $20,000 per year on her fixed rate mortgage. Her income is subject to a COLA clause. If the risk-free rate of interest is 3%, and the expected inflation rate is 2% per year, what is the spending power of her net income in 10 years, expressed in today’s dollars?

3. How would you find the present value of 10 years of Carla’s income without being given an inflation rate or interest rate? HINT: Use market data to determine your answer.

4. A corporate bond is rated BBB by S&P. It has a 10-year maturity, a 5% quarterly coupon, the YTM is 2.5%, the riskfree rate is 1.8% and the LGD is 40%. What is the implied risk premium demanded by bond owners?

Answer 1

1. Assuming semi-annual coupon payment

Par Value ($)	1000
Coupon rate	3.50%
Time to maturity (years)	5
Payment frequency	2
YTM	2.80%
Bond price ($)	1032

YTM	Bond price ($)
0%	1175
5.0%	934
10.0%	749
15.0%	605
20.0%	493
25.0%	405
30.0%	335
35.0%	279
40.0%	235
45.0%	199
50.0%	170
55.0%	146
60.0%	127
65.0%	111

2. (a) Since Carla's income is COLA adjusted, hence Carla's real income per year = $100,000
Out of this, she has to pay her mortgage of $20,000.

Carla's real net income = $80,000

Her net income can earn a real interest rate = Nominal risk-free rate - Inflation rate = 3% - 2% = 1%

Hence, discount rate (r) = 1%

Annuity payment (PMT) = $80,000

Time period (n) = 10 years

PV (Annuity) =

Hence, the spending power of her net income in 10 years, expressed in today’s dollars = PV(real net income in 10 years) = $757,704.36

2. (b) In the absence of nominal interest rate and inflation rate, we can use Consumer Price Index (CPI) data to convert a future cash flow to its present value.

(Salary)t+t1 = (Salary)t * [(CPI)t+t1 / (CPI)t]

3.

Default free rate = Risk free rate = 1.8%

Expected YTM = 2.5%

Or, Risk premium = Expected YTM - Risk free rate = 0.7%

(b) For a BBB rated bond with 10 year horizon, probability of default (Pd) = 3.23%

Loss given default (LGD) = 40%

Default premium = Expected Loss Rate = Pd*LGD = 3.23%*40% = 1.29%

(c) Implied risk premium = Risk premium + Default premium = 0.7% + 1.29% = 1.99%