1. Interest Rates Chapter 4 (part1)
Geng Niu

2. Interest rates
Money has a time value: the opportunity to invest it at some interest rate
An interest rate in a particular situation defines the amount of money a borrower promises to pay the lender.
There are many different types of interest rates.
Depend on the credit risk.

3. Fixed-income markets
Fixed-income securities are financial claims issued by governments, corporations, banks, and other financial intermediaries.
Cash flows promised to the buyers represent contractual obligations of the issues.
If obligations are not med, the buyers will have the right to take control of the firm.
Fixed-income securities (debt securities) are issued, traded, and invested in fixed-income markets (debt markets).
Account for 2/3 of the market value of all the securities.

4. Classification of debt securities
Treasury (Sovereign) securities.
Agency securities.
Corporate securities.
Mortgage-backed securities.
Asset-backed securities.
Municipal issues.
Emerging market securities.

5. Bonds in China

6. Types of Rates Treasury rates
Rates on instruments issued by a government in its own currency: treasury bills, treasury notes, and treasury bonds
LIBOR rates (London Interbank Offered Rate)
  Average of interest rates estimated by each of the leading banks in London that it would  pay to borrow from one another..
Repo

7. Libor
Each day, the ICE ( previously BBA )surveys a panel of banks (18 major global banks for the USD Libor), asking the question, "At what rate could you borrow funds…in a reasonable market size just prior to 11 am?"
The highest 4 and lowest 4 responses are thrown out, and the average of the remaining middle 10 is reported at 11:30 am.
Separate LIBOR rates reported for different maturities (overnight to 12 months) and currencies: USD, EUR, GBP, JPY, CHF
Libor rates serve as benchmarks, or reference rates from which commercial loans are set.

8. The Libor scandal
In 2012, an international investigation revealed a widespread plot by multiple banks—notably Deutsche Bank, Barclays, UBS, Rabobank, and the Royal Bank of Scotland—to manipulate Libor rates for profit starting as far back as 2003.
All told, global banks have paid over $9 billion in fines.
Wulin Suo

9. Repo Rates
Repurchase agreement is an agreement where a financial institution that owns securities agrees to sell them today for X and buy them bank in the future for a slightly higher price, Y
The financial institution obtains a loan.
The rate of interest is calculated from the difference between X and Y and is known as the repo rate

10. Repo Rates
Repurchase agreement (repo): an investor owns securities agrees to sell them to another company now and by them back later at a higher price.
Repo resembles a collateralized loan, but is
not subject to a bankruptcy “automatic stay”.
Overnight repo: most common
Term repos: longer-term arrangement
For the most part, when firms in the United States file for bankruptcy, a temporary hold is placed on the their assets known as automatic stay. The logic behind this bankruptcy provision is to prohibit creditors from collecting payments in a disorderly fashion, maintaining
the firm as a going concern.1 This provision does not exist for derivatives and repos, which are exempt
from automatic stay Infante, S. (2013). Repo collateral fire sales: the effects of exemption from automatic stay

11. China repo market Interbank repo market: Wholesale funding market
all participants are institutional investors
Trading operates on a private, one-to-one, over-the-counter platform
Each market maker can make bid and offer prices

12. China repo market Stock exchange repo market
The exchange acts as the counterparty to all buyers and sellers
All repo prices and volumes are observable
Continuous bid and offer prices are available
All stock exchange account holders can participate
Retail investors are restricted to reverse repo and a limited subset of bonds

14. Impact of Compounding
When we compound m times per year at rate R an amount A grows to A(1+R/m)m in one year
Compounding frequency
Value of $100 in one year at 10%
Annual (m=1)
110.00
Semiannual (m=2)
110.25
Quarterly (m=4)
110.38
Monthly (m=12)
110.47
Weekly (m=52)
110.51
Daily (m=365)
110.52

15. Impact of Compounding
10.25% with annual compounding is equivalent to 10% with semiannual compounding
An amount A is invested for n years at an interest rate of R per annum, terminal value:
A(1+R)n
If compounded m times per annum: A(1+R/m)mn

16. Continuous Compounding
In the limit as we compound more and more frequently we obtain continuously compounded interest rates

17. Continuous Compounding
$100 grows to $100eRT when invested at a continuously compounded rate R for time T
$100 received at time T discounts to $100e-RT at time zero when the continuously compounded discount rate is R

18. Conversion Formulas Define Rc : continuously compounded rate
Rm: same rate with compounding m times per year
Suppose : An amount A is invested for n years:
Terminal value using discrete compounding:
Terminal value using continuously compounding:

19. Conversion Formulas
If terminal values of the two investments are the same, Rc and Rm are equivalent:

20. Examples
10% with semiannual compounding is equivalent to 2ln(1.05)=9.758% with continuous compounding
8% with continuous compounding is equivalent to 4(e0.08/4 -1)=8.08% with quarterly compounding
Rates used in option pricing are nearly always expressed with continuous compounding

21. Zero Rates
A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T
All the interest and principal is realized at the end, no intermediate payments
A 5-year zero rate with continuous compounding is quoted as 5%, means $100, if invested for 5 years, grows to 100*e0.05*5 =128.40

22. Bond Pricing Most bonds pay coupons periodically.
Principal is paid at the end.
Price: present value of all cash flows
Use a different zero rate for each cash flow
Wulin Suo

23. Examples Maturity (years) Zero rate (cont. comp.) 0.5 5.0 1.0 5.8 1.5
6.4
2.0
6.8

24. Bond Pricing
To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate
In our example, the theoretical price of a two-year bond with a principal of $100 providing a 6% coupon semiannually is:
Note :half of the coupon (6%*100*0.5=3) is paid semiannually

25. Bond Yield
The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond
Suppose that the market price of the bond in our example equals its theoretical price of 98.39
The bond yield (continuously compounded) is given by solving
to get y= or 6.76%.

26. Par Yield
The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value.
In our example we solve

27. Par Yield Continued
In general if m is the number of coupon payments per year, d is the present value of $1 received at maturity and A is the present value of an annuity that pays $1 on each coupon date
(in our example, m = 2,
d = 1* e-0.068*2.0 =
A = e-0.05*0.5 +e-0.058*1.0 + e-0.064*1.5 + e-0.068*2.0 =

28. Data to Determine Zero Curve
Bond Principal
Time to Maturity (yrs)
Coupon per year ($)*
Bond price ($)
100
0.25
97.5
0.50
94.9
1.00
90.0
1.50 8
96.0
2.00 12
101.6
* Half the stated coupon is paid each year

29. The Bootstrap Method
An amount 2.5 can be earned on 97.5 during 3 months.
Because 100=97.5e ×0.25 the 3-month rate is % with continuous compounding
Similarly the 6 month and 1 year rates are % and % with continuous compounding

30. The Bootstrap Method continued
To calculate the 1.5 year rate we solve
to get R = or %
Similarly the two-year rate is %

31. Zero Curve Calculated from the Data (Figure 4.1, page 84)
Zero Rate (%)
10.808
10.681
10.469
10.536
10.127
Maturity (yrs)

32. Forward Rates
The forward rate is the future zero rate implied by today’s term structure of interest rates
Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded.
The forward rate, RF , is the rate such that:

33. 3. Invest $1 at rate R2, Maturing on Date T2.
Forward rates
Transaction
Cash Flow on T0
Cash Flow on T1
Cash Flow on T2
1. Invest $1 on T0 at rate R1 -1
𝑒 𝑇 1 𝑅 1
2. Sell forward the proceeds from Transaction 1 on date T1 at a forward rate RF, Maturing on Date T2.
- 𝑒 𝑇 1 𝑅 1
𝑒 𝑇 1 𝑅 1 𝑒 𝑅 𝐹 ( 𝑇 2 − 𝑇 1 )
Total
3. Invest $1 at rate R2, Maturing on Date T2.
𝑒 𝑇 2 𝑅 2

34. Formula for Forward Rates
The forward rate, RF ， for the period between times T1 and T2 is
This formula is only approximately true when rates are not expressed with continuous compounding

35. Application of the Formula
Year (n)
Zero rate for n-year investment
(% per annum)
Forward rate for nth year 1
3.0 2
4.0
5.0 3
4.6
5.8 4
6.2 5
5.3
6.5

36. Instantaneous Forward Rate
The instantaneous forward rate for a maturity T is the forward rate that applies for a very short time period starting at T.
When T2 approaches T1 , we have
RF =
where R is the T-year zero rate

37. Instantaneous Forward Rate
Define P(0,T) the price of a zero-coupon maturing at T, then P(0,T)=e-RT , the instantaneous forward rate can also be written as:

38. Instantaneous Forward Rate
Prove:
Note, R is a function of T
Use the chain rule in calculus

39. Lock in LIBOR forward rates
Case 1: borrow $100 at 3% for 1 year and invest the money at 4% for 2 years:
Cash outflow of 100*e0.03*1 = at the end of year 1
Cash inflow of 100*e0.04*2 = at the end of year 2
Since =103.05*e0.05 : a return equal to the forward rate (5%) is earned during year 2.

40. Lock in LIBOR forward rates
Case 2: borrow $100 at 5% for 4 year and invest the money at 4.6% for 3 years:
Cash inflow of 100*e0.046*3 = at the end of year 3
Cash outflow of 100*e0.05*4 = at the end of year 4
Since =114.80*e0.062 : money is being borrowed for year 4 at the forward rate of 6.2%