1. Interest Rate Markets

2. Measuring Interest Rates
The compounding frequency used for an interest rate is the unit of measurement.

3. Continuous Compounding
In the limit as we compound more and more frequently we obtain continuously compounded interest rates
Rs100 grows to Rs100eRT when invested at a continuously compounded rate R for time T
Rs100 received at time T discounts to Rs100e-RT at time zero when the continuously compounded discount rate is R

4. Conversion Formulas Define Rc : continuously compounded rate
Rm: same rate with compounding m times per year

5. Example
1.What is the equivalent rate for continuous compounding for an interest rate which is quoted as 10% per annum semi annual compounding.
2.A bank quotes you an interest rate of 14% per annum with quarterly compounding. What is the equivalent rate with (a) Continuous compounding (b) annual compounding.

6. Solution 1. 2ln(1+0.1/2)=0.09758 2.(a) 4ln (1+0.14/4)=13.76
(b)[ (1+0.14/4)^4]-1=14.75

7. Some More Questions
Q1. An Interest rate is quoted as 5% p.a. with semi annual compounding. Calculate the rate with:
Annual compounding
Monthly compounding
Continuous compounding

8. Solution (a) [(1+.05/2)^2]-1=5.0625% (b) 12[(1.050625)^1/12]-1=4.9485%
(c) 2ln[1+.025]=4.9385%

9. Types of Rates Bank Rates Deposit Rates Prime Lending Rates
Home Loan and Consumer Loan Rates.

10. Types of Rates
Types of interest rates important for understanding Futures and Options are
Treasury rates
LIBOR rates
Repo rates

11. Treasury Rates
Interest rates applicable to borrowing by a government in its own currency.
It is usually assumed that there is no chance that a government will default on an obligation denominated in its own currency.

12. LIBOR Rates Two types LIBOR and LIBID
Refers to rates at which International banks lend or borrow (accept deposit) from each other.
The duration is usually 1 month, 3 month, 6 month, and 12 months.
If bank quotes bid rates at 6% pa and offer rates at 6.5%pa it is willing to accept deposits from other banks at 6% and lend to another bank at 6.5%.

13. Repo Rates
A form of short term borrowing for dealers in government securities. The dealer sells the government securities to investors, usually on an overnight basis, and buys them back the following day. For the party selling the security (and agreeing to repurchase it in the future) it is a repo; for the party on the other end of the transaction (buying the security and agreeing to sell in the future) it is a reverse repurchase agreement

14. Zero Rates ( Zero Coupon Rate)
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
The n year zero rate is the rate of interest earned on an investment that starts today and lasts for n years.

15. Example
Suppose the five year treasury zero rate with continuous compounding is quoted as 5% per annum. It means that Rs 100 if invested at this risk free rate for five years would grow to 100e0.05*5 =
Most interest rates are not zero rates. Eg. A five year govt bond with 5% coupon has some returns realized before five years.

16. Example

17. 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 providing a 6% coupon semiannually is

18. 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 is given by solving
to get y= or 6.76%.

19. 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: (c/m)A +100d=100

20. Sample Data Bond Time to Annual Bond Principal Maturity Coupon Price
(Rupees)
(years)
(Rupees)
(Rupees)
100
0.25
97.5
100
0.50
94.9
100
1.00
90.0
100
1.50 8
96.0
100
2.00 12
101.6

21. The Bootstrap Method
An amount 2.5 can be earned on 97.5 during 3 months.
The 3-month rate is 4 times 2.5/97.5 or % with quarterly compounding
This is % with continuous compounding
Similarly the 6 month and 1 year rates are % and % with continuous compounding

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

23. Zero Curve Calculated from the Data
Zero Rate (%)
10.808
10.681
10.469
10.536
10.127
Maturity (yrs)

24. Question The following table gives the prices of bonds:
Calculate zero rates for 6,12,18 & 24 months. Also calculate fwd rates for 6to 12, 12 to 18, & 18to 24 months.
Principal
Time to maturity
Annual Coupon
Bond Price
100
0.5 98
1.0 95
1.5
6.2
101
2.0
8.0
104

25. Answer
Zero Rates Fwd Rates

26. Forward Rates
The forward rate is the future zero rate implied by today’s term structure of interest rates

27. Calculation of Forward Rates
Zero Rate for
Forward Rate an n
-year Investment
for n
th Year
Year ( n )
(% per annum)
(% per annum) 1
10.0 2
10.5
11.0 3
10.8
11.4 4
11.0
11.6 5
11.1
11.5

28. Formula for Forward Rates
Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded.
The forward rate for the period between times T1 and T2 is

29. Upward vs Downward Sloping Yield Curve (ZCYC)
For an upward sloping yield curve:
Fwd Rate > Zero Rate > Par Yield
For a downward sloping yield curve
Par Yield > Zero Rate > Fwd Rate

30. Question
Q1. The 6 month,12 month, 18 month and 24 month zero rates are 4%, 4.5%, 4.75% and 5% with semi annual compounding. What are rates with continuous compounding? What is the forward rate for 6 month period beginning in 18 months? Q2. What is the value of an FRA which promises to pay you 6% compd sa on a principal of $1million for 6 months period starting in 18 months

31. Solution Month Interest Rate (pa with sa compdg)% Interest Rate
(pa with continuous compdg) %
Fwd Rate
(4.94*2-4.69*1.5)/2-1.5
=( )/0.5
=5.68% 6 4
3.96 12
4.5
4.45 18
4.75
4.69
=5.76% pa with sa compdg 24 5
4.94

32. Answer for FRA L(Rk-Rm)(T2-T1)e^(-R2T2) Where Rk=6% Rm=5.76% T2=2

33. Question
What is the 2 year par yield when zero rates are as calculated by you? What is the yield on a 2 year bond that pays a coupon equal to par yield?
Hint: c=[( d)m]/A

34. Forward Rate Agreement
A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period

35. Bharti Instrument’s Interest Rate Risk
As an example, Bharti instruments has taken out a three-year, floating-rate loan in the amount of US$10 million (annual interest payments).
The loan is priced at US Dollars LIBOR +1.5%. The LIBOR base will be reset each year on an agreed upon date.
Although LIBOR is floating the spread of 1.5% is a fixed component of the loan.

36. Bharti Instrument’s Interest Rate Risk
Bharti will not know the actual cost of the loan until the loan has been completely repaid. Although the treasury managers may forecast what LIBOR will be for the life of the loan, they will not know with certainty until all payments have been calculated.
This uncertainty is not only an interest rate risk but also an actual cash flow risk associated with the interest payment.

37. Management of Interest Rate Risk through FRA’s
A forward rate agreement (FRA) is an interbank-traded contract to buy or sell interest rate payments on a notional principal.
These contracts are settled in cash.
The buyer of an FRA obtains the right to lock in an interest rate for a desired term that begins at a future date.

38. Management of Interest Rate Risk through FRA’s
The contract specifies that the seller of the FRA will pay the buyer the increased interest expense on a nominal sum (the notional principal) of money if interest rates rise above the agreed rate, but the buyer will pay the seller the differential interest expense if interest rates fall below the agreed rate.

39. FRA for Bharti
Bharti can lock in the first interest payment (due at the end of year 1), by buying an FRA that locks in a total interest payment of 6.5%.
If LIBOR rises above 5% by the end of year 1, Bharti would receive a cash payment from the FRA seller that would reduce the interest rate to 5%.
If LIBOR were to fall during the year below 5% ( not expected by treasury managers at Bharti), Bharti would make a cash payment to the seller of the FRA effectively raising its LIBOR payment to 5% and the total loan payment to 6.5%.

40. Example
A Corporate treasurer plans to raise Rs 10 Crore six months from now for three months. The current 3 month rate is 8% pa. The treasurer wants to hedge himself against a rise in interest rates using a 6/9 FRA that a bank is quoting at 8.1% pa. What will be effective cost for the firm if the three month MIBOR rate six months later is
8.5%
7.5%

41. Meaning of 6X9 & 3X6
The phrase “6 vs. 9” refers to a 3-month interest rate observed 6 months from the present, for a security with a maturity date 9 months from the present.
Similarly “3 vs 6” refers to a 3 month interest rate observed 3 months from the present for a security with a maturity date 6 months from the present.
What does 3 vs 12 FRA mean?

42. Solution( When interest rate rises)
The treasurer buys the 6/9 FRA at 8.1 percent per annum.
If the three month MIBOR rate six months later is 8.5% the bank compensates the firm for the difference in interest rates.
= ( )*10 Crores*90/360= Rs100,000
Since the actual payment is after maturity period three months discounted value of Rs 8.5% changes hands i.e. Rs (which when = Rs100,000)

43. Solution If Firm borrows Rs 10 Crores at 8.5% for three months.
=(0.085)*10Crores*90/360.=Rs 2,125,000
Net cost = 2,125, =2,025,000.
=(2,025,000)/10Cr* 360/90=8.1% pa.
Hence company has locked the cost at 8.1%.

44. Solution ( When the interest rate falls to 7.5%)
If the three month MIBOR six month later is 7.5%, the firm compensates the bank for the difference in interest rates.
The difference is ( )%.

45. Forward Rate Agreement continued
An FRA is equivalent to an agreement where interest at a predetermined rate, RK is exchanged for interest at the market rate
An FRA can be valued by assuming that the forward interest rate is certain to be realized

46. FRA’s Revised
A forward rate agreement (FRA) is an interbank-traded contract to buy or sell interest rate payments on a notional principal.
These contracts are settled in cash.
The buyer of an FRA obtains the right to lock in an interest rate for a desired term that begins at a future date.

47. FRA’s Revised
The contract specifies that the seller of the FRA will pay the buyer the increased interest expense on a nominal sum (the notional principal) of money if interest rates rise above the agreed rate, but the buyer will pay the seller the differential interest expense if interest rates fall below the agreed rate.

48. Forward Rate Agreements
For an FRA which agrees that a financial institution will earn an interest rate RK for the period of time between T1 and T2 on a principal L
RK : The rate of interest agreed to in FRA.
RF: The forward LIBOR interest rate between times T1and T2
RM:The actual LIBOR interest rate observed between time T1and T2.

49. Forward Rate Agreements
Assuming discrete compounding (i.e. T2-T1 =0.25 for s.a. compounding etc).
Normally a company X would earn RM from a LIBOR loan (investment).
The FRA means it will earn RK
The extra interest earned due to FRA will be RK-RM.

50. Forward Rate Agreements
The cash flow to company X will be:
L(RK-RM)(T2-T1).
To the company Y borrowing money from X the cash flow will be:
L(RM-RK)(T2-T1).
Thus we can also say that FRA is an agreement where company X receive interest on principal between T1 and T2 at the fixed rate RK and pay interest on the principal at the realized market rate of RM .
Similarly company Y will pay interest on principal between T1 and T2 at the fixed rate of RK and receive an interest of RM

51. Payoffs for X and Y from FRAs
Usually FRAs are settled at time T1 rather than T2. If settled at T1 they are discounted from T2 to T1 at the market rate RM.
Payoff for X:
{L(RK-RM)(T2-T1)}/(1+RM)(T2-T1)

52. Valuing FRAs
Assuming discrete compounding such that rates of interest are measured with a compounding frequency equal to their maturity the cash flows at two time periods will be:
Time T1= -L
Time T2=+L[1+RK(T2-T1)]

53. Valuing FRAs Remember K is delivery price in a forward contract &
F0 is forward price that would apply to the contract today
The value of a long forward contract, ƒ, is ƒ = (F0 – K )e–rT
Similarly, the value of a short forward contract is
(K – F0 )e–rT

54. Valuing FRAs Compare the two FRAs.
The first promises that the forward rate RF will be earned on a principal of L between times T1 and T2. The second promises that RK will be earned between the same two dates.
The contracts are the same except for the payments received at time T2.

55. Valuing FRAs
The excess of second contract over the first is therefore the present value of the difference between these interest payments or,
L(RK-RF)(T2-T1)e-R2T2 where R2 is the continuously compounded riskless zero rate for maturity T2.
As the value of FRA promising RF is zero the value of FRA promising RK is
V= L(RK-RF)(T2-T1)e-R2T2

56. Example
Suppose that the three month LIBOR rate is 5% and the 6 month LIBOR is 5.5% with continuous compounding. Consider an FRA where we will receive a rate of 7% measured with quarterly compounding on a principal of Rs1million between the end of month 3 and the end the month 6. What is the value of the FRA? The risk less zero rate is 5.5%for six months.

57. Solution
The forward rate is 6% with continuous compounding or % with quarterly compounding. (refer slide no. 24).
The value of the FRA is
*( )*0.25*e-0.055*0.5 =Rs2322

58. Theories of the Term Structure
Expectations Theory: forward rates equal expected future zero rates.
Market Segmentation: short, medium and long rates determined independently of each other.
Conjectures that there need be no relationship between short medium and long term interest rates.
Liquidity Preference Theory: forward rates higher than expected future zero rates. Asserts that the investors prefer liquidity and invest their funds for short term whereas borrowers prefer fixed interest rates for a long period of time.

59. Conclusions
If the interest rates offered by Banks and FIs follows the first theory long term interest rates would equal the average of expected future short term interest rates.
If not given incentives deposits would only be short term and borrowings would be long term causing excessive interest rate risks.
Thus most FIs and banks raise long term interest rates relative to expected future short term interest rates.

60. Conclusions
This strategy reduces the demand for long term fixed rate borrowings and encourages investors to deposit for long term.
The last theory leads to a situation in which forward rates are greater than expected future zero rates.
It is also consistent with the empirical result that yield curves tend to be upward sloping more often they are downward sloping.

61. Duration Duration of a bond that provides cash flow c i at time t i is
where B is its price and y is its yield (continuously compounded)
This leads to

62. How? ΔB=dB/dyΔy for small change in y.
Using the bond pricing relationship
B= Σ cie-yti substituting dB/dy we get
ΔB= -BDΔy.

63. Example
Calculate the duration for a 3 year 10% coupon bond with a face value of 100. The yield on the bond is 12% per annum with continuous compounding. The coupons are paid every six months.

64. Calculation of Duration
Time CF PV
Wt.
T * Wt
0.5 5
4.709
0.050
0.025
1.0
4.435
0.047
1.5
4.176
0.044
0.066
2.0
3.933
0.042
0.083
2.5
3.704
0.039
0.098
3.0
105
73.256
0.778
2.333
Total
130
94.213
1.000
2.653

65. Spreadsheet Solution
Duration Example.xls

66. Understanding Duration
For the bond considered the B= and D= Therefore
ΔB= * 2.653Δy or
ΔB= Δy thus when the yield increases by 0.1% the bond price goes down to
The same can be checked by using the conventional formula for price of the bond.

67. Duration Continued
When the yield y is expressed with compounding m times per year
The expression
is referred to as the “modified duration”

68. Example
What is the modified duration of the bond with price and duration The yield with semiannual compounding is %.

69. Solution
D=(2.653)/( /2) =2.499.

70. Situation
Portfolio A consists of a 1 year zero coupon bond with a face value of Rs 2000 and a 10 year zero coupon bond of Rs Portfolio B consists of a 5.95 year zero coupon bond with a face value of Rs The current yield on all bonds is 10% perannum.

71. Question 1.What is the duration of both portfolios?
2. What is the percentage change in value of each portfolio if yield goes up by 0.1% per annum and 5% for both?

72. Answer 1
Duration of portfolio A= weighted average of durations. (Dur. of B1* Price of B1+ Dur. of B2*Pr. Of B2)/(Price of A + Price of B Duration of Second portfolio is 5.95 years.

73. Answer 2
Discount the price of each bond by .101 instead of 0.1 and compare with values in previous slide.
Percentage decrease in value is .59%
Repeat the process for 5% values.