1. Interest Rates Chapter 4 (part 2)
Geng Niu

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

3. Forward Rate Agreement
An FRA contract: At T0 , Company X is agreeing to lend money to Y between T1 and T2 for a predetermined interest rate.
RK : the interest agreed to in the FRA
RF : forward LIBOR rate between T1 and T2
Rm : actual LIBOR rate observed at T1 for the period between T1 and T2 (reference rate)
L: principal underlying the contract

4. Forward Rate Agreement
Here we assume that the rates RK , RF , and RM , are all measured with a compounding frequency reflecting the length of the lending period to which they apply. Remember:
Here, RK , RF , and RM , are all measured with a compounding frequency (m) so that mn=1, and we have
𝐿 1+ 𝑅 𝑚 𝑚 𝑚𝑛 =𝐿+𝐿 𝑅 𝑚 𝑚 =𝐿+𝐿𝑛 𝑅 𝑚
For example, T2- T1 =0.5 indicates n=0.5, thus m=2 , interest rates in this case are expressed with semiannual compounding.
Thus, if you borrow L at time T1 for the rate RM, the interests you have to pay is LRM （T2 – T1 ) at time T2.

5. Forward Rate Agreement
FRA term
Contract made T0 T1
Settlement date T2
Maturity date
X (FRA seller ) (according to FRA ) –L L+LRK (T2-T1)
(borrow) L L-LRM (T2-T1)
Total L(RK - RM) (T2 –T1)
Y (FRA purchaser ) (according to FRA ) L L-LRK (T2-T1)
(invest) -L L+ LRM (T2-T1)
Total L(RM - RK) (T2 –T1)

6. Cash flow in Forward Rate Agreement
Suppose at T1 , X borrow from the market L dollars for period T1 to T2 , with market rate RM , and lend L to Y according to the FRA, with FRA rate RK
At T2 , the cash flow to company X is :
L(1+RK (T2 –T1))- L(1+RM (T2 –T1))= L(RK - RM) (T2 –T1)
Similarly, at T1 , Y lend the L dollars with market rate RM for period T1 to T2. At T2 , the cash flow to Y is :
L(1+RM (T2 –T1))- L(1+RK(T2 –T1)) = L(RM-RK) (T2 –T1)

7. Forward Rate Agreement
Usually FRAs are settled at time T1 , the payoff must then be discounted back
For X, the amount due at T1 is
For Y, the amount due T1 is

8. Settlement of FRA
The amount due is the only cash-flow that exists in an FRA and is due on the settlement date.
At settlement date, the principal just serves as the basis to calculate the difference between the two interest rates
The principal is not exchanged and there is no obligation by either party to borrow or lend capital.

9. Forward Rate Agreement
X will receive interest between T1 and T2 at the fixed rate RK and pay interest at the realized rate RM (which is not known at T0 when contract is made )
Y will pay interest between T1 and T2 at the fixed rate RK and receive interest at the realized rate RM
An FRA is equivalent to an agreement where interest at a predetermined rate, RK is exchanged for interest at the market rate, RM
An FRA can be purchased (sold) to speculate on rising (falling) interest rates or as a hedge for a future short (long) position in deposits and thus a protection against rising (falling) interest rates.

10. Value FRA FRA term Current time T1 Settlement date T2 Maturity date T0
X (FRA seller ) (according to FRA ) –L L+LRK (T2-T1)
Lend Le-R1T1 for T0 toT L
Borrow Le-R1T1 for T0 toT Le-R1T1eR2T2 =-LeRFc(T2-T1)
= -L-LRF (T2-T1)
Total L(RK - RF) (T2 –T1)
Note: RFc is cont. comp.; RF is dist. comp. with comp. freq m
we select m so that mn= m=1/(T2-T1)

11. Value Forward Rate Agreement
The FRA is always worth zero when RK=RF
When the FRA is first initiated, RK is set equal to RF so that FRA is worth zero for both parties.
Other wise, one party will for sure have a profit and the other a loss: impossible to make a contract!
After the initial date, RF will change.
What is the value of the contract to X (or Y) at some time point between initial data and settlement data?

12. Value FRA (for seller X)
FRA term
Initial date =T(-1)
Today=T0 T1 T2
RK=RF-1
RK(new)=RF
VT-1 (FRA1)=0
VT0 (FRA2)=0
What is the value of FRA1 at T0: VT0(FRA1) ?
VT2(FRA1)-VT2 (FRA2)=L(RK - RF)(T2-T1)
VT0(FRA1)-VT0 (FRA2)=L(RK - RF)(T2-T1)e-R2T2
VT0(FRA1)= L(RK - RF)(T2-T1)e-R2T2

13. Example
An FRA entered into some time ago ensures that a company will receive 4% (s.a.) on $100 million for six months starting in 1 year
Forward LIBOR for the period is 5% (s.a.)
The 1.5 year zero rate is 4.5% with continuous compounding
The value of the FRA (in $ millions) is

14. Example continued
If the six-month interest rate in one year turns out to be 5.5% (s.a.) there will be a payoff (in $ millions) of
in 1.5 years
The transaction might be settled at the one-year point for an equivalent payoff of

15. Duration
Suppose a bond provides the holder with cash flows ci at time ti (i=1,2,…n), the bond yield is y, then bond price B is

16. Duration Duration of a bond that provides cash flow ci at time ti is
where B is its price and y is its yield (continuously compounded)
Duration is a weighted average of the times when payments are made.

17. Duration Taylor expansion:
∆𝐵= 𝑑𝐵 𝑑𝑦 ∆𝑦+ 1 2! 𝑑 2 𝐵 𝑑 𝑦 2 ∆𝑦 ! 𝑑 3 𝐵 𝑑 𝑦 3 ∆𝑦 3 +⋯
With a small change Δy in the yield, it is approximately true that:

18. Duration

19. Key Duration Relationship
Duration is important because it leads to the following key relationship between (small) changes in a yield on the bond and the percentage changes in its price

20. Duration
Duration: an estimate of economic life of a bond measured by the weighted average time to receipt of cash flows
The shorter the duration, the less sensitive is a bond’s price to fluctuations in interest rates

21. Key Duration Relationship
When the yield y is expressed with compounding m times per year
The expression
is referred to as the “modified duration”
How to prove?

22. Duration of a bond portfolio
The duration of a portfolio is the average of the durations of the securities in the portfolio, weighted by each security’s price.

23. Convexity

24. Convexity The convexity, C, of a bond is defined as
This leads to a more accurate relationship
When used for bond portfolios it allows larger shifts in the yield curve to be considered, but the shifts still have to be parallel

25. Convexity
∆𝐵= 𝑑𝐵 𝑑𝑦 ∆𝑦+ 1 2! 𝑑 2 𝐵 𝑑 𝑦 2 ∆𝑦 2 𝑑 2 𝐵 𝑑 𝑦 2 =− 𝑖=1 𝑛 𝑡 𝑖 𝑐 𝑖 𝑒 𝑖 −𝑦 𝑡 𝑖 𝑑𝑦 = 𝑖=1 𝑛 𝑡 𝑖 2 𝑐 𝑖 𝑒 𝑖 −𝑦 𝑡 𝑖 𝐶= 1 𝐵 𝑑 2 𝐵 𝑑 𝑦 2 = 1 𝐵 𝑖=1 𝑛 𝑡 𝑖 2 𝑐 𝑖 𝑒 𝑖 −𝑦 𝑡 𝑖 ∆𝐵 𝐵 =−𝐷∆𝑦+ 1 2 𝐶 ∆𝑦 2

26. 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. Yield curve reflects the hedging and maturity needs of institutional investors
Liquidity Preference Theory: forward rates higher than expected future zero rates. Long term yield is greater because investors prefer the liquidity in short term issues.

27. Liquidity Preference Theory
Suppose that the outlook for rates is flat and you have been offered the following choices
Which would you choose as a depositor? Which for your mortgage?
Maturity
Deposit rate
Mortgage rate
1 year 3% 6%
5 year

28. Liquidity Preference Theory
The majority of a bank’s customers will opt for one—year deposits and five-year mortgages.
This creates an asset/liability mismatch for the bank.
No problem if interest rates fall.
If rates rise, the deposits will cost more in the future.

29. Liquidity Preference Theory cont
To match the maturities of borrowers and lenders a bank has to increase long rates above expected future short rates
In our example the bank might offer
Maturity
Deposit rate
Mortgage rate
1 year 3% 6%
5 year 4% 7%

30. Liquidity Preference Theory cont
The theory suggests that an investor demands a higher interest rate, or premium, on securities with long-term maturities, which carry greater risk, because all other factors being equal, investors prefer cash or other highly liquid holdings.

31. Patterns of yield curve/term structure

32. Normal yield curve Investors expect the economy to grow in the future.
This stronger growth will lead to higher inflation and higher interest rates.
Investors will not commit to purchasing longer-term securities without getting a higher long-term interest rate.
Typically occurs when central banks are “easing” monetary policy.

34. Inverted yield curve
Indicates that investors expect the economy to slow or decline in the future.
This slower growth may lead to lower inflation and lower interest rates for all maturities.
Typically indicates that central banks are “tightening” monetary policy.
An inverted yield curve has often historically been an indicator of an economic recession

36. Humped yield curve
Indicates an expectation of higher rates in the middle of the maturity periods covered.
Perhaps reflecting investor uncertainty about specific economic policies or conditions.
Or it may reflect a transition of the yield curve from a normal to inverted, or vice versa.