1. Lec 4: Interest Rates (Hull, Ch. 4)
There are 3 important ideas in this chapter:
1. Time Value of Money (Multipd Compounding and Discounting)
2. How to compute Forward interest rates.
3. Forward Rate Agreements (FRAs)
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2. 1. Multiperiod Compounding Buy a 3 year CD with $1,000 FV = ?
nominal interest rate of 10%: |–––––––––––––––––––––––|
If interest is compounded
▸ Yearly: FV = 1,000 ( )3 = $1,331
▸ Semiannually: FV = 1,000 ( /2 )6 = $1,340.1
▸ Quarterly: FV = 1,000 ( /4 )12 = $1,344.89
▸ Monthly: FV = 1,000 ( /12 )36 = $1,348.18
▸ Daily: FV = 1,000 ( /365 )365*3 = $1,349.80
▸ Continuous: FV = 1,000e0.10*3 = $1,349.86
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3. 2. Where do Forward interest rates come from?
(use simple interest rates)
Suppose the spot rates on zero coupon bonds are:
S1=10%/yr, S2=11%/yr, S3=12%/yr
Consider two investments A and B:
A: at t = 0, buy a 3 year zero coupon bond.
B: at t = 0, buy a 1 year zero.
Then, at t = 1 use proceeds to buy a 2-yr zero.
It seems intuitive that investing $1 should give the same FV:
[$1(1+S1)] (1+f13)2 = $1(1+S3)3 ➟ f13 ≈ 13%/yr
➟ the current yield curve predicts a 13%/yr interest rate on 2-year
bonds beginning 1-year from now.
➟ Using the same methodology, f12 = 12%/yr and f23 = 14%/yr
Graph:
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4. 2. How to compute Forward Rates Using c.c. interest rates
Suppose the (c.c.) spot rates are: S1=10%/yr, S2=11%/yr, S3=12%/yr
Consider two investments A and B:
A: at t = 0, buy a 3 year zero coupon bond.
B: at t = 0, buy a 1 year zero.
Then, at t = 1 use proceeds to buy a 2-yr zero.
We would expect that investing $1 should give the same FV:
FVB =$1(e0.1)(e2*f ) set = FVA =$1(e3*0.12)
➟ $1(e0.1)(e2*f ) = $1e3* Next, take the natural log of both sides:
➟ *f = 3*0.12 ➟ f13 = 13%/yr
➟ the current yield curve predicts a (c.c.) 13%/yr rate
on 2-year bonds beginning 1-year from now.
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5. Another example, consider two investments A and B:
A: at t = 0, buy a 3 year zero coupon bond.
B: at t = 0, buy a 2 year zero.
Then, at t = 2 use proceeds to buy a 1-yr zero.
Suppose we invest $100; the FV should be the same.
[$100(e2*0.11)](ef ) =$100(e3*0.12) and solve for f
2* f = 3*0.12 ➟ f23 = 14%/yr
➟ the current yield curve predicts a 14%/yr (c.c.) interest rate
on 1-year bonds beginning 2-year from now.
➟ f12 = 12%/yr
yield curve predicts a 12%/yr (c.c.) rate on 1-year bonds
beginning 1-year from now.
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6. 3. Forward Rate Agreements (FRAs). All you need to know about FRAs:
“forward rate agreement (FRA) is an over-the-counter derivative contract between parties that determines the rate of interest to be paid or received on an obligation beginning at a future start date.
The contract will determine the rates to be used along with the termination date and notional value. On this type of agreement, it is only the differential that is paid on the notional amount of the contract. It is paid on the effective date.
A FRA differs from a swap
Many banks and large corporations will use FRAs to hedge future interest rate exposure. The buyer hedges against the risk of rising interest rates” Got it?
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7. Translation: (by way of Example)
UTC will need to borrow $100M for 3 months beginning in June.
BoA has agreed to the loan.
Problem: In June, going rate of interest may ↓ (good) or ↑ (bad)
For example,
If the interest rate ↓ to 1.5%/quarter ➟ Interest Pmt = $1.5 M
If the interest rate ↑ to 2%/quarter ➟ Interest Pmt = $2 M
You work for UTC, as a financial risk manager.
Your bonus depends on minimizing risk for UTC. What to do?
Enter a FRA with GE capital (UTC is long the FRA).
The FRA interest rate is fixed at 1.8%/quarter.
Final Outcome (your bonus = $10,000) and
UTC will pay $1.8M in interest whether interest rates go ↑ or ↓
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8. Type 1: FRAs settled in arrears (in September)
There are 2 types of FRAs
Type 1: FRAs settled in arrears (in September)
t=0, (now): UTC enters a FRA with GE capital (UTC is long). CF = $0
Possible Outcomes in June:
➀ going rate of interest ↓ 1.5%/quarter
UTC borrows 1.5% from BoA. CF = +$100M
Then in September: UTC must pay back loan and settle the FRA.
▸ CF1 = -$100M (1.015) = -$101.5 M ➟ UTC pays this amount to BoA
▸ CF2 = $100M ( ) = -$300,000 ➟ from UTC to GE capital
Total CF = - $101.8 M
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9. Other Possible Outcome in June: ➁ going rate of interest ↑ 2%/quarter
UTC borrows 2% from BoA. ➟ CF = +$100M
Then in September: pay back loan and settle the FRA.
▸ CF1 = -$100M (1.02) = -$102 M ➟ UTC pays this amount to BoA
▸CF2=$100M( )=+$200,000 ➟ UTC receives from GE Capital
Total CF = - $101.8 M (can you see why this contract is called FRA?)
Moral of this story:
UTC pays 1.8% on the loan whether interest rates go up or down.
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10. Type 2: FRAs settled in June (time of Borrowing)
(Finance language: Tailing the Hedge)
t=0: UTC signs a FRA with GE capital (UTC is long). CF=$0
Possible Outcomes in June:
➀ If the interest rate = 1.5%/quarter
CF =+$100M( )= -$300,000 ➟ Pay the PV in June
CF to settle the FRA in June = - 300,000/{ }= -$295,566.50
A. UTC borrows 1.5% from BoA.
B. UTC borrows 1.5% from BoA.
C. UTC pays $295,566 to GE capital ➟ Total CF = + $100 M
Then in September: UTC must pay back both loans.
▸ CF1 = -$100M (1.015) = -$101.5 M ➟ UTC pays this amount to BoA
▸ CF2 = -$295,566.50(1.015) = -$300,000 ➟ UTC payment BoA
Total CF = - $101.8 M
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11. The Other Possible Outcome in June: ➁ interest rate = 2%/quarter
UTC borrows 2% from BoA. ➟ CF = +$100M
Also, settle the FRA in June
CF to settle the FRA =+$100M( )/1.02 = +$196,078
A. UTC borrows 2% from BoA.
B. UTC invests 2% with BoA.
C. UTC receives $196,078 from GE capital ➟ Total CF = + $100M
Then in September:
▸ CF1 = -$100M (1.02) = -$102 M ➟ UTC pays this amount to BoA
▸ CF2 = $196,078(1.02)= +$200,000 ➟ UTC receives this from BoA
Total CF = - $101.8 M
Moral of this story:
• UTC pays 1.8% on the loan whether interest rates go up or down.
• CFs are the same whether the FRA is settled in June or September.
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12. Thank you
(a Favara)
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