FINANCIEEL
RISICOMANAGEMENT
, 1. INTEREST RATES
What are interest rates?
= The amount of money a borrower promises to pay to the lender
Why do we need interest rates?
Page | 1 → Time value of money (= risk-free rate) (simply moving money across time will involve the risk-free
rate)
→ Credit risk = risk of default by the borrower, risk that the lender will not be paid
To determine the fair value of any financial asset.
Compounding = going forward in time with money
Discounting = going backwards in time with money
The higher the creditworthiness of the counterpart, the lower the credit risk. The lower the interest rate.
Lower bound = risk-free rate
𝐶𝐹
1 𝐶𝐹
2 𝐶𝐹
3
Discounted cash flow: 𝑉0 = (1+𝑟) 1 + (1+𝑟)2 + (1+𝑟)3 + ⋯
Discounted cash flow is fundamental to value:
- Bonds
- Equity
- Investments
- …
1.1 TREASURY RATES
= rates on instruments issued by a government in its own currency
→ Usually there is the assumption that there is no chance a government would default → Regarded a
risk-free
1.2 OVERNIGHT RATES
Banks have requirements of keeping reserves with the central bank.
At the end of the day, some banks have a surplus of funds and others have requirements for funds. This
leads to borrowing and lending overnight.
In the US the overnight rate is the federal funds rate. This rate is monitored by the FED, which may
intervene with its own transactions to raise or lower it.
In the Eurozone it is the euro short-term rate (ESTER). It replaces the euro overnight index average.
1.3 REPO RATE
Repurchase agreement:
A financial institution that owns securities agrees to sell them for X and buy them back in the future for a
slightly higher price Y.
The financial institution is obtaining a loan.
The repo rate is calculated from the difference between X and Y.
Very little credit risk:
- If the borrower does not honor the agreement → the lender keeps the securities
- If the lender does not keep to its side of the agreement → the original owner keeps the cash
Because: the security = collateral
,1.4 REFERENCE RATES
Frequently, the future interest rates are set tot equal to the value of a reference rate.
LIBOR = London interbank offered rate
→ Compiled by asking a panel of global banks to provide quotes estimating the unsecured rates of
interest at which they could borrow from other banks before 11 am. Page | 2
→ Problem:
o Not based on actual transactions
o Involves a certain amount of judgement
o Subject to manipulation
New reference rates
Plan is to base them on the overnight rates.
Longer rates can be determined from overnight rates by compounding them daily.
Risk-free as:
- One-day loans
- Creditworthy financial institutions
These new references are backward looking, while the LIBOR was forward looking and incorporated a credit
spread.
Problem:
- Credit spreads increase in stressed periods
- The spread between 3-month LIBOR and a 3-month overnight rates spiked to 364 basis points in
2008.
Why don’t we use the treasury rates as reference rates?
Because they are considered to be artificially low because banks are not required to keep capital for
Treasury instruments and in the US treasury instruments are given favorable tax treatment.
The new reference rates are considered as risk-free rates
1.5 IMPACT OF COMPOUNDING
𝑚
1
When we compound m times per year at rate R an amount A for a year, it grows to 𝐴 (1 + 𝑅 ) .
𝑀
A invested for n years at R interest rate per annum:
1 𝑛
- Annum 𝐴 (1 + 𝑅)
𝑚𝑛
1
- M times 𝐴 (1 + 𝑅 )
𝑀
- Equivalent interest rate = interest rate that are indifferent to compounding = the 2 interest rates
give the same amount
𝑚1 𝑛 𝑚2 𝑛
𝑚1
1 1 𝑅1 𝑚2
𝐴 (1 + ) = 𝐴 (1 + ) => 𝑅2 = 𝑚2 [(1 + ) − 1]
𝑅1 𝑅2 𝑚1
𝑀1 𝑀2
In the limit, as we compound more and more frequently → Continuously compounded Interest rates
𝐴𝑒 𝑅𝑛
𝑅𝑚 𝑚𝑛 𝑅𝑚 𝑅𝑐
Equivalent rate: 𝐴𝑒 𝑅𝑐 𝑛 = 𝐴 (1 + 𝑚
) => 𝑅𝑐 = 𝑚 × ln (1 + 𝑚
) => 𝑅𝑚 = 𝑚 (𝑒 𝑚 − 1)
With continuous compounding, interest are generating interest continuously.
, 1.6 ZERO RATES
A zero rate, for maturity T is the rate of interest earned on an investment that provides a payoff only at
time T.
During the period T there is only one cashflow.
bond pricing
Page | 3
Most bond pay coupons to the holder periodically. The value of a bond is computed as the present value of
all cash flows that will be received by the owner of the bond.
The most appropriate could be to use a different zero rate for each cash flow.
Example: Suppose that a 2-year bond with a principal of €100 provides coupons at the
rate of 6% per annum semiannually.
3𝑒 −0.05×0.5 + 3𝑒 −0.058×1 + 3𝑒 −0.064×1.5 + 103𝑒 −0.068×2
= 98.39
It is the single discount rate that makes the present value of the cash flows on the bond equal to the
market price of the bond. In the example, suppose that the market price of the bond equals the value of
98.39.
The bond yield is given by: 3𝑒 −𝑦×0.5 + 3𝑒 −𝑦×1 + 3𝑒 −𝑦×1.5 + 103𝑒 −𝑦×2 = 98.39
𝑦 = 0.0676 𝑜𝑟 6.76%
Par yield
It is the coupon rate that causes the bond price to equal its face value.
𝑐 𝑐 𝑐
In the example, we solve 2 𝑒 −0.05×0.5 + 2 𝑒 −0.058×1 + 2 𝑒 −0.064×1.5 + (100 + 𝑐2 )𝑒 −0.068×2 = 100
𝑐 = 6.87
More generally (with d the present value of €1 received at maturity and A the present value of an annuity
of 1€ on each coupon date), (FVd = discounted face value)
𝑐 (100 − 100𝑑)𝑚
100 = 𝐴 + 100𝑑 => 𝑐 =
𝑚 𝐴
The bootstrap method
The 3-month bond has the effect of turning an investment of
99.6 into 100 in 3 months.
100 = 99.6 𝑒 𝑅×0.25
That is, 1.603%.
The 6-month continuously compounded rate
100 = 99.0 𝑒 𝑅×0.5
That is, 2.010%.
The 1-year continuously compounded rate
100 = 97.8 𝑒 𝑅×1.0
That is, 2.225%.
To calculate the 1.5-year rate we solve:
2𝑒 −0.02010×0.5 + 2𝑒 −0.2225×1 + 102𝑒 −𝑅×1.5 = 102.5
𝑒 −1.5𝑅 = 0.96631
ln(0.96631)
𝑅=− 1.5
= 0.02284
To calculate the 2-year rate we solve:
2.5𝑒 −0.02010×0.5 + 2.5𝑒 −0.2225×1 + 2.5𝑒 −0.2284×1.5 + 102.5𝑒 −𝑅×2.0 = 105
𝑅 = 0.02416
RISICOMANAGEMENT
, 1. INTEREST RATES
What are interest rates?
= The amount of money a borrower promises to pay to the lender
Why do we need interest rates?
Page | 1 → Time value of money (= risk-free rate) (simply moving money across time will involve the risk-free
rate)
→ Credit risk = risk of default by the borrower, risk that the lender will not be paid
To determine the fair value of any financial asset.
Compounding = going forward in time with money
Discounting = going backwards in time with money
The higher the creditworthiness of the counterpart, the lower the credit risk. The lower the interest rate.
Lower bound = risk-free rate
𝐶𝐹
1 𝐶𝐹
2 𝐶𝐹
3
Discounted cash flow: 𝑉0 = (1+𝑟) 1 + (1+𝑟)2 + (1+𝑟)3 + ⋯
Discounted cash flow is fundamental to value:
- Bonds
- Equity
- Investments
- …
1.1 TREASURY RATES
= rates on instruments issued by a government in its own currency
→ Usually there is the assumption that there is no chance a government would default → Regarded a
risk-free
1.2 OVERNIGHT RATES
Banks have requirements of keeping reserves with the central bank.
At the end of the day, some banks have a surplus of funds and others have requirements for funds. This
leads to borrowing and lending overnight.
In the US the overnight rate is the federal funds rate. This rate is monitored by the FED, which may
intervene with its own transactions to raise or lower it.
In the Eurozone it is the euro short-term rate (ESTER). It replaces the euro overnight index average.
1.3 REPO RATE
Repurchase agreement:
A financial institution that owns securities agrees to sell them for X and buy them back in the future for a
slightly higher price Y.
The financial institution is obtaining a loan.
The repo rate is calculated from the difference between X and Y.
Very little credit risk:
- If the borrower does not honor the agreement → the lender keeps the securities
- If the lender does not keep to its side of the agreement → the original owner keeps the cash
Because: the security = collateral
,1.4 REFERENCE RATES
Frequently, the future interest rates are set tot equal to the value of a reference rate.
LIBOR = London interbank offered rate
→ Compiled by asking a panel of global banks to provide quotes estimating the unsecured rates of
interest at which they could borrow from other banks before 11 am. Page | 2
→ Problem:
o Not based on actual transactions
o Involves a certain amount of judgement
o Subject to manipulation
New reference rates
Plan is to base them on the overnight rates.
Longer rates can be determined from overnight rates by compounding them daily.
Risk-free as:
- One-day loans
- Creditworthy financial institutions
These new references are backward looking, while the LIBOR was forward looking and incorporated a credit
spread.
Problem:
- Credit spreads increase in stressed periods
- The spread between 3-month LIBOR and a 3-month overnight rates spiked to 364 basis points in
2008.
Why don’t we use the treasury rates as reference rates?
Because they are considered to be artificially low because banks are not required to keep capital for
Treasury instruments and in the US treasury instruments are given favorable tax treatment.
The new reference rates are considered as risk-free rates
1.5 IMPACT OF COMPOUNDING
𝑚
1
When we compound m times per year at rate R an amount A for a year, it grows to 𝐴 (1 + 𝑅 ) .
𝑀
A invested for n years at R interest rate per annum:
1 𝑛
- Annum 𝐴 (1 + 𝑅)
𝑚𝑛
1
- M times 𝐴 (1 + 𝑅 )
𝑀
- Equivalent interest rate = interest rate that are indifferent to compounding = the 2 interest rates
give the same amount
𝑚1 𝑛 𝑚2 𝑛
𝑚1
1 1 𝑅1 𝑚2
𝐴 (1 + ) = 𝐴 (1 + ) => 𝑅2 = 𝑚2 [(1 + ) − 1]
𝑅1 𝑅2 𝑚1
𝑀1 𝑀2
In the limit, as we compound more and more frequently → Continuously compounded Interest rates
𝐴𝑒 𝑅𝑛
𝑅𝑚 𝑚𝑛 𝑅𝑚 𝑅𝑐
Equivalent rate: 𝐴𝑒 𝑅𝑐 𝑛 = 𝐴 (1 + 𝑚
) => 𝑅𝑐 = 𝑚 × ln (1 + 𝑚
) => 𝑅𝑚 = 𝑚 (𝑒 𝑚 − 1)
With continuous compounding, interest are generating interest continuously.
, 1.6 ZERO RATES
A zero rate, for maturity T is the rate of interest earned on an investment that provides a payoff only at
time T.
During the period T there is only one cashflow.
bond pricing
Page | 3
Most bond pay coupons to the holder periodically. The value of a bond is computed as the present value of
all cash flows that will be received by the owner of the bond.
The most appropriate could be to use a different zero rate for each cash flow.
Example: Suppose that a 2-year bond with a principal of €100 provides coupons at the
rate of 6% per annum semiannually.
3𝑒 −0.05×0.5 + 3𝑒 −0.058×1 + 3𝑒 −0.064×1.5 + 103𝑒 −0.068×2
= 98.39
It is the single discount rate that makes the present value of the cash flows on the bond equal to the
market price of the bond. In the example, suppose that the market price of the bond equals the value of
98.39.
The bond yield is given by: 3𝑒 −𝑦×0.5 + 3𝑒 −𝑦×1 + 3𝑒 −𝑦×1.5 + 103𝑒 −𝑦×2 = 98.39
𝑦 = 0.0676 𝑜𝑟 6.76%
Par yield
It is the coupon rate that causes the bond price to equal its face value.
𝑐 𝑐 𝑐
In the example, we solve 2 𝑒 −0.05×0.5 + 2 𝑒 −0.058×1 + 2 𝑒 −0.064×1.5 + (100 + 𝑐2 )𝑒 −0.068×2 = 100
𝑐 = 6.87
More generally (with d the present value of €1 received at maturity and A the present value of an annuity
of 1€ on each coupon date), (FVd = discounted face value)
𝑐 (100 − 100𝑑)𝑚
100 = 𝐴 + 100𝑑 => 𝑐 =
𝑚 𝐴
The bootstrap method
The 3-month bond has the effect of turning an investment of
99.6 into 100 in 3 months.
100 = 99.6 𝑒 𝑅×0.25
That is, 1.603%.
The 6-month continuously compounded rate
100 = 99.0 𝑒 𝑅×0.5
That is, 2.010%.
The 1-year continuously compounded rate
100 = 97.8 𝑒 𝑅×1.0
That is, 2.225%.
To calculate the 1.5-year rate we solve:
2𝑒 −0.02010×0.5 + 2𝑒 −0.2225×1 + 102𝑒 −𝑅×1.5 = 102.5
𝑒 −1.5𝑅 = 0.96631
ln(0.96631)
𝑅=− 1.5
= 0.02284
To calculate the 2-year rate we solve:
2.5𝑒 −0.02010×0.5 + 2.5𝑒 −0.2225×1 + 2.5𝑒 −0.2284×1.5 + 102.5𝑒 −𝑅×2.0 = 105
𝑅 = 0.02416
