1. Théorie Financière 2. Valeur actuelle
Professeur André Farber

2. Present Value: general formula
Cash flows: C1, C2, C3, … ,Ct, … CT
Discount factors: DF1, DF2, … ,DFt, … , DFT
Present value: PV = C1 × DF1 + C2 × DF2 + … + CT × DFT
An example:
Year
Cash flow
Discount factor
Present value
NPV = = 23.13
We now have a general formula to compute present values. The formula is very simple. You simply multiply each cash flow by the corresponding discount factor and you take the sum.
Keep in mind that the discount factors are market prices for zero-coupon.
An analogy is helpful to understand the logic.
Suppose that you are offered the following deal. You pay 100 EUR and you receive 30 USD (US dollar), 40 GBP (British pound) and 60 CHF (Swiss franc). Is this a good deal?
Before accepting, you will convert all cash flows into one unit: the euro using current spot exchange rates. If the spot rates are:
1 USD = 1 EUR 1 GBP = 1.50 EUR 1 CHF = 0.70 EUR
the net present value is NPV = * * * 0.70
= = +32
This is a good deal.
Present value calculation is similar. Instead of cash flows being denominated in different currencies, they take place at different points in time. The NPV is obtained by valuing each cash flow at the appropriate “exchange rate”.
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3. Using prices of U.S. Treasury STRIPS
Separate Trading of Registered Interest and Principal of Securities
Prices of zero-coupons
Example: Suppose you observe the following prices
Maturity Price for $100 face value
The market price of $1 in 5 years is DF5 = 0.80
NPV = * 0.80 = = +20
A zero-coupon is a pure discount bond. You pay a price today and you receive one single payment (the face value of the zero-coupon) at one future date (the maturity of the zero-coupon).
In the early 1980s, several U.S. investment banks started “stripping” U.S. Treasury issues. They would buy a bond (consisting of a number of coupons and the principal) and sell each element separately as a zero-coupon using feline names as CATS, TIGRS, COUGARS and LIONS1. The idea was taken over by the U.S. Treasury in 1985 who started offering its own zero-coupon instruments under the denomination STRIPS (nothing to do with strip tease, at least as far as I know..).
Prices of U.S. Treasury strips are regularly published in the WSJ. Note that prices are quoted with colons that represent 32nds. For instance, 93:16 means /32 =
Strips were later on introduced in various European markets. In the UK, Gilt strips exist (a gilt is a long term British government bond) and quotations can be found in the FT. France introduced stripped OATs (OAT means Obligation Assimilée du Trésor). In Belgium, strips on Belgian government bonds are known as “obligations démambrées”.
1 CATS: Certificates of Accrual on Treasury Securities (Salomon Brothers)
TIGRS: Treasury Investment Growth Receipts (Merrill Lynch)
COUGARS: Certificates on Government Receipts (A. G. Beckers Paribas)
LIONS: Lehman Investment Opportunity Notes (Lehman Brothers)
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4. Present value and discounting
How much would an investor pay today to receive €Ct in t years given market interest rate rt?
We know that 1 €0 => (1+rt)t €t
Hence PV  (1+rt)t = Ct => PV = Ct/(1+rt)t = Ct  DFt
The process of calculating the present value of future cash flows is called discounting.
The present value of a future cash flow is obtained by multiplying this cash flow by a discount factor (or present value factor) DFt
The general formula for the t-year discount factor is:
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5. Spot interest rates
Back to STRIPS. Suppose that the price of a 5-year zero-coupon with face value equal to 100 is 75.
What is the underlying interest rate?
The yield-to-maturity on a zero-coupon is the discount rate such that the market value is equal to the present value of future cash flows.
We know that 75 = 100 * DF and DF5 = 1/(1+r5)5
The YTM r5 is the solution of:
The solution is:
This is the 5-year spot interest rate
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6. Term structure of interest rate
Relationship between spot interest rate and maturity.
Example:
Maturity Price for €100 face value YTM (Spot rate)
r1 = 2.00%
r2 = 2.79%
r3 = 3.41%
r4 = 3.70%
r5 = 4.56%
Term structure is:
Upward sloping if rt > rt-1 for all t
Flat if rt = rt-1 for all t
Downward sloping (or inverted) if rt < rt-1 for all t
Starting from the prices of zero-coupons with different maturities, we calculate the yield-to-maturity of each of them to obtain the various spot interest rates.
In general, spot rates vary with maturity. The term structure of interest rates is the relationship between spot interest rates (yields of zero-coupons) and maturity.
The term structure of interest rates for the European Monetary Union is available of the web site of the European Commission: Click on Euro yield curve.
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8. Zero coupon yield curve Euro 5-aug-2005 Source:http://epp. eurostat
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9. Using one single discount rate
When analyzing risk-free cash flows, it is important to capture the current term structure of interest rates: discount rates should vary with maturity.
When dealing with risky cash flows, the term structure is often ignored.
Present value are calculated using a single discount rate r, the same for all maturities.
Remember: this discount rate represents the expected return.
= Risk-free interest rate + Risk premium
This simplifying assumption leads to a few useful formulas for:
Perpetuities (constant or growing at a constant rate)
Annuities (constant or growing at a constant rate)
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10. Constant perpetuity Ct =C for t =1, 2, 3, .....
Proof:
PV = C d + C d² + C d3 + …
PV(1+r) = C + C d + C d² + …
PV(1+r)– PV = C
PV = C/r
Ct =C for t =1, 2, 3, .....
Examples: Preferred stock (Stock paying a fixed dividend)
Suppose r =10% Yearly dividend = 50
Market value P0?
Note: expected price next year =
Expected return =
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11. Growing perpetuity Ct =C1 (1+g)t-1 for t=1, 2, 3, ..... r>g
Example: Stock valuation based on:
Next dividend div1, long term growth of dividend g
If r = 10%, div1 = 50, g = 5%
Note: expected price next year =
Expected return =
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12. Constant annuity
A level stream of cash flows for a fixed numbers of periods
C1 = C2 = … = CT = C
Examples:
Equal-payment house mortgage
Installment credit agreements
PV = C * DF1 + C * DF2 + … + C * DFT +
= C * [DF1 + DF2 + … + DFT]
= C * Annuity Factor
Annuity Factor = present value of €1 paid at the end of each T periods.
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13. Constant Annuity Ct = C for t = 1, 2, …,T
Difference between two annuities:
Starting at t = 1 PV=C/r
Starting at t = T+1 PV = C/r ×[1/(1+r)T]
Example: 20-year mortgage
Annual payment = €25,000
Borrowing rate = 10%
PV =( 25,000/0.10)[1-1/(1.10)20] = 25,000 * 10 *(1 – )
= 25,000 *
= € 212,839
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14. Growing annuity Ct = C1 (1+g)t-1 for t = 1, 2, …, T r ≠ g
This is again the difference between two growing annuities:
Starting at t = 1, first cash flow = C1
Starting at t = T+1 with first cash flow = C1 (1+g)T
Example: What is the NPV of the following project if r = 10%?
Initial investment = 100, C1 = 20, g = 8%, T = 10
NPV= – [20/(10% - 8%)]*[1 – (1.08/1.10)10]
= – =
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15. Useful formulas: summary
Constant perpetuity: Ct = C for all t
Growing perpetuity: Ct = Ct-1(1+g)
r>g t = 1 to ∞
Constant annuity: Ct=C t=1 to T
Growing annuity: Ct = Ct-1(1+g)
t = 1 to T
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16. Compounding interval Up to now, interest paid annually
If n payments per year, compounded value after 1 year :
Example: Monthly payment :
r = 12%, n = 12
Compounded value after 1 year : ( /12)12=
Effective Annual Interest Rate: 12.68%
Continuous compounding:
[1+(r/n)]n→er (e= )
Example : r = 12% e12 =
Effective Annual Interest Rate : 12.75%
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17. Juggling with compounding intervals
The effective annual interest rate is 10%
Consider a perpetuity with annual cash flow C = 12
If this cash flow is paid once a year: PV = 12 / 0.10 = 120
Suppose know that the cash flow is paid once a month (the monthly cash flow is 12/12 = 1 each month). What is the present value?
Solution 1:
Calculate the monthly interest rate (keeping EAR constant)
(1+rmonthly)12 = 1.10 → rmonthly = %
Use perpetuity formula:
PV = 1 / =
Solution 2:
Calculate stated annual interest rate = % * 12 = 9.568%
Use perpetuity formula: PV = 12 / =
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18. Interest rates and inflation: real interest rate
Nominal interest rate = 10% Date 0 Date 1
Individual invests $ 1,000
Individual receives $ 1,100
Hamburger sells for $1 $1.06
Inflation rate = 6%
Purchasing power (# hamburgers) H1,000 H1,038
Real interest rate = 3.8%
(1+Nominal interest rate) = (1+Real interest rate)×(1+Inflation rate)
Approximation:
Real interest rate ≈ Nominal interest rate - Inflation rate
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19. Bond Valuation Objectives for this session :
1.Introduce the main categories of bonds
2.Understand bond valuation
3.Analyse the link between interest rates and bond prices
4.Introduce the term structure of interest rates
5.Examine why interest rates might vary according to maturity
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20. Zero-coupon bond Pure discount bond - Bullet bond
The bondholder has a right to receive:
one future payment (the face value) F
at a future date (the maturity) T
Example : a 10-year zero-coupon bond with face value $1,000
Value of a zero-coupon bond:
Example :
If the 1-year interest rate is 5% and is assumed to remain constant
the zero of the previous example would sell for
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21. Level-coupon bond Periodic interest payments (coupons)
Europe : most often once a year
US : every 6 months
Coupon usually expressed as % of principal
At maturity, repayment of principal
Example : Government bond issued on March 31,2000
Coupon 6.50%
Face value 100
Final maturity 2005
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22. Valuing a level coupon bond
Example: If r = 5%
Note: If P0 > F: the bond is sold at a premium
If P0 <F: the bond is sold at a discount
Expected price one year later P1 =
Expected return: [ ( – )]/ = 5%
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23. When does a bond sell at a premium?
Notations: C = coupon, F = face value, P = price
Suppose C / F > r
1-year to maturity:
2-years to maturity:
As: P1 > F
with
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24. A level coupon bond as a portfolio of zero-coupons
« Cut » level coupon bond into 5 zero-coupon
Face value Maturity Value
Zero
Zero
Zero
Zero
Zero
Total
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25. Law of one price
Suppose that you observe the following data:
What are the underlying discount factors? Bootstrap method
= DF = DF DF = DF DF DF
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26. Bond prices and interest rates
Bond prices fall with a
rise in interest rates and rise with a fall in
interest rates
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27. Sensitivity of zero-coupons to interest rate
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28. Duration for Zero-coupons
Consider a zero-coupon with t years to maturity:
What happens if r changes?
For given P, the change is proportional to the maturity.
As a first approximation (for small change of r):
Duration = Maturity
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29. Duration for coupon bonds
Consider now a bond with cash flows: C1, ...,CT
View as a portfolio of T zero-coupons.
The value of the bond is: P = PV(C1) + PV(C2) PV(CT)
Fraction invested in zero-coupon t: wt = PV(Ct) / P •
Duration : weighted average maturity of zero-coupons
D= w1 × 1 + w2 × 2 + w3 × 3+…+wt × t +…+ wT ×T
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30. Duration - example Back to our 5-year 6.50% coupon bond.
Face value Value wt
Zero %
Zero %
Zero %
Zero %
Zero %
Total
Duration D = .0581× × × × ×5
= 4.44
For coupon bonds, duration < maturity
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31. Price change calculation based on duration
General formula:
In example: Duration = 4.44 (when r=5%)
If Δr =+1% : Δ ×4.44 × 1% = %
Check: If r = 6%, P =
ΔP/P = ( – )/ = %
Difference due to convexity
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32. Duration -mathematics
If the interest rate changes:
Divide both terms by P to calculate a percentage change:
As:
we get:
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33. Yield to maturity Suppose that the bond price is known.
Yield to maturity = implicit discount rate
Solution of following equation:
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