Topics to be covered
Concept of Duration Measuring Duration g Modified Duration Eff ti D Effective Duration ti Portfolio Duration Benefits of Duration Bond Portfolio Immunization Bond Portfolio Immunization Limitations of Duration Numerical Questions
Concept of Duration
Duration is a measure which considers both price and
reinvestment risk and plays a very crucial role on understanding price – yield relationships Duration measures the time taken by a bond to repay the original price of the bond through its internal cash flows Duration is always lesser than or equal to the maturity of y q y the bond. It can never be greater than the maturity of the bond Bonds where Duration is equal to the maturity of the bond are called Zero Coupon Bonds, or Deep Discount Bonds p , p where no coupon is paid
Measuring Duration
Step 1 – Find the present value of each annual coupon
or principal payment. Use the prevailing YTM on the bond as the discount rate Step 2 ‐ Divide this present value by the current Step 2 Divide this present value by the current market price of the bond Step 3 Multiply the relative value by the year in Step 3 – Multiply the relative value by the year in which the cash flow is to be received Step 4 – Add up the values calculated in step 3
Modified Duration
Modified Duration = Duration in Years / (1 + YTM) To determine the bond’s percentage price change p g p g
resulting due to increase in market interest rates or decrease in market interest rates can be computed by decrease in market interest rates can be computed by the below mentioned formula: ‐
% change in price of a bond = (‐1) Modified Duration % change in price of a bond = (‐1)* Modified Duration * %change in interest rates
Bond prices and interest rates movement have an Bond prices and interest rates movement have an
inverse relationship and thus if interest rates go up, bond prices fall and vice versa b d i f ll d i
Effective Duration
Effective Duration is a measure of price sensitivity
calculated from actual bond prices associated with different interest rates Effective Duration (Pa Pb) / (Po (Yb Effective Duration = (Pa – Pb) / (Po (Yb – Ya))
Portfolio Duration
The duration of a portfolio of bonds can be computed
in two ways: ‐
Map the cash flows of the bond into various term
buckets, when they are due, and using yield of the portfolio, discount the total cash flows of the portfolio. Compute duration with the usual formula, treating the aggregate cash flows s if they were a single bond Compute the weighted duration of a portfolio, using the market value of the bond as the weightage
Benefits of Duration
Duration is especially useful in determining the
relative riskiness of two or more bonds when visual inspection of their characteristics makes it unclear g g which is most vulnerable to changing interest rates Duration helps in Structuring of bond portfolios Duration helps in bond portfolio immunization Duration helps in bond portfolio immunization
Limitations of Duration
Duration is not a static property of a bond. Duration of a
bond changes over time, and with changes in market yields. Any strategy based on duration values of a bond will, therefore, require continuous and active evaluation Computing duration involves the discounting cash flows of a bond. It is common to use the YTM of the bond, as the rate at which cash flows are discounted. Therefore, the limitations of YTM extend to the computation of duration The results obtained by using duration to measure price change are only an approximation of the actual price yield relationship, which is not linear, but convex
Numerical Questions
E Example 1 – C l l l 1 Calculate Macaulay Duration and Modified M l D i d M difi d
Duration of a bond for company A, if the coupon rate is given to be 8% and YTM is 6% and the time to maturity is 5 given to be 8% and YTM is 6% and the time to maturity is 5 years. Face Value of the bond is INR 1,00,000 and interest payments are made annually. Also, calculate % change in the price of the bond if the YT falls by 100 basis points of the price of the bond if the YT falls by 100 basis points of 1% from 6% to 5%. Also, calculate % change in bond price y if YTM increases by 2% Solution: ‐
Step 1 – We have to calculate the Market Price of the bond,
for which we will use the following formula as the bond is f hi h ill th f ll i f l th b d i maturing at Par value: ‐
YTM = (((Maturity Value – Purchase Price)/ Years to Maturity) + Coupon)/ ((Maturity Value*0.4 + Purchase Price*0.6) 0.06 = (((100000 – Purchase Price) / 5) + 8000) / ((100000*0.4+Purchase Price*0.6) Purchase Price = 108475
Numerical Questions
S l i Solution: ‐
Step 2 – Calculation of Duration: ‐
Year (1) 1 2 3 4 5 Annual Cash Flows (2) 8000 8000 8000 8000 108000 Present Value of INR 1 at 6% (3) 0.94 0.89 0.84 0 84 0.79 0.75 Present Value of Annual Cash Flows (4) = (2) * (3) 7,520 7,120 6,720 6 720 6,320 81,000 Duration Present Value of Annual Cash Flows divided by Current Market Price (5) = (4) / 1,08,696 0.0692 0.0655 0.0618 0 0618 0.0581 0.7452 Time Relative or Duration (6) = (5) * (1) 0.0692 0.1310 0.1855 0 1855 0.2326 3.726 4.3442
Numerical Questions
Solution: ‐ Step 4 – Calculating % changes in Price if YTM changes
% change in bond prices = (‐1) * Modified Duration * % change in
YTM If YTM falls by 1% = (‐1) * 4.0983 * ‐1% = 4.0983% or bond price will increase by 4.0983% 4 0983% b d i ill i b 4 0983% If YTM increases by 2% = (‐1) 4 0983 2% = ( 1) * 4.0983 * 2% = ‐8.1967% or bond price will decrease by 8.1967%
Numerical Questions
Example 2 – Mr. Lakshya is considering the purchase of the
following debenture: Par Value – IN 100 V l INR 100 Maturity – 3 years Coupon – 11%
a) If the investor requires a YTM of 13% on debentures of
equivalent risk and maturity, what does he believe is a fair i l t i k d t it h td h b li i f i market price? b) If the debenture is selling for a price of INR 97.59, what is b) If the debenture is selling for a price of INR 97 59 what is its promised YTM? c) What is the duration of this debenture? d) If an investor has a horizon date of 4 years, why is this debenture risky? e) If an investor has a horizon date of 2 years, why is this ) If i h h i d f2 h i hi debenture risky?
Numerical Questions
Example 3 – Mr. Gupta recently purchased a 10% bond
with FV INR 1000 and 4 years to maturity. Interest payments are paid annually by the company. Mr. p p Gupta paid INR 1032.40 for the bond.
a) What is bond’s YTM?
Numerical Questions
Example 4 – Calculate duration of a 5 year, 11% bond at a YTM of 9% (coupon f f (
payments are received semi‐annually). Current market price of the bond is INR 107.91 Solution –
Period (1) 1 2 3 4 5 6 7 8 9 10 Semi Annual Cash Flows (2) 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 105.5 Present Value of INR 1 at 4.5% (YTM / 2) (3) 0.9569 0.9158 0.8763 0.8386 0.8024 0.7679 0.7348 0.7032 0.6729 0.6439 Present Value of Semi Annual Cash Flows (4) = (2) * (3) (4) (2) * (3) 5.26 5.04 4.82 4.61 4.41 4.22 4.04 3.87 3.70 67.93 Duration Present Value of Semi Annual Cash Flows divided by Current Market Price (5) = (4) / 107.91 (5) (4) / 107 91 0.049 0.047 0.045 0.043 0.041 0.039 0.037 0.036 0.034 0.630 Time Relative or Duration (6) = (5) * (1) (6) (5) * (1) 0.049 0.093 0.135 0.171 0.205 0.234 0.259 0.288 0.306 6.300 8.04 8 04
Duration = 8.04 half years or 4.02 years
Numerical Questions
Example 5 – X Ltd issued a 5 year 10% bond (face
value INR 100). Yield to maturity of the bond is 12%. Calculate the market price of the bond. Also, calculate y y duration of the bond. If the yield falls by 100 basis points, calculate the change in the bond price Solution – Solution
Step 1 – We have to calculate the Market Price of the
bond, for which we will use the following formula as the bond for which we will use the following formula as the bond is maturing at Par value: ‐
YTM = (((Maturity Value Purchase Price)/ Years to Maturity) + YTM = (((Maturity Value – Purchase Price)/ Years to Maturity) + Coupon)/ ((Maturity Value*0.4 + Purchase Price*0.6) 0.12 = (((100 – Purchase Price) / 5) + 10) / ((100*0.4+Purchase Price*0.6) Purchase Price = 92.65
Numerical Questions
S l i Solution: ‐
Step 2 – Calculation of Duration: ‐
Year (1) 1 2 3 4 5 Annual Cash Flows (2) 10 10 10 10 110 Present Value of INR 1 at 12% (3) 0.89 0.80 0.71 0 71 0.64 0.57 Present Value of Annual Cash Flows (4) = (2) * (3) 8.93 7.97 7.12 7 12 6.36 62.42 Duration Present Value of Annual Cash Flows divided by Current Market Price (5) = (4) / 92.65 0.10 0.09 0.08 0 08 0.07 0.67 Time Relative or Duration (6) = (5) * (1) 0.10 0.17 0.23 0 23 0.27 3.37 4.14
Numerical Questions
Solution: ‐ Step 4 – Calculating % changes in Price if YTM changes
% change in bond prices = (‐1) * Modified Duration * % change in
YTM If YTM falls by 1% = (‐1) * 3.70 * ‐1% = 3.70% or bond price will increase by 3.70% or it will become 3 70% b d i ill i b 3 70% it ill b 96.08 from 92.65