Present Value of Bond Depends –Time to Maturity(Duration) –Yield to Maturity or Market Interest Rate: Interest rate fluctuate depending on risk –Face Value –Coupon Payment or Coupon Interest

There are Two segments when we are working to find out the Present Value of Bond –Coupon Payments –Principle Repayments

EXAMPLE  A Bond is issued for 10 years with a coupon payments of Rs.80 per year. Market rate is 8% for similar risk. Face value is Rs. 1000/- What should be the selling price of the bond?

Solution: There are two components need valuation: 1 – Annuity: Rs. 80/yr for 10 years 2 – Principal repayment after 10 years

PV Of Annuity = 80 x [{(1-1/(1.08) 10 }/0.08] PV Of Annuity = 80 x [{(1-1/(1.08) 10 }/0.08] = 80 x = 80 x = or 537 = or 537 PV of Principal = 1000/(1.08) 10 = = Adding both components (Selling Price) = 1000 = 1000 The reason was the YTM of Market Interest Rate of this type of Bond and Coupon Payment Rate is the same which is 8%.

HOW TO VALUE A BOND: AFTER ONE YEAR AFTER ONE YEAR –Time to maturity = 9 years –YTM: Risen to 10% –Other terms & conditions unchanged

PV of Principal = 1000/(1.10) 9 = = PV of Annuity = 80 x [1 – 1/(1.10) 9 ]/0.10 = = Adding both components PV of Bond = (rounded off to nearest rupee) (rounded off to nearest rupee)

Why 885? Market rate of YTM move up to 10% or 100 per year. Current coupon payment is 80 per year. Investor would be getting 20 per year less for the rest of nine years.

Fitting 20 per year in formula returns: = 20 x ((1 – 1/(1.10) 9 )/0.1 = 20 x ((1 – 1/(1.10) 9 )/0.1 = 115.xx = 115.xx This is the amount of discount the investor will get at maturity. Let’s see another variation

 Time to Maturity = 9 years  YTM: Drops to 6%  Coupon Rate is 8%  Other terms unchanged  What is the value of Bond?

Present Value= 1000/(1.06) 9 = = PV Of Annuity = 80 x(1-1/(1.06) 9 /0.06 = = Adding both components PV of bond = over & above the face value. 136 is basically is premium, which is demanded in market on face value.

Again, why 136?  This can be found: =(80-60) x [(1-1(1.06) 9 ]/0.06 =(80-60) x [(1-1(1.06) 9 ]/0.06 = 136 = 136

Summary: –YTM & Coupon Rate were same  Result PV of Bond was exactly equal to the FV –YTM greater than Coupon Rate  Results PV of the Bond less than the FV –YTM lower than Coupon Rate  Result PV of the Bond was greater than FV

CONCLUSION: –A Bond will be sold on a discount when YTM is greater than coupon rate. –A bond will be sold on premium when YTM is lower than the coupon rate.

–Current Yield Vs YTM –For a bond selling above the face value is said to sell at premium. It means investor who buys it at a premium face a capital loss over the life of bond. So return on bond will be less than the current yield. –For a bond selling below the face value is said to sell at discount. This means capital gain at maturity. The return on this bond is greater than its current yield.

 EFFECTIVE YIELD A bond pays semi-annual interest payments i.e., twice a year. Face value is Rs.1000/- and coupon rate is 12%. This means two six-monthly payments of Rs. 60/- each. Bond matures in 7 years and yield to maturity is 14%. What is the effective annual yield on this bond? A bond pays semi-annual interest payments i.e., twice a year. Face value is Rs.1000/- and coupon rate is 12%. This means two six-monthly payments of Rs. 60/- each. Bond matures in 7 years and yield to maturity is 14%. What is the effective annual yield on this bond?

1-PV = 1000/(1.07) 14 = 1000 / = 1000 / = = PV of annuity = = 60 x (1 – 1/(1.07) 14 /0.07 = 60 x (1 – 1/(1.07) 14 /0.07 = 60 x = 60 x = = Total PV of bond = = = Effective annual Yield = (1+i/m) m - 1 Effective annual Yield = (1 +.14/2) 2 -1 = 14.49% = 14.49%

NOMINAL & REAL INTEREST RATE  Interest Rate: –Inflation adverse effects on valuation –Inflation persistent increase in general price level  Real Interest rate: –Nominal Interest Rate adjusted for inflation becomes Real Interest Rate Relationship between Nominal and Real Interest Rates is known as Fisher Effect

Example – Fisher Effect –Today you can buy one unit of a product at Rs. 5/-. It means you can buy 20 units in Rs. 100/-. Inflation rate is 5%. And nominal interest rate is 15.5%. What is real rate of return?

 Solution: –Your buying power at the end of one year is: – =105/20 = 5.25 –Your investment of Rs 100 after one year is: –100 x (1.1550) = –Then: – /5.25= 22 Real increase: A year ago you could buy 20 units and now you can buy 22 units – increase of 10% (22-20)/20.

Solution with Formula Fisher’s Formula 1 + R = (1+r) x (1+h) Where: R= Nominal interest rate r = real interest rate h = inflation rate Putting Values

 Solution with formula: 1 + R= (1+r) x (1+h) 1 + R= (1+r) x (1+h) = (1+r) x (1+0.05) (1 + r) = /1.05 = 1.10 (1 + r) = /1.05 = 1.10 r =.10 or 10% r =.10 or 10%

Example – Fisher Effect You need to invest an amount today to produce Rs. 100/- after a year. Nominal interest rate is 10% and inflation rate is 7%. What is the “exact” real interest rate?

Solution: PV of Rs. 100= 100/(1.10) PV of Rs. 100= 100/(1.10) =90.91 =90.91 If inflation rate is 7%, real value of Rs. 100 is therefore = 100 / 1.07 = 100 / 1.07 = = Real Interest Rate =1 + Nominal/1+Inflation = 1.10 / 1.07 = 1.10 / 1.07 = or 2.80% = or 2.80%

 If we discount real value of our Rs. 100 investment (93.46) by 2.8%, we get PV = 93.46/1.028 PV = 93.46/1.028 = = 90.91

 Point to Remember –Current Cash Flow must be discounted by NOMINAL INTEREST RATE –Real Cash Flow must be discounted by REAL INTEREST RATE REAL INTEREST RATE

Finding Nominal Rate  Example: An investor requires 10% real interest rate. Inflation rate is 8%. What is exact nominal interest rate? Solution: 1 + R = (1+r) x (1+h) =1.10 x 1.08 = or 18.80%