# Yield to Maturity-YTM and Yield to Call-YTC

**Yield to Maturity-YTM**

**Yield to maturity (YTM)** is the most widely used measure of return on the bond. It is the compounded rate of return an investor expects to receive from a bond purchased at the current market price which he holds till maturity. On the other hand, it may be termed as an internal rate of return or discount rate that makes the present value of all the futures cash inflows from the bond equal to the purchase price of the bond.

**The relationship among the cash outflow, the cash inflow, and the YTM of a bond can be expressed as:**

_{n}

PM =∑ [C_{t }/ (1 + YTM)_{t} + TV / (1 + YTM)^{n}]

_{ t = 1}

**where,**

PM = market price of the bond,

C_{t }= cash inflow from the bond during the whole life of the bond,

YTM = yield to maturity,

TV = terminal value of the bond,

n = total maturity period of the bond.

Hence, it is possible to estimate the YTM equating both sides of the equation by trial and error method. Let us consider a bond with a face value of Tk. 1000 had a 15 percent coupon rate which will mature at par. Five years of maturity bond can be purchased at Tk. 800 in the market.

**The YTM of the said bond can be determined as under:**

**The relationship among the cash outflow, the cash inflow, and the YTM of a bond can be expressed as:**

_{n}

PM =∑ [C_{t }/ (1 + YTM)_{t} + TV / (1 + YTM)^{n}]

_{ t = 1}

_{5}

800 =∑ [150 / (1 + YTM)_{t} + 1000 / (1 + YTM)^{5}]

_{ t = 1}

Since the market price is lower than the face value, the YTM would be higher than the coupon interest rate. This can be estimated by the trial and error method. Firstly, we may consider 20 percent as YTM.

**Then the right hand side of the equation becomes-**

^{1}+150/(1 + .20)

^{2}+150/(1 + .20)

^{3}+150/(1 + .20)

^{4}+

150/(1 + .20)^{5}] + [1000/(1 + .20)^{5}

= [125 + 104.17 + 86.81 + 72.34 + 60.28] + [401.88]

= 448.60 + 401.88 = 850.48

Since the estimated value Tk. 850.48 is higher than the desired value Tk. 800, we should try again by a higher discount rate. Taking 25 per cent as YTM, the right hand side of the equation would become-

[150/(1 + .25)^{1}+150/(1 + .25)

^{2}+150/(1 + .25)

^{3}+150/(1 + .25)

^{4}+

150/(1 + .25)^{5}] + [1000/(1 + .25)^{5}

= [120 + 96 + 76.80 + 61.44 + 49.15] + [327.68]

= 403.39 + 327.68 = 731.07

This value is lower than our desired value Tk. 800. Hence, the desired YTM lies between 20 percent and 25 percent.

**It can be estimated by using interpolation as shown below:**

YTM = Lower YTM + [(Value at lower YTM – Desired Value) / (Value at lower YTM – Value at higher YTM)] (Higher YTM – Lower TYM)

- = 20 + [ ( 850.48 – 800 ) / ( 850.48 – 731.07 ) ] ( 25 – 20 )
- = 20 + ( 0.4227 ) ( 5 ) = 20 + 2.1135 = 22.11 per cent.

Alternatively, the present values of the future cash inflows can be estimated by using present value tables like:

Tk. 150 (present value annuity factor for 5 yrs., 20%) + Tk. 1000 (present value factor for 5 yrs., 20%)

= (150 × 2.9906) + (1000 × 0.4019) = 448.59 + 401.90 = 850.49

Again,

Tk. 150 (present value annuity factor for 5 yrs., 25%) + Tk. 1000 (present value factor for 5 yrs., 25%)

= (150 × 2.689) + (1000 × 0.328) = 403.35 + 328 = 731.35

Therefore,

YTM

= 20 + [ ( 850.49 – 800) / ( 850.49 – 731.35 ) ] ( 25 – 20 )

= 20 + ( 0.4238 ) ( 5 ) = 20 + 2.119 = 22.12 per cent.

**Yield to Call-YTC**

At the option of the issuer or of the investor, some bonds may be redeemable before their maturity period. If such an option is executed, the subject bond would be called for redemption at the specific call price on the specified call date. For bonds likely to be called, the yield to maturity calculation is unrealistic. Hence, the better calculation here is termed as a yield to call (YTC). The end of the deferred call period, when a **bond** can first be called, is often used for the yield-to-call calculation.

This is appropriate for the bonds selling at a premium.

**In the case of a redeemable bond, two yields are to be calculated as **

**Yield to maturity:**It asserts that the bond will be redeemed only at the end of the full maturity period.**Yield to call:**It implies that the bond will be redeemed at the call date before the full maturity. Yield-to-call is the discount rate that makes the present value of cash inflows to call equal to the bond’s current market price.

**Let us consider an example.**

A bond with a face value of Tk. 100 having a 15 percent coupon rate will mature at par in 15 years. The bond is callable in 5 years at Tk. 115. It currently sells for Tk. 105 in the market.

**The YTC of the said bond can be determined as under:**

**The relationship among the cash outflow, the cash inflow, and the YTC of a bond can be expressed as:**

_{n}

PM =∑ [C_{t }/ (1 + YTC)_{t} + TV / (1 + YTC)^{n}]

_{ t = 1}

By putting the values, we have

_{5}

105 =∑ [15 / (1 + YTC)_{t} + 115 / (1 + YTC)^{5}]

_{ t = 1}

We have to find the value of YTC that makes the right hand side of the equation equal to Tk. 105. This can be done by trial and error method. Since the market price is lower than the face value, the YTC would be higher than the coupon interest rate. Firstly, we may consider 15 per cent as YTC.

**Then the right hand side of the equation becomes-**

^{1}+15/(1+.15)

^{2}+15/(1+.15)

^{3}+15/(1+.15)

^{4}+15/(1+.15)

^{5}]+ [115/(1+.15)

^{5}]

= [13.04 + 11.34 + 9.86 + 8.58 + 7.46] + [57.17]

= 50.28 + 57.17 = 107.45

Since the estimated value, Tk. 107.45 is higher than the desired value, Tk. 105; we should try again by a higher discount rate. Taking 18 per cent as YTC, the right hand side of the equation would become-

[15/(1+.18)^{1}+15/(1+.18)

^{2}+15/(1+.18)

^{3}+15/(1+.18)

^{4}+15/(1+.18)

^{5}]+ [115/(1+.18)

^{5}]

= [12.71 + 10.77 + 9.13 + 7.74 + 6.56] + [50.27]

= 46.91 + 50.27 = 97.18

This value is lower than our desired value, Tk. 105. Hence, desired YTC lies between 15 per cent and 18 per cent.

**It can be estimated by using interpolation as shown below:**

YTC = Lower YTC + [(Value at lower YTC – Desired Value) / (Value at lower YTC – Value at higher YTC)] (Higher YTC -Lower TYC)

= 15 + [ ( 107.45 – 105 ) / ( 107.45 – 97.18 ) ] ( 18 – 15 )

= 15 + (0.2386)(3) = 15 + 0.7157 = 15.72 per cent.

**Yield to Maturity-YTM and Yield to Call-YTC**

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